Finite Group Databasebeta

← #10897672

Subgroups — #10897672

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83811C211yesmaximal, characteristic, complemented
2#78649C211yesmaximal, characteristic, complemented
3C3×#1881C211yesmaximal, characteristic, complemented
4#72121C211yesmaximal, characteristic
5#72114C211yesmaximal, characteristic, complemented
6#72026C211yesmaximal, characteristic, complemented
7#70405C211yesmaximal, characteristic, complemented
8#70395C211yesmaximal, characteristic, complemented
9#70296C211yesmaximal, characteristic, complemented
10#70288C211yesmaximal, characteristic, complemented
11#70012C211yesmaximal, characteristic, complemented
12#69034C211yesmaximal, characteristic, complemented
13#69017C211yesmaximal, characteristic, complemented
14#68936C211yesmaximal, characteristic, complemented
15#68919C211yesmaximal, characteristic, complemented
16#29387—31nomaximal, Sylow
17C3×#575C2211yescharacteristic
18C3×#568C2211yescharacteristic
19Dic3×#41C2211yescharacteristic
20#5822C2211yescharacteristic
21#5822C2211yescharacteristic
22#5745C2211yescharacteristic
23C3×#421C2211yescharacteristic
24C3×#416C2211yescharacteristic
25C3×#408C2211yescharacteristic
26#5240C2211yescharacteristic
27#5236C2211yescharacteristic
28C2×#837C2211yescharacteristic
29#5234C2211yescharacteristic
30#5234C2211yescharacteristic
31#5145—22no—
32#5144—22no—
33C2×#825C2211yescharacteristic
34#5129C2211yescharacteristic
35#5124C2211yescharacteristic
36#5119C2211yescharacteristic
37#5119C2211yescharacteristic
38Dic3×M16C2211yescharacteristic
39#5113C2211yescharacteristic
40#5102C2211yescharacteristic
41C2×#815C2211yescharacteristic
42#5006C2211yescharacteristic
43#4999C2211yescharacteristic
44#4720—22no—
45#4719—22no—
46C3×#344C2211yescharacteristic
47C3×#341C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4559C2211yescharacteristic
50#4556C2211yescharacteristic
51#4556C2211yescharacteristic
52#4555C2211yescharacteristic
53#4552C2211yescharacteristic
54#4532C2211yescharacteristic
55#4531C2211yescharacteristic
56#2956—31no—
57#2923—31no—
58#2103—31no—
59#2098—31no—
60#2053—31no—
61#2017—31no—
62#1938—31no—
63#1938—31no—
64#1906—31no—
65#1906—31no—
66#1881—31no—
67#1881D311yescharacteristic, complemented
68#1868—31no—
69#1868—31no—
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C2×C3×#41C2311yescharacteristic
73C2×#235C2311yescharacteristic
74C3×#137—21no—
75C3×#137—21no—
76C3×AGL(1,ℤ/8)—21no—
77C3×AGL(1,ℤ/8)—21no—
78C3×#135—41no—
79C3×#131—22no—
80C3×#131—41no—
81C2×C3×M16—41no—
82C2×C3×M16C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#117C2311yescharacteristic
87#951—21no—
88#951—21no—
89#950—22no—
90#950—41no—
91#949—21no—
92#949—21no—
93Q8×Dic3—21no—
94#942—21no—
95#942—21no—
96D4×Dic3—21no—
97D4×Dic3—22no—
98#932—21no—
99#932—21no—
100#926—21no—
101#926C2311yescharacteristic
102#926C2311yescharacteristic
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×#208C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109#907—41no—
110D3×M16—41no—
111#901—22no—
112C3×#104—41no—
113C3×#103—21no—
114C3×#102—21no—
115C3×C22⋊C8—22no—
116#837—21no—
117#837—21no—
118#837—41no—
119#834—22no—
120#830—22no—
121#830—41no—
122#825—21no—
123#825—21no—
124#822—22no—
125#821—21no—
126(C2×D3)⋊C8—41no—
127#816—21no—
128#815—21no—
129#815—21no—
130#815D412yes—
131Dic3⋊C8—22no—
132Dic3⋊C8—22no—
133Dic3⋊C8—41no—
134C8×Dic3D412yes—
135#806—22no—
136C4⋊(C3⋊C8)—41no—
137#575—31no—
138#575—31no—
139#575C2×D311yescharacteristic
140#568—31no—
141#568C2×D311yescharacteristic
142C4×#41—31no—
143#444—62no—
144#443—62no—
145#433—62no—
146#432—62no—
147#430—31no—
148#424—31no—
149#424—31no—
150#424—31no—
151#424—31no—
152#421—31no—
153#421C2×D311yescharacteristic
154C2×#104—31no—
155C2×#104—31no—
156#416—31no—
157#416C2×D311yescharacteristic
158#410—31no—
159#410—31no—
160#408—31no—
161#408—31no—
162#408—31no—
163#408C2×D311yescharacteristic
164C4×M16—31no—
165C2×#97—31no—
166#344—31no—
167#344—31no—
168#344—31no—
169#344C2×D311yescharacteristic
170#341—31no—
171#341C2×D311yescharacteristic
172#339—31no—
173#339—31no—
174#338—31no—
175H(3,ℤ/4)—31no—
176C3×#41—21no—
177C3×#41—21no—
178C3×#41—21no—
179C3×#41—21no—
180C3×#41—21no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C2×C3×Q8#4111yescharacteristic
185C2×C3×Q8#4111yescharacteristic
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4#4111yescharacteristic
189C2×C3×D4#4111yescharacteristic
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2411yescharacteristic
193C2×#67—21no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196#235—21no—
197#235—21no—
198#235—21no—
199#235—21no—
200C2×D12#4111yescharacteristic
201C2×C4×D3—21no—
202C2×Dic6#4111yescharacteristic
203C3×Q16—41no—
204C3×SD16—41no—
205C3×SD16—41no—
206C3×D8—41no—
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—21no—
218C2×C3×C8—21no—
219C2×C3×C8—42no—
220C2×C3×C8C2×D412yes—
221C3×C4⋊C4—21no—
222C3×C22⋊C4—21no—
223C3×C42—21no—
224#217—22no—
225#217—21no—
226#217#4112yes—
227#216—22no—
228#216—41no—
229#215—22no—
230#215—41no—
231#214—22no—
232#214—41no—
233#213—22no—
234#213—41no—
235C4⋊Dic3—22no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4×Dic3—22no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3C2×D411yescharacteristic
243C4×Dic3C2×D411yescharacteristic
244#208—21no—
245#208—21no—
246#208—21no—
247#208—21no—
248#208—21no—
249#208—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—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
260C2×#41C22×D311yescharacteristic
261#137—21no—
262#137—21no—
263#137—61no—
264#137—61no—
265#137—61no—
266#137—61no—
267AGL(1,ℤ/8)—21no—
268AGL(1,ℤ/8)—21no—
269AGL(1,ℤ/8)—61no—
270AGL(1,ℤ/8)—61no—
271AGL(1,ℤ/8)—61no—
272AGL(1,ℤ/8)—61no—
273#135—41no—
274#135—62no—
275#135—121no—
276#131—22no—
277#131—41no—
278#131—62no—
279#131—121no—
280C2×M16—31no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—41no—
284C2×M16—121no—
285C2×M16C22×D311yescharacteristic
286C2×M16C22×D311yescharacteristic
287C2×M16C22×D311yescharacteristic
288C22×C8—31no—
289C22×C8C22×D311yescharacteristic
290C4×Q8—61no—
291C4×D4—62no—
292C4×D4—61no—
293#117—31no—
294#117—31no—
295#117—61no—
296#117C22×D311yescharacteristic
297C2×C42—31no—
298C2×C42—61no—
299#108—62no—
300C4⋊C8—62no—
301C4⋊C8—62no—
302C4⋊C8—121no—
303C4⋊C8—121no—
304#104—41no—
305#104—61no—
306#104—61no—
307#104—61no—
308#104—61no—
309#104—121no—
310#103—21no—
311#103—61no—
312#102—21no—
313#102—61no—
314#99—62no—
315#99—62no—
316C22⋊C8—22no—
317C22⋊C8—62no—
318C22⋊C8—121no—
319C22⋊C8—121no—
320#97—32no—
321#97—61no—
322#97—61no—
323C4×C8—32no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3C2×#4111yescharacteristic
327C22×D3—21no—
328C3×Q8—21no—
329C3×Q8—21no—
330C3×Q8—21no—
331C3×Q8—21no—
332C3×D4—21no—
333C3×D4—21no—
334C3×D4—21no—
335C3×D4—21no—
336C3×D4—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
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×C4C22×D411yescharacteristic
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4H(5,𝔽2)11yescharacteristic
352C2×C3×C4H(5,𝔽2)11yescharacteristic
353#67—41no—
354#67—41no—
355C2×Dic3—21no—
356C2×Dic3—22no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—41no—
360C2×Dic3—81no—
361C2×Dic3—82no—
362D12—21no—
363D12—21no—
364C4×D3—41no—
365C4×D3—41no—
366Dic6—21no—
367Dic6—21no—
368C3×C8—22no—
369C3×C8—41no—
370C3×C8—41no—
371C3×C8—41no—
372C3⋊C8—22no—
373C3⋊C8—41no—
374C3⋊C8—41no—
375C3⋊C8—41no—
376#41—21no—
377#41—21no—
378#41—21no—
379#41—21no—
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#23511yescharacteristic
390C2×Q8#23711yescharacteristic
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—31no—
394C2×D4—61no—
395C2×D4#23511yescharacteristic
396C2×D4#23711yescharacteristic
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—62no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4C23×D311yescharacteristic
403Q16—41no—
404Q16—62no—
405Q16—121no—
406SD16—41no—
407SD16—41no—
408SD16—62no—
409SD16—62no—
410SD16—121no—
411SD16—121no—
412D8—41no—
413D8—62no—
414D8—121no—
415M16—21no—
416M16—21no—
417M16—21no—
418M16—21no—
419M16—21no—
420M16—21no—
421M16—42no—
422M16—61no—
423M16—61no—
424M16—61no—
425M16—61no—
426M16—61no—
427M16—61no—
428M16—122no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—32no—
435C2×C8—42no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—122no—
442C2×C8D3×D412yes—
443C4⋊C4—21no—
444C4⋊C4—62no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C22⋊C4—21no—
448C22⋊C4—32no—
449C22⋊C4—62no—
450C22⋊C4—61no—
451C42—21no—
452C42—31no—
453C42—31no—
454C42—62no—
455C42—61no—
456C42—61no—
457C42—61no—
458C22×C3—41no—
459C22×C3—42no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—81no—
463C22×C3#52511yescharacteristic
464C22×C3#55311yescharacteristic
465C22×C3#55311yescharacteristic
466C2×D3—41no—
467C2×D3—81no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—81no—
471C3×C4#52511yescharacteristic
472C3×C4#53511yescharacteristic
473C3×C4#55111yescharacteristic
474C3×C4#55111yescharacteristic
475Dic3—42no—
476Dic3—41no—
477Dic3—41no—
478Dic3—41no—
479Dic3—81no—
480C23—21no—
481C23—21no—
482C23—61no—
483C23D3×#4111yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—21no—
488Q8—61no—
489Q8—61no—
490D4—21no—
491D4—21no—
492D4—21no—
493D4—21no—
494D4—41no—
495D4—41no—
496D4—41no—
497D4—41no—
498D4—61no—
499D4—61no—
500D4—121no—
501D4—121no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—62no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—241no—
517C2×C4—242no—
518C2×C4C2×#23511yescharacteristic
519C2×C4C2×D3×D411yescharacteristic
520C2×C4C2×#23711yescharacteristic
521C2×C4#100411yescharacteristic
522C2×C4#100411yescharacteristic
523C2×C4D3×#4111yescharacteristic
524C8—22no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—62no—
529C8—121no—
530C8—121no—
531C8—121no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—81no—
536C2×C3—81no—
537C2×C3#259811yescharacteristic
538D3—81no—
539C22—41no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—81no—
544C22—121no—
545C22—241no—
546C22#554711yescharacteristic
547C22#560111yescharacteristic
548C22#566111yescharacteristic
549C4—41no—
550C4—41no—
551C4—81no—
552C4—121no—
553C4—121no—
554C4—122no—
555C4—121no—
556C4—241no—
557C4#549511yescharacteristic
558C4#555611yescharacteristic
559C4#559211yescharacteristic
560C4#560611yescharacteristic
561C3#2938711yescharacteristic, complemented, Sylow
562C2—21no—
563C2—21no—
564C2—21no—
565C2—81no—
566C2—81no—
567C2—241no—
568C2#7477611yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1881C2
socleC2×C3#2598
prime radical#72121C2
hypercenter#1881D3
hypocenterC3#29387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1881

Complements

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

NH ≅ G/Ntype
1. #83811C2semidirect
2. #78649C2semidirect
3. C3×#1881C2semidirect
5. #72114C2semidirect
6. #72026C2semidirect
7. #70405C2semidirect
8. #70395C2semidirect
9. #70296C2semidirect
10. #70288C2semidirect
11. #70012C2semidirect
12. #69034C2semidirect
13. #69017C2semidirect
14. #68936C2semidirect
15. #68919C2semidirect
67. #1881D3semidirect
561. C3#29387semidirect

Maximal subgroups

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

subgroupindex
#838112
#786492
C3×#18812
#721212
#721142
#720262
#704052
#703952
#702962
#702882
#700122
#690342
#690172
#689362
#689192
#293873