Finite Group Databasebeta

← #29696911

Subgroups — #29696911

Order 15616.

1150 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
1#21459416C211yesmaximal, characteristic
2M16×Dic122C211yesmaximal, characteristic, complemented
3C4×#14519742C211yesmaximal, characteristic, complemented
4#21446736C212yesmaximal, complemented
5#21446648C211yesmaximal, characteristic, complemented
6#21446645C212yesmaximal, complemented
7#21445840C211yesmaximal, characteristic, complemented
8#21445837C212yesmaximal, complemented
9C4×#14519703C211yesmaximal, characteristic, complemented
10#104×Dic61C212yesmaximal, complemented
11C61×#1980C211yesmaximal, characteristic, complemented
12#16840498C411yescharacteristic, complemented
13#16840494C412yescomplemented
14#16840487C411yescharacteristic, complemented
15#16840486C411yescharacteristic, complemented
16#16840481C412yescomplemented
17#16840464—21no—
18#16840455—22no—
19#16840432C411yescharacteristic, complemented
20#16840418—21no—
21#16840308C2212yes—
22#16840294—21no—
23#16840292—22no—
24#41×Dic61C2212yes—
25C4×#11758222C2211yescharacteristic
26C4×#11758224—21no—
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519742—21no—
29#16839892—21no—
30#16839890—22no—
31C8×Dic122—21no—
32#16839786C2211yescharacteristic
33#16839785C2212yes—
34C2×#14519705C2212yes—
35C2×#14519704C2212yes—
36C2×#14519704C2212yes—
37C2×#14519703C2211yescharacteristic
38C2×#14519703C2211yescharacteristic
39M16×Dic61C2212yes—
40C4⋊C4×Dic61—21no—
41C4×#11758196C2212yes—
42#16839613—22no—
43C42×Dic61C2212yes—
44C42×Dic61C2211yescharacteristic
45#16839523C2211yescharacteristic
46#16839523C2211yescharacteristic
47#16839522C2212yes—
48C61×#532C2211yescharacteristic
49C2×#104×C61C2212yes—
50C2×#104×C61C2212yes—
51C4×M16×C61C2211yescharacteristic
52C4×C4⋊C4×C61C2211yescharacteristic
53C2×#11758222C2311yescharacteristic
54C2×#11758224—21no—
55C22×Dic122D411yescharacteristic
56C2×#11758194C2312yes—
57#14519775C2×C412yes—
58#14519773C2×C412yes—
59#14519772—22no—
60#14519770—22no—
61#14519770—22no—
62#14519768—21no—
63#14519767—22no—
64#14519766C2×C412yes—
65#14519764—21no—
66#14519764C2×C412yes—
67#14519763—22no—
68#14519761—22no—
69#14519759—22no—
70#14519758—22no—
71#14519757—21no—
72#14519754—22no—
73#14519749—22no—
74#14519749—22no—
75#14519749—22no—
76#14519748—21no—
77Q8×Dic61C2×C412yes—
78Q8×Dic61C2×C412yes—
79C4×Dic122C2×C411yescharacteristic
80C4×Dic122C2×C411yescharacteristic
81C4×Dic122C2×C412yes—
82C4×Dic122C2×C411yescharacteristic
83C4×Dic122C2×C411yescharacteristic
84C4×Dic122D411yescharacteristic
85C4×Dic122D412yes—
86C4×Dic122D411yescharacteristic
87#14519742—28no—
88#14519742—21no—
89C4×D244C2×C411yescharacteristic
90C4×D244C2×C411yescharacteristic
91D4×Dic61—22no—
92D4×Dic61—22no—
93D4×Dic61—21no—
94D4×Dic61C2×C412yes—
95D4×Dic61C2×C412yes—
96C4×C22⋊D61—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100#14519737—22no—
101#14519736—22no—
102#14519736C2311yescharacteristic
103#14519735—21no—
104#14519734—21no—
105#14519733—22no—
106C2×C4⋊Dic61D411yescharacteristic
107C2×#11758196—22no—
108C2×#11758196D412yes—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61—22no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61—22no—
114C2×C4×Dic61C2311yescharacteristic
115C2×C4×Dic61C2311yescharacteristic
116C2×C4×Dic61C2312yes—
117C42×D61—21no—
118C4⋊#92418—21no—
119C4⋊#92418—21no—
120#14519706—22no—
121#14519706—22no—
122#14519705—42no—
123#14519704—24no—
124#14519704—42no—
125#14519703—22no—
126#14519703—22no—
127#14519687—22no—
128C8×Dic61—22no—
129#14519684—22no—
130C2×#41×C61C2312yes—
131C2×M16×C61C2311yescharacteristic
132C2×M16×C61C2311yescharacteristic
133C4⋊Q8×C61C2×C411yescharacteristic
134#127×C61C2×C411yescharacteristic
135#124×C61C2×C412yes—
136#121×C61—21no—
137#121×C61—21no—
138C4×D4×C61—21no—
139C4×D4×C61—21no—
140C2×C4⋊C4×C61—21no—
141C2×C42×C61C2312yes—
142C2×C42×C61C2311yescharacteristic
143#104×C61—24no—
144#104×C61—42no—
145#97×C61—21no—
146C4×C8×C61—21no—
147#95×C61—21no—
148#41×C61—22no—
149#41×C61—22no—
150#41×C61—22no—
151#41×C61—22no—
152C2×Q8×C61C22×C412yes—
153C2×D4×C61—22no—
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61C22×C412yes—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—21no—
163C2×C22⋊D61—22no—
164C2×C22⋊D61—22no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—22no—
168C22×Dic61—22no—
169C22×Dic61C2×D412yes—
170C22×Dic61C2×D412yes—
171#11758224—44no—
172#11758222—21no—
173#11758222—21no—
174#11758222—21no—
175#11758222—21no—
176C2×D244C22×C411yescharacteristic
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×Dic122—24no—
181C2×Dic122C22⋊C412yes—
182C2×Dic122C22⋊C411yescharacteristic
183C2×Dic122C22⋊C411yescharacteristic
184C2×Dic122C22×C411yescharacteristic
185M16×C61—21no—
186M16×C61—21no—
187M16×C61—21no—
188M16×C61#4112yescomplemented
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C4⋊C4×C61—24no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C42×C61—22no—
197C42×C61—22no—
198C42×C61C22×C411yescharacteristic
199C42×C61C22×C411yescharacteristic
200C42×C61C2×D411yescharacteristic
201C42×C61C2×D411yescharacteristic
202C22⋊Dic61—21no—
203C22⋊Dic61—22no—
204C22⋊Dic61—22no—
205C22⋊Dic61—22no—
206C22⋊Dic61—22no—
207#11758198—21no—
208#11758198—21no—
209#11758198—22no—
210#11758198—22no—
211C4⋊Dic61—22no—
212C4⋊Dic61—22no—
213C4⋊Dic61C22⋊C411yescharacteristic
214C4⋊Dic61C22⋊C411yescharacteristic
215C4⋊Dic61C22⋊C412yes—
216C4⋊Dic61C22×C411yescharacteristic
217C4⋊Dic61C22×C411yescharacteristic
218#11758196—28no—
219#11758196—21no—
220#11758196—21no—
221#11758196—22no—
222#11758196—22no—
223#11758196C22⋊C412yes—
224#11758196C22⋊C412yes—
225#11758196C22⋊C412yes—
226#11758196C22⋊C412yes—
227C4×Dic61—28no—
228C4×Dic61—24no—
229C4×Dic61—21no—
230C4×Dic61—21no—
231C4×Dic61—21no—
232C4×Dic61—21no—
233C4×Dic61—22no—
234C4×Dic61—22no—
235C4×Dic61—21no—
236C4×Dic61—21no—
237C4×Dic61—44no—
238C4×Dic61—42no—
239C4×Dic61C22×C412yes—
240C4×Dic61C22×C412yes—
241C4×Dic61C2×D412yes—
242C4×Dic61C2×D412yes—
243C4×Dic61#4112yescomplemented
244#11758194—22no—
245#11758194—22no—
246C2×#92418—22no—
247C23×C61—22no—
248C23×C61C22×D411yescharacteristic
249C22×D61—21no—
250Q8×C61—22no—
251Q8×C61—22no—
252D4×C61—22no—
253D4×C61—22no—
254D4×C61—42no—
255D4×C61—42no—
256D4×C61—42no—
257D4×C61—81no—
258C2×C4×C61—22no—
259C2×C4×C61—24no—
260C2×C4×C61—22no—
261C2×C4×C61—41no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61C2×C22⋊C411yescharacteristic
267C2×C4×C61C2×C22⋊C411yescharacteristic
268C2×C4×C61C2×C22⋊C411yescharacteristic
269C2×C4×C61C2×C22⋊C411yescharacteristic
270C2×C4×C61C23×C411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C2×C4×C612−1+411yescharacteristic
276C22⋊D61—41no—
277C22⋊D61—41no—
278C22⋊D61—44no—
279C22⋊D61—82no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—24no—
284C2×Dic61—22no—
285C2×Dic61—21no—
286C2×Dic61—21no—
287C2×Dic61—21no—
288C2×Dic61—42no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—44no—
292C2×Dic61—82no—
293C2×Dic61C2×C22⋊C412yes—
294C2×Dic61C2×C22⋊C412yes—
295C2×Dic61C2×C22⋊C412yes—
296C2×Dic61C2×C22⋊C412yes—
297D244—21no—
298D244—21no—
299C4×D61—41no—
300C4×D61—41no—
301C4×D61—42no—
302C4×D61—81no—
303Dic122—22no—
304Dic122—24no—
305Dic122—21no—
306Dic122—21no—
307Dic122#10414yescomplemented
308C8×C61—22no—
309C8×C61—41no—
310#92418—42no—
311#22550—611nomaximal, Sylow
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—42no—
315C22×C61—42no—
316C22×C61—42no—
317C22×C61C22×C22⋊C411yescharacteristic
318C22×C61#54811yescharacteristic
319C22×C61#55011yescharacteristic
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—41no—
323C4×C61—21no—
324C4×C61—21no—
325C4×C61—21no—
326C4×C61—44no—
327C4×C61—42no—
328C4×C61C2×#10412yes—
329C4×C61C4×#4111yescharacteristic
330C4×C61#52011yescharacteristic
331C4×C61#54811yescharacteristic
332C4×C61#55011yescharacteristic
333Dic61—24no—
334Dic61—22no—
335Dic61—22no—
336Dic61—22no—
337Dic61—42no—
338Dic61—41no—
339Dic61—41no—
340Dic61—41no—
341Dic61C2×#10414yes—
342#3480—611no—
343Q8×M16—611no—
344C42×Q8—611no—
345#1980—612no—
346#1980D6111yescharacteristic, complemented
347#1962—612no—
348#1962—611no—
349#1806—612no—
350#1806—611no—
351C4×#104—612no—
352C4×#104—611no—
353C2×C61—21no—
354C2×C61—21no—
355C2×C61—82no—
356C2×C61#234111yescharacteristic
357C2×C61C22×#10411yescharacteristic
358C2×C61#290111yescharacteristic
359D61—81no—
360#559—611no—
361#559—612no—
362#557—611no—
363#556—611no—
364#556—612no—
365#553—1222no—
366#553—1221no—
367#550—611no—
368#548—1221no—
369#532—612no—
370#532C2×D6111yescharacteristic
371#529—1222no—
372#529—1221no—
373C4×#41—612no—
374C4×#41—611no—
375C4×#41—1221no—
376C2×C4×Q8—611no—
377C2×C4×Q8—1221no—
378#446—1221no—
379#446—1222no—
380C8×Q8—1221no—
381#423—612no—
382#423—611no—
383C2×#104—612no—
384C2×#104—611no—
385C2×#104—611no—
386C2×#104—612no—
387C2×#104—612no—
388C2×#104C2×D6112yes—
389C2×#104C2×D6112yes—
390C4×M16—612no—
391C4×M16C2×D6111yescharacteristic
392C4×C4⋊C4—612no—
393C4×C4⋊C4—1222no—
394C4×C4⋊C4—1221no—
395C4×C4⋊C4C2×D6111yescharacteristic
396C43—612no—
397C43—611no—
398#339—611no—
399#339—611no—
400#339—612no—
401C61#2255011yescharacteristic, complemented, Sylow
402C2×#41—611no—
403C2×#41—1221no—
404C2×#41C22×D6112yes—
405C22×Q8—611no—
406C2×M16—612no—
407C2×M16C22×D6111yescharacteristic
408C2×M16C22×D6111yescharacteristic
409C4⋊Q8—612no—
410C4⋊Q8C4×D6111yescharacteristic
411#127—612no—
412#127C4×D6111yescharacteristic
413#126—1222no—
414#126—1222no—
415#126—1222no—
416#125—1222no—
417#125—1221no—
418#124—612no—
419#124—612no—
420#124—1222no—
421#124—1221no—
422#124C4×D6112yes—
423#123—1222no—
424#123—1222no—
425#123—1222no—
426C22⋊Q8—1222no—
427C22⋊Q8—1221no—
428#121—21no—
429#121—21no—
430#121—1221no—
431#121—1222no—
432#121—1222no—
433#121—1222no—
434C4×Q8—611no—
435C4×Q8—611no—
436C4×Q8—612no—
437C4×Q8—611no—
438C4×Q8—612no—
439C4×Q8—611no—
440C4×Q8—612no—
441C4×Q8—612no—
442C4×Q8—611no—
443C4×Q8—611no—
444C4×Q8—1221no—
445C4×Q8—1228no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—612no—
449C4×D4—612no—
450C4×D4—611no—
451C4×D4—611no—
452C4×D4—1222no—
453C4×D4—1222no—
454C4×D4—1222no—
455C4×D4—1221no—
456C4×D4—1222no—
457C4×D4—1222no—
458C4×D4—1221no—
459C4×D4—1221no—
460#117—611no—
461#117—1222no—
462#117—1222no—
463#117—1221no—
464#117—1221no—
465C2×C4⋊C4—21no—
466C2×C4⋊C4—612no—
467C2×C4⋊C4—611no—
468C2×C4⋊C4—1222no—
469C2×C42—612no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—1222no—
473C2×C42—1222no—
474C2×C42—1221no—
475C2×C42—1221no—
476C2×C42—1222no—
477C2×C42—1221no—
478C2×C42C22×D6112yes—
479C2×C42C22×D6111yescharacteristic
480C4⋊C8—1222no—
481C4⋊C8—1222no—
482C4⋊C8—1221no—
483C4⋊C8—1221no—
484#104—24no—
485#104—42no—
486#104—1222no—
487#104—1224no—
488#104—1222no—
489#104—2442no—
490#104—2442no—
491#97—21no—
492#97—1222no—
493C4×C8—21no—
494C4×C8—1222no—
495#95—21no—
496#95—1222no—
497#41—22no—
498#41—22no—
499#41—22no—
500#41—22no—
501#41—1221no—
502#41—1221no—
503#41—1221no—
504#41—1221no—
505#41—2444no—
506C2×Q8—611no—
507C2×Q8—612no—
508C2×Q8—611no—
509C2×Q8—611no—
510C2×Q8—1224no—
511C2×Q8C2×C4×D6112yes—
512C2×D4—22no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—611no—
516C2×D4—1222no—
517C2×D4—1222no—
518C2×D4—1221no—
519C2×D4C2×C4×D6112yes—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—612no—
523C22×C4—612no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4D4×D6111yescharacteristic
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534M16—21no—
535M16—21no—
536M16—21no—
537M16—1222no—
538M16—1222no—
539M16#1175822412yescomplemented
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—1222no—
543C4⋊C4—24no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—612no—
547C4⋊C4—612no—
548C4⋊C4—612no—
549C4⋊C4—612no—
550C4⋊C4—611no—
551C4⋊C4—611no—
552C4⋊C4—612no—
553C4⋊C4—611no—
554C4⋊C4—611no—
555C4⋊C4—1222no—
556C4⋊C4—1222no—
557C4⋊C4—1222no—
558C4⋊C4—1222no—
559C4⋊C4—1228no—
560C4⋊C4—1221no—
561C4⋊C4—1221no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—1222no—
565C22⋊C4—1222no—
566C22⋊C4—1222no—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C22⋊C4—1222no—
570C22⋊C4—1221no—
571C22⋊C4—1221no—
572C22⋊C4—1221no—
573C42—22no—
574C42—22no—
575C42—612no—
576C42—612no—
577C42—612no—
578C42—612no—
579C42—612no—
580C42—1222no—
581C42—1222no—
582C42—1224no—
583C42—1228no—
584C42—1221no—
585C42—1221no—
586C42—1221no—
587C42—1221no—
588C42—1221no—
589C42—1221no—
590C42—2442no—
591C42—2444no—
592C42C2×C4×D6111yescharacteristic
593C42C2×C4×D6111yescharacteristic
594C42D4×D6111yescharacteristic
595C42D4×D6111yescharacteristic
596C23—22no—
597C23—1221no—
598C23C2×D4×D6111yescharacteristic
599Q8—22no—
600Q8—22no—
601Q8—614no—
602Q8—1224no—
603Q8—1222no—
604Q8—1221no—
605Q8—1221no—
606D4—22no—
607D4—22no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—81no—
612D4—1221no—
613D4—1221no—
614D4—4882no—
615D4—2444no—
616D4—2441no—
617D4—2441no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—612no—
627C2×C4—612no—
628C2×C4—612no—
629C2×C4—612no—
630C2×C4—1224no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1222no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—4881no—
639C2×C4—4882no—
640C2×C4—2444no—
641C2×C4—2442no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2442no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4C22⋊C4×D6111yescharacteristic
648C2×C4C22⋊C4×D6111yescharacteristic
649C2×C4C22⋊C4×D6111yescharacteristic
650C2×C4C22⋊C4×D6111yescharacteristic
651C2×C4C22×C4×D6111yescharacteristic
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4#41×D6111yescharacteristic
654C2×C4C2×#1175822411yescharacteristic
655C2×C4#1451982211yescharacteristic
656C2×C4#1451982411yescharacteristic
657C8—22no—
658C8—41no—
659C8—2442no—
660C22—21no—
661C22—21no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—2441no—
666C22—2441no—
667C22—2441no—
668C22C2×C22⋊C4×D6111yescharacteristic
669C22#1684042011yescharacteristic
670C22#1684043211yescharacteristic
671C4—21no—
672C4—21no—
673C4—21no—
674C4—44no—
675C4—42no—
676C4—614no—
677C4—1224no—
678C4—1222no—
679C4—1222no—
680C4—1222no—
681C4—2442no—
682C4—2441no—
683C4—2441no—
684C4—2441no—
685C4#104×D6112yes—
686C4C4×#1175822411yescharacteristic
687C4#1684024111yescharacteristic
688C4#1684041811yescharacteristic
689C4#1684043011yescharacteristic
690C2—21no—
691C2—21no—
692C2—82no—
693C2—4881no—
694C2#2144865811yescharacteristic
695C2C2×#104×D6111yescharacteristic
696C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1980C2
socleC22×C61C22×C22⋊C4
prime radical#21459416C2
cosocleC22×C4×C61C24
hypercenter#1980D61
hypocenterC61#22550
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. M16×Dic122C2semidirect
3. C4×#14519742C2semidirect
4. #21446736C2semidirect
5. #21446648C2semidirect
6. #21446645C2semidirect
7. #21445840C2semidirect
8. #21445837C2semidirect
9. C4×#14519703C2semidirect
10. #104×Dic61C2semidirect
11. C61×#1980C2semidirect
12. #16840498C4semidirect
13. #16840494C4semidirect
14. #16840487C4semidirect
15. #16840486C4semidirect
16. #16840481C4semidirect
19. #16840432C4semidirect
188. M16×C61#41semidirect
243. C4×Dic61#41semidirect
307. Dic122#104semidirect
346. #1980D61semidirect
401. C61#22550semidirect
539. M16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214594162
M16×Dic1222
C4×#145197422
#214467362
#214466482
#214466452
#214458402
#214458372
C4×#145197032
#104×Dic612
C61×#19802
#2255061