Finite Group Databasebeta

← #10793773

Subgroups — #10793773

Order 768.

1014 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4717C211yesmaximal, characteristic
2#78494C211yesmaximal, characteristic
3#78486C212yesmaximal, complemented
4C2×#4700C211yesmaximal, characteristic
5C3×C4×#97C211yesmaximal, characteristic, complemented
6#69840C211yesmaximal, characteristic, complemented
7#69839C212yesmaximal, complemented
8#69829C211yesmaximal, characteristic, complemented
9#69789C212yesmaximal, complemented
10#69779C211yesmaximal, characteristic, complemented
11C4×#805C212yesmaximal, complemented
12#20512—31nomaximal, Sylow
13C2×#904C2211yescharacteristic
14C2×#901C2212yes—
15C2×#901C2212yes—
16C4×#235C2212yes—
17C2×C3×#97C2211yescharacteristic
18C2×C3×#97C2211yescharacteristic
19C2×C3×#97C2211yescharacteristic
20C2×C3×#97C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C3×C43C2211yescharacteristic
24#5113C2212yes—
25C2×#819C2211yescharacteristic
26C2×#819C2211yescharacteristic
27#5106C2211yescharacteristic
28#5101C2212yes—
29#5101C2212yes—
30C2×#805C2212yes—
31C2×#805C2212yes—
32C4×#208C2212yes—
33#4717C418yescomplemented
34#4716C412yescomplemented
35#4711C412yescomplemented
36#4710C414yescomplemented
37#4708—22no—
38#4708—22no—
39#4707—22no—
40D3×#97—22no—
41#4700C418yescomplemented
42#4552C2212yes—
43#4523C2212yes—
44#4523C2212yes—
45C2×#444—31no—
46C2×#443—31no—
47#2943—31no—
48#2875—32no—
49#1778—31no—
50#1773—31no—
51#1772—31no—
52#1765—32no—
53#1764—32no—
54C4×#104—32no—
55C4×#97D311yescharacteristic, complemented
56C2×#235C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C2×C3×C42—22no—
62C2×C3×C42C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C4×#67—24no—
66#926C2312yes—
67C2×#208C2312yes—
68C2×#208C2312yes—
69C2×Dic12C2×C411yescharacteristic
70#904C2×C414yes—
71#904C2×C414yes—
72C2×D24C2×C411yescharacteristic
73C2×#204C2×C412yes—
74#901—24no—
75#901—24no—
76#901C2×C418yescomplemented
77#901C2×C418yescomplemented
78C2×ΓL(1,𝔽25)—22no—
79C2×ΓL(1,𝔽25)—22no—
80C2×D3×C8—22no—
81C2×D3×C8—22no—
82C4×D12C2×C414yes—
83#872—22no—
84C42×D3—22no—
85C4×Dic6C2×C414yes—
86C3×#97—22no—
87C3×#97C2×C412yes—
88C3×#97C2×C412yes—
89C3×#97C2×C414yescomplemented
90C3×#97D412yes—
91C3×#97D412yes—
92C3×C4×C8—22no—
93C3×C4×C8C2×C414yescomplemented
94#821C2×C414yes—
95#819C2×C414yescomplemented
96#819C2×C414yescomplemented
97#818C2×C412yes—
98#817C2×C412yes—
99#816C2×C414yes—
100#815—22no—
101#815—22no—
102C8×Dic3—22no—
103C8×Dic3—22no—
104#805C2×C418yescomplemented
105#805C2×C418yescomplemented
106#803—22no—
107C4×C3⋊C8—22no—
108C2×#135—31no—
109C2×#131—32no—
110C2×#131—32no—
111C4×#41—32no—
112#444—38no—
113#443—38no—
114#442—32no—
115#441—32no—
116#440—34no—
117C2×#108—31no—
118C2×#108—31no—
119#427—31no—
120C2×#104—32no—
121C2×#104—32no—
122#416—32no—
123#405—32no—
124#405—32no—
125#405—62no—
126#405—62no—
127C4×M16—32no—
128C4×M16—62no—
129C2×#97—62no—
130C2×#97C2×D311yescharacteristic
131C2×#97C2×D311yescharacteristic
132C2×#97C2×D311yescharacteristic
133C2×#97C2×D311yescharacteristic
134C2×C4×C8C2×D311yescharacteristic
135C2×C4×C8C2×D311yescharacteristic
136C43C2×D311yescharacteristic
137#341—32no—
138#339—32no—
139#339—32no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4#4112yes—
143C22×C3×C4C2411yescharacteristic
144C2×#67—22no—
145#235—22no—
146#235—22no—
147#235C22×C414yes—
148#235C22×C414yes—
149C2×D12C22×C412yes—
150C2×C4×D3—22no—
151C2×C4×D3—24no—
152C2×Dic6C22×C412yes—
153C2×C3×C8—21no—
154C2×C3×C8—21no—
155C2×C3×C8C22×C412yes—
156C2×C3×C8C22×C411yescharacteristic
157C2×C3×C8C22×C411yescharacteristic
158C2×C3×C8C22×C412yes—
159C2×C3×C8C22×C412yes—
160C2×C3×C8C22×C412yes—
161C2×C3×C8C22×C412yes—
162C2×C3×C8C2×D411yescharacteristic
163C2×C3×C8C2×D412yes—
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8#4112yes—
166C2×C3×C8#4112yes—
167C3×C42—21no—
168C3×C42—21no—
169C3×C42—44no—
170C3×C42C22×C412yes—
171C3×C42C22×C412yes—
172C3×C42C22×C412yes—
173C3×C42C2×D412yes—
174#217—22no—
175(C2×D3)⋊C4—24no—
176C4⋊Dic3C22×C414yes—
177Dic3⋊C4—24no—
178C4×Dic3—24no—
179C4×Dic3—22no—
180#208—22no—
181#208—22no—
182#208C22×C414yes—
183#208C22×C414yes—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—22no—
186Dic12C4214yescomplemented
187D24C4214yescomplemented
188#204C4218yescomplemented
189ΓL(1,𝔽25)—28no—
190ΓL(1,𝔽25)—28no—
191D3×C8—28no—
192D3×C8—28no—
193C2×#41—32no—
194#135—34no—
195#135—34no—
196C2×Q16—31no—
197C2×SD16—32no—
198C2×D8—31no—
199#131—38no—
200#131—38no—
201#131—64no—
202#131—64no—
203C2×M16—32no—
204C2×M16—32no—
205C2×M16—62no—
206C2×M16—62no—
207C22×C8—62no—
208C22×C8—62no—
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211C22×C8C22×D311yescharacteristic
212C22×C8C22×D311yescharacteristic
213C4×Q8—34no—
214C4×D4—34no—
215C4×D4—64no—
216#117—32no—
217#117—62no—
218C2×C42—22no—
219C2×C42—62no—
220C2×C42C22×D311yescharacteristic
221C2×C42C22×D311yescharacteristic
222C2×C42C22×D311yescharacteristic
223#108—34no—
224#108—34no—
225#107—32no—
226#106—32no—
227#104—38no—
228#104—38no—
229#103—34no—
230#102—34no—
231#97—22no—
232#97—62no—
233#97—62no—
234#97—62no—
235#97C4×D312yes—
236#97C4×D312yes—
237#97C4×D314yescomplemented
238#97D1212yes—
239#97D1212yes—
240C4×C8—22no—
241C4×C8—62no—
242C4×C8—62no—
243C4×C8—62no—
244C4×C8C4×D314yescomplemented
245C23×C3C2×#4111yescharacteristic
246C22×D3—22no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4C4×D414yes—
253C2×C3×C4C23×C411yescharacteristic
254C2×C3×C4C23×C411yescharacteristic
255C2×C3×C4C23×C411yescharacteristic
256C2×C3×C4C22×D411yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4111yescharacteristic
259#67—24no—
260#67—24no—
261C2×Dic3—22no—
262C2×Dic3—24no—
263D12—22no—
264D12C2×C4214yes—
265C4×D3—24no—
266C4×D3—24no—
267C4×D3—84no—
268Dic6—22no—
269Dic6C2×C4214yes—
270C3×C8—22no—
271C3×C8C2×C4212yes—
272C3×C8C2×C4212yes—
273C3×C8C4×D412yes—
274C3×C8C4×D414yes—
275C3×C8C4×D412yes—
276C3⋊C8—24no—
277C3⋊C8—24no—
278#41—34no—
279#41—34no—
280#41—62no—
281#41—62no—
282C2×Q8—32no—
283C2×D4—32no—
284C2×D4—62no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—62no—
288C22×C4—64no—
289C22×C4#23512yes—
290C22×C4C23×D311yescharacteristic
291Q16—34no—
292SD16—38no—
293D8—34no—
294M16—34no—
295M16—34no—
296M16—62no—
297M16—62no—
298M16—68no—
299M16—68no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—62no—
303C2×C8—62no—
304C2×C8—68no—
305C2×C8—68no—
306C2×C8C2×C4×D312yes—
307C2×C8C2×C4×D312yes—
308C2×C8C2×C4×D311yescharacteristic
309C2×C8C2×C4×D311yescharacteristic
310C2×C8C2×C4×D312yes—
311C2×C8C2×C4×D312yes—
312C2×C8C2×C4×D312yes—
313C2×C8C2×D1211yescharacteristic
314C2×C8C2×D1212yes—
315C2×C8C2×D1211yescharacteristic
316C2×C8#23512yes—
317C2×C8#23512yes—
318C4⋊C4—34no—
319C4⋊C4—64no—
320C22⋊C4—62no—
321C22⋊C4—64no—
322C42—21no—
323C42—21no—
324C42—44no—
325C42—64no—
326C42—62no—
327C42C2×C4×D312yes—
328C42C2×C4×D312yes—
329C42C2×C4×D312yes—
330C42C2×D1212yes—
331C22×C3—22no—
332C22×C3C2×C4×D411yescharacteristic
333C22×C3C4×#4111yescharacteristic
334C22×C3C4×#4111yescharacteristic
335C2×D3—24no—
336C2×D3—44no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—42no—
340C3×C4C22×C4211yescharacteristic
341C3×C4C2×C4×D411yescharacteristic
342C3×C4C2×C4×D411yescharacteristic
343C3×C4C4×#4111yescharacteristic
344Dic3—24no—
345Dic3—44no—
346C23—62no—
347C23C2×#23511yescharacteristic
348Q8—34no—
349Q8—62no—
350D4—34no—
351D4—64no—
352D4—64no—
353D4—62no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—62no—
360C2×C4—64no—
361C2×C4—64no—
362C2×C4—64no—
363C2×C4—244no—
364C2×C4C4×D1214yes—
365C2×C4C22×C4×D311yescharacteristic
366C2×C4C22×C4×D311yescharacteristic
367C2×C4C22×C4×D311yescharacteristic
368C2×C4C22×D1211yescharacteristic
369C2×C4C2×#23511yescharacteristic
370C2×C4C2×#23511yescharacteristic
371C8—22no—
372C8—64no—
373C8—64no—
374C8C42×D312yes—
375C8C42×D312yes—
376C8C4×D1212yes—
377C8C4×D1214yes—
378C8C4×D1212yes—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3C42×D411yescharacteristic
382C2×C3#295612yes—
383D3—44no—
384C22—22no—
385C22—64no—
386C22—124no—
387C22C2×C4×D1211yescharacteristic
388C22C4×#23511yescharacteristic
389C22C4×#23511yescharacteristic
390C4—24no—
391C4—24no—
392C4—42no—
393C4—64no—
394C4—124no—
395C4C2×C42×D311yescharacteristic
396C4C2×C4×D1211yescharacteristic
397C4C2×C4×D1211yescharacteristic
398C4C4×#23511yescharacteristic
399C3#2051211yescharacteristic, complemented, Sylow
400C2—21no—
401C2—21no—
402C2—124no—
403C2C42×D1211yescharacteristic
404C2#7854112yes—
subgroupG/H
centerC2×C4C22×D12
commutatorC3×C4C22×C42
FrattiniC22×C4C23×D3
FittingC3×C4×#97C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenterC4×#97D3
hypocenterC3#20512
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ C4×#97

Complements

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

NH ≅ G/Ntype
3. #78486C2semidirect
5. C3×C4×#97C2semidirect
6. #69840C2semidirect
7. #69839C2semidirect
8. #69829C2semidirect
9. #69789C2semidirect
10. #69779C2semidirect
11. C4×#805C2semidirect
33. #4717C4semidirect
34. #4716C4semidirect
35. #4711C4semidirect
36. #4710C4semidirect
41. #4700C4semidirect
55. C4×#97D3semidirect
76. #901C2×C4semidirect
77. #901C2×C4semidirect
89. C3×#97C2×C4semidirect
93. C3×C4×C8C2×C4semidirect
95. #819C2×C4semidirect
96. #819C2×C4semidirect
104. #805C2×C4semidirect
105. #805C2×C4semidirect
186. Dic12C42semidirect
187. D24C42semidirect
188. #204C42semidirect
237. #97C4×D3semidirect
244. C4×C8C4×D3semidirect
399. C3#20512semidirect

Maximal subgroups

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

subgroupindex
C2×#47172
#784942
#784862
C2×#47002
C3×C4×#972
#698402
#698392
#698292
#697892
#697792
C4×#8052
#205123