Finite Group Databasebeta

← #10897650

Subgroups — #10897650

Order 768.

662 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83811C211yesmaximal, characteristic, complemented
2#78620C211yesmaximal, characteristic, complemented
3C3×#1891C211yesmaximal, characteristic, complemented
4#72028C211yesmaximal, characteristic, complemented
5#71715C211yesmaximal, characteristic
6#71712C211yesmaximal, characteristic, complemented
7#70144C211yesmaximal, characteristic, complemented
8#70130C211yesmaximal, characteristic, complemented
9#69992C211yesmaximal, characteristic, complemented
10#69987C211yesmaximal, characteristic, complemented
11#69966C211yesmaximal, characteristic, complemented
12#69033C211yesmaximal, characteristic, complemented
13#69024C211yesmaximal, characteristic, complemented
14#69017C211yesmaximal, characteristic, complemented
15#69012C211yesmaximal, characteristic, complemented
16#29385—31nomaximal, Sylow
17C3×#568C2211yescharacteristic
18C3×#525C2211yescharacteristic
19#5822C2211yescharacteristic
20#5791C2211yescharacteristic
21C2×#926C2211yescharacteristic
22#5745C2211yescharacteristic
23C3×#413C2211yescharacteristic
24C3×#407C2211yescharacteristic
25C3×#407C2211yescharacteristic
26#5236C2211yescharacteristic
27C2×#837C2211yescharacteristic
28C2×#837C2211yescharacteristic
29#5207C2211yescharacteristic
30#5207C2211yescharacteristic
31#5145—22no—
32#5144—22no—
33C2×#825C2211yescharacteristic
34C2×#825C2211yescharacteristic
35#5132C2211yescharacteristic
36#5125C2211yescharacteristic
37#5124C2211yescharacteristic
38#5124C2211yescharacteristic
39#5118C2211yescharacteristic
40#5118C2211yescharacteristic
41Dic3×M16C2211yescharacteristic
42Dic3×M16C2211yescharacteristic
43#5005C2211yescharacteristic
44#4720—22no—
45#4720—22no—
46C3×#344C2211yescharacteristic
47C3×#344C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4559C2211yescharacteristic
50#4556C2211yescharacteristic
51#4555C2211yescharacteristic
52#4555C2211yescharacteristic
53#4532C2211yescharacteristic
54#4532C2211yescharacteristic
55#4531C2211yescharacteristic
56#2956—31no—
57#2921—31no—
58#2102—31no—
59#2091—31no—
60#2050—31no—
61#2010—31no—
62#1932—31no—
63#1932—31no—
64#1897—31no—
65#1897—31no—
66#1891—31no—
67#1891D311yescharacteristic, complemented
68#1861—31no—
69#1861—31no—
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C2×#235C2311yescharacteristic
74C3×#131—22no—
75C3×#131—41no—
76C2×C3×M16—41no—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C3×#123—21no—
82C3×C22⋊Q8—21no—
83C3×#121—21no—
84C3×#120—21no—
85C3×C4×D4—41no—
86C3×#117C2311yescharacteristic
87#953—21no—
88C2×#217—21no—
89#951—21no—
90#950—22no—
91#949—21no—
92#942—21no—
93#932—21no—
94#930—21no—
95#929—21no—
96C4×#67—41no—
97#926—21no—
98#926—21no—
99#926—22no—
100#926C2311yescharacteristic
101C2×C4⋊Dic3—21no—
102#924—21no—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109#907—41no—
110D3×M16—41no—
111#901—22no—
112C3×#101—21no—
113C3×#100—21no—
114C3×C22⋊C8—22no—
115C3×C22⋊C8—41no—
116#837—21no—
117#837—21no—
118#837—22no—
119#837—41no—
120#830—22no—
121#830—41no—
122#825—21no—
123#825—21no—
124#825—22no—
125#824—21no—
126#823—21no—
127#822—22no—
128#822—41no—
129#819—41no—
130#815—21no—
131#815D412yescomplemented
132Dic3⋊C8—22no—
133Dic3⋊C8—22no—
134C8×Dic3—21no—
135C8×Dic3D412yescomplemented
136#805—41no—
137#575—31no—
138#568—31no—
139#568C2×D311yescharacteristic
140#525—31no—
141#525C2×D311yescharacteristic
142C2×#117—31no—
143#444—62no—
144#443—62no—
145#433—62no—
146#433—62no—
147#430—31no—
148#430—31no—
149#430—31no—
150#423—31no—
151#423—31no—
152#421—31no—
153#421—31no—
154C2×#104—31no—
155C2×#104—31no—
156C2×#104—31no—
157C2×#104—31no—
158#413—31no—
159#413C2×D311yescharacteristic
160#407—31no—
161#407—31no—
162#407C2×D311yescharacteristic
163#407C2×D311yescharacteristic
164C4×M16—31no—
165C4×M16—31no—
166#344—31no—
167#344—31no—
168#344C2×D311yescharacteristic
169#344C2×D311yescharacteristic
170#339—31no—
171#339—31no—
172#339—31no—
173#339—31no—
174#338—31no—
175H(3,ℤ/4)—31no—
176C24×C3#4111yescharacteristic
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C3×#41—21no—
181C2×C3×Q8#4111yescharacteristic
182C2×C3×D4—21no—
183C2×C3×D4—41no—
184C2×C3×D4#4111yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4#4111yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C2×#67—21no—
195C2×#67—41no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198#235—21no—
199#235—21no—
200#235—21no—
201#235—21no—
202C2×D12#4111yescharacteristic
203C2×C4×D3—21no—
204C2×Dic6#4111yescharacteristic
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—21no—
208C3×M16—21no—
209C3×M16—21no—
210C3×M16—21no—
211C3×M16—21no—
212C3×M16—21no—
213C3×M16—42no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—42no—
218C2×C3×C8C2×D412yes—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—41no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—41no—
223C3×C22⋊C4—41no—
224C3×C42—21no—
225#217—22no—
226#217—22no—
227#217—21no—
228#217—41no—
229#216—22no—
230#215—22no—
231#214—22no—
232#213—22no—
233(C2×D3)⋊C4—41no—
234C4⋊Dic3—22no—
235C4⋊Dic3—22no—
236C4⋊Dic3—21no—
237Dic3⋊C4—41no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3C2×D411yescharacteristic
242C4×Dic3C2×D411yescharacteristic
243C4×Dic3#4112yes—
244#208—21no—
245#208—21no—
246#208—21no—
247#208—21no—
248#208—21no—
249#208—21no—
250#208—21no—
251#208—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8C2×D412yes—
256ΓL(1,𝔽25)—42no—
257D3×C8—42no—
258C2×#41—31no—
259C2×#41C22×D311yescharacteristic
260C23×C4C22×D311yescharacteristic
261#137—61no—
262#137—61no—
263AGL(1,ℤ/8)—61no—
264AGL(1,ℤ/8)—61no—
265#135—62no—
266#131—22no—
267#131—41no—
268#131—62no—
269#131—121no—
270C2×M16—31no—
271C2×M16—31no—
272C2×M16—31no—
273C2×M16—31no—
274C2×M16—41no—
275C2×M16—121no—
276C2×M16C22×D311yescharacteristic
277C2×M16C22×D311yescharacteristic
278C2×M16C22×D311yescharacteristic
279C2×M16C22×D311yescharacteristic
280#123—21no—
281#123—61no—
282C22⋊Q8—21no—
283C22⋊Q8—61no—
284#121—21no—
285#121—61no—
286#120—21no—
287#120—61no—
288C4×D4—41no—
289C4×D4—121no—
290#117—31no—
291#117—62no—
292#117—61no—
293#117—61no—
294#117C22×D311yescharacteristic
295C2×C4⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299#108—62no—
300#108—121no—
301#108—121no—
302C4⋊C8—62no—
303C4⋊C8—62no—
304#104—62no—
305#104—61no—
306#104—61no—
307#104—62no—
308#104—61no—
309#104—61no—
310#104—121no—
311#104—121no—
312#101—21no—
313#101—61no—
314#100—21no—
315#100—61no—
316C22⋊C8—22no—
317C22⋊C8—41no—
318C22⋊C8—62no—
319C22⋊C8—121no—
320#97—32no—
321#97—61no—
322C4×C8—32no—
323C4×C8—61no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—81no—
329C23×C3C2×#4111yescharacteristic
330C22×D3—21no—
331C3×Q8—21no—
332C3×Q8—21no—
333C3×D4—21no—
334C3×D4—21no—
335C3×D4—41no—
336C3×D4—41no—
337C3×D4—82no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4H(5,𝔽2)11yescharacteristic
354C2×C3×C4H(5,𝔽2)11yescharacteristic
355#67—41no—
356#67—41no—
357#67—82no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—41no—
363C2×Dic3—81no—
364C2×Dic3—81no—
365D12—21no—
366D12—21no—
367C4×D3—41no—
368C4×D3—41no—
369Dic6—21no—
370Dic6—21no—
371C3×C8—22no—
372C3×C8—41no—
373C3×C8—41no—
374C3×C8—41no—
375C3⋊C8—22no—
376C3⋊C8—41no—
377C3⋊C8—41no—
378C3⋊C8—41no—
379C24#23511yescharacteristic
380#41—21no—
381#41—21no—
382#41—21no—
383#41—21no—
384#41—61no—
385#41—61no—
386#41—61no—
387#41—61no—
388C2×Q8—31no—
389C2×Q8#23711yescharacteristic
390C2×D4—21no—
391C2×D4—31no—
392C2×D4—41no—
393C2×D4—61no—
394C2×D4—121no—
395C2×D4#23711yescharacteristic
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4#23511yescharacteristic
407C22×C4C23×D311yescharacteristic
408Q16—62no—
409SD16—62no—
410SD16—62no—
411D8—62no—
412M16—21no—
413M16—21no—
414M16—21no—
415M16—21no—
416M16—21no—
417M16—21no—
418M16—21no—
419M16—21no—
420M16—42no—
421M16—61no—
422M16—61no—
423M16—61no—
424M16—61no—
425M16—61no—
426M16—61no—
427M16—61no—
428M16—61no—
429M16—122no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—32no—
434C2×C8—42no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—122no—
439C2×C8D3×D412yes—
440C4⋊C4—21no—
441C4⋊C4—41no—
442C4⋊C4—62no—
443C4⋊C4—62no—
444C4⋊C4—61no—
445C4⋊C4—121no—
446C22⋊C4—21no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—62no—
450C22⋊C4—62no—
451C22⋊C4—61no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C42—21no—
455C42—31no—
456C42—31no—
457C42—32no—
458C42—61no—
459C42—61no—
460C42—61no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—42no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3—41no—
469C22×C3—81no—
470C22×C3#52511yescharacteristic
471C22×C3#55111yescharacteristic
472C22×C3#55111yescharacteristic
473C2×D3—41no—
474C2×D3—81no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4#52511yescharacteristic
479C3×C4#53511yescharacteristic
480C3×C4#55311yescharacteristic
481C3×C4#55311yescharacteristic
482Dic3—41no—
483Dic3—41no—
484Dic3—42no—
485Dic3—41no—
486Dic3—81no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—61no—
492C23—81no—
493C23C2×#23511yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—61no—
497Q8—61no—
498D4—21no—
499D4—21no—
500D4—41no—
501D4—41no—
502D4—61no—
503D4—61no—
504D4—82no—
505D4—121no—
506D4—121no—
507D4—242no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—62no—
522C2×C4—61no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—241no—
527C2×C4—241no—
528C2×C4C2×D3×D411yescharacteristic
529C2×C4C2×#23711yescharacteristic
530C2×C4#100411yescharacteristic
531C2×C4#100411yescharacteristic
532C2×C4D3×#4111yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C8—22no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—62no—
539C8—121no—
540C8—121no—
541C8—121no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—81no—
546C2×C3—81no—
547C2×C3#259811yescharacteristic
548D3—81no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—81no—
558C22—121no—
559C22—241no—
560C22#549511yescharacteristic
561C22#554711yescharacteristic
562C22#555611yescharacteristic
563C4—41no—
564C4—41no—
565C4—81no—
566C4—121no—
567C4—121no—
568C4—122no—
569C4—121no—
570C4—241no—
571C4#559211yescharacteristic
572C4#560111yescharacteristic
573C4#560611yescharacteristic
574C4#566111yescharacteristic
575C3#2938511yescharacteristic, complemented, Sylow
576C2—21no—
577C2—21no—
578C2—21no—
579C2—81no—
580C2—81no—
581C2—241no—
582C2#7477611yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1891C2
socleC2×C3#2598
prime radical#71715C2
hypercenter#1891D3
hypocenterC3#29385
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1891

Complements

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

NH ≅ G/Ntype
1. #83811C2semidirect
2. #78620C2semidirect
3. C3×#1891C2semidirect
4. #72028C2semidirect
6. #71712C2semidirect
7. #70144C2semidirect
8. #70130C2semidirect
9. #69992C2semidirect
10. #69987C2semidirect
11. #69966C2semidirect
12. #69033C2semidirect
13. #69024C2semidirect
14. #69017C2semidirect
15. #69012C2semidirect
67. #1891D3semidirect
131. #815D4semidirect
135. C8×Dic3D4semidirect
575. C3#29385semidirect

Maximal subgroups

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

subgroupindex
#838112
#786202
C3×#18912
#720282
#717152
#717122
#701442
#701302
#699922
#699872
#699662
#690332
#690242
#690172
#690122
#293853