Finite Group Databasebeta

← #29646912

Subgroups — #29646912

Order 15616.

1216 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#11758222C211yesmaximal, characteristic
2#21454780C211yesmaximal, characteristic, complemented
3#21454487C211yesmaximal, characteristic, complemented
4#21444286C212yesmaximal, complemented
5#21444269C211yesmaximal, characteristic, complemented
6#21444260C212yesmaximal, complemented
7#21444234C212yesmaximal, complemented
8#21444219C211yesmaximal, characteristic, complemented
9#21444208C212yesmaximal, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#1486C211yesmaximal, characteristic, complemented
12C22×#11758222—21no—
13#16840433C411yescharacteristic
14D4×Dic122C411yescharacteristic
15#16840421—21no—
16#16840417C411yescharacteristic
17#16840413C412yes—
18#16840408—21no—
19#16840402—22no—
20D4×D244C411yescharacteristic
21D4×C22⋊D61C412yes—
22#16840312C2211yescharacteristic
23#16840309C2212yes—
24#16840273—21no—
25#16840271—22no—
26C4×#11758222C2211yescharacteristic
27C4×D4×D61C2212yes—
28#16839889—21no—
29#16839839—21no—
30#16839830—22no—
31#16839822—22no—
32#16839818C2212yescomplemented
33#16839814C2212yescomplemented
34#16839813C2212yescomplemented
35C2×C4⋊#92418C2211yescharacteristic, complemented
36#16839730—21no—
37#16839720—21no—
38C4×#11758189C2212yescomplemented
39C2×C4×#92418C2211yescharacteristic, complemented
40#16839491C2212yescomplemented
41#16839491C2212yescomplemented
42D4⋊#92418C2211yescharacteristic
43D4⋊#92418C2211yescharacteristic
44D4⋊#92418C2211yescharacteristic, complemented
45D4⋊#92418C2211yescharacteristic, complemented
46#16839489C2212yescomplemented
47#16839489C2212yescomplemented
48C2×C4×D4×C61C2211yescharacteristic
49C61×#431C2211yescharacteristic, complemented
50C61×#423C2211yescharacteristic, complemented
51C61×D4⋊C8C2212yes—
52C61×D4⋊C8C2212yescomplemented
53C2×#11758222—21no—
54C2×#11758222—22no—
55C2×#11758222—21no—
56C2×#11758222—22no—
57C2×#11758222—21no—
58C2×#11758222—44no—
59C2×#11758222—44no—
60C2×#11758222D411yescharacteristic
61C22×Dic122—21no—
62C22×D244—21no—
63C22×C22⋊D61—22no—
64C2×D4×D61C2×C412yes—
65C22×C4×D61—22no—
66C2×#11758189—22no—
67C22×#92418—21no—
68C22×#92418—21no—
69#14519777C2×C411yescharacteristic
70#14519776C2×C412yes—
71#14519774C2×C411yescharacteristic
72#14519773C2×C412yes—
73#14519765—21no—
74#14519763—22no—
75#14519762—22no—
76#14519760—22no—
77#14519759—22no—
78#14519756—21no—
79#14519756—21no—
80#14519755—22no—
81#14519754—22no—
82#14519752C2×C412yes—
83#14519751—21no—
84#14519751—21no—
85#14519751—21no—
86#14519751C2×C412yes—
87#14519750—22no—
88#14519750—22no—
89#14519749—22no—
90#14519748C2×C412yes—
91#14519747—22no—
92#14519746—22no—
93#14519745—22no—
94C4×Dic122C2×C411yescharacteristic
95C4×Dic122D411yescharacteristic
96C4×D244—21no—
97C4×D244C2×C411yescharacteristic
98C4×D244D411yescharacteristic
99C4×D244C2312yes—
100D4×Dic61C2×C412yes—
101#14519739C2×C412yes—
102C4×C22⋊D61—22no—
103C4×C22⋊D61—22no—
104C4×C22⋊D61C2×C412yes—
105C4×C22⋊D61D412yes—
106#14519737—22no—
107#14519736D411yescharacteristic
108#14519734D412yes—
109C4⋊C4×D61C2×C412yes—
110C22⋊C4×D61—22no—
111C42×D61C2312yes—
112C4⋊#92418—21no—
113C4⋊#92418C2311yescharacteristic
114C4⋊#92418C2311yescharacteristic
115#14519706—22no—
116#14519689—22no—
117#14519689—22no—
118#14519689—22no—
119#14519687—22no—
120#14519687—22no—
121C4×#92418—21no—
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C8×Dic61—22no—
125C22×D4×C61C2×C411yescharacteristic
126C23×C4×C61—21no—
127C2×M16×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2×C411yescharacteristic
131C4×D4×C61C2×C411yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C4×D4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C22⋊C4×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2312yes—
140C22⋊C8×C61—21no—
141C4×C8×C61C2312yes—
142C24×C61—21no—
143C23×D61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—41no—
147C2×D4×C61C22×C411yescharacteristic
148C2×D4×C61C22×C411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—42no—
155C22×C4×C61—42no—
156C22×C4×C61C22×C411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D411yescharacteristic
159C2×C22⋊D61—22no—
160C2×C22⋊D61—22no—
161C2×C22⋊D61—22no—
162C2×C22⋊D61—22no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—48no—
165C2×C22⋊D61C22⋊C412yes—
166C22×Dic61—22no—
167D4×D61—24no—
168D4×D61—42no—
169#11758222—22no—
170#11758222—22no—
171#11758222—44no—
172#11758222—44no—
173#11758222—44no—
174#11758222—44no—
175#11758222—48no—
176#11758222—42no—
177#11758222—42no—
178C2×D244—22no—
179C2×D244—21no—
180C2×D244—21no—
181C2×D244—44no—
182C2×D244C22⋊C411yescharacteristic
183C2×D244C22×C412yes—
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—22no—
187C2×C4×D61—44no—
188C2×C4×D61—44no—
189C2×C4×D61C22×C412yes—
190C2×C4×D61C2×D412yes—
191C2×C4×D61C2×D412yes—
192C2×Dic122—21no—
193C2×Dic122—22no—
194C2×Dic122—44no—
195C2×Dic122C22⋊C411yescharacteristic
196M16×C61—42no—
197C2×C8×C61—22no—
198C2×C8×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61C22×C411yescharacteristic
201C4⋊C4×C61C22×C411yescharacteristic
202C22⋊C4×C61—21no—
203C22⋊C4×C61—21no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C42×C61C22×C411yescharacteristic
207C42×C61C2×D411yescharacteristic
208C42×C61C2×D411yescharacteristic
209C42×C61C2411yescharacteristic
210C22⋊Dic61—22no—
211C22⋊Dic61—22no—
212C22⋊Dic61C22⋊C412yes—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—22no—
217#11758198C22⋊C412yes—
218#11758198C22⋊C412yes—
219C4⋊Dic61—21no—
220C4⋊Dic61C22⋊C411yescharacteristic
221C4⋊Dic61C22⋊C411yescharacteristic
222C4⋊Dic61C22×C412yes—
223#11758196—22no—
224#11758196—22no—
225#11758196C22⋊C412yes—
226#11758196C22⋊C412yes—
227C4×Dic61C22×C412yes—
228C4×Dic61C2×D412yes—
229C4×Dic61C2×D412yes—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—42no—
235C2×#92418—42no—
236#11758189—44no—
237C23×C61—21no—
238C23×C61—21no—
239C23×C61—42no—
240C23×C61—42no—
241C23×C61—81no—
242C23×C61C2×C22⋊C411yescharacteristic
243C22×D61—22no—
244C22×D61—44no—
245C22×D61—44no—
246C22×D61—82no—
247C22×D61C2×C22⋊C412yes—
248D4×C61—21no—
249D4×C61—21no—
250D4×C61—21no—
251D4×C61—42no—
252D4×C61#13112yes—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61C2×C22⋊C411yescharacteristic
266C2×C4×C61C2×C22⋊C411yescharacteristic
267C2×C4×C61C2×C22⋊C411yescharacteristic
268C2×C4×C61AGL(1,ℤ/8)12yes—
269C2×C4×C61AGL(1,ℤ/8)12yes—
270C2×C4×C61C23×C411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C22⋊D61—24no—
274C22⋊D61—44no—
275C22⋊D61—44no—
276C22⋊D61—44no—
277C22⋊D61—44no—
278C22⋊D61—82no—
279C2×Dic61—22no—
280C2×Dic61—44no—
281C2×Dic61C2×C22⋊C412yes—
282C2×Dic61C2×C22⋊C412yes—
283C2×Dic61C2×C22⋊C412yes—
284D244—22no—
285D244—22no—
286D244—22no—
287D244—44no—
288D244—42no—
289D244—81no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—42no—
295C4×D61—44no—
296C4×D61—44no—
297C4×D61—42no—
298C4×D61—42no—
299Dic122—22no—
300Dic122—44no—
301Dic122—42no—
302C8×C61—42no—
303C8×C61—42no—
304#92418—22no—
305#92418—41no—
306#92418—41no—
307#92418—41no—
308#17511—611nomaximal, Sylow
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—22no—
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—41no—
315C22×C61—41no—
316C22×C61—41no—
317C22×C61—84no—
318C22×C61#41812yes—
319C22×C61C22×C22⋊C411yescharacteristic
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—22no—
323C2×D61—42no—
324C2×D61—42no—
325C2×D61—42no—
326C2×D61—88no—
327C4×C61—21no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—41no—
331C4×C61—41no—
332C4×C61#41812yes—
333C4×C61C2×#13111yescharacteristic
334C4×C61#56811yescharacteristic
335C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
336C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
337Dic61—22no—
338Dic61—22no—
339Dic61—22no—
340Dic61—42no—
341D4×#41—611no—
342#2965—611no—
343#2925—611no—
344#1496—612no—
345#1492—611no—
346#1490—612no—
347#1486D6111yescharacteristic, complemented
348#1484—612no—
349#1480—611no—
350#1478—612no—
351C2×D4⋊C8—611no—
352C2×C61—22no—
353C2×C61—42no—
354C2×C61—81no—
355C2×C61#288011yescharacteristic
356C2×C61C2×#41811yescharacteristic
357C2×C61#290111yescharacteristic
358D61—42no—
359D61—82no—
360C22×#41—1221no—
361#550—611no—
362D4×Q8—611no—
363#548—1221no—
364#547—611no—
365#547—612no—
366#546—1222no—
367#546—1221no—
368D42—612no—
369D42—611no—
370#532—612no—
371#532—611no—
372#525—1222no—
373#525—1221no—
374C4×#41—611no—
375C2×C4×D4—612no—
376C2×C4×D4C2×D6111yescharacteristic
377#446—1221no—
378#436—1221no—
379#435—1222no—
380#433—1222no—
381#432—612no—
382#431—612no—
383#431—612no—
384#431C2×D6111yescharacteristic, complemented
385#423C2×D6111yescharacteristic, complemented
386C2×C4⋊C8—611no—
387#408—1221no—
388#405—1221no—
389C4×M16—612no—
390C2×C4×C8—611no—
391D4⋊C8—611no—
392D4⋊C8—611no—
393D4⋊C8—611no—
394D4⋊C8—611no—
395D4⋊C8—612no—
396D4⋊C8—612no—
397D4⋊C8—612no—
398D4⋊C8—612no—
399D4⋊C8C2×D6112yes—
400D4⋊C8C2×D6112yescomplemented
401C61#1751111yescharacteristic, complemented, Sylow
402C2×#41—611no—
403C2×#41—1222no—
404C2×#41—1222no—
405C2×#41—1221no—
406C2×#41—1221no—
407C2×#41—1221no—
408C2×#41—2444no—
409C2×#41—2444no—
410C22×Q8—1221no—
411C22×D4—612no—
412C22×D4—1221no—
413C22×D4—1222no—
414C22×D4C4×D6111yescharacteristic
415C23×C4—21no—
416C23×C4—1222no—
417C2×M16—21no—
418C2×M16—1222no—
419C22×C8—1221no—
420C22×C8—1221no—
421C4⋊Q8—612no—
422C4⋊Q8—611no—
423#127—612no—
424#127—611no—
425#124—1222no—
426#124—1221no—
427#123—1222no—
428#123—1222no—
429#123—1222no—
430C22⋊Q8—1222no—
431C22⋊Q8—1222no—
432C22⋊Q8—1221no—
433C22⋊Q8—1221no—
434#121—612no—
435#121—612no—
436#121—612no—
437#121—1222no—
438#121—1222no—
439#121—1221no—
440#121—1221no—
441#121—1222no—
442#121—1221no—
443#120—1222no—
444#120—1222no—
445#120—1222no—
446C4×Q8—611no—
447C4×Q8—611no—
448C4×D4—21no—
449C4×D4—21no—
450C4×D4—612no—
451C4×D4—612no—
452C4×D4—612no—
453C4×D4—612no—
454C4×D4—611no—
455C4×D4—611no—
456C4×D4—612no—
457C4×D4—1222no—
458C4×D4—1222no—
459C4×D4—1222no—
460C4×D4—1221no—
461C4×D4C4×D6111yescharacteristic
462C4×D4C4×D6111yescharacteristic
463C4×D4C22×D6111yescharacteristic
464C4×D4C22×D6111yescharacteristic
465#117—611no—
466#117—612no—
467C2×C4⋊C4—612no—
468C2×C4⋊C4C4×D6111yescharacteristic
469C2×C22⋊C4—21no—
470C2×C22⋊C4—1222no—
471C2×C42—612no—
472C2×C42C22×D6111yescharacteristic
473C4⋊C8—21no—
474C4⋊C8—21no—
475C4⋊C8—611no—
476C4⋊C8—611no—
477C4⋊C8—1222no—
478C4⋊C8—1221no—
479C4⋊C8C22×D6112yes—
480C22⋊C8—21no—
481C22⋊C8—1222no—
482C22⋊C8—1222no—
483C22⋊C8—1222no—
484#97—1222no—
485#97—1222no—
486C4×C8—611no—
487C4×C8—611no—
488C4×C8—1222no—
489C4×C8—1221no—
490C4×C8C22×D6112yes—
491C24—21no—
492C24—1222no—
493#41—1222no—
494#41—1222no—
495#41—2444no—
496#41—2444no—
497#41—2448no—
498#41—2444no—
499#41—2444no—
500#41—2442no—
501#41—2442no—
502C2×Q8—611no—
503C2×Q8—1222no—
504C2×Q8—1221no—
505C2×Q8—2444no—
506C2×D4—22no—
507C2×D4—21no—
508C2×D4—41no—
509C2×D4—612no—
510C2×D4—612no—
511C2×D4—611no—
512C2×D4—1222no—
513C2×D4—1222no—
514C2×D4—1222no—
515C2×D4—1222no—
516C2×D4—1224no—
517C2×D4—1222no—
518C2×D4—1222no—
519C2×D4—1221no—
520C2×D4—1221no—
521C2×D4—2444no—
522C2×D4—2442no—
523C2×D4—2448no—
524C2×D4C2×C4×D6111yescharacteristic
525C2×D4C2×C4×D6111yescharacteristic
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—42no—
532C22×C4—42no—
533C22×C4—612no—
534C22×C4—612no—
535C22×C4—612no—
536C22×C4—1222no—
537C22×C4—1222no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—2444no—
541C22×C4—2444no—
542C22×C4C2×C4×D6111yescharacteristic
543C22×C4D4×D6111yescharacteristic
544C22×C4D4×D6111yescharacteristic
545M16—42no—
546M16—2444no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—2442no—
554C2×C8—2442no—
555C4⋊C4—21no—
556C4⋊C4—612no—
557C4⋊C4—612no—
558C4⋊C4—611no—
559C4⋊C4—611no—
560C4⋊C4—612no—
561C4⋊C4—1222no—
562C4⋊C4—1222no—
563C4⋊C4—1221no—
564C4⋊C4C2×C4×D6111yescharacteristic
565C4⋊C4C2×C4×D6111yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—21no—
570C22⋊C4—612no—
571C22⋊C4—612no—
572C22⋊C4—612no—
573C22⋊C4—1222no—
574C22⋊C4—1222no—
575C22⋊C4—1222no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C22⋊C4—1222no—
579C42—612no—
580C42—612no—
581C42—612no—
582C42C2×C4×D6111yescharacteristic
583C42D4×D6111yescharacteristic
584C42D4×D6111yescharacteristic
585C42C23×D6111yescharacteristic
586C23—21no—
587C23—21no—
588C23—42no—
589C23—42no—
590C23—81no—
591C23—612no—
592C23—1222no—
593C23—4882no—
594C23—2444no—
595C23—2444no—
596C23C22⋊C4×D6111yescharacteristic
597Q8—1222no—
598Q8—2444no—
599Q8—2442no—
600D4—21no—
601D4—21no—
602D4—21no—
603D4—42no—
604D4—1224no—
605D4—1222no—
606D4—1222no—
607D4—1222no—
608D4—4881no—
609D4—4882no—
610D4—2444no—
611D4—2444no—
612D4—2444no—
613D4—2444no—
614D4—2444no—
615D4—2442no—
616D4#1451978512yes—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—612no—
630C2×C4—612no—
631C2×C4—612no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1222no—
636C2×C4—1222no—
637C2×C4—2442no—
638C2×C4—2444no—
639C2×C4—2444no—
640C2×C4—2444no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4C22⋊C4×D6111yescharacteristic
644C2×C4C22⋊C4×D6111yescharacteristic
645C2×C4C22⋊C4×D6111yescharacteristic
646C2×C4#1451980112yes—
647C2×C4#1451980412yes—
648C2×C4C22×C4×D6111yescharacteristic
649C2×C4C2×D4×D6111yescharacteristic
650C2×C4C2×D4×D6111yescharacteristic
651C8—42no—
652C8—42no—
653C8—1222no—
654C8—2441no—
655C8—2441no—
656C8—2441no—
657C22—21no—
658C22—21no—
659C22—22no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—84no—
666C22—1222no—
667C22—1222no—
668C22—1222no—
669C22—4888no—
670C22—2442no—
671C22—2442no—
672C22—2442no—
673C22#1683976512yes—
674C22C2×C22⋊C4×D6111yescharacteristic
675C4—21no—
676C4—21no—
677C4—21no—
678C4—41no—
679C4—41no—
680C4—1222no—
681C4—1222no—
682C4—1222no—
683C4—2442no—
684C4#1683976512yes—
685C4C2×#1451978511yescharacteristic
686C4#1684052811yescharacteristic
687C4C2×#1451980111yescharacteristic
688C4C2×#1451980411yescharacteristic
689C2—22no—
690C2—42no—
691C2—81no—
692C2—4882no—
693C2—2442no—
694C2#2145426911yescharacteristic
695C2C2×#1683976511yescharacteristic
696C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1486C2
socleC22×C61C22×C22⋊C4
prime radicalD4×#11758222C2
cosocleC42×C61C24
hypercenter#1486D61
hypocenterC61#17511
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454780C2semidirect
3. #21454487C2semidirect
4. #21444286C2semidirect
5. #21444269C2semidirect
6. #21444260C2semidirect
7. #21444234C2semidirect
8. #21444219C2semidirect
9. #21444208C2semidirect
10. C2×D4⋊#92418C2semidirect
11. C61×#1486C2semidirect
32. #16839818C22semidirect
33. #16839814C22semidirect
34. #16839813C22semidirect
35. C2×C4⋊#92418C22semidirect
38. C4×#11758189C22semidirect
39. C2×C4×#92418C22semidirect
40. #16839491C22semidirect
41. #16839491C22semidirect
44. D4⋊#92418C22semidirect
45. D4⋊#92418C22semidirect
46. #16839489C22semidirect
47. #16839489C22semidirect
49. C61×#431C22semidirect
50. C61×#423C22semidirect
52. C61×D4⋊C8C22semidirect
347. #1486D61semidirect
384. #431C2×D61semidirect
385. #423C2×D61semidirect
400. D4⋊C8C2×D61semidirect
401. C61#17511semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
D4×#117582222
#214547802
#214544872
#214442862
#214442692
#214442602
#214442342
#214442192
#214442082
C2×D4⋊#924182
C61×#14862
#1751161