Finite Group Databasebeta

← #10950893

Subgroups — #10950893

Order 768.

516 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×#3168C211yesmaximal, characteristic, complemented
2#78469C211yesmaximal, characteristic, complemented
3#71354C211yesmaximal, characteristic, complemented
4#71351C211yesmaximal, characteristic, complemented
5#71320C211yesmaximal, characteristic, complemented
6#71318C211yesmaximal, characteristic, complemented
7#71084C211yesmaximal, characteristic, complemented
8#71083C211yesmaximal, characteristic, complemented
9#71074C211yesmaximal, characteristic, complemented
10#71073C211yesmaximal, characteristic
11#70913C211yesmaximal, characteristic, complemented
12#70909C211yesmaximal, characteristic, complemented
13#70885C211yesmaximal, characteristic, complemented
14#70884C211yesmaximal, characteristic, complemented
15C4×#812C211yesmaximal, characteristic, complemented
16#33750—31nomaximal, Sylow
17C3×#529—21no—
18#5479—21no—
19C3×#499C2211yescharacteristic
20C3×#495C2211yescharacteristic
21C3×#488—21no—
22C3×#467C2211yescharacteristic
23C3×#466C2211yescharacteristic
24C3×#465—21no—
25C3×C4×Q16C2211yescharacteristic
26C3×C4×SD16—21no—
27C3×C4×D8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C2×#829C2211yescharacteristic
30C2×#828C2211yescharacteristic
31C2×#827C2211yescharacteristic
32C2×#826C2211yescharacteristic
33#5116C2211yescharacteristic
34#5115C2211yescharacteristic
35#5114—21no—
36C2×#812C2211yescharacteristic
37C2×#812C2211yescharacteristic
38C4×Dic12C2211yescharacteristic
39#4697—21no—
40#4696C2211yescharacteristic
41C4×D24C2211yescharacteristic
42C4×#204—21no—
43#4683C2211yescharacteristic
44#4564C2211yescharacteristic
45#4564C2211yescharacteristic
46#4562C2211yescharacteristic
47#4562C2211yescharacteristic
48#4492C2211yescharacteristic
49#4492C2211yescharacteristic
50#4491C2211yescharacteristic
51#4491C2211yescharacteristic
52#4490C2211yescharacteristic
53#4490C2211yescharacteristic
54#4489C2211yescharacteristic
55#4489C2211yescharacteristic
56#4461C2211yescharacteristic
57#4461C2211yescharacteristic
58C4×C3⋊C16C2211yescharacteristic
59C4×C3⋊C16C2211yescharacteristic
60#3168—31no—
61#3168D311yescharacteristic, complemented
62#2253—31no—
63#2249—31no—
64#2242—31no—
65#2242—31no—
66#2220—31no—
67#2220—31no—
68#2219—31no—
69#2218—31no—
70#2177—31no—
71#2176—31no—
72#2175—31no—
73#2175—31no—
74C4×M32—31no—
75C2×C3×Q16C2311yescharacteristic
76C2×C3×SD16—21no—
77C2×C3×D8C2311yescharacteristic
78C22×C3×C8D411yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×#126—41no—
81C3×#125—21no—
82C3×#124—21no—
83C3×#123—41no—
84C3×C22⋊Q8—21no—
85C3×#121—21no—
86C3×C4×Q8—21no—
87C3×C4×D4—21no—
88C3×C4×D4—41no—
89C3×#117—41no—
90C2×C3×C42D411yescharacteristic
91#930—21no—
92#929—41no—
93C4×#67—41no—
94#924—21no—
95C2×Dic12C2311yescharacteristic
96C2×D24C2311yescharacteristic
97C2×#204—21no—
98#876—41no—
99#875—21no—
100C4×D12—21no—
101#872—41no—
102#870—21no—
103C4×Dic6—21no—
104C3×#107C2311yescharacteristic
105C3×#107C2311yescharacteristic
106C3×#106—21no—
107C3×#103—21no—
108C3×#103—21no—
109C3×#102—21no—
110C3×#102—21no—
111C3×C4×C8D411yescharacteristic
112C3×C4×C8D411yescharacteristic
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#818C2311yescharacteristic
122#818C2311yescharacteristic
123#817—21no—
124#816—21no—
125#816—21no—
126#812—22no—
127#812—22no—
128C2×C3⋊C16C2311yescharacteristic
129C2×C3⋊C16C2311yescharacteristic
130C2×C3⋊C16C2311yescharacteristic
131C2×C3⋊C16C2311yescharacteristic
132#529—21no—
133#529—61no—
134C2×Q32—31no—
135C2×SD32—31no—
136C2×SD32—31no—
137C2×D16—31no—
138C2×M32—31no—
139C2×M32—31no—
140#499—31no—
141#499C2×D311yescharacteristic
142#495—31no—
143#495C2×D311yescharacteristic
144#488—21no—
145#488—61no—
146#467—31no—
147#467C2×D311yescharacteristic
148#466—31no—
149#466C2×D311yescharacteristic
150#465—21no—
151#465—61no—
152C4×Q16—31no—
153C4×Q16C2×D311yescharacteristic
154C4×SD16—21no—
155C4×SD16—61no—
156C4×D8—31no—
157C4×D8C2×D311yescharacteristic
158C2×C4×C8C2×D311yescharacteristic
159#367—31no—
160#367—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×D4—21no—
177C2×C3×D4—41no—
178C22×C3×C4—41no—
179C22×C3×C4D811yescharacteristic
180C22×C3×C4D811yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C2×#67—41no—
183C2×D12—21no—
184C2×C4×D3—41no—
185C2×Dic6—21no—
186C3×Q16—21no—
187C3×Q16—21no—
188C3×SD16—42no—
189C3×D8—21no—
190C3×D8—21no—
191C2×C3×C8—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⋊C4—41no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207C3×C22⋊C4—41no—
208C3×C22⋊C4—41no—
209C3×C42—41no—
210C3×C42D811yescharacteristic
211C3×C42D811yescharacteristic
212C3×C42C2×D411yescharacteristic
213C3×C42C2×D411yescharacteristic
214#217—41no—
215(C2×D3)⋊C4—41no—
216(C2×D3)⋊C4—41no—
217(C2×D3)⋊C4—41no—
218(C2×D3)⋊C4—41no—
219C4⋊Dic3—21no—
220C4⋊Dic3—21no—
221Dic3⋊C4—41no—
222Dic3⋊C4—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
241C22×C8#6711yescharacteristic
242C22×C8C22×D311yescharacteristic
243#126—41no—
244#126—121no—
245#125—21no—
246#125—61no—
247#124—21no—
248#124—61no—
249#123—41no—
250#123—121no—
251C22⋊Q8—21no—
252C22⋊Q8—61no—
253#121—21no—
254#121—61no—
255C4×Q8—21no—
256C4×Q8—61no—
257C4×D4—21no—
258C4×D4—41no—
259C4×D4—61no—
260C4×D4—121no—
261#117—41no—
262#117—121no—
263C2×C42#6711yescharacteristic
264Q32—62no—
265SD32—62no—
266SD32—62no—
267D16—62no—
268M32—62no—
269M32—62no—
270C2×C16—31no—
271C2×C16—31no—
272C2×C16—31no—
273C2×C16—31no—
274#107—31no—
275#107—31no—
276#107C22×D311yescharacteristic
277#107C22×D311yescharacteristic
278#106—21no—
279#106—61no—
280#103—21no—
281#103—21no—
282#103—61no—
283#103—61no—
284#102—21no—
285#102—21no—
286#102—61no—
287#102—61no—
288C4×C8#6711yescharacteristic
289C4×C8#6711yescharacteristic
290C4×C8C22×D311yescharacteristic
291C4×C8C22×D311yescharacteristic
292C23×C3—41no—
293C23×C3C2×D811yescharacteristic
294C22×D3—41no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—82no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—82no—
308C2×C3×C4C2×D811yescharacteristic
309C2×C3×C4C2×D811yescharacteristic
310C2×C3×C4C2×D811yescharacteristic
311C2×C3×C4C2×D811yescharacteristic
312C2×C3×C4C2×D811yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314#67—82no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318D12—41no—
319D12—41no—
320C4×D3—82no—
321Dic6—41no—
322Dic6—41no—
323C3×C8—21no—
324C3×C8—21no—
325C3×C8#13512yes—
326C3×C8C2×#4111yescharacteristic
327C3×C8C2×#4111yescharacteristic
328C2×Q8—21no—
329C2×Q8—61no—
330C2×D4—21no—
331C2×D4—41no—
332C2×D4—61no—
333C2×D4—121no—
334C22×C4—41no—
335C22×C4—121no—
336C22×C4#21311yescharacteristic
337C22×C4#21311yescharacteristic
338C22×C4C2×#6711yescharacteristic
339Q16—21no—
340Q16—21no—
341Q16—61no—
342Q16—61no—
343SD16—42no—
344SD16—122no—
345D8—21no—
346D8—21no—
347D8—61no—
348D8—61no—
349C2×C8—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—41no—
362C4⋊C4—61no—
363C4⋊C4—61no—
364C4⋊C4—121no—
365C4⋊C4—121no—
366C4⋊C4—121no—
367C4⋊C4—121no—
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—
378C42—41no—
379C42—121no—
380C42#21311yescharacteristic
381C42#21311yescharacteristic
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—
418D4—41no—
419D4—41no—
420D4—82no—
421D4—121no—
422D4—121no—
423D4—242no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—82no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—121no—
435C2×C4—242no—
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#341411yescharacteristic
451C2×C3#341411yescharacteristic
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#3375011yescharacteristic, complemented, Sylow
477C2—41no—
478C2—161no—
479C2—481no—
480C2#8329111yescharacteristic
481C2#8389411yescharacteristic
482C2#8389411yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3168C2
socleC22×C3C22×D8
prime radical#71073C2
hypercenter#3168D3
hypocenterC3#33750
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. C3×#3168C2semidirect
2. #78469C2semidirect
3. #71354C2semidirect
4. #71351C2semidirect
5. #71320C2semidirect
6. #71318C2semidirect
7. #71084C2semidirect
8. #71083C2semidirect
9. #71074C2semidirect
11. #70913C2semidirect
12. #70909C2semidirect
13. #70885C2semidirect
14. #70884C2semidirect
15. C4×#812C2semidirect
61. #3168D3semidirect
476. C3#33750semidirect

Maximal subgroups

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

subgroupindex
C3×#31682
#784692
#713542
#713512
#713202
#713182
#710842
#710832
#710742
#710732
#709132
#709092
#708852
#708842
C4×#8122
#337503