Finite Group Databasebeta

← #10896462

Subgroups — #10896462

Order 768.

736 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊Q8C211yesmaximal, characteristic
2C4×#869C211yesmaximal, characteristic
3#71780C212yesmaximal
4#69611C212yesmaximal
5#69604C212yesmaximal
6#69567C212yesmaximal
7#69458C214yesmaximal
8#69392C211yesmaximal, characteristic
9#29272—31nomaximal, Sylow
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×Q8—21no—
12C2×#869C2211yescharacteristic
13C2×C4×Dic6—21no—
14C3×#391—21no—
15C3×#389—21no—
16C3×#389C2212yes—
17C3×#384C2211yescharacteristic
18C3×C4×C4⋊C4—21no—
19C3×C4×C4⋊C4C2212yes—
20C3×C43C2211yescharacteristic
21C2×#807C2214yes—
22#4935C2211yescharacteristic
23C4×C4⋊Dic3—21no—
24C4×C4⋊Dic3C2212yes—
25#4929—21no—
26#4928—21no—
27#4928C2212yes—
28C2×C4⋊(C3⋊C8)C2212yes—
29#4529C2214yes—
30#4529C2214yes—
31#4529C2214yes—
32#4524C2212yes—
33#4524C2212yes—
34C4×C4⋊Q8—31no—
35C4×C4⋊Q8D311yescharacteristic
36#2099—32no—
37#2057—32no—
38#1988—32no—
39#1947—32no—
40#1918—34no—
41#1831—31no—
42C22×C3×Q8—21no—
43C22×Dic6—21no—
44C3×C4⋊Q8—22no—
45C3×C4⋊Q8Q814yescomplemented
46C3×C4×Q8—44no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4C2312yes—
52C2×C3×C4⋊C4C2312yes—
53C2×C3×C42—22no—
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C3×C42D411yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×C3×C42C2311yescharacteristic
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×C4⋊Dic3C2312yes—
65C2×C4⋊Dic3C2312yes—
66C2×Dic3⋊C4—22no—
67C2×C4×Dic3—22no—
68C22×C3⋊C8C2314yes—
69#869—24no—
70C4×Dic6—44no—
71C3×#95—22no—
72C3×#95—22no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75#807—28no—
76C4⋊(C3⋊C8)—24no—
77C2×C4⋊Q8—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4×Q8—21no—
80C2×C4×Q8—61no—
81C2×#107—34no—
82C2×C4⋊C8—32no—
83#391—21no—
84#391—61no—
85#389—21no—
86#389—32no—
87#389—61no—
88#389C2×D312yes—
89#384—31no—
90#384C2×D311yescharacteristic
91C4×C4⋊C4—21no—
92C4×C4⋊C4—32no—
93C4×C4⋊C4—61no—
94C4×C4⋊C4C2×D312yes—
95C43C2×D311yescharacteristic
96#340—34no—
97#340—34no—
98#340—34no—
99#336—32no—
100#336—32no—
101C2×C3×Q8—24no—
102C2×C3×Q8—42no—
103C2×C3×Q8—42no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2411yescharacteristic
113C22×Dic3—22no—
114C22×Dic3—22no—
115C2×Dic6—24no—
116C2×Dic6—42no—
117C2×Dic6—42no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4—42no—
122C3×C4⋊C4—44no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4C2×Q814yes—
125C3×C42—22no—
126C3×C42—22no—
127C3×C42—22no—
128C3×C42—22no—
129C3×C42—41no—
130C3×C42—44no—
131C3×C42D814yes—
132C3×C42C2×Q812yes—
133C3×C42#4112yes—
134C4⋊Dic3—24no—
135C4⋊Dic3—22no—
136C4⋊Dic3—24no—
137C4⋊Dic3—42no—
138C4⋊Dic3—42no—
139C4⋊Dic3#4114yes—
140Dic3⋊C4—44no—
141C4×Dic3—44no—
142C2×C3⋊C8—24no—
143C2×C3⋊C8—24no—
144C2×C3⋊C8—24no—
145C22×Q8—21no—
146C22×Q8—61no—
147C22×C8—34no—
148C4⋊Q8—22no—
149C4⋊Q8—64no—
150C4⋊Q8Dic614yescomplemented
151C4×Q8—44no—
152C4×Q8—124no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—32no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4—62no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4C22×D312yes—
164C2×C4⋊C4C22×D312yes—
165C2×C42—22no—
166C2×C42—62no—
167C2×C42#6711yescharacteristic
168C2×C42#6711yescharacteristic
169C2×C42#6711yescharacteristic
170C2×C42#6711yescharacteristic
171C2×C42C22×D311yescharacteristic
172C2×C42C22×D311yescharacteristic
173C2×C42C22×D311yescharacteristic
174#107—68no—
175C4⋊C8—64no—
176#95—22no—
177#95—22no—
178#95—62no—
179#95—62no—
180C23×C3C22×D411yescharacteristic
181C3×Q8—48no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—24no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4C22⋊Q812yes—
200C2×C3×C4C22⋊Q812yes—
201C2×C3×C4C2×D812yes—
202C2×C3×C4C2×D812yes—
203C2×C3×C4C2×D812yes—
204C2×C3×C4#13712yes—
205C2×C3×C4#13712yes—
206C2×C3×C4C22×Q811yescharacteristic
207C2×C3×C4C2×#4111yescharacteristic
208C2×C3×C4C2×#4112yes—
209C2×C3×C42−1+412yes—
210C2×Dic3—24no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—42no—
216Dic6—48no—
217C3⋊C8—44no—
218C2×Q8—24no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—64no—
222C2×Q8—122no—
223C2×Q8—122no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4C2×#6711yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232C22×C4C2×#6711yescharacteristic
233C22×C4C2×#6711yescharacteristic
234C22×C4C23×D311yescharacteristic
235C2×C8—64no—
236C2×C8—64no—
237C2×C8—64no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—34no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—64no—
246C4⋊C4—62no—
247C4⋊C4—64no—
248C4⋊C4—124no—
249C4⋊C4—122no—
250C4⋊C4—122no—
251C4⋊C4C2×Dic614yes—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—41no—
257C42—44no—
258C42—124no—
259C42#21314yes—
260C42C2×Dic612yes—
261C42#23512yes—
262C22×C3C2×C22⋊Q811yescharacteristic
263C22×C3#52511yescharacteristic
264C22×C3#53611yescharacteristic
265C22×C3C22×D811yescharacteristic
266C22×C3C2×#13711yescharacteristic
267C22×C3C2×#13711yescharacteristic
268C22×C3#57511yescharacteristic
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—41no—
274C3×C4—44no—
275C3×C4—82no—
276C3×C4#47414yes—
277C3×C4#55612yes—
278C3×C4#55712yes—
279Dic3—44no—
280Dic3—82no—
281C23C22×#6711yescharacteristic
282Q8—48no—
283Q8—128no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—24no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—64no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4—122no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4#92412yes—
308C2×C4#92412yes—
309C2×C4C2×#21312yes—
310C2×C4C2×#21312yes—
311C2×C4C2×#21312yes—
312C2×C4#94212yes—
313C2×C4#95112yes—
314C2×C4C22×Dic611yescharacteristic
315C2×C4C2×#23511yescharacteristic
316C2×C4#100712yes—
317C2×C4D3×#4112yes—
318C8—124no—
319C2×C3#258111yescharacteristic
320C2×C3C2×#47411yescharacteristic
321C2×C3#307811yescharacteristic
322C2×C3#310011yescharacteristic
323C2×C3#311311yescharacteristic
324C2×C3#322611yescharacteristic
325C2×C3#323711yescharacteristic
326C22C2×#92411yescharacteristic
327C22C22×#21311yescharacteristic
328C22C2×#94211yescharacteristic
329C22#581611yescharacteristic
330C22#582211yescharacteristic
331C22C2×#95111yescharacteristic
332C22#582811yescharacteristic
333C4—22no—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—44no—
338C4—41no—
339C4—82no—
340C4—124no—
341C4—242no—
342C4#500714yes—
343C4#556812yes—
344C4#556912yes—
345C3#2927211yescharacteristic, complemented, Sylow
346C2C2×#500711yescharacteristic
347C2#8243111yescharacteristic
348C2#8248111yescharacteristic
349C2#8321511yescharacteristic
350C2#8329811yescharacteristic
351C2#8336911yescharacteristic
352C2#8337211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊Q8C2
socle, prime radicalC23×C3C22×D4
hypercenterC4×C4⋊Q8D3
hypocenterC3#29272
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
45. C3×C4⋊Q8Q8semidirect
150. C4⋊Q8Dic6semidirect
345. C3#29272semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊Q82
C4×#8692
#717802
#696112
#696042
#695672
#694582
#693922
#292723