Finite Group Databasebeta

← #10819955

Subgroups — #10819955

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79169C211yesmaximal, characteristic, complemented
2#79168C211yesmaximal, characteristic
3#78573C211yesmaximal, characteristic, complemented
4#78572C211yesmaximal, characteristic
5C2×#4715C211yesmaximal, characteristic
6C3×#1773C211yesmaximal, characteristic, complemented
7#70710C211yesmaximal, characteristic, complemented
8#70708C211yesmaximal, characteristic
9#69904C211yesmaximal, characteristic, complemented
10#69902C211yesmaximal, characteristic
11#69829C211yesmaximal, characteristic, complemented
12#68766C211yesmaximal, characteristic
13#68765C211yesmaximal, characteristic, complemented
14#68765C211yesmaximal, characteristic, complemented
15#68759C211yesmaximal, characteristic
16#23203—31nomaximal, Sylow
17C22×Dic12C2211yescharacteristic
18C22×Dic12C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C22×#204C2211yescharacteristic
21#5521C2211yescharacteristic
22#5516C2211yescharacteristic
23#5516C2211yescharacteristic
24#5516C2211yescharacteristic
25C2×#875C2211yescharacteristic
26C2×#869C2211yescharacteristic
27C2×C3×#108C2211yescharacteristic
28C3×#427C2211yescharacteristic
29C3×#405C2211yescharacteristic
30C3×#405C2211yescharacteristic
31C2×C3×#97C2211yescharacteristic
32C2×#824C2211yescharacteristic
33C2×#824C2211yescharacteristic
34C2×#824C2211yescharacteristic
35#5131C2211yescharacteristic
36C2×#823C2211yescharacteristic
37#5120C2211yescharacteristic
38#5120C2211yescharacteristic
39#5120C2211yescharacteristic
40C2×#819C2211yescharacteristic
41#5106C2211yescharacteristic
42#5101C2211yescharacteristic
43#5101C2211yescharacteristic
44#5004C2211yescharacteristic
45#5004C2211yescharacteristic
46#5004C2211yescharacteristic
47#5001C2211yescharacteristic
48#4889—22no—
49#4873—22no—
50#4853—22no—
51#4852—22no—
52#4716—22no—
53#4715C418yescomplemented
54#4710—22no—
55C4×Dic12—22no—
56C4×#204—22no—
57C3×#339C2211yescharacteristic
58C3×#339C2211yescharacteristic
59#4523C2211yescharacteristic
60#4523C2211yescharacteristic
61C2×#497—31no—
62#2950—31no—
63#2950—31no—
64#2949—31no—
65#2949—31no—
66#1979—31no—
67#1979—31no—
68#1977—31no—
69#1977—31no—
70#1888—31no—
71#1886—31no—
72#1886—31no—
73#1885—31no—
74#1773—31no—
75#1773D311yescharacteristic, complemented
76C22×D12C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C2×C3×M16C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#117C2311yescharacteristic
85C3×#117C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×(C2×D3)⋊C4—41no—
88#926C2311yescharacteristic
89#926C2311yescharacteristic
90C2×C4⋊Dic3—41no—
91C2×#208C2311yescharacteristic
92C2×#208C2311yescharacteristic
93C2×Dic12—21no—
94C2×Dic12—21no—
95C2×Dic12—21no—
96C2×Dic12—21no—
97C2×Dic12—22no—
98C2×Dic12—22no—
99C2×Dic12C2×C414yescomplemented
100C2×Dic12C2×C414yescomplemented
101C2×#204—21no—
102C2×#204—21no—
103C2×#204—21no—
104C2×#204—21no—
105C2×#204—22no—
106C2×#204—22no—
107C2×#204C2×C414yes—
108C2×#204C2×C414yes—
109#893—22no—
110#887—22no—
111#887—22no—
112#887—22no—
113D3×C22⋊C4—41no—
114#877—41no—
115#877—41no—
116#877—41no—
117#875—41no—
118#875C2×C414yes—
119C4×D12—22no—
120#869—41no—
121#869C2×C414yescomplemented
122C4×Dic6—22no—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C3×#108—22no—
126C3×#107D412yes—
127C3×#106D412yes—
128C3×#97—21no—
129C3×#97—21no—
130C3×#97C2×C414yescomplemented
131C3×C4×C8—21no—
132C3×C4×C8—21no—
133#824—41no—
134#824—41no—
135#824—41no—
136#823—41no—
137#821—22no—
138#819—22no—
139#818D412yes—
140#817D412yes—
141#816—22no—
142#816—22no—
143#816—22no—
144#815—21no—
145#815—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148#810—22no—
149#810—22no—
150#810—22no—
151#809—22no—
152C22×Q16—31no—
153C22×Q16—31no—
154C22×SD16—31no—
155C22×SD16—31no—
156C2×C4⋊Q8—31no—
157C2×#124—31no—
158#519—31no—
159#519—31no—
160#519—31no—
161#518—31no—
162#497—38no—
163#441—62no—
164#441—62no—
165#440—62no—
166#440—62no—
167C4×Q16—62no—
168C4×Q16—62no—
169C4×SD16—62no—
170C4×SD16—62no—
171C2×#108—31no—
172C2×#108C2×D311yescharacteristic
173#427—31no—
174#427C2×D311yescharacteristic
175#419—31no—
176#419—31no—
177#419—31no—
178#419—31no—
179#419—31no—
180#419—31no—
181#418—31no—
182#418—31no—
183C2×#101—31no—
184C2×#101—31no—
185C2×#101—31no—
186C2×#100—31no—
187#405—31no—
188#405—31no—
189#405C2×D311yescharacteristic
190#405C2×D311yescharacteristic
191C2×#97C2×D311yescharacteristic
192#339—31no—
193#339—31no—
194#339C2×D311yescharacteristic
195#339C2×D311yescharacteristic
196C23×D3—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12C22×C412yes—
208C2×C4×D3—42no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6C22×C412yes—
225C2×Dic6C22×C412yes—
226C2×Dic6C22×C412yes—
227C3×M16—21no—
228C3×M16—21no—
229C3×M16—21no—
230C3×M16—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8C22×C412yes—
234C2×C3×C8C22×C412yes—
235C2×C3×C8C2×D411yescharacteristic
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8#4112yes—
240C2×C3×C8#4112yes—
241C3×C4⋊C4C2×D412yes—
242C3×C4⋊C4C2×D412yes—
243C3×C22⋊C4—21no—
244C3×C22⋊C4—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42C22×C412yes—
249#217—21no—
250#217—21no—
251(C2×D3)⋊C4—44no—
252(C2×D3)⋊C4—42no—
253C4⋊Dic3—44no—
254C4⋊Dic3C2×D412yes—
255C4⋊Dic3C2×D412yes—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—42no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264#208—21no—
265#208—21no—
266#208—21no—
267#208—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270Dic12—24no—
271Dic12—24no—
272Dic12—24no—
273Dic12—24no—
274#204—24no—
275#204—24no—
276#204—24no—
277#204—24no—
278C22×Q8—31no—
279C22×Q8—31no—
280C22×Q8—31no—
281C22×D4—31no—
282C2×Q16—34no—
283C2×Q16—34no—
284C2×Q16—61no—
285C2×Q16—61no—
286C2×Q16—62no—
287C2×Q16—61no—
288C2×Q16—61no—
289C2×Q16—62no—
290C2×SD16—34no—
291C2×SD16—34no—
292C2×SD16—61no—
293C2×SD16—61no—
294C2×SD16—62no—
295C2×SD16—61no—
296C2×SD16—61no—
297C2×SD16—62no—
298C2×M16—31no—
299C2×M16—31no—
300C2×M16C22×D311yescharacteristic
301C2×M16C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303C22×C8C22×D311yescharacteristic
304C4⋊Q8—34no—
305C4⋊Q8—121no—
306#124—34no—
307#124—121no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×Q8—62no—
313C4×Q8—62no—
314C4×D4—62no—
315C4×D4—62no—
316#117—31no—
317#117—31no—
318#117—121no—
319#117—121no—
320#117—121no—
321#117C22×D311yescharacteristic
322#117C22×D311yescharacteristic
323C2×C4⋊C4—121no—
324C2×C22⋊C4—121no—
325C2×C22⋊C4—121no—
326C2×C42C22×D311yescharacteristic
327#108—22no—
328#108—62no—
329#107—32no—
330#107D1212yes—
331#106—32no—
332#106D1212yes—
333#103—62no—
334#103—62no—
335#103—62no—
336#103—62no—
337#103—62no—
338#103—62no—
339#102—62no—
340#102—62no—
341#101—121no—
342#101—121no—
343#101—121no—
344#100—121no—
345#97—21no—
346#97—21no—
347#97—61no—
348#97—61no—
349#97C4×D314yescomplemented
350C4×C8—21no—
351C4×C8—21no—
352C4×C8—61no—
353C4×C8—61no—
354C23×C3H(5,𝔽2)11yescharacteristic
355C22×D3—22no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—82no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—42no—
363C2×C3×C4C23×C411yescharacteristic
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380D12—22no—
381D12—22no—
382D12—41no—
383D12—41no—
384C4×D3—162no—
385Dic6—22no—
386Dic6—22no—
387Dic6—22no—
388Dic6—22no—
389Dic6—22no—
390Dic6—22no—
391Dic6—41no—
392Dic6—41no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396Dic6—41no—
397C3×C8—41no—
398C3×C8—41no—
399C3×C8C4×D412yes—
400C3×C8C4×D412yes—
401C3×C8C4×D412yes—
402C3×C8C4×D412yes—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C24—61no—
406C2×Q8—32no—
407C2×Q8—32no—
408C2×Q8—32no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×D4—32no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C22×C4—21no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—122no—
435C22×C4C23×D311yescharacteristic
436Q16—64no—
437Q16—64no—
438Q16—64no—
439Q16—64no—
440SD16—64no—
441SD16—64no—
442SD16—64no—
443SD16—64no—
444M16—21no—
445M16—21no—
446M16—21no—
447M16—21no—
448M16—61no—
449M16—61no—
450M16—61no—
451M16—61no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—61no—
455C2×C8—61no—
456C2×C8C2×C4×D312yes—
457C2×C8C2×C4×D312yes—
458C2×C8C2×D1211yescharacteristic
459C2×C8C2×D1211yescharacteristic
460C2×C8#23512yes—
461C2×C8D3×D411yescharacteristic
462C2×C8D3×D411yescharacteristic
463C2×C8#23912yes—
464C4⋊C4—32no—
465C4⋊C4—32no—
466C4⋊C4—124no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4C2×D1212yes—
471C4⋊C4C2×D1212yes—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—124no—
477C22⋊C4—122no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—61no—
482C42—61no—
483C42—122no—
484C42—122no—
485C42—122no—
486C42C2×C4×D312yes—
487C22×C3—22no—
488C22×C3#51811yescharacteristic
489C22×C3D4211yescharacteristic
490C22×C3#55911yescharacteristic
491C2×D3—42no—
492C2×D3—82no—
493C2×D3—82no—
494C2×D3—161no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4C2×C4×D411yescharacteristic
499C3×C4C2×C4×D411yescharacteristic
500C3×C4#55011yescharacteristic
501C3×C4#55011yescharacteristic
502Dic3—42no—
503Dic3—42no—
504Dic3—42no—
505Dic3—42no—
506Dic3—42no—
507C23—62no—
508C23—121no—
509C23—121no—
510C23—242no—
511C23#100911yescharacteristic
512Q8—62no—
513Q8—62no—
514Q8—62no—
515Q8—62no—
516Q8—62no—
517Q8—62no—
518Q8—121no—
519Q8—121no—
520Q8—121no—
521Q8—121no—
522Q8—121no—
523Q8—121no—
524D4—62no—
525D4—62no—
526D4—121no—
527D4—121no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—42no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—482no—
544C2×C4C22×C4×D311yescharacteristic
545C2×C4C22×D1211yescharacteristic
546C2×C4C2×#23511yescharacteristic
547C2×C4C2×D3×D411yescharacteristic
548C2×C4C2×#23911yescharacteristic
549C2×C4#100911yescharacteristic
550C8—41no—
551C8—41no—
552C8—121no—
553C8—121no—
554C8C4×D1212yes—
555C8C4×D1212yes—
556C8#89312yes—
557C8#89312yes—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3#236611yescharacteristic
561C2×C3#328912yes—
562D3—82no—
563C22—22no—
564C22—122no—
565C22—242no—
566C22—242no—
567C22—481no—
568C22#552111yescharacteristic
569C22D4×D1211yescharacteristic
570C22#558511yescharacteristic
571C4—42no—
572C4—42no—
573C4—42no—
574C4—122no—
575C4—122no—
576C4—122no—
577C4—122no—
578C4—122no—
579C4C2×C4×D1211yescharacteristic
580C4C2×#89311yescharacteristic
581C4#555311yescharacteristic
582C4#557711yescharacteristic
583C3#2320311yescharacteristic, complemented, Sylow
584C2—21no—
585C2—21no—
586C2—242no—
587C2#7365311yescharacteristic
588C2#7990112yes—
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1773C2
socleC22×C3D42
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1773D3
hypocenterC3#23203
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1773

Complements

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

NH ≅ G/Ntype
1. #79169C2semidirect
3. #78573C2semidirect
6. C3×#1773C2semidirect
7. #70710C2semidirect
9. #69904C2semidirect
11. #69829C2semidirect
13. #68765C2semidirect
14. #68765C2semidirect
53. #4715C4semidirect
75. #1773D3semidirect
99. C2×Dic12C2×C4semidirect
100. C2×Dic12C2×C4semidirect
121. #869C2×C4semidirect
130. C3×#97C2×C4semidirect
349. #97C4×D3semidirect
583. C3#23203semidirect

Maximal subgroups

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

subgroupindex
#791692
#791682
#785732
#785722
C2×#47152
C3×#17732
#707102
#707082
#699042
#699022
#698292
#687662
#687652
#687592
#232033