Finite Group Databasebeta

← #10795551

Subgroups — #10795551

Order 768.

616 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#901C211yesmaximal, characteristic
2C3×#1772C211yesmaximal, characteristic, complemented
3#70718C211yesmaximal, characteristic
4#70698C211yesmaximal, characteristic, complemented
5#70664C211yesmaximal, characteristic
6#70657C211yesmaximal, characteristic, complemented
7#69898C212yesmaximal
8#69895C212yesmaximal, complemented
9#69779C211yesmaximal, characteristic, complemented
10#68749C212yesmaximal, complemented
11#68737C212yesmaximal
12#20787—31nomaximal, Sylow
13C2×#901C2211yescharacteristic
14C2×#901C2211yescharacteristic
15C4×#235C2211yescharacteristic
16C2×C3×#108C2211yescharacteristic
17C3×#427C2211yescharacteristic
18C3×#405C2212yes—
19C2×C3×C4×C8C2211yescharacteristic
20#5132C2212yes—
21C2×#819C2211yescharacteristic
22#5106C2211yescharacteristic
23#5101C2212yes—
24#5101C2211yescharacteristic
25#5101C2211yescharacteristic
26#5006C2212yes—
27C4×#208C2211yescharacteristic
28#4689C412yescomplemented
29#4689C412yescomplemented
30C4×ΓL(1,𝔽25)C412yescomplemented
31C4×D3×C8C412yescomplemented
32C3×#339C2212yes—
33C3⋊#338C2212yes—
34#4559C2212yes—
35#4557C2212yes—
36#4533C2212yes—
37#4531C2212yes—
38#4531C2212yes—
39#4523C2212yes—
40C4×#131—31no—
41#1841—32no—
42#1833—31no—
43#1833—31no—
44#1827—32no—
45#1813—31no—
46#1812—31no—
47#1772—31no—
48#1772D311yescharacteristic, complemented
49#1757—32no—
50#1747—32no—
51C2×#235C2311yescharacteristic
52C2×C3×M16C2312yes—
53C22×C3×C8C2311yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C3×#117C2312yes—
56C2×C3×C42C2311yescharacteristic
57C4×#67—22no—
58#926C2312yes—
59#926C2311yescharacteristic
60C2×#208C2312yes—
61C2×#208C2311yescharacteristic
62C2×#208C2311yescharacteristic
63#901—21no—
64#901—21no—
65#901—21no—
66#901—21no—
67#901—21no—
68#901—21no—
69#901—21no—
70#901—21no—
71C2×ΓL(1,𝔽25)C2×C412yes—
72C2×ΓL(1,𝔽25)C2×C412yes—
73C2×D3×C8C2×C412yes—
74C2×D3×C8C2×C412yes—
75C4×D12—21no—
76#872C2×C412yes—
77C42×D3C2×C412yes—
78C4×Dic6—21no—
79C3×#108—21no—
80C3×#108—21no—
81C3×#107—21no—
82C3×#106—21no—
83C3×#97—22no—
84C3×C4×C8—22no—
85C3×C4×C8C2×C411yescharacteristic
86C3×C4×C8C2×C411yescharacteristic
87C3×C4×C8C2×C411yescharacteristic
88C3×C4×C8C2×C411yescharacteristic
89#824—22no—
90#823—22no—
91#819—21no—
92#819—21no—
93#818—21no—
94#817—21no—
95#815—22no—
96#815C2×C412yes—
97#815C2×C412yes—
98C8×Dic3—22no—
99C8×Dic3C2×C412yes—
100C8×Dic3C2×C412yes—
101#806—24no—
102#803C2×C412yes—
103C4×C3⋊C8C2×C412yes—
104C2×#131—31no—
105C2×#131—31no—
106C4×#41—31no—
107C2×#108—31no—
108C2×#108C2×D311yescharacteristic
109#427—31no—
110#427C2×D311yescharacteristic
111#413—32no—
112#410—32no—
113#405—31no—
114#405—31no—
115#405—32no—
116#405—32no—
117#405—32no—
118#405C2×D312yes—
119C4×M16—31no—
120C4×M16—32no—
121C2×C4×C8—32no—
122C2×C4×C8C2×D311yescharacteristic
123#344—32no—
124#344—32no—
125#339—32no—
126#339C2×D312yes—
127#338—32no—
128H(3,ℤ/4)—32no—
129H(3,ℤ/4)—32no—
130H(3,ℤ/4)—32no—
131C22×C3×C4—21no—
132C22×C3×C4C2411yescharacteristic
133C2×#67#4112yes—
134#235—21no—
135#235—21no—
136#235—21no—
137#235—21no—
138C2×D12#4111yescharacteristic
139C2×C4×D3—22no—
140C2×C4×D3C22×C412yes—
141C2×Dic6#4111yescharacteristic
142C3×M16—22no—
143C3×M16—22no—
144C2×C3×C8—22no—
145C2×C3×C8C22×C411yescharacteristic
146C2×C3×C8C22×C411yescharacteristic
147C2×C3×C8C22×C411yescharacteristic
148C2×C3×C8C22×C411yescharacteristic
149C2×C3×C8#4111yescharacteristic
150C2×C3×C8#4111yescharacteristic
151C2×C3×C8#4111yescharacteristic
152C2×C3×C8#4111yescharacteristic
153C2×C3×C8#4111yescharacteristic
154C2×C3×C8#4111yescharacteristic
155C2×C3×C8#4111yescharacteristic
156C2×C3×C8#4111yescharacteristic
157C3×C4⋊C4—22no—
158C3×C22⋊C4—22no—
159C3×C42—21no—
160C3×C42—22no—
161C3×C42C22×C411yescharacteristic
162C3×C42C22×C411yescharacteristic
163#217—21no—
164#217—22no—
165(C2×D3)⋊C4—22no—
166C4⋊Dic3—21no—
167C4⋊Dic3—22no—
168Dic3⋊C4—22no—
169C4×Dic3—22no—
170C4×Dic3—22no—
171C4×Dic3C22×C412yes—
172#208—21no—
173#208—21no—
174#208—21no—
175#208—21no—
176#208—22no—
177#208—22no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8C22×C412yes—
180C2×C3⋊C8C22×C412yes—
181ΓL(1,𝔽25)—22no—
182ΓL(1,𝔽25)—22no—
183ΓL(1,𝔽25)—22no—
184ΓL(1,𝔽25)—22no—
185D3×C8—22no—
186D3×C8—22no—
187D3×C8—22no—
188D3×C8—22no—
189C2×#41—31no—
190#131—61no—
191#131—61no—
192#131—61no—
193#131—61no—
194#131—61no—
195#131—61no—
196#131—61no—
197#131—61no—
198C2×M16—31no—
199C2×M16—31no—
200C2×M16—32no—
201C2×M16—32no—
202C2×M16—32no—
203C2×M16C22×D312yes—
204C22×C8—32no—
205C22×C8—32no—
206C22×C8C22×D311yescharacteristic
207C22×C8C22×D311yescharacteristic
208C4×Q8—61no—
209C4×D4—61no—
210C4×D4—62no—
211#117—31no—
212#117—32no—
213#117—32no—
214#117C22×D312yes—
215C2×C42—32no—
216C2×C42C22×D311yescharacteristic
217#108—21no—
218#108—21no—
219#108—61no—
220#108—61no—
221#107—21no—
222#107—61no—
223#106—21no—
224#106—61no—
225#101—62no—
226#100—62no—
227#99—64no—
228#97—22no—
229#97—32no—
230#97—32no—
231#97—32no—
232#97—62no—
233C4×C8—22no—
234C4×C8—32no—
235C4×C8—32no—
236C4×C8—32no—
237C4×C8—62no—
238C4×C8C4×D311yescharacteristic
239C4×C8C4×D311yescharacteristic
240C4×C8C4×D311yescharacteristic
241C4×C8C4×D311yescharacteristic
242C23×C3C2×#4111yescharacteristic
243C22×D3#11712yes—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4C23×C411yescharacteristic
249C2×C3×C4C2×#4111yescharacteristic
250C2×C3×C4C2×#4111yescharacteristic
251C2×C3×C4C2×#4111yescharacteristic
252C2×C3×C4C2×#4111yescharacteristic
253C2×C3×C4C2×#4111yescharacteristic
254#67—22no—
255#67—22no—
256C2×Dic3—22no—
257C2×Dic3—42no—
258C2×Dic3#11712yes—
259D12—21no—
260D12—21no—
261C4×D3—22no—
262C4×D3—22no—
263C4×D3—44no—
264Dic6—21no—
265Dic6—21no—
266C3×C8—42no—
267C3×C8#11711yescharacteristic
268C3×C8#11711yescharacteristic
269C3×C8#11711yescharacteristic
270C3×C8#11711yescharacteristic
271C3×C8#11711yescharacteristic
272C3×C8#11711yescharacteristic
273C3×C8#11711yescharacteristic
274C3×C8#11711yescharacteristic
275C3⋊C8—22no—
276C3⋊C8—22no—
277C3⋊C8—42no—
278#41—61no—
279#41—61no—
280#41—61no—
281#41—61no—
282C2×Q8—31no—
283C2×D4—32no—
284C2×D4—31no—
285C22×C4—21no—
286C22×C4—32no—
287C22×C4—62no—
288C22×C4C23×D311yescharacteristic
289M16—22no—
290M16—22no—
291M16—61no—
292M16—61no—
293M16—61no—
294M16—61no—
295M16—62no—
296M16—62no—
297M16—62no—
298M16—62no—
299M16—62no—
300M16—62no—
301C2×C8—22no—
302C2×C8—32no—
303C2×C8—32no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—62no—
308C2×C8—62no—
309C2×C8C2×C4×D311yescharacteristic
310C2×C8C2×C4×D311yescharacteristic
311C2×C8C2×C4×D311yescharacteristic
312C2×C8C2×C4×D311yescharacteristic
313C2×C8#23511yescharacteristic
314C2×C8#23511yescharacteristic
315C2×C8#23511yescharacteristic
316C2×C8#23511yescharacteristic
317C2×C8#23711yescharacteristic
318C2×C8#23711yescharacteristic
319C2×C8#23911yescharacteristic
320C2×C8#23911yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—62no—
323C4⋊C4—61no—
324C4⋊C4—62no—
325C22⋊C4—22no—
326C22⋊C4—62no—
327C22⋊C4—61no—
328C22⋊C4—62no—
329C42—21no—
330C42—22no—
331C42—32no—
332C42—62no—
333C42—62no—
334C42C2×C4×D311yescharacteristic
335C42C2×C4×D311yescharacteristic
336C22×C3—41no—
337C22×C3C2×#11711yescharacteristic
338C22×C3#52911yescharacteristic
339C22×C3#52911yescharacteristic
340C2×D3—22no—
341C2×D3—42no—
342C3×C4—22no—
343C3×C4—41no—
344C3×C4—82no—
345C3×C4C2×#11711yescharacteristic
346C3×C4C2×#11711yescharacteristic
347C3×C4#52911yescharacteristic
348C3×C4#52911yescharacteristic
349Dic3—22no—
350Dic3—42no—
351Dic3—82no—
352C23—32no—
353C23D3×#4111yescharacteristic
354Q8—61no—
355Q8—61no—
356D4—62no—
357D4—62no—
358D4—61no—
359D4—61no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—32no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—62no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—124no—
369C2×C4—122no—
370C2×C4C22×C4×D311yescharacteristic
371C2×C4C2×#23511yescharacteristic
372C2×C4C2×#23511yescharacteristic
373C2×C4C2×#23711yescharacteristic
374C2×C4C2×#23911yescharacteristic
375C2×C4D3×#4111yescharacteristic
376C8—42no—
377C8—62no—
378C8—62no—
379C8—122no—
380C8#87211yescharacteristic
381C8#87211yescharacteristic
382C8#87211yescharacteristic
383C8#87211yescharacteristic
384C8#89211yescharacteristic
385C8#89211yescharacteristic
386C8#89211yescharacteristic
387C8#89211yescharacteristic
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3#228311yescharacteristic
392D3—42no—
393C22—41no—
394C22—62no—
395C22—122no—
396C22D3×#11711yescharacteristic
397C22#555711yescharacteristic
398C22#558011yescharacteristic
399C4—22no—
400C4—41no—
401C4—62no—
402C4—82no—
403C4—122no—
404C4—242no—
405C4C2×#87211yescharacteristic
406C4C2×#89211yescharacteristic
407C4#555711yescharacteristic
408C4#558011yescharacteristic
409C3#2078711yescharacteristic, complemented, Sylow
410C2—21no—
411C2—21no—
412C2—21no—
413C2—122no—
414C2#7375211yescharacteristic
subgroupG/H
centerC8#892
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1772C2
socleC2×C3#2283
prime radicalC2×#235C23
cosocleC22×C3×C4C24
hypercenter#1772D3
hypocenterC3#20787
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C8 ◁ C22×C8 ◁ #1772

Complements

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

NH ≅ G/Ntype
2. C3×#1772C2semidirect
4. #70698C2semidirect
6. #70657C2semidirect
8. #69895C2semidirect
9. #69779C2semidirect
10. #68749C2semidirect
28. #4689C4semidirect
29. #4689C4semidirect
30. C4×ΓL(1,𝔽25)C4semidirect
31. C4×D3×C8C4semidirect
48. #1772D3semidirect
409. C3#20787semidirect

Maximal subgroups

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

subgroupindex
C4×#9012
C3×#17722
#707182
#706982
#706642
#706572
#698982
#698952
#697792
#687492
#687372
#207873