Finite Group Databasebeta

← #10896372

Subgroups — #10896372

Order 768.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2327C211yesmaximal, characteristic
2#73391C211yesmaximal, characteristic
3#71778C212yesmaximal
4#71629C212yesmaximal
5#71628C212yesmaximal
6#69686C212yesmaximal
7#69668C212yesmaximal
8#69540C212yesmaximal
9#69437C211yesmaximal, characteristic
10#29264—31nomaximal, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C4×Q8C2212yes—
13C2×#869C2211yescharacteristic
14C3×#391C2212yes—
15C3×#387—22no—
16C3×#387—22no—
17C3×#382C2211yescharacteristic
18C3×#376C2211yescharacteristic
19C2×#835C2212yes—
20C2×#835C2212yes—
21C2×#835C2212yes—
22C2×#835C2212yes—
23C2×#808C2212yes—
24C2×#807C2212yes—
25#4936C2211yescharacteristic
26C4×C4⋊Dic3C2212yes—
27#4928C2212yes—
28C2×C4⋊(C3⋊C8)C2212yes—
29#4654—22no—
30#4646—22no—
31#4529C2212yes—
32#4529C2212yes—
33#4524C2212yes—
34#4524C2211yescharacteristic
35#4524C2211yescharacteristic
36#2331—31no—
37#2327D311yescharacteristic
38#2097—32no—
39#2024—32no—
40#1981—32no—
41#1949—32no—
42#1865—32no—
43#1863—32no—
44#1832—31no—
45C22×C3×Q8C2312yes—
46C22×Dic6—21no—
47C3×C4⋊Q8—22no—
48C3×C4⋊Q8Q814yescomplemented
49C3×C4×Q8—42no—
50C3×C4×Q8—42no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4C2312yes—
54C2×C3×C42—22no—
55C2×C3×C42C2311yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×C4⋊Dic3—21no—
59C2×C4⋊Dic3C2312yes—
60C2×C4⋊Dic3C2312yes—
61C2×Dic3⋊C4—22no—
62C2×C4×Dic3—22no—
63C22×C3⋊C8C2312yes—
64C22×C3⋊C8C2312yes—
65#869—24no—
66C3×#95—22no—
67C3×#95—22no—
68C3×#95—22no—
69C3×#95D412yes—
70C3×#95D412yes—
71#835—24no—
72#835—24no—
73#835—42no—
74#835—42no—
75C3⋊#95—21no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78C3⋊#95—21no—
79#808—42no—
80#807—42no—
81C4⋊(C3⋊C8)—24no—
82C2×C4⋊Q8—31no—
83C2×C4⋊Q8C2×D311yescharacteristic
84C2×C4×Q8C2×D312yes—
85C2×#107—32no—
86C2×#106—32no—
87C2×C4⋊C8—32no—
88C2×#103—32no—
89C2×#103—32no—
90C2×#103—32no—
91C2×#103—32no—
92#391—62no—
93#391C2×D312yes—
94#389—32no—
95#387—22no—
96#387—22no—
97#387—62no—
98#382—31no—
99#382C2×D311yescharacteristic
100C4×C4⋊C4—32no—
101#376C2×D311yescharacteristic
102#340—32no—
103#340—32no—
104#336—32no—
105#336—31no—
106#336—31no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—22no—
111C2×C3×Q8—22no—
112C2×C3×Q8C2×Q814yes—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2411yescharacteristic
120C22×Dic3—22no—
121C22×Dic3—22no—
122C2×Dic6—24no—
123C2×Dic6—82no—
124C3×C4⋊C4—28no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—44no—
128C3×C42—22no—
129C3×C42—22no—
130C3×C42—44no—
131C3×C42C2×Q812yes—
132C3×C42#4112yes—
133C4⋊Dic3—24no—
134C4⋊Dic3—22no—
135C4⋊Dic3—22no—
136C4⋊Dic3—22no—
137C4⋊Dic3#4114yes—
138Dic3⋊C4—82no—
139C4×Dic3—82no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—22no—
144C2×C3⋊C8—22no—
145C2×C3⋊C8—22no—
146C22×Q8—61no—
147C22×Q8C22×D312yes—
148C22×C8—32no—
149C22×C8—32no—
150C4⋊Q8—22no—
151C4⋊Q8—64no—
152C4⋊Q8Dic614yescomplemented
153C4×Q8—42no—
154C4×Q8—42no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—32no—
158C2×C4⋊C4—32no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4C22×D312yes—
162C2×C42—22no—
163C2×C42—62no—
164C2×C42C22×D311yescharacteristic
165C2×C42C22×D311yescharacteristic
166C2×C42C22×D311yescharacteristic
167#107—122no—
168#106—122no—
169C4⋊C8—64no—
170#103—64no—
171#103—64no—
172#103—122no—
173#103—122no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—62no—
178#95—61no—
179#95—61no—
180#95—62no—
181#95#6712yes—
182#95#6712yes—
183C23×C3C22×D411yescharacteristic
184C3×Q8—24no—
185C3×Q8—24no—
186C3×Q8—42no—
187C3×Q8—42no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—24no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4C22⋊Q814yes—
204C2×C3×C4C22×Q811yescharacteristic
205C2×C3×C4C2×#4111yescharacteristic
206C2×C3×C4C2×#4112yes—
207C2×C3×C4H(5,𝔽2)12yes—
208C2×Dic3—24no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214Dic6—84no—
215C3⋊C8—42no—
216C3⋊C8—42no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—64no—
223C2×Q8—242no—
224C2×Q8C2×Dic614yes—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C22×C4C2×#6712yes—
232C22×C4C2×#6712yes—
233C22×C4C23×D311yescharacteristic
234C2×C8—62no—
235C2×C8—62no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—62no—
240C4⋊C4—28no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—34no—
244C4⋊C4—44no—
245C4⋊C4—64no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—242no—
250C42—22no—
251C42—22no—
252C42—44no—
253C42—242no—
254C42C2×Dic612yes—
255C42#23512yes—
256C22×C3C2×C22⋊Q811yescharacteristic
257C22×C3#52511yescharacteristic
258C22×C3#53511yescharacteristic
259C22×C3#57712yes—
260C22×C3#57812yes—
261C3×C4—24no—
262C3×C4—44no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—42no—
266C3×C4—82no—
267C3×C4#55412yes—
268C3×C4#55912yes—
269Dic3—44no—
270Dic3—82no—
271C23C22×#6711yescharacteristic
272Q8—24no—
273Q8—24no—
274Q8—42no—
275Q8—42no—
276Q8—244no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—64no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4#92414yes—
299C2×C4C22×Dic611yescharacteristic
300C2×C4C2×#23511yescharacteristic
301C2×C4#100412yes—
302C2×C4D3×#4112yes—
303C8—122no—
304C8—122no—
305C2×C3#258011yescharacteristic
306C2×C3#308212yes—
307C2×C3#312311yescharacteristic
308C2×C3#312711yescharacteristic
309C2×C3#322512yes—
310C22C2×#92411yescharacteristic
311C22#580511yescharacteristic
312C22#582811yescharacteristic
313C22#583712yes—
314C22#583812yes—
315C4—24no—
316C4—42no—
317C4—44no—
318C4—41no—
319C4—41no—
320C4—82no—
321C4—124no—
322C4—242no—
323C4#553912yes—
324C4#555912yes—
325C3#2926411yescharacteristic, complemented, Sylow
326C2#8238811yescharacteristic
327C2#8255012yes—
328C2#8341012yes—
329C2#8344711yescharacteristic
330C2#8345111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2327C2
socle, prime radicalC23×C3C22×D4
hypercenter#2327D3
hypocenterC3#29264
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2327

Complements

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

NH ≅ G/Ntype
48. C3×C4⋊Q8Q8semidirect
152. C4⋊Q8Dic6semidirect
325. C3#29264semidirect

Maximal subgroups

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

subgroupindex
C3×#23272
#733912
#717782
#716292
#716282
#696862
#696682
#695402
#694372
#292643