Finite Group Databasebeta

← #10950776

Subgroups — #10950776

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3175C211yesmaximal, characteristic, complemented
2#78568C211yesmaximal, characteristic, complemented
3#71384C211yesmaximal, characteristic, complemented
4#71383C211yesmaximal, characteristic, complemented
5#71320C211yesmaximal, characteristic, complemented
6#71319C211yesmaximal, characteristic, complemented
7#71084C211yesmaximal, characteristic, complemented
8#71081C211yesmaximal, characteristic, complemented
9#71074C211yesmaximal, characteristic, complemented
10#71071C211yesmaximal, characteristic
11#70913C211yesmaximal, characteristic, complemented
12C4×#829C211yesmaximal, characteristic, complemented
13#70884C211yesmaximal, characteristic, complemented
14C4×#826C211yesmaximal, characteristic, complemented
15#69881C211yesmaximal, characteristic, complemented
16#33739—31nomaximal, Sylow
17C3×#542—21no—
18#5534—21no—
19C3×#500C2211yescharacteristic
20C3×#495C2211yescharacteristic
21C3×#489—21no—
22C3×#468—21no—
23C3×#467C2211yescharacteristic
24C3×#466C2211yescharacteristic
25C3×#440—21no—
26C3×C4×Q16C2211yescharacteristic
27C3×C4×D8C2211yescharacteristic
28C3×#405C2211yescharacteristic
29C2×#829C2211yescharacteristic
30C2×#828C2211yescharacteristic
31C2×#827C2211yescharacteristic
32C2×#826C2211yescharacteristic
33#5134—21no—
34#5116C2211yescharacteristic
35#5115C2211yescharacteristic
36C2×#812C2211yescharacteristic
37C22×C3⋊C16C2211yescharacteristic
38#4714—21no—
39#4710—21no—
40C4×Dic12C2211yescharacteristic
41#4696C2211yescharacteristic
42C4×D24C2211yescharacteristic
43#4682C2211yescharacteristic
44#4564C2211yescharacteristic
45#4564C2211yescharacteristic
46#4562C2211yescharacteristic
47#4562C2211yescharacteristic
48#4492C2211yescharacteristic
49#4492C2211yescharacteristic
50#4491C2211yescharacteristic
51#4491C2211yescharacteristic
52#4489C2211yescharacteristic
53#4489C2211yescharacteristic
54#4489C2211yescharacteristic
55#4489C2211yescharacteristic
56#4461C2211yescharacteristic
57#4461C2211yescharacteristic
58C4×C3⋊C16C2211yescharacteristic
59C4×C3⋊C16C2211yescharacteristic
60#3175—31no—
61#3175D311yescharacteristic, complemented
62#2252—31no—
63#2249—31no—
64#2243—31no—
65#2243—31no—
66#2220—31no—
67#2220—31no—
68#2216—31no—
69#2215—31no—
70#2175—31no—
71#2175—31no—
72C4×Q32—31no—
73C4×D16—31no—
74#2108—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×SD16—21no—
77C2×C3×D8C2311yescharacteristic
78C2×C3×M16D411yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C4⋊Q8—21no—
81C3×#126—41no—
82C3×#124—21no—
83C3×#124—41no—
84C3×#123—41no—
85C3×C22⋊Q8—21no—
86C3×C22⋊Q8—41no—
87C3×#121—21no—
88C3×C4×Q8—21no—
89C3×C4×D4—21no—
90C3×#117D411yescharacteristic
91#930—21no—
92#924—21no—
93C2×Dic12C2311yescharacteristic
94C2×D24C2311yescharacteristic
95C2×#204—21no—
96#898—41no—
97#896—41no—
98#885—41no—
99#883—41no—
100#875—21no—
101C4×D12—21no—
102#869—21no—
103C4×Dic6—21no—
104C3×#107—21no—
105C3×#107C2311yescharacteristic
106C3×#107C2311yescharacteristic
107C3×#103—21no—
108C3×#103—21no—
109C3×#102—21no—
110C3×#102—21no—
111C3×#97D411yescharacteristic
112C3×#97D411yescharacteristic
113C3×C4×C8C2311yescharacteristic
114C3×C4×C8C2311yescharacteristic
115#829—22no—
116#828—22no—
117#827—22no—
118#826—22no—
119#821—21no—
120#821—21no—
121#818—21no—
122#818C2311yescharacteristic
123#818C2311yescharacteristic
124#816—21no—
125#816—21no—
126#812—22no—
127C2×C3⋊C16—22no—
128C2×C3⋊C16C2311yescharacteristic
129C2×C3⋊C16C2311yescharacteristic
130C2×C3⋊C16C2311yescharacteristic
131C2×C3⋊C16C2311yescharacteristic
132#542—21no—
133#542—61no—
134C2×Q32—31no—
135C2×SD32—31no—
136C2×SD32—31no—
137C2×D16—31no—
138C2×M32—31no—
139C22×C16—31no—
140#500—31no—
141#500C2×D311yescharacteristic
142#495—31no—
143#495C2×D311yescharacteristic
144#489—21no—
145#489—61no—
146#468—21no—
147#468—61no—
148#467—31no—
149#467C2×D311yescharacteristic
150#466—31no—
151#466C2×D311yescharacteristic
152#440—21no—
153#440—61no—
154C4×Q16—31no—
155C4×Q16C2×D311yescharacteristic
156C4×D8—31no—
157C4×D8C2×D311yescharacteristic
158#405C2×D311yescharacteristic
159#366—31no—
160#366—31no—
161#366—31no—
162#366—31no—
163#358—31no—
164#358—31no—
165#358—31no—
166#358—31no—
167#357—31no—
168#357—31no—
169#357—31no—
170#357—31no—
171#346—31no—
172#346—31no—
173C4×C16—31no—
174C4×C16—31no—
175C2×C3×Q8—21no—
176C2×C3×Q8—41no—
177C2×C3×D4—21no—
178C2×C3×D4—41no—
179C22×C3×C4—41no—
180C22×C3×C4C2×D411yescharacteristic
181C2×#67—41no—
182C2×D12—21no—
183C2×C4×D3—41no—
184C2×Dic6—21no—
185C2×Dic6—41no—
186C3×Q16—21no—
187C3×Q16—21no—
188C3×SD16—42no—
189C3×D8—21no—
190C3×D8—21no—
191C3×M16—22no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8C2×D411yescharacteristic
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2411yescharacteristic
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—41no—
201C3×C4⋊C4—41no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4D811yescharacteristic
204C3×C4⋊C4D811yescharacteristic
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—41no—
210C3×C22⋊C4D811yescharacteristic
211C3×C22⋊C4D811yescharacteristic
212C3×C42—41no—
213C3×C42C2×D411yescharacteristic
214C3×C42C2×D411yescharacteristic
215#217—41no—
216(C2×D3)⋊C4—41no—
217(C2×D3)⋊C4—41no—
218(C2×D3)⋊C4—41no—
219(C2×D3)⋊C4—41no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—41no—
223Dic3⋊C4—41no—
224Dic3⋊C4—41no—
225C4×Dic3—41no—
226Dic12—21no—
227Dic12—21no—
228D24—21no—
229D24—21no—
230#204—42no—
231C3⋊C16—21no—
232C3⋊C16—21no—
233C3⋊C16#4112yes—
234C3⋊C16#4112yes—
235C2×Q16—31no—
236C2×Q16C22×D311yescharacteristic
237C2×SD16—21no—
238C2×SD16—61no—
239C2×D8—31no—
240C2×D8C22×D311yescharacteristic
241C2×M16#6711yescharacteristic
242C22×C8C22×D311yescharacteristic
243C4⋊Q8—21no—
244C4⋊Q8—61no—
245#126—41no—
246#126—121no—
247#124—21no—
248#124—41no—
249#124—61no—
250#124—121no—
251#123—41no—
252#123—121no—
253C22⋊Q8—21no—
254C22⋊Q8—41no—
255C22⋊Q8—61no—
256C22⋊Q8—121no—
257#121—21no—
258#121—61no—
259C4×Q8—21no—
260C4×Q8—61no—
261C4×D4—21no—
262C4×D4—61no—
263#117#6711yescharacteristic
264Q32—62no—
265SD32—62no—
266SD32—62no—
267D16—62no—
268M32—62no—
269C2×C16—31no—
270C2×C16—31no—
271C2×C16—31no—
272C2×C16—31no—
273C2×C16—62no—
274#107—21no—
275#107—31no—
276#107—31no—
277#107—61no—
278#107C22×D311yescharacteristic
279#107C22×D311yescharacteristic
280#103—21no—
281#103—21no—
282#103—61no—
283#103—61no—
284#102—21no—
285#102—21no—
286#102—61no—
287#102—61no—
288#97#6711yescharacteristic
289#97#6711yescharacteristic
290C4×C8C22×D311yescharacteristic
291C4×C8C22×D311yescharacteristic
292C23×C3—41no—
293C23×C3C2×D811yescharacteristic
294C22×D3—41no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—161no—
298C3×D4—41no—
299C3×D4—41no—
300C3×D4—161no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—161no—
307C2×C3×C4C2×D811yescharacteristic
308C2×C3×C4C2×D811yescharacteristic
309C2×C3×C4C2×D811yescharacteristic
310C2×C3×C4C2×D811yescharacteristic
311C2×C3×C4C2×D811yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313#67—161no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317D12—41no—
318D12—41no—
319C4×D3—161no—
320Dic6—41no—
321Dic6—41no—
322Dic6—161no—
323C3×C8—21no—
324C3×C8—21no—
325C3×C8#13512yes—
326C3×C8C2×#4111yescharacteristic
327C3×C8C2×#4111yescharacteristic
328C2×Q8—21no—
329C2×Q8—41no—
330C2×Q8—61no—
331C2×Q8—121no—
332C2×D4—21no—
333C2×D4—41no—
334C2×D4—61no—
335C2×D4—121no—
336C22×C4—41no—
337C22×C4—121no—
338C22×C4C2×#6711yescharacteristic
339Q16—21no—
340Q16—21no—
341Q16—61no—
342Q16—61no—
343SD16—42no—
344SD16—122no—
345D8—21no—
346D8—21no—
347D8—61no—
348D8—61no—
349M16—22no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8C2×#6711yescharacteristic
353C2×C8C2×#6711yescharacteristic
354C2×C8C2×#6711yescharacteristic
355C2×C8C23×D311yescharacteristic
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—61no—
362C4⋊C4—61no—
363C4⋊C4—121no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4#21311yescharacteristic
367C4⋊C4#21311yescharacteristic
368C22⋊C4—41no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—121no—
374C22⋊C4—121no—
375C22⋊C4—121no—
376C22⋊C4—121no—
377C22⋊C4—121no—
378C22⋊C4#21311yescharacteristic
379C22⋊C4#21311yescharacteristic
380C42—41no—
381C42—121no—
382C42C2×#6711yescharacteristic
383C42C2×#6711yescharacteristic
384C16—32no—
385C16—32no—
386C16—61no—
387C16—61no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3C22×D811yescharacteristic
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—81no—
404C3×C4—81no—
405C3×C4—81no—
406C3×C4#53211yescharacteristic
407C3×C4C2×#13511yescharacteristic
408Dic3—81no—
409Dic3—81no—
410Dic3—81no—
411C23—41no—
412C23—121no—
413C23C2×#21311yescharacteristic
414Q8—41no—
415Q8—41no—
416Q8—121no—
417Q8—121no—
418Q8—161no—
419Q8—481no—
420D4—41no—
421D4—41no—
422D4—121no—
423D4—121no—
424D4—161no—
425D4—481no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—161no—
435C2×C4—481no—
436C2×C4C2×#21311yescharacteristic
437C2×C4C2×#21311yescharacteristic
438C2×C4C2×#21311yescharacteristic
439C2×C4C2×#21311yescharacteristic
440C2×C4C2×#21311yescharacteristic
441C2×C4C22×#6711yescharacteristic
442C8—21no—
443C8—21no—
444C8#95012yes—
445C8D3×#4111yescharacteristic
446C8D3×#4111yescharacteristic
447C2×C3—41no—
448C2×C3—161no—
449C2×C3#316911yescharacteristic
450C2×C3#341511yescharacteristic
451C2×C3#341711yescharacteristic
452D3—161no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22C22×#21311yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—241no—
472C4—241no—
473C4—241no—
474C4C2×#95011yescharacteristic
475C4#583211yescharacteristic
476C3#3373911yescharacteristic, complemented, Sylow
477C2—41no—
478C2—161no—
479C2—481no—
480C2#8329111yescharacteristic
481C2#8400711yescharacteristic
482C2#8401711yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3175C2
socleC22×C3C22×D8
prime radical#71071C2
hypercenter#3175D3
hypocenterC3#33739
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. C3×#3175C2semidirect
2. #78568C2semidirect
3. #71384C2semidirect
4. #71383C2semidirect
5. #71320C2semidirect
6. #71319C2semidirect
7. #71084C2semidirect
8. #71081C2semidirect
9. #71074C2semidirect
11. #70913C2semidirect
12. C4×#829C2semidirect
13. #70884C2semidirect
14. C4×#826C2semidirect
15. #69881C2semidirect
61. #3175D3semidirect
476. C3#33739semidirect

Maximal subgroups

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

subgroupindex
C3×#31752
#785682
#713842
#713832
#713202
#713192
#710842
#710812
#710742
#710712
#709132
C4×#8292
#708842
C4×#8262
#698812
#337393