Finite Group Databasebeta

← #29693916

Subgroups — #29693916

Order 15616.

796 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456296C211yesmaximal, characteristic
2#21454736C211yesmaximal, characteristic, complemented
3#21454677C211yesmaximal, characteristic, complemented
4#21446722C211yesmaximal, characteristic, complemented
5#21446353C211yesmaximal, characteristic, complemented
6#21446348C211yesmaximal, characteristic, complemented
7#21446344C211yesmaximal, characteristic, complemented
8#21446339C211yesmaximal, characteristic, complemented
9#21446316C211yesmaximal, characteristic, complemented
10#21446309C211yesmaximal, characteristic, complemented
11#21446297C211yesmaximal, characteristic, complemented
12#21446290C211yesmaximal, characteristic, complemented
13#21445942C211yesmaximal, characteristic, complemented
14#21445574C211yesmaximal, characteristic, complemented
15C61×#1976C211yesmaximal, characteristic, complemented
16C2×#14519807C2211yescharacteristic
17C2×#14519802C2211yescharacteristic
18C22×#11758190C2211yescharacteristic
19#16840529C2211yescharacteristic
20#16840529C2211yescharacteristic
21#16840333C2211yescharacteristic
22#16840326C2211yescharacteristic
23#16840239C2211yescharacteristic
24#16840236C2211yescharacteristic
25#16840155C412yescomplemented
26#16840150C412yescomplemented
27#16840143C412yescomplemented
28#16840133C412yescomplemented
29#16839884—22no—
30#16839879—22no—
31#16839861—22no—
32#16839849—22no—
33C2×#14519713C2211yescharacteristic
34#16839779C2211yescharacteristic
35#16839779C2211yescharacteristic
36#16839770C2211yescharacteristic
37#16839770C2211yescharacteristic
38#16839766C2211yescharacteristic
39#16839766C2211yescharacteristic
40C2×#14519695C2211yescharacteristic
41C2×#14519693C2211yescharacteristic
42#16839717C2211yescharacteristic
43#16839717C2211yescharacteristic
44#16839532C2211yescharacteristic
45#16839532C2211yescharacteristic
46#16839520C2211yescharacteristic
47#16839519C2211yescharacteristic
48#16839518C2211yescharacteristic
49#16839518C2211yescharacteristic
50#16839518C2211yescharacteristic
51#16839518C2211yescharacteristic
52C2×#135×C61C2211yescharacteristic
53C61×#427C2211yescharacteristic
54C61×#421C2211yescharacteristic
55C61×#421C2211yescharacteristic
56C61×#410C2211yescharacteristic
57C61×#410C2211yescharacteristic
58C61×#405C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×D244C2311yescharacteristic
61#14519807—41no—
62#14519807—41no—
63#14519802—41no—
64#14519802—41no—
65C2×#11758202C2×C412yescomplemented
66C2×#11758190—21no—
67C2×#11758190—21no—
68C2×#11758190—21no—
69C2×#11758190—21no—
70C2×#11758190C2×C412yes—
71C2×#11758190C2×C412yes—
72C2×#11758200C2×C412yescomplemented
73C2×#11758201C2×C412yes—
74C2×#11758199C2×C412yes—
75#14519784—22no—
76#14519784—22no—
77#14519784—41no—
78#14519784—41no—
79C2×#11758194—41no—
80C2×#11758194—41no—
81C2×#11758194C2311yescharacteristic
82C2×#11758194C2311yescharacteristic
83C2×#11758194C2311yescharacteristic
84C2×#11758194C2311yescharacteristic
85#14519775C2×C412yescomplemented
86#14519773C2×C412yes—
87#14519766C2×C412yes—
88#14519764C2×C412yescomplemented
89#14519758—41no—
90#14519756—41no—
91#14519751—41no—
92#14519745—41no—
93C4×Dic122—41no—
94#14519742—22no—
95C4×D244—41no—
96#14519739—22no—
97#14519736C2311yescharacteristic
98#14519736C2311yescharacteristic
99#14519733—41no—
100C22⋊C4×D61—41no—
101#14519713—21no—
102#14519713—21no—
103#14519704—22no—
104#14519704—22no—
105#14519700—22no—
106#14519700—41no—
107#14519697—22no—
108#14519697—41no—
109#14519695—41no—
110#14519693—41no—
111#14519687C2×C412yescomplemented
112#14519687D412yescomplemented
113C8×Dic61C2×C412yescomplemented
114C8×Dic61D412yescomplemented
115C2×#41×C61C2311yescharacteristic
116C2×#41×C61C2311yescharacteristic
117#135×C61—21no—
118#135×C61—21no—
119#135×C61—21no—
120#135×C61—21no—
121C2×Q16×C61C2×C411yescharacteristic, complemented
122C2×SD16×C61C2×C411yescharacteristic, complemented
123C2×SD16×C61C2×C411yescharacteristic, complemented
124C2×D8×C61C2×C411yescharacteristic, complemented
125C2×M16×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127#117×C61C2311yescharacteristic
128#117×C61C2311yescharacteristic
129#117×C61C2311yescharacteristic
130#107×C61—21no—
131#106×C61—21no—
132#104×C61—41no—
133#104×C61—41no—
134#99×C61—41no—
135#99×C61—41no—
136#97×C61—21no—
137C4×C8×C61—21no—
138C23×D61—21no—
139#41×C61—21no—
140#41×C61—21no—
141#41×C61—21no—
142#41×C61—21no—
143#41×C61—21no—
144#41×C61—21no—
145#41×C61—21no—
146#41×C61—21no—
147C2×Q8×C61C22×C411yescharacteristic
148C2×Q8×C61C22×C411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C22×C411yescharacteristic
152C2×D4×C61C22×C411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2411yescharacteristic
156C2×C22⋊D61—42no—
157C22×Dic61—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—42no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C22×C412yes—
171Q16×C61—21no—
172Q16×C61—21no—
173SD16×C61—21no—
174SD16×C61—21no—
175SD16×C61—21no—
176SD16×C61—21no—
177D8×C61—21no—
178D8×C61—21no—
179M16×C61—21no—
180M16×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C22×C411yescharacteristic
183C2×C8×C61C22×C411yescharacteristic
184C2×C8×C61C2×D411yescharacteristic
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61#4111yescharacteristic
187C2×C8×C61#4111yescharacteristic
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C42×C61—21no—
195C42×C61—21no—
196C42×C61—21no—
197C22⋊Dic61—21no—
198C22⋊Dic61—21no—
199C22⋊Dic61—42no—
200#11758202—42no—
201#11758201—42no—
202#11758200—42no—
203#11758199—42no—
204#11758198—42no—
205#11758198—42no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208#11758196—42no—
209#11758196—42no—
210C4×Dic61—42no—
211C4×Dic61C22×C412yes—
212C4×Dic61C2×D412yes—
213#11758194—21no—
214#11758194—21no—
215#11758194—21no—
216#11758194—21no—
217#11758194—21no—
218#11758194—21no—
219#11758194—21no—
220#11758194—21no—
221#11758194—42no—
222#11758194—42no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—42no—
226C2×#92418—42no—
227C2×#92418C22×C412yes—
228C2×#92418C2×D412yes—
229#11758190—22no—
230#11758190—22no—
231#11758190—22no—
232#11758190—22no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61H(5,𝔽2)11yescharacteristic
236C22×D61—22no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—161no—
240Q8×C61—21no—
241Q8×C61—21no—
242Q8×C61—21no—
243Q8×C61—21no—
244D4×C61—21no—
245D4×C61—21no—
246D4×C61—21no—
247D4×C61—21no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—41no—
251D4×C61—41no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61H(5,𝔽2)11yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C22⋊D61—162no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273D244—22no—
274D244—22no—
275D244—41no—
276D244—41no—
277C4×D61—162no—
278Dic122—22no—
279Dic122—22no—
280Dic122—41no—
281Dic122—41no—
282C8×C61—41no—
283C8×C61C4×D411yescharacteristic
284C8×C61C4×D411yescharacteristic
285C8×C61C4×D411yescharacteristic
286C8×C61C4×D411yescharacteristic
287#92418—22no—
288#92418—22no—
289#92418—41no—
290#92418—41no—
291#22284—611nomaximal, Sylow
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—81no—
296C22×C61—81no—
297C22×C61#51811yescharacteristic
298C22×C61#53511yescharacteristic
299C22×C61#54011yescharacteristic
300C2×D61—42no—
301C2×D61—82no—
302C2×D61—82no—
303C2×D61—161no—
304C4×C61—41no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61C2×C4×D411yescharacteristic
310C4×C61#51811yescharacteristic
311C4×C61#53511yescharacteristic
312C4×C61#54011yescharacteristic
313Dic61—42no—
314Dic61—42no—
315Dic61—42no—
316#3154—611no—
317#2957—611no—
318#2949—611no—
319#1977—611no—
320#1976D6111yescharacteristic, complemented
321#1912—611no—
322#1911—611no—
323#1910—611no—
324#1909—611no—
325#1905—611no—
326#1904—611no—
327#1903—611no—
328#1902—611no—
329#1833—611no—
330#1747—611no—
331C2×C61—21no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61—81no—
335C2×C61—81no—
336C2×C61#232811yescharacteristic
337D61—82no—
338C2×#137—611no—
339C2×AGL(1,ℤ/8)—611no—
340C2×#135C2×D6111yescharacteristic
341C22×SD16—611no—
342#568—611no—
343#568—611no—
344#536—611no—
345#535—611no—
346#519—611no—
347#518—611no—
348#497—612no—
349#496—612no—
350#495—612no—
351#492—612no—
352#444—1222no—
353#443—1222no—
354#440—1222no—
355C4×SD16—1222no—
356C2×#108—611no—
357#427C2×D6111yescharacteristic
358#421—611no—
359#421—611no—
360#421C2×D6111yescharacteristic
361#421C2×D6111yescharacteristic
362#419—611no—
363#419—611no—
364#418—611no—
365#418—611no—
366C2×#101—611no—
367C2×#100—611no—
368#410C2×D6111yescharacteristic
369#410C2×D6111yescharacteristic
370#405—611no—
371#405—611no—
372#405C2×D6111yescharacteristic
373#344—611no—
374#344—611no—
375#338—611no—
376H(3,ℤ/4)—611no—
377H(3,ℤ/4)—611no—
378H(3,ℤ/4)—611no—
379H(3,ℤ/4)—611no—
380H(3,ℤ/4)—611no—
381C61#2228411yescharacteristic, complemented, Sylow
382C2×#41C22×D6111yescharacteristic
383C2×#41C22×D6111yescharacteristic
384C22×Q8—611no—
385C22×D4—611no—
386#137—2441no—
387#137—2441no—
388AGL(1,ℤ/8)—2441no—
389AGL(1,ℤ/8)—2441no—
390#135—21no—
391#135—21no—
392#135—21no—
393#135—21no—
394C2×Q16—612no—
395C2×Q16C4×D6111yescharacteristic, complemented
396C2×SD16—612no—
397C2×SD16—612no—
398C2×SD16—612no—
399C2×SD16—612no—
400C2×SD16—1221no—
401C2×SD16—1221no—
402C2×SD16—1221no—
403C2×SD16—1221no—
404C2×SD16C4×D6111yescharacteristic, complemented
405C2×SD16C4×D6111yescharacteristic, complemented
406C2×D8—612no—
407C2×D8C4×D6111yescharacteristic, complemented
408#131—1222no—
409#131—1222no—
410#131—2441no—
411#131—2441no—
412C2×M16—611no—
413C2×M16—611no—
414C2×M16—611no—
415C2×M16—611no—
416C2×M16—2441no—
417C2×M16—2441no—
418C2×M16C22×D6111yescharacteristic
419C22×C8C22×D6111yescharacteristic
420C4⋊Q8—612no—
421#127—612no—
422#124—612no—
423#124—612no—
424#123—2441no—
425C22⋊Q8—2441no—
426#121—2441no—
427#120—2441no—
428C4×Q8—1222no—
429C4×Q8—2441no—
430C4×D4—1222no—
431C4×D4—2441no—
432#117—611no—
433#117—611no—
434#117—2441no—
435#117C22×D6111yescharacteristic
436#117C22×D6111yescharacteristic
437#117C22×D6111yescharacteristic
438C2×C22⋊C4—2441no—
439#108—1221no—
440#108—1221no—
441#107—21no—
442#106—21no—
443#104—41no—
444#104—41no—
445#104—1222no—
446#104—1222no—
447#103—1222no—
448#103—2441no—
449#102—1222no—
450#102—2441no—
451#101—2441no—
452#100—2441no—
453#99—41no—
454#99—41no—
455#97—21no—
456#97—612no—
457#97—612no—
458C4×C8—21no—
459C4×C8—612no—
460C4×C8—612no—
461C24—1221no—
462#41—21no—
463#41—21no—
464#41—21no—
465#41—21no—
466#41—21no—
467#41—21no—
468#41—21no—
469#41—21no—
470C2×Q8—612no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8C2×C4×D6111yescharacteristic
477C2×Q8C2×C4×D6111yescharacteristic
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—612no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—1221no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—2442no—
487C2×D4C2×C4×D6111yescharacteristic
488C2×D4C2×C4×D6111yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—1221no—
492C22×C4—2442no—
493C22×C4C23×D6111yescharacteristic
494Q16—21no—
495Q16—21no—
496Q16—2442no—
497SD16—21no—
498SD16—21no—
499SD16—21no—
500SD16—21no—
501SD16—1222no—
502SD16—1222no—
503SD16—1222no—
504SD16—1222no—
505SD16—2442no—
506SD16—2442no—
507D8—21no—
508D8—21no—
509D8—2442no—
510M16—21no—
511M16—21no—
512M16—1221no—
513M16—1221no—
514M16—1221no—
515M16—1221no—
516M16—1221no—
517M16—1221no—
518M16—1221no—
519M16—1221no—
520M16—2442no—
521M16—2442no—
522C2×C8—21no—
523C2×C8—612no—
524C2×C8—612no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—2442no—
528C2×C8—2442no—
529C2×C8C2×C4×D6111yescharacteristic
530C2×C8C2×C4×D6111yescharacteristic
531C2×C8D4×D6111yescharacteristic
532C2×C8D4×D6111yescharacteristic
533C2×C8#1175822411yescharacteristic
534C2×C8#1175822411yescharacteristic
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—1221no—
546C22⋊C4—1221no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C22⋊C4—2442no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—612no—
554C42—612no—
555C42—2442no—
556C23—21no—
557C23—21no—
558C23—1222no—
559C23—9761no—
560C23—2441no—
561C23—2441no—
562C23#1451982211yescharacteristic
563Q8—21no—
564Q8—21no—
565Q8—21no—
566Q8—21no—
567Q8—1222no—
568Q8—1222no—
569Q8—2441no—
570Q8—2441no—
571D4—21no—
572D4—21no—
573D4—21no—
574D4—21no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—1222no—
580D4—1222no—
581D4—9762no—
582D4—2441no—
583D4—2441no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—9762no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4C22×C4×D6111yescharacteristic
600C2×C4C2×D4×D6111yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982311yescharacteristic
604C2×C4#1451982311yescharacteristic
605C8—41no—
606C8—1222no—
607C8—1222no—
608C8—2441no—
609C8—2441no—
610C8#1451973711yescharacteristic
611C8#1451973711yescharacteristic
612C8#1451973711yescharacteristic
613C8#1451973711yescharacteristic
614C22—41no—
615C22—41no—
616C22—41no—
617C22—81no—
618C22—81no—
619C22—9761no—
620C22—4882no—
621C22—4882no—
622C22—2442no—
623C22#1684023411yescharacteristic
624C22#1684032511yescharacteristic
625C22#1684036511yescharacteristic
626C4—41no—
627C4—41no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—2442no—
632C4—2442no—
633C4—2442no—
634C4C2×#1451973711yescharacteristic
635C4#1684023411yescharacteristic
636C4#1684033011yescharacteristic
637C4#1684036511yescharacteristic
638C2—21no—
639C2—21no—
640C2—21no—
641C2—81no—
642C2—81no—
643C2—4882no—
644C2#2144854311yescharacteristic
subgroupG/H
centerC2#21448543
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1976C2
socleC2×C61#2328
prime radical#21456296C2
cosocleC22×C4×C61C24
hypercenter#1976D61
hypocenterC61#22284
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. #21454736C2semidirect
3. #21454677C2semidirect
4. #21446722C2semidirect
5. #21446353C2semidirect
6. #21446348C2semidirect
7. #21446344C2semidirect
8. #21446339C2semidirect
9. #21446316C2semidirect
10. #21446309C2semidirect
11. #21446297C2semidirect
12. #21446290C2semidirect
13. #21445942C2semidirect
14. #21445574C2semidirect
15. C61×#1976C2semidirect
25. #16840155C4semidirect
26. #16840150C4semidirect
27. #16840143C4semidirect
28. #16840133C4semidirect
65. C2×#11758202C2×C4semidirect
72. C2×#11758200C2×C4semidirect
85. #14519775C2×C4semidirect
88. #14519764C2×C4semidirect
111. #14519687C2×C4semidirect
112. #14519687D4semidirect
113. C8×Dic61C2×C4semidirect
114. C8×Dic61D4semidirect
121. C2×Q16×C61C2×C4semidirect
122. C2×SD16×C61C2×C4semidirect
123. C2×SD16×C61C2×C4semidirect
124. C2×D8×C61C2×C4semidirect
320. #1976D61semidirect
381. C61#22284semidirect
395. C2×Q16C4×D61semidirect
404. C2×SD16C4×D61semidirect
405. C2×SD16C4×D61semidirect
407. C2×D8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214562962
#214547362
#214546772
#214467222
#214463532
#214463482
#214463442
#214463392
#214463162
#214463092
#214462972
#214462902
#214459422
#214455742
C61×#19762
#2228461