Finite Group Databasebeta

← #10782820

Subgroups — #10782820

Order 768.

590 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#83217C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic, complemented
3#79028C211yesmaximal, characteristic
4#79017C211yesmaximal, characteristic, complemented
5C2×C3×#333C211yesmaximal, characteristic, complemented
6#70433C211yesmaximal, characteristic, complemented
7#70423C211yesmaximal, characteristic, complemented
8#68514C211yesmaximal, characteristic, complemented
9#68513C211yesmaximal, characteristic, complemented
10#68510C211yesmaximal, characteristic, complemented
11#68509C211yesmaximal, characteristic, complemented
12#68502C211yesmaximal, characteristic, complemented
13#68501C211yesmaximal, characteristic, complemented
14#68498C211yesmaximal, characteristic, complemented
15#68497C211yesmaximal, characteristic, complemented
16#19591—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18#5479C2211yescharacteristic
19#5477C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22#5115—21no—
23#5114—21no—
24#5111—21no—
25C8×#67—21no—
26#5091C2211yescharacteristic
27#5091C2211yescharacteristic
28#5089—21no—
29#5088C2211yescharacteristic
30#5088C2211yescharacteristic
31#5087—21no—
32#5087—21no—
33#5084C2211yescharacteristic
34#5084C2211yescharacteristic
35#5081—21no—
36#5078C2211yescharacteristic
37#5078C2211yescharacteristic
38#4971—21no—
39#4967—21no—
40#4967—21no—
41#4967—21no—
42#4926C2211yescharacteristic
43#4925C2211yescharacteristic
44#4850C2211yescharacteristic
45#4848C2211yescharacteristic
46#4847—21no—
47#4844—21no—
48#4843C2211yescharacteristic
49#4842C2211yescharacteristic
50#4840—21no—
51#4839—21no—
52#4838C2211yescharacteristic
53#4834C2211yescharacteristic
54#4831C2211yescharacteristic
55#4830C2211yescharacteristic
56C3×#333C2211yescharacteristic
57C3×#333C2211yescharacteristic
58C3×#333C2211yescharacteristic
59C3×#333C2211yescharacteristic
60#4485C2211yescharacteristic
61#4485C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3247—31no—
69#3238—31no—
70#3226—31no—
71#2994—31no—
72#1698—31no—
73#1697—31no—
74#1696—31no—
75#1695—31no—
76#1678—31no—
77#1677—31no—
78#1676—31no—
79#1675—31no—
80#1573—31no—
81#1551—31no—
82C2×#333D311yescharacteristic, complemented
83C22×C3×Q8—21no—
84C2×#235—21no—
85C22×C3×C8—21no—
86C22×C3×C8—21no—
87C3×C4⋊Q8—21no—
88C3×C4⋊Q8—21no—
89C3×C4⋊Q8—21no—
90C3×C4⋊Q8C2311yescharacteristic
91C3×C4⋊Q8C2311yescharacteristic
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95#930—21no—
96#930—21no—
97#930D411yescharacteristic
98#929—21no—
99C4×#67—21no—
100#924D411yescharacteristic
101C2×D24—21no—
102C2×#204—21no—
103C2×ΓL(1,𝔽25)—21no—
104C2×D3×C8—21no—
105#876—21no—
106#875—21no—
107#875D411yescharacteristic
108#874C2311yescharacteristic
109C4×D12—21no—
110C4×D12—21no—
111C4×D12C2311yescharacteristic
112#872—21no—
113#870D411yescharacteristic
114#869C2311yescharacteristic
115C4×Dic6C2311yescharacteristic
116C3×C4⋊C8—21no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122#830—21no—
123#830D411yescharacteristic
124#830D411yescharacteristic
125#821—21no—
126#821—21no—
127#821—21no—
128(C2×D3)⋊C8—21no—
129(C2×D3)⋊C8—21no—
130#818—21no—
131#817—21no—
132#816—21no—
133#815—21no—
134Dic3⋊C8—21no—
135Dic3⋊C8—21no—
136C8×Dic3—21no—
137#810—21no—
138#810—21no—
139#809—21no—
140#809—21no—
141#809—21no—
142#809—21no—
143#809—21no—
144#809—21no—
145#808—21no—
146#808—21no—
147#807—21no—
148#807—21no—
149#807—21no—
150#807—21no—
151#807—21no—
152#807—21no—
153C4⋊(C3⋊C8)D411yescharacteristic
154C4⋊(C3⋊C8)D411yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157#803—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4⋊Q8C2×D311yescharacteristic
162#529—31no—
163#501—61no—
164#500—31no—
165#500—31no—
166#489—61no—
167#486—31no—
168#486—31no—
169#483—61no—
170#480—61no—
171#480—61no—
172#480—61no—
173#478—61no—
174#477—31no—
175#477—31no—
176#477—31no—
177#476—31no—
178#476—31no—
179#476—31no—
180#474—61no—
181#474—61no—
182#466—61no—
183#465—61no—
184#463—61no—
185#460—31no—
186#459—31no—
187#446—31no—
188C8×Q8—31no—
189#436—31no—
190#435—61no—
191C8×D4—31no—
192C8×D4—61no—
193#433—61no—
194#432—31no—
195#431—31no—
196#424—61no—
197C2×C4⋊C8C2×D311yescharacteristic
198C2×C4⋊C8C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333C2×D311yescharacteristic
206#333C2×D311yescharacteristic
207#333C2×D311yescharacteristic
208#333C2×D311yescharacteristic
209#331—31no—
210#331—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—41no—
214C2×C3×Q8—82no—
215C22×C3×C4—41no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4#4111yescharacteristic
219C2×#67—21no—
220C2×#67—41no—
221#235—82no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C2×D411yescharacteristic
226C2×C4×D3—21no—
227C2×C4×D3—41no—
228C2×Dic6—21no—
229C2×Dic6C2×D411yescharacteristic
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—81no—
235C2×C3×C8—81no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—41no—
241C3×C4⋊C4—41no—
242C3×C42C2×D411yescharacteristic
243C3×C42C2×D411yescharacteristic
244C3×C42#4111yescharacteristic
245C3×C42C2411yescharacteristic
246#217—21no—
247(C2×D3)⋊C4—21no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—41no—
252(C2×D3)⋊C4—41no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×D411yescharacteristic
256C4⋊Dic3C2×D411yescharacteristic
257Dic3⋊C4—21no—
258Dic3⋊C4—21no—
259Dic3⋊C4—21no—
260Dic3⋊C4—21no—
261C4×Dic3—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8C2×D411yescharacteristic
265C2×C3⋊C8C2×D411yescharacteristic
266C2×C3⋊C8C2×D411yescharacteristic
267C2×C3⋊C8C2×D411yescharacteristic
268D24—81no—
269#204—81no—
270ΓL(1,𝔽25)—81no—
271D3×C8—81no—
272C2×#41—61no—
273C22×Q8—21no—
274C2×SD16—61no—
275C2×D8—61no—
276C2×M16—61no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—61no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—31no—
284C4⋊Q8C22×D311yescharacteristic
285C4⋊Q8C22×D311yescharacteristic
286#127—31no—
287#126—61no—
288#125—31no—
289#124—31no—
290#124—61no—
291#123—61no—
292C22⋊Q8—31no—
293#121—31no—
294#121—61no—
295#121—61no—
296C4×Q8—31no—
297C4×D4—31no—
298C4×D4—61no—
299C4×D4—61no—
300C4×D4—61no—
301#117—61no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C42C22×D311yescharacteristic
305#107—61no—
306#107—61no—
307#107—61no—
308#107—61no—
309#107—61no—
310#107—61no—
311#107—61no—
312#106—61no—
313#106—61no—
314#106—61no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8C22×D311yescharacteristic
324C4⋊C8C22×D311yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—61no—
328#103—61no—
329#103—61no—
330#102—61no—
331#102—61no—
332#102—61no—
333#102—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339C22⋊C8—31no—
340C22⋊C8—31no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344#97—61no—
345#97—61no—
346C4×C8—31no—
347C4×C8—31no—
348C4×C8—61no—
349C23×C3#12111yescharacteristic
350C22×D3—21no—
351C22×D3—41no—
352C3×Q8—81no—
353C3×Q8—81no—
354C3×Q8—82no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—82no—
363C2×C3×C4#12111yescharacteristic
364C2×C3×C4#12111yescharacteristic
365C2×C3×C4#12111yescharacteristic
366C2×C3×C4C22×D411yescharacteristic
367C2×C3×C4C22×D411yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369#67—42no—
370#67—82no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
375D12—21no—
376D12—21no—
377D12—41no—
378D12—41no—
379D12—41no—
380D12—41no—
381D12—81no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—82no—
385Dic6—21no—
386Dic6—21no—
387Dic6—81no—
388C3×C8—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—242no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—31no—
402C2×Q8—41no—
403C2×Q8—61no—
404C2×Q8—82no—
405C2×D4—31no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—121no—
411C22×C4—41no—
412C22×C4—61no—
413C22×C4—121no—
414C22×C4#23511yescharacteristic
415C22×C4D3×D411yescharacteristic
416C22×C4D3×D411yescharacteristic
417SD16—241no—
418D8—241no—
419M16—241no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—31no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—81no—
431C2×C8—81no—
432C2×C8—241no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—31no—
438C4⋊C4—31no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C22⋊C4—61no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C22⋊C4—61no—
452C22⋊C4—121no—
453C22⋊C4—121no—
454C42—61no—
455C42#23511yescharacteristic
456C42D3×D411yescharacteristic
457C42D3×D411yescharacteristic
458C42C23×D311yescharacteristic
459C22×C3—21no—
460C22×C3—21no—
461C22×C3#47212yes—
462C22×C3C2×#12111yescharacteristic
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—41no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4#57711yescharacteristic
476C3×C4#57711yescharacteristic
477C3×C4#57811yescharacteristic
478C3×C4#57811yescharacteristic
479Dic3—41no—
480Dic3—41no—
481Dic3—41no—
482Dic3—81no—
483C23—61no—
484C23—121no—
485C23#88411yescharacteristic
486Q8—61no—
487Q8—61no—
488Q8—81no—
489Q8—81no—
490Q8—82no—
491Q8—241no—
492D4—61no—
493D4—61no—
494D4—122no—
495D4—121no—
496D4—121no—
497D4—121no—
498D4—121no—
499D4—241no—
500D4—242no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—82no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—242no—
516C2×C4#88411yescharacteristic
517C2×C4#88411yescharacteristic
518C2×C4#88411yescharacteristic
519C2×C4C2×#23511yescharacteristic
520C2×C4C2×D3×D411yescharacteristic
521C2×C4C2×D3×D411yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—61no—
527C8—61no—
528C8—61no—
529C8—61no—
530C8—121no—
531C8—121no—
532C2×C3—22no—
533C2×C3#304611yescharacteristic
534C2×C3#306311yescharacteristic
535C2×C3C2×#47211yescharacteristic
536D3—81no—
537D3—161no—
538C22—21no—
539C22—21no—
540C22—121no—
541C22—121no—
542C22—121no—
543C22—241no—
544C22—241no—
545C22—241no—
546C22#474912yes—
547C22C2×#88411yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—81no—
553C4—81no—
554C4—121no—
555C4—121no—
556C4—121no—
557C4—241no—
558C4#576211yescharacteristic
559C4#576211yescharacteristic
560C4#576611yescharacteristic
561C4#576611yescharacteristic
562C3#1959111yescharacteristic, complemented, Sylow
563C2—22no—
564C2—241no—
565C2—481no—
566C2C2×#474911yescharacteristic
567C2#7880011yescharacteristic
568C2#7881511yescharacteristic
subgroupG/H
centerC22C2×#884
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC2×C3×#333C2
socleC22×C3C2×#121
prime radical#79028C2
hypercenterC2×#333D3
hypocenterC3#19591
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
1. #83217C2semidirect
2. #83215C2semidirect
4. #79017C2semidirect
5. C2×C3×#333C2semidirect
6. #70433C2semidirect
7. #70423C2semidirect
8. #68514C2semidirect
9. #68513C2semidirect
10. #68510C2semidirect
11. #68509C2semidirect
12. #68502C2semidirect
13. #68501C2semidirect
14. #68498C2semidirect
15. #68497C2semidirect
82. C2×#333D3semidirect
562. C3#19591semidirect

Maximal subgroups

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

subgroupindex
#832172
#832152
#790282
#790172
C2×C3×#3332
#704332
#704232
#685142
#685132
#685102
#685092
#685022
#685012
#684982
#684972
#195913