Finite Group Databasebeta

← #30610775

Subgroups — #30610775

Order 15616.

1284 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462624C211yesmaximal, characteristic, complemented
2#21462464C211yesmaximal, characteristic, complemented
3#21461166C211yesmaximal, characteristic, complemented
4#21461164C212yesmaximal, complemented
5#21461146C212yesmaximal, complemented
6#21461145C211yesmaximal, characteristic, complemented
7#21461126C211yesmaximal, characteristic, complemented
8#21461124C212yesmaximal, complemented
9C2×#16840610C212yesmaximal, complemented
10D61⋊(D4⋊C8)C211yesmaximal, characteristic, complemented
11#21459306C211yesmaximal, characteristic
12C2×#14519864—21no—
13#16840811—21no—
14#16840789—22no—
15D4×#92420—21no—
16#16840781C2211yescharacteristic, complemented
17#16840779C2212yescomplemented
18#16840769—21no—
19#16840768—22no—
20#16840766C2211yescharacteristic, complemented
21#16840765C2212yescomplemented
22D61⋊(C4⋊C8)C2211yescharacteristic, complemented
23C2×C4⋊#92420C2212yescomplemented
24C4×D61⋊C8C2211yescharacteristic, complemented
25C2×C4×#92420C2212yescomplemented
26#16840610C2211yescharacteristic
27#16840610C2211yescharacteristic
28#16840610C2212yes—
29#16840610C2212yescomplemented
30#16840610C2212yescomplemented
31#16840610C2212yescomplemented
32#16840610C2212yescomplemented
33D4⋊#92420C2211yescharacteristic, complemented
34D4⋊#92420C2211yescharacteristic, complemented
35D4⋊#92420C2212yescomplemented
36C2×#41×D61—21no—
37#16840433C411yescharacteristic
38#16840427C411yescharacteristic
39#16840422—21no—
40#16840413C412yes—
41#16840404—22no—
42#16840400—21no—
43D4×D244C411yescharacteristic
44#16840396C412yes—
45#16840313C2211yescharacteristic
46#16840307C2212yes—
47#16840275—21no—
48#16840270—22no—
49C4×#11758224C2211yescharacteristic
50C2×#14519739C2212yes—
51C4×D4×D61C2211yescharacteristic
52C61×#547C411yescharacteristic
53#14519864—44no—
54C2×#11758213—21no—
55C2×#11758217—22no—
56C2×D61⋊C8—21no—
57C2×D61⋊C8—21no—
58C22×#92420—22no—
59C22×#92420—22no—
60#14519844—21no—
61#14519844—22no—
62#14519844—22no—
63C4⋊#92420—22no—
64C4⋊#92420—21no—
65C4⋊#92420—21no—
66C4⋊#92420C2311yescharacteristic
67C4⋊#92420C2312yes—
68C4⋊#92420C2311yescharacteristic
69#14519834—21no—
70C22⋊#92420—22no—
71C4×#92420—22no—
72C4×#92420—21no—
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C4×#92420C2311yescharacteristic
76C2×#11758226—21no—
77C2×#11758222—22no—
78C2×#11758222—21no—
79C2×#11758224—22no—
80C2×#11758224D411yescharacteristic
81#41×D61—44no—
82#41×D61—44no—
83C2×Q8×D61—21no—
84C22×D244C2×C412yes—
85C2×D4×D61—21no—
86C2×D4×D61—22no—
87C2×D4×D61C2×C411yescharacteristic
88C22×C4×D61—22no—
89C22×C4×D61—21no—
90#14519776C2×C412yes—
91#14519774C2×C411yescharacteristic
92#14519773C2×C412yes—
93#14519766—21no—
94#14519763—22no—
95#14519762—22no—
96#14519760—22no—
97#14519755—22no—
98#14519755—22no—
99#14519753—21no—
100#14519752—21no—
101#14519752—21no—
102#14519752C2×C412yes—
103#14519751—21no—
104#14519751C2×C412yes—
105#14519750—22no—
106#14519750—22no—
107#14519749—22no—
108#14519747—22no—
109#14519747—22no—
110#14519745—22no—
111C4×Dic122C2×C411yescharacteristic
112#14519742D411yescharacteristic
113C4×D244C2×C411yescharacteristic
114D4×Dic61D411yescharacteristic
115D4×Dic61C2311yescharacteristic
116#14519739—22no—
117#14519739—22no—
118#14519739—21no—
119#14519739C2×C412yes—
120#14519739C2×C412yes—
121#14519739C2311yescharacteristic
122#14519739C2312yes—
123C4×C22⋊D61—21no—
124C4×C22⋊D61C2×C412yes—
125#14519737—21no—
126#14519737—22no—
127#14519737D412yes—
128#14519735D411yescharacteristic
129#14519733D412yes—
130C4⋊C4×D61C2×C411yescharacteristic
131C2×#11758198—22no—
132C22⋊C4×D61—21no—
133C2×C4×Dic61C2312yes—
134C42×D61C2311yescharacteristic
135C2×#41×C61—21no—
136C4⋊Q8×C61C2×C411yescharacteristic
137#123×C61—22no—
138C22⋊Q8×C61—21no—
139#121×C61C2×C412yes—
140C4×D4×C61C2×C411yescharacteristic
141C4×D4×C61C2×C411yescharacteristic
142C2×C4⋊C4×C61C2×C412yes—
143C23×D61—22no—
144C23×D61—21no—
145#41×C61—44no—
146C2×Q8×C61—21no—
147C2×D4×C61—22no—
148C2×D4×C61C22⋊C411yescharacteristic
149C2×D4×C61C22×C411yescharacteristic
150C22×C4×C61—22no—
151C22×C4×C61—21no—
152C22×C4×C61C22×C412yes—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—21no—
156C2×C22⋊D61—22no—
157C2×C22⋊D61C22⋊C412yes—
158C22×Dic61—21no—
159C22×Dic61C2×D412yes—
160C22×Dic61C2×D412yes—
161#11758226—42no—
162#11758226—42no—
163Q8×D61—44no—
164#11758224—44no—
165#11758224—44no—
166#11758224D814yes—
167#11758224SD1614yes—
168D4×D61—22no—
169D4×D61—41no—
170D4×D61—44no—
171D4×D61—48no—
172#11758222—48no—
173#11758222—44no—
174C2×D244—24no—
175C2×D244—22no—
176C2×D244—21no—
177C2×D244—22no—
178C2×D244—21no—
179C2×D244—42no—
180C2×D244C22×C412yes—
181C2×D244C22×C411yescharacteristic
182C2×C4×D61—21no—
183C2×C4×D61—21no—
184C2×C4×D61—22no—
185C2×C4×D61—22no—
186C2×C4×D61—21no—
187C2×C4×D61—22no—
188C2×C4×D61—44no—
189C2×C4×D61—44no—
190C2×C4×D61—42no—
191C2×C4×D61—42no—
192C2×C4×D61C22×C411yescharacteristic
193C2×C4×D61C2×D411yescharacteristic
194C2×C4×D61C2×D411yescharacteristic
195C2×Dic122—22no—
196C2×Dic122C22⋊C411yescharacteristic
197#11758217—44no—
198C2×#92420—21no—
199C2×#92420—21no—
200C2×#92420—22no—
201C2×#92420—21no—
202C2×#92420—22no—
203C2×#92420—21no—
204C2×#92420—44no—
205C2×#92420—44no—
206#11758213—42no—
207D61⋊C8—42no—
208D61⋊C8—42no—
209C4⋊C4×C61—22no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61C22⋊C411yescharacteristic
212C4⋊C4×C61C22×C411yescharacteristic
213C4⋊C4×C61C22×C412yes—
214C22⋊C4×C61—21no—
215C22⋊C4×C61—22no—
216C22⋊C4×C61C22⋊C412yes—
217C42×C61C22×C411yescharacteristic
218C22⋊Dic61—21no—
219C22⋊Dic61C22⋊C412yes—
220#11758198—21no—
221#11758198—22no—
222#11758198—22no—
223#11758198—22no—
224#11758198—22no—
225#11758198—21no—
226#11758198—22no—
227#11758198C22⋊C412yes—
228C4⋊Dic61C22⋊C411yescharacteristic
229C4⋊Dic61C22×C411yescharacteristic
230#11758196—21no—
231#11758196—22no—
232#11758196C22⋊C412yes—
233#11758196C22⋊C412yes—
234C4×Dic61C22×C412yes—
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D412yes—
237C4×Dic61C2×D411yescharacteristic
238C4×Dic61C2×D412yes—
239C4×Dic61C2411yescharacteristic
240C23×C61—21no—
241C23×C61C2×C22⋊C412yes—
242C22×D61—21no—
243C22×D61—22no—
244C22×D61—42no—
245C22×D61—44no—
246C22×D61—44no—
247C22×D61—42no—
248C22×D61—81no—
249C22×D61—82no—
250C22×D61C2×C22⋊C411yescharacteristic
251Q8×C61—42no—
252D4×C61—21no—
253D4×C61—21no—
254D4×C61—44no—
255D4×C61#10214yes—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—22no—
259C2×C4×C61—21no—
260C2×C4×C61—44no—
261C2×C4×C61—42no—
262C2×C4×C61C2×C22⋊C412yes—
263C2×C4×C61C2×C22⋊C411yescharacteristic
264C2×C4×C61C2×C22⋊C411yescharacteristic
265C2×C4×C61C23×C411yescharacteristic
266C22⋊D61—42no—
267C22⋊D61—44no—
268C22⋊D61—44no—
269C22⋊D61—82no—
270C22⋊D61#10218yes—
271C2×Dic61—21no—
272C2×Dic61—42no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C412yes—
278C2×Dic61C2×D814yes—
279C2×Dic61C2×SD1614yes—
280C2×Dic61C22×D411yescharacteristic
281C2×Dic61C22×D411yescharacteristic
282D244—22no—
283D244—21no—
284D244—22no—
285D244—44no—
286D244—44no—
287D244—42no—
288D244—81no—
289D244#13112yescomplemented
290C4×D61—21no—
291C4×D61—21no—
292C4×D61—41no—
293C4×D61—42no—
294C4×D61—42no—
295C4×D61—41no—
296C4×D61—42no—
297C4×D61—41no—
298C4×D61—42no—
299C4×D61—44no—
300C4×D61—44no—
301C4×D61C2×D812yes—
302C4×D61C2×SD1612yes—
303Dic122—44no—
304Dic122#10214yes—
305#92420—22no—
306#92420—41no—
307#92420—42no—
308#92420—41no—
309#92420—41no—
310#92420—42no—
311#17507—611nomaximal, Sylow
312C22×C61—22no—
313C22×C61—22no—
314C22×C61—41no—
315C22×C61—41no—
316C22×C61—41no—
317C22×C61C2×#10214yes—
318C22×C61C22×C22⋊C411yescharacteristic
319C2×D61—21no—
320C2×D61—21no—
321C2×D61—22no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—42no—
325C2×D61—42no—
326C2×D61—42no—
327C2×D61—84no—
328C2×D61—88no—
329C2×D61C2×#10212yes—
330C4×C61—22no—
331C4×C61—21no—
332C4×C61—42no—
333C4×C61—41no—
334C4×C61C2×#10212yes—
335C4×C61C2×#13111yescharacteristic
336C4×C61#56811yescharacteristic
337Dic61—21no—
338Dic61—21no—
339Dic61—22no—
340Dic61—41no—
341Dic61C2×#10214yes—
342Dic61C22×D811yescharacteristic
343Dic61C22×SD1611yescharacteristic
344D4×#41—611no—
345C8×#41—611no—
346#2906—611no—
347#1492—612no—
348#1492—611no—
349#1486—611no—
350#1486—612no—
351#1480—612no—
352#1480—611no—
353C2×D4⋊C8—611no—
354C2×D4⋊C8—612no—
355C2×C61—24no—
356C2×C61—81no—
357C2×C61#288011yescharacteristic
358C2×C61C22×#10211yescharacteristic
359C2×C61#290111yescharacteristic
360D61—22no—
361D61—42no—
362D61—82no—
363C22×#41—1221no—
364C2×#131—1221no—
365#550—611no—
366D4×Q8—611no—
367#548—1221no—
368#547—612no—
369#547#692311yescharacteristic
370#546—1222no—
371#546—1221no—
372D42—612no—
373D42—611no—
374#532—611no—
375#532—612no—
376#525—1221no—
377#525—1222no—
378C4×#41—611no—
379C2×C4×D4—611no—
380C2×C4×D4—612no—
381C8×Q8—1221no—
382C8×D4—1221no—
383C8×D4—1222no—
384#431—612no—
385#431—611no—
386#424—1222no—
387#424—1221no—
388#423—612no—
389#423—611no—
390C2×C4⋊C8—612no—
391C2×C4⋊C8—611no—
392C2×C4×C8—612no—
393C2×C4×C8—611no—
394D4⋊C8—611no—
395D4⋊C8—611no—
396D4⋊C8—612no—
397D4⋊C8—611no—
398D4⋊C8—611no—
399D4⋊C8—612no—
400D4⋊C8—612no—
401D4⋊C8—612no—
402D4⋊C8—612no—
403D4⋊C8—612no—
404C61#1750711yescharacteristic, complemented, Sylow
405C2×#41—21no—
406C2×#41—611no—
407C2×#41—1222no—
408C2×#41—1221no—
409C2×#41—1222no—
410C2×#41—1221no—
411C2×#41—2444no—
412C2×#41—2444no—
413C22×Q8—1221no—
414C22×D4—611no—
415C22×D4—612no—
416C22×D4—1222no—
417C22×D4—1221no—
418C23×C4—1222no—
419C23×C4—1221no—
420#131—2444no—
421C2×M16—1222no—
422C2×M16—1221no—
423C22×C8—1222no—
424C22×C8—1222no—
425C22×C8—1221no—
426C22×C8—1221no—
427C4⋊Q8—612no—
428C4⋊Q8C2×#692311yescharacteristic
429#127—612no—
430#127—611no—
431#124—1222no—
432#124—1221no—
433#123—22no—
434#123—1222no—
435#123—1222no—
436C22⋊Q8—21no—
437C22⋊Q8—1222no—
438C22⋊Q8—1222no—
439C22⋊Q8—1221no—
440#121—612no—
441#121—612no—
442#121—1222no—
443#121—1222no—
444#121—1221no—
445#121—1221no—
446#121—1222no—
447#121—1221no—
448#121C2×#692312yes—
449#120—1222no—
450#120—1222no—
451#120—1222no—
452C4×Q8—611no—
453C4×Q8—611no—
454C4×D4—611no—
455C4×D4—612no—
456C4×D4—612no—
457C4×D4—611no—
458C4×D4—611no—
459C4×D4—612no—
460C4×D4—612no—
461C4×D4—612no—
462C4×D4—611no—
463C4×D4—1221no—
464C4×D4—1222no—
465C4×D4—1222no—
466C4×D4—1222no—
467C4×D4—1221no—
468C4×D4—1221no—
469C4×D4C2×#692311yescharacteristic
470C4×D4C2×#692311yescharacteristic
471#117—611no—
472#117—612no—
473C2×C4⋊C4—611no—
474C2×C4⋊C4C2×#692312yes—
475C2×C22⋊C4—1221no—
476C2×C22⋊C4—1222no—
477C2×C42—612no—
478C2×C42—611no—
479C4⋊C8—611no—
480C4⋊C8—612no—
481C4⋊C8—611no—
482C4⋊C8—1222no—
483C4⋊C8—1221no—
484C4⋊C8—1221no—
485C4⋊C8—1221no—
486C4⋊C8—1222no—
487C4⋊C8—1222no—
488C22⋊C8—1222no—
489C22⋊C8—1221no—
490C4×C8—611no—
491C4×C8—612no—
492C4×C8—611no—
493C4×C8—1222no—
494C4×C8—1221no—
495C24—1221no—
496C24—1222no—
497#41—44no—
498#41—614no—
499#41—614no—
500#41—2448no—
501#41—2444no—
502#41—2444no—
503#41—2444no—
504#41—2442no—
505#41—2442no—
506C2×Q8—21no—
507C2×Q8—611no—
508C2×Q8—1222no—
509C2×Q8—2444no—
510C2×D4—22no—
511C2×D4—612no—
512C2×D4—611no—
513C2×D4—612no—
514C2×D4—1224no—
515C2×D4—1222no—
516C2×D4—1222no—
517C2×D4—1221no—
518C2×D4—1222no—
519C2×D4—1222no—
520C2×D4—1222no—
521C2×D4—1221no—
522C2×D4—1222no—
523C2×D4—1221no—
524C2×D4—2442no—
525C2×D4—2441no—
526C2×D4—2448no—
527C2×D4—2444no—
528C2×D4C22⋊#692311yescharacteristic
529C2×D4C22×#692311yescharacteristic
530C22×C4—22no—
531C22×C4—21no—
532C22×C4—612no—
533C22×C4—611no—
534C22×C4—612no—
535C22×C4—611no—
536C22×C4—611no—
537C22×C4—1221no—
538C22×C4—1222no—
539C22×C4—1222no—
540C22×C4—1222no—
541C22×C4—1221no—
542C22×C4—1221no—
543C22×C4—1221no—
544C22×C4—2444no—
545C22×C4—2444no—
546C22×C4—2442no—
547C22×C4—2442no—
548C22×C4C22×#692312yes—
549M16—2442no—
550M16—2444no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1222no—
554C2×C8—1221no—
555C2×C8—1222no—
556C2×C8—1221no—
557C2×C8—2442no—
558C2×C8—2444no—
559C2×C8—2442no—
560C2×C8—2444no—
561C4⋊C4—22no—
562C4⋊C4—21no—
563C4⋊C4—612no—
564C4⋊C4—612no—
565C4⋊C4—611no—
566C4⋊C4—611no—
567C4⋊C4—1221no—
568C4⋊C4—1222no—
569C4⋊C4C22⋊#692311yescharacteristic
570C4⋊C4C22×#692311yescharacteristic
571C4⋊C4C22×#692312yes—
572C22⋊C4—21no—
573C22⋊C4—22no—
574C22⋊C4—612no—
575C22⋊C4—612no—
576C22⋊C4—1221no—
577C22⋊C4—1222no—
578C22⋊C4—1222no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C22⋊C4—1221no—
582C22⋊C4—1221no—
583C22⋊C4—1222no—
584C22⋊C4C22⋊#692312yes—
585C42—612no—
586C42—611no—
587C42—612no—
588C42—611no—
589C42—611no—
590C42—612no—
591C42C22×#692311yescharacteristic
592C23—21no—
593C23—611no—
594C23—1221no—
595C23—1222no—
596C23—4882no—
597C23—4881no—
598C23—2444no—
599C23—2442no—
600C23—2442no—
601C23—2444no—
602C23C2×C22⋊#692312yes—
603Q8—42no—
604Q8—614no—
605Q8—2444no—
606D4—21no—
607D4—21no—
608D4—44no—
609D4—618no—
610D4—612no—
611D4—1222no—
612D4—1221no—
613D4—1222no—
614D4—4881no—
615D4—4882no—
616D4—2444no—
617D4—2444no—
618D4—2444no—
619D4—2442no—
620D4—2444no—
621D4—2442no—
622D4#1451983914yes—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—614no—
630C2×C4—614no—
631C2×C4—612no—
632C2×C4—611no—
633C2×C4—612no—
634C2×C4—612no—
635C2×C4—611no—
636C2×C4—612no—
637C2×C4—611no—
638C2×C4—1221no—
639C2×C4—1221no—
640C2×C4—1221no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2444no—
645C2×C4—2444no—
646C2×C4—2442no—
647C2×C4—2442no—
648C2×C4—2442no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2442no—
652C2×C4—2442no—
653C2×C4C2×C22⋊#692312yes—
654C2×C4C2×C22⋊#692311yescharacteristic
655C2×C4C2×C22⋊#692311yescharacteristic
656C2×C4C23×#692311yescharacteristic
657C8—1222no—
658C8—2441no—
659C8—2441no—
660C8—2442no—
661C8—2441no—
662C8—2442no—
663C22—22no—
664C22—22no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—612no—
669C22—1221no—
670C22—1221no—
671C22—1222no—
672C22—4888no—
673C22—4884no—
674C22—2442no—
675C22—2442no—
676C22—2442no—
677C22—2441no—
678C22—2441no—
679C22C2×#1451983914yes—
680C22C22×C22⋊#692311yescharacteristic
681C4—22no—
682C4—21no—
683C4—41no—
684C4—42no—
685C4—611no—
686C4—611no—
687C4—614no—
688C4—1221no—
689C4—1221no—
690C4—1222no—
691C4—2441no—
692C4C2×#1451983912yes—
693C4D61⋊#13111yescharacteristic
694C4#1684084111yescharacteristic
695C2—24no—
696C2—81no—
697C2—1222no—
698C2—4882no—
699C2—2442no—
700C2#2146240911yescharacteristic
701C2C22×#1451983911yescharacteristic
702C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#547C4
socleC22×C61C22×C22⋊C4
prime radical#21459306C2
cosocleC4×Dic61C24
hypercenter#547#6923
hypocenterC61#17507
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #21462624C2semidirect
2. #21462464C2semidirect
3. #21461166C2semidirect
4. #21461164C2semidirect
5. #21461146C2semidirect
6. #21461145C2semidirect
7. #21461126C2semidirect
8. #21461124C2semidirect
9. C2×#16840610C2semidirect
10. D61⋊(D4⋊C8)C2semidirect
16. #16840781C22semidirect
17. #16840779C22semidirect
20. #16840766C22semidirect
21. #16840765C22semidirect
22. D61⋊(C4⋊C8)C22semidirect
23. C2×C4⋊#92420C22semidirect
24. C4×D61⋊C8C22semidirect
25. C2×C4×#92420C22semidirect
29. #16840610C22semidirect
30. #16840610C22semidirect
31. #16840610C22semidirect
32. #16840610C22semidirect
33. D4⋊#92420C22semidirect
34. D4⋊#92420C22semidirect
35. D4⋊#92420C22semidirect
289. D244#131semidirect
404. C61#17507semidirect

Maximal subgroups

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

subgroupindex
#214626242
#214624642
#214611662
#214611642
#214611462
#214611452
#214611262
#214611242
C2×#168406102
D61⋊(D4⋊C8)2
#214593062
#1750761