Finite Group Databasebeta

← #10897663

Subgroups — #10897663

Order 768.

654 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#79070C211yesmaximal, characteristic, complemented
3C3×#1868C211yesmaximal, characteristic, complemented
4#72020C211yesmaximal, characteristic, complemented
5#70549C211yesmaximal, characteristic, complemented
6#70546C211yesmaximal, characteristic, complemented
7#70535C211yesmaximal, characteristic, complemented
8#70055C211yesmaximal, characteristic, complemented
9#70043C211yesmaximal, characteristic, complemented
10#70013C211yesmaximal, characteristic
11#69997C211yesmaximal, characteristic, complemented
12#69039C211yesmaximal, characteristic, complemented
13#69035C211yesmaximal, characteristic, complemented
14#69017C211yesmaximal, characteristic, complemented
15#69016C211yesmaximal, characteristic, complemented
16#29386—31nomaximal, Sylow
17C3×#568C2211yescharacteristic
18#5822C2211yescharacteristic
19#5822C2211yescharacteristic
20#5748C2211yescharacteristic
21#5745C2211yescharacteristic
22#5522C2211yescharacteristic
23C3×#424C2211yescharacteristic
24C3×#424C2211yescharacteristic
25C3×#410C2211yescharacteristic
26C3×#408C2211yescharacteristic
27C2×#837C2211yescharacteristic
28#5233C2211yescharacteristic
29#5145—22no—
30#5144—22no—
31#5139C2211yescharacteristic
32C2×#825C2211yescharacteristic
33#5129C2211yescharacteristic
34#5124C2211yescharacteristic
35Dic3×M16C2211yescharacteristic
36#5110C2211yescharacteristic
37#5110C2211yescharacteristic
38#5102C2211yescharacteristic
39C2×C8×Dic3C2211yescharacteristic
40#5006C2211yescharacteristic
41#5005C2211yescharacteristic
42#5002C2211yescharacteristic
43#4999C2211yescharacteristic
44#4840—22no—
45#4833—22no—
46C3×#344C2211yescharacteristic
47C3×#344C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4558C2211yescharacteristic
50#4558C2211yescharacteristic
51#4556C2211yescharacteristic
52#4555C2211yescharacteristic
53#4533C2211yescharacteristic
54#4532C2211yescharacteristic
55#4531C2211yescharacteristic
56#2956—31no—
57#2922—31no—
58#2103—31no—
59#2098—31no—
60#2053—31no—
61#2017—31no—
62#1937—31no—
63#1937—31no—
64#1906—31no—
65#1906—31no—
66#1881—31no—
67#1881—31no—
68#1868—31no—
69#1868D311yescharacteristic, complemented
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C2×#235C2311yescharacteristic
73C2×#235C2311yescharacteristic
74C3×#131—22no—
75C3×#131—41no—
76C2×C3×M16—41no—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#117C2311yescharacteristic
82C3×#117C2311yescharacteristic
83#951—21no—
84#951—21no—
85#950—22no—
86#950—41no—
87#949—21no—
88#949—21no—
89#942—21no—
90#942—21no—
91#932—21no—
92#932—21no—
93#926C2311yescharacteristic
94C2×C4×Dic3C2311yescharacteristic
95C2×#208C2311yescharacteristic
96C2×#208C2311yescharacteristic
97C2×#208C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99#909—21no—
100#909—21no—
101#908—21no—
102#908—21no—
103#907—41no—
104D3×M16—41no—
105#904—41no—
106#901—22no—
107#893—21no—
108#887—21no—
109#67⋊C4—22no—
110#872—21no—
111C42×D3—21no—
112C3×C4⋊C8—22no—
113C3×C4⋊C8—41no—
114C3×#99—22no—
115C3×C22⋊C8—41no—
116#837—21no—
117#837—21no—
118#835—21no—
119#832—21no—
120#825—21no—
121#825—21no—
122#825—41no—
123#822—22no—
124(C2×D3)⋊C8—22no—
125(C2×D3)⋊C8—22no—
126(C2×D3)⋊C8—41no—
127#815D412yes—
128Dic3⋊C8—41no—
129C8×Dic3—21no—
130C8×Dic3—21no—
131C8×Dic3D412yes—
132#810—21no—
133#809—21no—
134#806—22no—
135#805—41no—
136C4⋊(C3⋊C8)—22no—
137#575—31no—
138#575—31no—
139#575—31no—
140#568—31no—
141#568C2×D311yescharacteristic
142C4×#41—31no—
143#444—62no—
144#443—62no—
145#433—62no—
146#431—62no—
147#430—31no—
148#424—31no—
149#424—31no—
150#424C2×D311yescharacteristic
151#424C2×D311yescharacteristic
152#421—31no—
153#421—31no—
154C2×#104—31no—
155C2×#104—31no—
156#416—31no—
157#416—31no—
158#410—31no—
159#410C2×D311yescharacteristic
160#408—31no—
161#408—31no—
162#408—31no—
163#408C2×D311yescharacteristic
164C4×M16—31no—
165C2×C4×C8—31no—
166#344—31no—
167#344—31no—
168#344C2×D311yescharacteristic
169#344C2×D311yescharacteristic
170#341—31no—
171#341—31no—
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—
180C2×C3×Q8#4111yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4#4111yescharacteristic
183C22×C3×C4—21no—
184C22×C3×C4C2411yescharacteristic
185C2×#67—21no—
186C2×#67—21no—
187C22×Dic3—21no—
188#235—21no—
189#235—21no—
190#235—21no—
191#235—21no—
192#235—21no—
193#235—21no—
194#235—21no—
195#235—21no—
196C2×D12#4111yescharacteristic
197C2×D12#4111yescharacteristic
198C2×C4×D3—22no—
199C2×C4×D3—21no—
200C2×C4×D3—21no—
201C2×Dic6#4111yescharacteristic
202C2×Dic6#4111yescharacteristic
203C3×M16—21no—
204C3×M16—21no—
205C3×M16—21no—
206C3×M16—21no—
207C3×M16—21no—
208C3×M16—21no—
209C3×M16—42no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—42no—
216C2×C3×C8C2×D412yes—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4#4112yes—
221C3×C42—21no—
222C3×C42—21no—
223#217—21no—
224#216—22no—
225#216—41no—
226#215—22no—
227#215—41no—
228#214—22no—
229#214—41no—
230#213—22no—
231#213—41no—
232(C2×D3)⋊C4—22no—
233C4⋊Dic3—21no—
234Dic3⋊C4—22no—
235C4×Dic3—22no—
236C4×Dic3—21no—
237C4×Dic3—21no—
238C4×Dic3C2×D411yescharacteristic
239C4×Dic3C2×D411yescharacteristic
240#208—21no—
241#208—21no—
242#208—21no—
243#208—21no—
244#208—21no—
245#208—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8C2×D412yes—
252Dic12—41no—
253D24—41no—
254#204—41no—
255#204—41no—
256ΓL(1,𝔽25)—42no—
257D3×C8—42no—
258C2×#41—31no—
259C2×#41—31no—
260C2×#41C22×D311yescharacteristic
261#137—61no—
262#137—61no—
263#137—61no—
264#137—61no—
265#137—61no—
266#137—61no—
267AGL(1,ℤ/8)—61no—
268AGL(1,ℤ/8)—61no—
269AGL(1,ℤ/8)—61no—
270AGL(1,ℤ/8)—61no—
271AGL(1,ℤ/8)—61no—
272AGL(1,ℤ/8)—61no—
273#135—62no—
274#135—121no—
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—61no—
292C4×D4—62no—
293#117—31no—
294#117—61no—
295#117C22×D311yescharacteristic
296#117C22×D311yescharacteristic
297C2×C42—31no—
298C2×C42—61no—
299#108—62no—
300C4⋊C8—22no—
301C4⋊C8—41no—
302C4⋊C8—62no—
303C4⋊C8—121no—
304#104—61no—
305#104—61no—
306#104—61no—
307#104—61no—
308#104—121no—
309#104—121no—
310#103—61no—
311#103—61no—
312#102—61no—
313#102—61no—
314#99—22no—
315#99—62no—
316C22⋊C8—41no—
317C22⋊C8—62no—
318C22⋊C8—62no—
319C22⋊C8—121no—
320#97—32no—
321C4×C8—32no—
322C4×C8—61no—
323C4×C8—61no—
324C23×C3—21no—
325C23×C3C2×#4111yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C3×Q8—21no—
329C3×Q8—21no—
330C3×D4—21no—
331C3×D4—21no—
332C3×D4—41no—
333C3×D4—41no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4C2×#4111yescharacteristic
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4H(5,𝔽2)11yescharacteristic
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345#67—41no—
346#67—41no—
347#67—41no—
348#67—41no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—42no—
354C2×Dic3—41no—
355D12—21no—
356D12—21no—
357D12—21no—
358D12—21no—
359C4×D3—44no—
360C4×D3—41no—
361C4×D3—41no—
362C4×D3—41no—
363C4×D3—41no—
364Dic6—21no—
365Dic6—21no—
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—61no—
381#41—61no—
382#41—61no—
383#41—61no—
384#41—61no—
385#41—61no—
386#41—61no—
387#41—61no—
388C2×Q8—31no—
389C2×Q8—31no—
390C2×Q8#23911yescharacteristic
391C2×D4—21no—
392C2×D4—31no—
393C2×D4—31no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4#23911yescharacteristic
397C22×C4—21no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4C23×D311yescharacteristic
403Q16—62no—
404Q16—121no—
405Q16—121no—
406SD16—62no—
407SD16—62no—
408SD16—121no—
409SD16—121no—
410SD16—121no—
411SD16—121no—
412D8—62no—
413D8—121no—
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—21no—
445C4⋊C4—62no—
446C4⋊C4—61no—
447C22⋊C4—21no—
448C22⋊C4—62no—
449C22⋊C4—61no—
450C22⋊C4#23512yes—
451C42—21no—
452C42—21no—
453C42—31no—
454C42—31no—
455C42—62no—
456C42—61no—
457C42—61no—
458C22×C3—41no—
459C22×C3—41no—
460C22×C3—81no—
461C22×C3#52511yescharacteristic
462C22×C3#55311yescharacteristic
463C22×C3#55311yescharacteristic
464C2×D3—42no—
465C2×D3—41no—
466C2×D3—41no—
467C2×D3—81no—
468C3×C4—42no—
469C3×C4—41no—
470C3×C4—81no—
471C3×C4#52511yescharacteristic
472C3×C4#53511yescharacteristic
473C3×C4#55111yescharacteristic
474C3×C4#55111yescharacteristic
475Dic3—22no—
476Dic3—41no—
477Dic3—41no—
478Dic3—41no—
479Dic3—81no—
480C23—21no—
481C23—61no—
482C23—61no—
483C23C2×#23511yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—61no—
487Q8—61no—
488Q8—61no—
489Q8—61no—
490D4—21no—
491D4—21no—
492D4—41no—
493D4—41no—
494D4—61no—
495D4—61no—
496D4—61no—
497D4—61no—
498D4—121no—
499D4—121no—
500D4—121no—
501D4—121no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—124no—
512C2×C4—122no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4C2×D3×D411yescharacteristic
519C2×C4#100411yescharacteristic
520C2×C4#100411yescharacteristic
521C2×C4C2×#23911yescharacteristic
522C2×C4D3×#4111yescharacteristic
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#259811yescharacteristic
537D3—81no—
538D3—81no—
539C22—41no—
540C22—41no—
541C22—81no—
542C22—122no—
543C22—121no—
544C22—121no—
545C22—241no—
546C22#551411yescharacteristic
547C22#554711yescharacteristic
548C22#555811yescharacteristic
549C4—42no—
550C4—41no—
551C4—62no—
552C4—81no—
553C4—121no—
554C4—121no—
555C4—121no—
556C4—241no—
557C4#559211yescharacteristic
558C4#562711yescharacteristic
559C4#563711yescharacteristic
560C4#565911yescharacteristic
561C3#2938611yescharacteristic, complemented, Sylow
562C2—21no—
563C2—21no—
564C2—21no—
565C2—81no—
566C2—241no—
567C2—241no—
568C2#7449611yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1868C2
socleC2×C3#2598
prime radical#70013C2
hypercenter#1868D3
hypocenterC3#29386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1868

Complements

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

NH ≅ G/Ntype
1. #83811C2semidirect
2. #79070C2semidirect
3. C3×#1868C2semidirect
4. #72020C2semidirect
5. #70549C2semidirect
6. #70546C2semidirect
7. #70535C2semidirect
8. #70055C2semidirect
9. #70043C2semidirect
11. #69997C2semidirect
12. #69039C2semidirect
13. #69035C2semidirect
14. #69017C2semidirect
15. #69016C2semidirect
69. #1868D3semidirect
561. C3#29386semidirect

Maximal subgroups

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

subgroupindex
#838112
#790702
C3×#18682
#720202
#705492
#705462
#705352
#700552
#700432
#700132
#699972
#690392
#690352
#690172
#690162
#293863