Finite Group Databasebeta

← #10950868

Subgroups — #10950868

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3170C211yesmaximal, characteristic, complemented
2#78469C211yesmaximal, characteristic
3#71386C211yesmaximal, characteristic, complemented
4#71383C211yesmaximal, characteristic, complemented
5#71314C211yesmaximal, characteristic, complemented
6#71311C211yesmaximal, characteristic, complemented
7#71082C212yesmaximal
8#71081C212yesmaximal, complemented
9#70913C212yesmaximal, complemented
10#70909C212yesmaximal, complemented
11C2×#4461C211yesmaximal, characteristic, complemented
12#33747—31nomaximal, Sylow
13C3×#533—21no—
14#5479—21no—
15C3×#500C2211yescharacteristic
16C3×#494C2211yescharacteristic
17C3×#487—21no—
18C3×#467—21no—
19C3×#467C2212yes—
20C3×C4×Q16—21no—
21C3×C4×Q16C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23C2×#829C2212yes—
24C2×#828C2212yes—
25#5116C2211yescharacteristic
26#5115C2211yescharacteristic
27#5114—21no—
28C22×C3⋊C16C2212yes—
29C4×Dic12C2211yescharacteristic
30#4697—21no—
31#4696C2211yescharacteristic
32C4×D24C2211yescharacteristic
33C4×#204—21no—
34#4683C2211yescharacteristic
35#4564C2212yes—
36#4564C2212yes—
37#4492C2212yes—
38#4491C2212yes—
39#4490C2212yes—
40#4489C2212yes—
41#4461C2212yes—
42#4461C2211yescharacteristic
43#4461C2211yescharacteristic
44#3170D311yescharacteristic, complemented
45#3168—31no—
46#2255—31no—
47#2251—31no—
48#2245—31no—
49#2243—31no—
50#2217—32no—
51#2216—32no—
52#2176—32no—
53#2175—32no—
54C2×#346—31no—
55C2×C3×Q16—21no—
56C2×C3×Q16C2312yes—
57C22×C3×C8D411yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×#125—41no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—41no—
64C3×C4×Q8—22no—
65C3×C4×Q8—41no—
66C3×#117—41no—
67C2×C3×C42D411yescharacteristic
68#930—21no—
69#929—41no—
70C4×#67—41no—
71#924—21no—
72C2×Dic12C2311yescharacteristic
73C2×D24C2311yescharacteristic
74C2×#204—21no—
75#876—41no—
76#875—21no—
77C4×D12—21no—
78#872—41no—
79#870—21no—
80C4×Dic6—21no—
81C3×#107—21no—
82C3×#107C2312yes—
83C3×#103—22no—
84C3×#103—22no—
85C3×C4×C8D411yescharacteristic
86C3×C4×C8D411yescharacteristic
87C3×C4×C8C2311yescharacteristic
88C3×C4×C8C2311yescharacteristic
89#829—24no—
90#828—24no—
91#821—21no—
92#821—21no—
93#818C2311yescharacteristic
94#818C2311yescharacteristic
95#817—21no—
96#816—21no—
97#816—21no—
98C2×C3⋊C16—24no—
99C2×C3⋊C16C2312yes—
100C2×C3⋊C16C2312yes—
101#533—21no—
102#529—61no—
103C2×Q32—32no—
104C2×SD32—32no—
105C22×C16—32no—
106#500C2×D311yescharacteristic
107#499—31no—
108#495—31no—
109#494C2×D311yescharacteristic
110#488—61no—
111#487—21no—
112#467—21no—
113#467—31no—
114#467C2×D312yes—
115#466—31no—
116#465—61no—
117C4×Q16—21no—
118C4×Q16—31no—
119C4×Q16C2×D312yes—
120C4×SD16—61no—
121C4×D8—31no—
122C2×C4×C8C2×D311yescharacteristic
123#367—32no—
124#366—32no—
125#358—32no—
126#358—32no—
127#358—32no—
128#357—32no—
129#346—32no—
130#346—31no—
131#346—31no—
132C2×C3×Q8—22no—
133C2×C3×Q8—41no—
134C22×C3×C4D811yescharacteristic
135C22×C3×C4D811yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C2×#67—41no—
138C2×D12—21no—
139C2×C4×D3—41no—
140C2×Dic6—21no—
141C3×Q16—22no—
142C3×Q16—22no—
143C3×Q16—42no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8C2×D411yescharacteristic
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8C2×D411yescharacteristic
149C2×C3×C8C2411yescharacteristic
150C3×C4⋊C4—22no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—42no—
154C3×C4⋊C4—42no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—42no—
157C3×C42—42no—
158C3×C42D811yescharacteristic
159C3×C42D811yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42C2×D411yescharacteristic
162#217—41no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172Dic3⋊C4—41no—
173C4×Dic3—41no—
174Dic12—22no—
175D24—22no—
176#204—42no—
177C3⋊C16—22no—
178C3⋊C16#4114yes—
179C2×Q16—21no—
180C2×Q16—31no—
181C2×Q16C22×D312yes—
182C2×SD16—61no—
183C2×D8—31no—
184C22×C8#6711yescharacteristic
185C22×C8C22×D311yescharacteristic
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188#126—121no—
189#125—41no—
190#125—61no—
191#124—61no—
192#123—121no—
193C22⋊Q8—22no—
194C22⋊Q8—41no—
195C22⋊Q8—61no—
196#121—61no—
197C4×Q8—22no—
198C4×Q8—41no—
199C4×Q8—61no—
200C4×D4—61no—
201C4×D4—121no—
202#117—41no—
203#117—121no—
204C2×C42#6711yescharacteristic
205Q32—64no—
206SD32—64no—
207C2×C16—32no—
208C2×C16—32no—
209C2×C16—64no—
210#107—21no—
211#107—31no—
212#107—31no—
213#107C22×D312yes—
214#106—61no—
215#103—22no—
216#103—22no—
217#103—61no—
218#103—61no—
219#102—61no—
220#102—61no—
221C4×C8#6711yescharacteristic
222C4×C8#6711yescharacteristic
223C4×C8C22×D311yescharacteristic
224C4×C8C22×D311yescharacteristic
225C23×C3C2×D811yescharacteristic
226C22×D3—41no—
227C3×Q8—42no—
228C3×Q8—42no—
229C3×Q8—82no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4C2×D811yescharacteristic
237C2×C3×C4C2×D811yescharacteristic
238C2×C3×C4C2×D811yescharacteristic
239C2×C3×C4C2×D811yescharacteristic
240C2×C3×C4C2×D811yescharacteristic
241C2×C3×C4C22×D411yescharacteristic
242#67—82no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246D12—41no—
247D12—41no—
248C4×D3—82no—
249Dic6—41no—
250Dic6—41no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8#13512yes—
254C3×C8C2×#4112yes—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—61no—
258C2×D4—61no—
259C2×D4—121no—
260C22×C4—121no—
261C22×C4#21311yescharacteristic
262C22×C4#21311yescharacteristic
263C22×C4C2×#6711yescharacteristic
264Q16—22no—
265Q16—22no—
266Q16—42no—
267Q16—62no—
268SD16—122no—
269D8—62no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8C2×#6711yescharacteristic
273C2×C8C2×#6711yescharacteristic
274C2×C8C2×#6711yescharacteristic
275C2×C8C23×D311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—42no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C42—42no—
295C42—121no—
296C42#21311yescharacteristic
297C42#21311yescharacteristic
298C42C2×#6711yescharacteristic
299C42C2×#6711yescharacteristic
300C16—34no—
301C16—62no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3#51012yes—
305C22×C3C22×D811yescharacteristic
306C2×D3—81no—
307C2×D3—81no—
308C2×D3—81no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—82no—
315C3×C4—82no—
316C3×C4#53211yescharacteristic
317C3×C4C2×#13511yescharacteristic
318Dic3—81no—
319Dic3—81no—
320Dic3—81no—
321C23—121no—
322C23C2×#21311yescharacteristic
323Q8—42no—
324Q8—42no—
325Q8—82no—
326Q8—121no—
327Q8—121no—
328D4—121no—
329D4—121no—
330D4—242no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—121no—
338C2×C4—121no—
339C2×C4—121no—
340C2×C4—242no—
341C2×C4C2×#21311yescharacteristic
342C2×C4C2×#21311yescharacteristic
343C2×C4C2×#21311yescharacteristic
344C2×C4C2×#21311yescharacteristic
345C2×C4C2×#21311yescharacteristic
346C2×C4C22×#6711yescharacteristic
347C8—21no—
348C8—21no—
349C8#95012yes—
350C8D3×#4112yes—
351C2×C3—22no—
352C2×C3#316911yescharacteristic
353C2×C3C2×#51011yescharacteristic
354C2×C3#341411yescharacteristic
355D3—161no—
356C22—21no—
357C22—21no—
358C22—241no—
359C22—241no—
360C22—241no—
361C22#517912yes—
362C22C22×#21311yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—82no—
370C4—241no—
371C4—241no—
372C4—241no—
373C4C2×#95011yescharacteristic
374C4#583211yescharacteristic
375C3#3374711yescharacteristic, complemented, Sylow
376C2—22no—
377C2—481no—
378C2#8329111yescharacteristic
379C2C2×#517911yescharacteristic
380C2#8389411yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3170C2
socleC22×C3C22×D8
prime radical#5115C22
hypercenter#3170D3
hypocenterC3#33747
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3170

Complements

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

NH ≅ G/Ntype
1. C3×#3170C2semidirect
3. #71386C2semidirect
4. #71383C2semidirect
5. #71314C2semidirect
6. #71311C2semidirect
8. #71081C2semidirect
9. #70913C2semidirect
10. #70909C2semidirect
11. C2×#4461C2semidirect
44. #3170D3semidirect
375. C3#33747semidirect

Maximal subgroups

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

subgroupindex
C3×#31702
#784692
#713862
#713832
#713142
#713112
#710822
#710812
#709132
#709092
C2×#44612
#337473