Finite Group Databasebeta

← #10847337

Subgroups — #10847337

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2962C211yesmaximal, characteristic, complemented
2#83350C211yesmaximal, characteristic
3#83342C211yesmaximal, characteristic, complemented
4C2×#4847C211yesmaximal, characteristic
5#70542C211yesmaximal, characteristic
6#70542C211yesmaximal, characteristic
7#70533C211yesmaximal, characteristic, complemented
8#70532C211yesmaximal, characteristic, complemented
9#70474C211yesmaximal, characteristic, complemented
10#69894C211yesmaximal, characteristic, complemented
11#69893C211yesmaximal, characteristic, complemented
12#69892C211yesmaximal, characteristic
13#69892C211yesmaximal, characteristic
14#69880C211yesmaximal, characteristic, complemented
15C4×#823C211yesmaximal, characteristic
16#25493—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×#870C2211yescharacteristic
20C3×#446—21no—
21C3×#446—21no—
22C3×#446—21no—
23C3×C8×Q8—21no—
24C3×#424C2211yescharacteristic
25C3×#424C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C3×#405C2211yescharacteristic
28C3×#405C2211yescharacteristic
29C3×C4×M16C2211yescharacteristic
30C2×#823C2211yescharacteristic
31C2×#823C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#819C2211yescharacteristic
35C2×#819C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#817C2211yescharacteristic
38#5087—21no—
39#5084—21no—
40#5081—21no—
41#5081—21no—
42#5066—21no—
43#5066—21no—
44#5065—21no—
45#5065—21no—
46#5001C2211yescharacteristic
47#5001C2211yescharacteristic
48#4995C2211yescharacteristic
49#4995C2211yescharacteristic
50#4995C2211yescharacteristic
51#4995C2211yescharacteristic
52#4995C2211yescharacteristic
53#4995C2211yescharacteristic
54#4924C2211yescharacteristic
55C4×#208C2211yescharacteristic
56#4847—21no—
57#4847—21no—
58#4847—21no—
59#4847—21no—
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64#4554C2211yescharacteristic
65#4554C2211yescharacteristic
66#4552C2211yescharacteristic
67#4552C2211yescharacteristic
68#3166—31no—
69#3077—31no—
70C2×#478—31no—
71#2962D311yescharacteristic, complemented
72#2090—31no—
73#2089—31no—
74#2063—31no—
75#2063—31no—
76#1998—31no—
77#1952—31no—
78#1951—31no—
79#1950—31no—
80#1931—31no—
81#1931—31no—
82C4×#100—31no—
83C22×D12C2311yescharacteristic
84C22×C4×D3—21no—
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4⋊Q8D411yescharacteristic
90C3×C4⋊Q8D411yescharacteristic
91C3×#125D411yescharacteristic
92C3×#125D411yescharacteristic
93C3×C22⋊Q8—21no—
94C3×C22⋊Q8—21no—
95C3×C4×Q8—21no—
96C3×C4×Q8—21no—
97C3×#117C2311yescharacteristic
98C3×#117C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C2×#208C2311yescharacteristic
110C4×D12—21no—
111C4×D12—21no—
112C4×D12—21no—
113C4×D12—21no—
114#870—21no—
115#870—21no—
116#870—21no—
117#870—21no—
118C3×C4⋊C8—21no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8—21no—
121C3×C4⋊C8—21no—
122C3×C4⋊C8D411yescharacteristic
123C3×C4⋊C8D411yescharacteristic
124C3×C4⋊C8D411yescharacteristic
125C3×C4⋊C8D411yescharacteristic
126C3×#97—21no—
127C3×#97—21no—
128C3×#97—21no—
129C3×C4×C8—21no—
130C3×C4×C8—21no—
131C3×C4×C8—21no—
132#823—22no—
133#823Q814yescomplemented
134#821—21no—
135#821—21no—
136#821—21no—
137#821—21no—
138#819—22no—
139#819—22no—
140#818—21no—
141#818—21no—
142#817—21no—
143#817—21no—
144#809—21no—
145#809—21no—
146#809—21no—
147#809—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#808—21no—
152#808—21no—
153#808—21no—
154#807—21no—
155#807—21no—
156#807—21no—
157#807—21no—
158#807—21no—
159#807—21no—
160C4⋊(C3⋊C8)—21no—
161C4⋊(C3⋊C8)—21no—
162#803—21no—
163C4×C3⋊C8—21no—
164#533C2×D311yescharacteristic
165C2×#125—31no—
166C2×C4×D4—31no—
167#501—61no—
168#501—61no—
169#499—61no—
170#499—61no—
171#478—61no—
172#478—61no—
173#478—61no—
174#478—61no—
175#478—61no—
176#478—61no—
177#476—61no—
178#474—61no—
179#446—21no—
180#446—21no—
181#446—21no—
182C8×Q8—21no—
183C2×#108—31no—
184C2×#108—31no—
185#428—31no—
186#428—31no—
187#428—31no—
188#428—31no—
189#428—31no—
190#428—31no—
191C2×#107—31no—
192C2×#106—31no—
193#424C2×D311yescharacteristic
194#424C2×D311yescharacteristic
195#423—31no—
196C2×C4⋊C8C2×D311yescharacteristic
197#418—31no—
198#418—31no—
199C2×#102—31no—
200C2×#102—31no—
201C2×#100—31no—
202C2×#100—31no—
203#405C2×D311yescharacteristic
204#405C2×D311yescharacteristic
205C4×M16—31no—
206C4×M16C2×D311yescharacteristic
207#343—31no—
208#343—31no—
209#343—31no—
210#343—31no—
211#343—31no—
212#343—31no—
213#341—31no—
214#341—31no—
215C23×D3—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—21no—
218C22×C3×C4#4111yescharacteristic
219C22×C3×C4#4111yescharacteristic
220C22×C3×C4C2411yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12C2×Q812yes—
230C2×C4×D3—41no—
231C2×C4×D3—41no—
232C2×C4×D3—82no—
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16C2×Q812yes—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D411yescharacteristic
244C2×C3×C8C2×D411yescharacteristic
245C2×C3×C8C2×D411yescharacteristic
246C2×C3×C8C2×D411yescharacteristic
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4C2×D411yescharacteristic
254C3×C4⋊C4C2×D411yescharacteristic
255C3×C4⋊C4C2×D411yescharacteristic
256C3×C4⋊C4C2×D411yescharacteristic
257C3×C22⋊C4—21no—
258C3×C22⋊C4—21no—
259C3×C42—21no—
260C3×C42—21no—
261C3×C42C2×D411yescharacteristic
262C3×C42C2×D411yescharacteristic
263C3×C42C2×D411yescharacteristic
264C3×C42C2×D411yescharacteristic
265(C2×D3)⋊C4—41no—
266(C2×D3)⋊C4—41no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273Dic3⋊C4—41no—
274Dic3⋊C4—41no—
275Dic3⋊C4—41no—
276Dic3⋊C4—41no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208—21no—
281#208—21no—
282#208—21no—
283#208C2×Q812yes—
284#208#4112yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C22×D4—31no—
290C23×C4—61no—
291C2×M16—31no—
292C2×M16—31no—
293C2×M16—31no—
294C2×M16—31no—
295C2×M16C22×D311yescharacteristic
296C2×M16C22×D311yescharacteristic
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299C4⋊Q8D1211yescharacteristic
300C4⋊Q8D1211yescharacteristic
301#125—61no—
302#125—61no—
303#125—61no—
304#125—61no—
305#125D1211yescharacteristic
306#125D1211yescharacteristic
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—61no—
312C4×D4—61no—
313C4×D4—61no—
314C4×D4—61no—
315#117C22×D311yescharacteristic
316#117C22×D311yescharacteristic
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C42C22×D311yescharacteristic
324#108—62no—
325#108—62no—
326#107—61no—
327#107—61no—
328#107—61no—
329#107—61no—
330#107—61no—
331#107—61no—
332#107—61no—
333#107—61no—
334#106—61no—
335#106—61no—
336#106—61no—
337#106—61no—
338#106—61no—
339#106—61no—
340#106—61no—
341#106—61no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—61no—
347C4⋊C8—61no—
348C4⋊C8#6711yescharacteristic
349C4⋊C8#6711yescharacteristic
350C4⋊C8#6711yescharacteristic
351C4⋊C8#6711yescharacteristic
352#102—61no—
353#102—61no—
354#102—61no—
355#102—61no—
356#102—61no—
357#102—61no—
358#102—61no—
359#102—61no—
360#100—34no—
361#100—62no—
362#97—21no—
363#97—21no—
364#97—21no—
365#97—61no—
366C4×C8—21no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—61no—
370C23×C3C2×#4111yescharacteristic
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—81no—
375C22×D3—81no—
376C3×Q8—42no—
377C3×Q8—81no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4#12111yescharacteristic
385C2×C3×C4#12111yescharacteristic
386C2×C3×C4#12111yescharacteristic
387C2×C3×C4#12111yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×Q811yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—81no—
398C2×Dic3—81no—
399C2×Dic3—81no—
400D12—22no—
401D12—22no—
402D12—41no—
403D12—41no—
404C4×D3—82no—
405C4×D3—82no—
406C3×C8—21no—
407C3×C8—21no—
408C3×C8—21no—
409C3×C8—21no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—22no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—61no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×D4—32no—
420C2×D4—61no—
421C2×D4—61no—
422C2×D4—61no—
423C2×D4—61no—
424C2×D4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—121no—
429C22×C4—121no—
430C22×C4—242no—
431C22×C4#23511yescharacteristic
432C22×C4#23511yescharacteristic
433C22×C4C23×D311yescharacteristic
434M16—21no—
435M16—21no—
436M16—21no—
437M16—32no—
438M16—32no—
439M16—61no—
440M16—61no—
441M16—61no—
442M16—61no—
443M16—61no—
444M16—61no—
445M16D3×Q812yes—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—61no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8C2×#6711yescharacteristic
457C2×C8C2×#6711yescharacteristic
458C2×C8C2×#6711yescharacteristic
459C2×C8C2×#6711yescharacteristic
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—61no—
467C4⋊C4—61no—
468C4⋊C4—61no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—121no—
473C4⋊C4—121no—
474C4⋊C4—121no—
475C4⋊C4—121no—
476C4⋊C4C2×D1211yescharacteristic
477C4⋊C4C2×D1211yescharacteristic
478C4⋊C4C2×D1211yescharacteristic
479C4⋊C4C2×D1211yescharacteristic
480C22⋊C4—21no—
481C22⋊C4—21no—
482C22⋊C4—121no—
483C22⋊C4—121no—
484C42—21no—
485C42—21no—
486C42C2×D1211yescharacteristic
487C42C2×D1211yescharacteristic
488C42C2×#6711yescharacteristic
489C42C2×#6711yescharacteristic
490C22×C3—21no—
491C22×C3—21no—
492C22×C3C2×#12111yescharacteristic
493C22×C3#53311yescharacteristic
494C22×C3#54111yescharacteristic
495C2×D3—42no—
496C2×D3—82no—
497C2×D3—81no—
498C2×D3—81no—
499C2×D3—82no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4#54711yescharacteristic
509C3×C4#54811yescharacteristic
510C3×C4D4×Q811yescharacteristic
511C3×C4D4×Q811yescharacteristic
512Dic3—81no—
513Dic3—81no—
514Dic3—81no—
515C23—62no—
516C23—121no—
517C23—121no—
518C23—241no—
519C23—241no—
520C23C2×#23511yescharacteristic
521Q8—42no—
522Q8—81no—
523D4—62no—
524D4—62no—
525D4—121no—
526D4—121no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—121no—
534C2×C4—121no—
535C2×C4—121no—
536C2×C4—242no—
537C2×C4—242no—
538C2×C4—241no—
539C2×C4—241no—
540C2×C4—241no—
541C2×C4#93011yescharacteristic
542C2×C4#93011yescharacteristic
543C2×C4#93011yescharacteristic
544C2×C4#93011yescharacteristic
545C2×C4C22×D1211yescharacteristic
546C2×C4C2×D3×Q811yescharacteristic
547C2×C4#100711yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C2×C4#101011yescharacteristic
550C2×C4C22×#6711yescharacteristic
551C8—22no—
552C8—21no—
553C8—21no—
554C8—21no—
555C8—21no—
556C8—41no—
557C8—62no—
558C8—62no—
559C8—121no—
560C8—121no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3#242911yescharacteristic
564C2×C3#306711yescharacteristic
565C2×C3#306811yescharacteristic
566D3—82no—
567C22—21no—
568C22—21no—
569C22—122no—
570C22—242no—
571C22—242no—
572C22—241no—
573C22—241no—
574C22#557811yescharacteristic
575C22#558311yescharacteristic
576C22C2×#93011yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—241no—
586C4—241no—
587C4—241no—
588C4Q8×D1211yescharacteristic
589C4#557611yescharacteristic
590C4Q8×#6711yescharacteristic
591C4#582411yescharacteristic
592C3#2549311yescharacteristic, complemented, Sylow
593C2—21no—
594C2—21no—
595C2—242no—
596C2#8249611yescharacteristic
597C2#8380511yescharacteristic
598C2#8381411yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2962C2
socleC22×C3C2×#121
prime radicalC22×D12C23
hypercenter#2962D3
hypocenterC3#25493
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. C3×#2962C2semidirect
3. #83342C2semidirect
7. #70533C2semidirect
8. #70532C2semidirect
9. #70474C2semidirect
10. #69894C2semidirect
11. #69893C2semidirect
14. #69880C2semidirect
71. #2962D3semidirect
133. #823Q8semidirect
592. C3#25493semidirect

Maximal subgroups

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

subgroupindex
C3×#29622
#833502
#833422
C2×#48472
#705422
#705332
#705322
#704742
#698942
#698932
#698922
#698802
C4×#8232
#254933