Finite Group Databasebeta

← #10836686

Subgroups — #10836686

Order 768.

690 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79061C211yesmaximal, characteristic, complemented
2C2×#4847C211yesmaximal, characteristic, complemented
3#79000C211yesmaximal, characteristic, complemented
4C4×#908C211yesmaximal, characteristic
5C3×#1950C211yesmaximal, characteristic, complemented
6#70714C211yesmaximal, characteristic
7#70714C211yesmaximal, characteristic
8#70701C211yesmaximal, characteristic, complemented
9#70700C211yesmaximal, characteristic, complemented
10#70527C211yesmaximal, characteristic, complemented
11#69880C211yesmaximal, characteristic, complemented
12#68931C211yesmaximal, characteristic, complemented
13#68931C211yesmaximal, characteristic, complemented
14#68922C211yesmaximal, characteristic, complemented
15#68921C211yesmaximal, characteristic, complemented
16#24724—31nomaximal, Sylow
17C2×#908C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×#870C2211yescharacteristic
21C2×C3×#108C2211yescharacteristic
22C2×C3×#108C2211yescharacteristic
23C3×#423C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×C4×M16C2211yescharacteristic
26#5133C2211yescharacteristic
27#5133C2211yescharacteristic
28#5133C2211yescharacteristic
29#5131C2211yescharacteristic
30C2×#823C2211yescharacteristic
31C2×#823C2211yescharacteristic
32#5121C2211yescharacteristic
33#5121C2211yescharacteristic
34#5121C2211yescharacteristic
35C2×#821C2211yescharacteristic
36C2×#821C2211yescharacteristic
37#5110C2211yescharacteristic
38#5110C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#817C2211yescharacteristic
41#5101C2211yescharacteristic
42#5101C2211yescharacteristic
43#4924C2211yescharacteristic
44#4847—21no—
45#4847—21no—
46#4847—21no—
47#4847—21no—
48#4847—21no—
49#4847—21no—
50#4840—21no—
51#4840—21no—
52#4835—21no—
53#4833—21no—
54#4829—21no—
55#4829—21no—
56#4711—21no—
57#4710—21no—
58C4×D24—21no—
59C4×#204—21no—
60C3×#341C2211yescharacteristic
61C3×#341C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4554C2211yescharacteristic
67#4554C2211yescharacteristic
68#3080—31no—
69C2×#478—31no—
70C4×AGL(1,ℤ/8)—31no—
71#2925—31no—
72#2079—31no—
73#2078—31no—
74#2051—31no—
75#2051—31no—
76#1998—31no—
77#1952—31no—
78#1951—31no—
79#1950D311yescharacteristic, complemented
80#1931—31no—
81#1931—31no—
82#1800—31no—
83C2×#235C2311yescharacteristic
84C22×D12C2311yescharacteristic
85C22×C4×D3—21no—
86C2×C3×M16C2311yescharacteristic
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C4×#67—21no—
94C4×#67—21no—
95C2×(C2×D3)⋊C4—21no—
96#926C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×#208C2311yescharacteristic
103C2×#208C2311yescharacteristic
104#908—22no—
105#908—22no—
106C2×D24—21no—
107C2×#204—21no—
108C4×D12—21no—
109C4×D12—21no—
110C4×D12—21no—
111C4×D12—21no—
112C4×D12D411yescharacteristic
113C4×D12D411yescharacteristic
114#872—21no—
115C42×D3—21no—
116#870—21no—
117#870—21no—
118#870—21no—
119#870—21no—
120C4×Dic6D411yescharacteristic
121C4×Dic6D411yescharacteristic
122C3×#108—22no—
123C3×#108—22no—
124C3×C4⋊C8—21no—
125C3×C4⋊C8—21no—
126C3×C4⋊C8D411yescharacteristic
127C3×C4⋊C8D411yescharacteristic
128C3×C4⋊C8D411yescharacteristic
129C3×C4⋊C8D411yescharacteristic
130C3×#97—21no—
131C3×C4×C8—21no—
132#823—22no—
133#823—22no—
134#821—21no—
135#821—21no—
136#821—21no—
137#821—21no—
138#821—21no—
139#821—21no—
140#821—21no—
141#821—21no—
142#821—21no—
143(C2×D3)⋊C8—21no—
144(C2×D3)⋊C8—21no—
145(C2×D3)⋊C8—21no—
146(C2×D3)⋊C8—21no—
147#818—21no—
148#818—21no—
149#818—21no—
150#818—21no—
151#818—21no—
152#817—21no—
153#817—21no—
154#817—21no—
155#817—21no—
156#817—21no—
157#816—21no—
158#816—21no—
159#816—21no—
160#815—21no—
161#815—21no—
162C8×Dic3—21no—
163C8×Dic3—21no—
164C4⋊(C3⋊C8)—21no—
165C4⋊(C3⋊C8)—21no—
166C2×AGL(1,ℤ/8)—31no—
167C2×#125—31no—
168C4×#41—31no—
169C2×C4×D4—31no—
170#479—61no—
171#479—61no—
172#478—61no—
173#478—61no—
174#478—61no—
175#478—61no—
176#478—61no—
177#478—61no—
178#442—61no—
179#440—61no—
180C4×SD16—61no—
181C4×D8—61no—
182#433—61no—
183#433—61no—
184#432—61no—
185#431—61no—
186C2×#108C2×D311yescharacteristic
187C2×#108C2×D311yescharacteristic
188#428—31no—
189#428—31no—
190#428—31no—
191C2×#107—31no—
192C2×#106—31no—
193#423—31no—
194#423C2×D311yescharacteristic
195C2×C4⋊C8C2×D311yescharacteristic
196#418—31no—
197#417—31no—
198#417—31no—
199#417—31no—
200C2×#102—31no—
201C2×#102—31no—
202C2×#100—31no—
203C2×#100—31no—
204#408—31no—
205#408—31no—
206#405—31no—
207#405—31no—
208C4×M16C2×D311yescharacteristic
209#343—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#343—31no—
214#343—31no—
215#341C2×D311yescharacteristic
216#341C2×D311yescharacteristic
217C23×D3—21no—
218C22×C3×C4#4111yescharacteristic
219C22×C3×C4#4111yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C2×#67—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C22×Dic3—21no—
225#235—21no—
226#235—21no—
227#235—21no—
228#235—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12C2×D411yescharacteristic
235C2×D12#4112yes—
236C2×C4×D3—21no—
237C2×C4×D3—21no—
238C2×C4×D3—21no—
239C2×C4×D3—41no—
240C2×C4×D3—41no—
241C2×C4×D3—82no—
242C2×Dic6C2×D411yescharacteristic
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16—21no—
248C3×M16—21no—
249C3×M16—21no—
250C3×M16#4112yes—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8C2×D411yescharacteristic
260C2×C3×C8C2×D411yescharacteristic
261C2×C3×C8C2×D411yescharacteristic
262C2×C3×C8C2×D411yescharacteristic
263C3×C42C2×D411yescharacteristic
264C3×C42C2×D411yescharacteristic
265C3×C42C2×D411yescharacteristic
266C3×C42C2×D411yescharacteristic
267#217—21no—
268(C2×D3)⋊C4—21no—
269(C2×D3)⋊C4—21no—
270(C2×D3)⋊C4—41no—
271(C2×D3)⋊C4—41no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3C2×D411yescharacteristic
279C4⋊Dic3C2×D411yescharacteristic
280Dic3⋊C4—21no—
281Dic3⋊C4—21no—
282Dic3⋊C4—41no—
283Dic3⋊C4—41no—
284Dic3⋊C4—41no—
285Dic3⋊C4—41no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289#208—21no—
290#208—21no—
291#208—21no—
292#208—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295D24—24no—
296#204—24no—
297C2×#41—31no—
298C22×D4—31no—
299C23×C4—61no—
300AGL(1,ℤ/8)—62no—
301AGL(1,ℤ/8)—62no—
302C2×SD16—61no—
303C2×D8—61no—
304C2×M16—31no—
305C2×M16—31no—
306C2×M16C22×D311yescharacteristic
307C2×M16C22×D311yescharacteristic
308C2×M16C22×D311yescharacteristic
309C2×M16C22×D311yescharacteristic
310C22×C8C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312#125—61no—
313#125—61no—
314#125—61no—
315#125—61no—
316C4×Q8—31no—
317C4×Q8—31no—
318C4×D4—31no—
319C4×D4—31no—
320C4×D4—61no—
321C4×D4—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326#117—31no—
327#117—61no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C42—61no—
335C2×C42C22×D311yescharacteristic
336#108—22no—
337#108—22no—
338#107—61no—
339#107—61no—
340#107—61no—
341#107—61no—
342#107—61no—
343#106—61no—
344#106—61no—
345#106—61no—
346#106—61no—
347#106—61no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—61no—
351C4⋊C8—61no—
352C4⋊C8D1211yescharacteristic
353C4⋊C8D1211yescharacteristic
354C4⋊C8D1211yescharacteristic
355C4⋊C8D1211yescharacteristic
356#103—61no—
357#103—61no—
358#103—61no—
359#102—61no—
360#102—61no—
361#102—61no—
362#102—61no—
363#102—61no—
364#102—61no—
365#102—61no—
366#102—61no—
367#102—61no—
368#100—62no—
369#100—62no—
370C22⋊C8—61no—
371C22⋊C8—61no—
372C22⋊C8—61no—
373C22⋊C8—61no—
374#97—21no—
375#97—61no—
376#97—61no—
377C4×C8—21no—
378C4×C8—61no—
379C4×C8—61no—
380C23×C3C2×#4111yescharacteristic
381C22×D3—21no—
382C22×D3—22no—
383C22×D3—41no—
384C22×D3—41no—
385C22×D3—81no—
386C22×D3—81no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4#12111yescharacteristic
390C2×C3×C4#12111yescharacteristic
391C2×C3×C4#12111yescharacteristic
392C2×C3×C4#12111yescharacteristic
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D411yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C42−1+411yescharacteristic
398C2×C3×C42−1+411yescharacteristic
399#67—41no—
400#67—41no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—81no—
408C2×Dic3—81no—
409C2×Dic3—81no—
410D12—22no—
411D12—21no—
412D12—21no—
413D12—22no—
414D12—41no—
415D12—41no—
416C4×D3—42no—
417C4×D3—41no—
418C4×D3—41no—
419C4×D3—81no—
420C4×D3—82no—
421C4×D3—82no—
422Dic6—21no—
423Dic6—21no—
424C3×C8—22no—
425C3×C8—21no—
426C3×C8—21no—
427C3×C8—21no—
428C3×C8—21no—
429C3×C8—41no—
430C3×C8—41no—
431C3×C8—41no—
432C3⋊C8—41no—
433C3⋊C8—41no—
434C24—61no—
435#41—61no—
436#41—61no—
437#41—61no—
438#41—61no—
439C2×Q8—31no—
440C2×D4—31no—
441C2×D4—32no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—121no—
455C22×C4—121no—
456C22×C4—242no—
457C22×C4#23911yescharacteristic
458C22×C4#23911yescharacteristic
459C22×C4C23×D311yescharacteristic
460SD16—64no—
461D8—64no—
462M16—21no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16#23712yes—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8C2×D1211yescharacteristic
485C2×C8C2×D1211yescharacteristic
486C2×C8C2×D1211yescharacteristic
487C2×C8C2×D1211yescharacteristic
488C4⋊C4—31no—
489C4⋊C4—31no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C4⋊C4—61no—
498C4⋊C4—121no—
499C4⋊C4—121no—
500C4⋊C4—121no—
501C4⋊C4—121no—
502C22⋊C4—61no—
503C22⋊C4—61no—
504C22⋊C4—61no—
505C22⋊C4—121no—
506C22⋊C4—121no—
507C42—61no—
508C42—61no—
509C42—61no—
510C42C2×D1211yescharacteristic
511C42C2×D1211yescharacteristic
512C42D3×D411yescharacteristic
513C42D3×D411yescharacteristic
514C22×C3—21no—
515C22×C3—21no—
516C22×C3C2×#12111yescharacteristic
517C22×C3#55211yescharacteristic
518C22×C3#55611yescharacteristic
519C2×D3—41no—
520C2×D3—42no—
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—82no—
524C2×D3—81no—
525C2×D3—81no—
526C2×D3—82no—
527C3×C4—21no—
528C3×C4—21no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4#52511yescharacteristic
532C3×C4#53611yescharacteristic
533C3×C4#54711yescharacteristic
534C3×C4#54811yescharacteristic
535Dic3—41no—
536Dic3—41no—
537Dic3—41no—
538Dic3—81no—
539Dic3—81no—
540Dic3—81no—
541C23—61no—
542C23—62no—
543C23—121no—
544C23—121no—
545C23—241no—
546C23—241no—
547C23C2×#23911yescharacteristic
548Q8—61no—
549Q8—61no—
550D4—62no—
551D4—61no—
552D4—62no—
553D4—61no—
554D4—121no—
555D4—121no—
556D4—121no—
557D4—121no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—122no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—241no—
570C2×C4—242no—
571C2×C4—242no—
572C2×C4—241no—
573C2×C4—241no—
574C2×C4—241no—
575C2×C4#89511yescharacteristic
576C2×C4#89511yescharacteristic
577C2×C4#89511yescharacteristic
578C2×C4#89511yescharacteristic
579C2×C4C22×D1211yescharacteristic
580C2×C4C2×D3×D411yescharacteristic
581C2×C4C2×#23711yescharacteristic
582C2×C4#100711yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4#101011yescharacteristic
585C8—21no—
586C8—21no—
587C8—21no—
588C8—21no—
589C8—22no—
590C8—41no—
591C8—41no—
592C8—41no—
593C8—121no—
594C8—121no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3#241511yescharacteristic
598C2×C3#306711yescharacteristic
599C2×C3#306811yescharacteristic
600D3—81no—
601D3—82no—
602C22—21no—
603C22—21no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—122no—
608C22—242no—
609C22—242no—
610C22—241no—
611C22—241no—
612C22C2×#89511yescharacteristic
613C22#569411yescharacteristic
614C22#573211yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—121no—
620C4—121no—
621C4—121no—
622C4—241no—
623C4—241no—
624C4—241no—
625C4#555511yescharacteristic
626C4#557611yescharacteristic
627C4#560611yescharacteristic
628C4#562811yescharacteristic
629C3#2472411yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—241no—
633C2—242no—
634C2#7621011yescharacteristic
635C2#7925611yescharacteristic
636C2#7926111yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1950C2
socleC22×C3C2×#121
prime radicalC2×#908C22
hypercenter#1950D3
hypocenterC3#24724
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1950

Complements

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

NH ≅ G/Ntype
1. #79061C2semidirect
2. C2×#4847C2semidirect
3. #79000C2semidirect
5. C3×#1950C2semidirect
8. #70701C2semidirect
9. #70700C2semidirect
10. #70527C2semidirect
11. #69880C2semidirect
12. #68931C2semidirect
13. #68931C2semidirect
14. #68922C2semidirect
15. #68921C2semidirect
79. #1950D3semidirect
629. C3#24724semidirect

Maximal subgroups

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

subgroupindex
#790612
C2×#48472
#790002
C4×#9082
C3×#19502
#707142
#707012
#707002
#705272
#698802
#689312
#689222
#689212
#247243