Finite Group Databasebeta

← #10694895

Subgroups — #10694895

Order 768.

496 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5116C211yesmaximal, characteristic
2C3×C8×C22⋊C4C211yesmaximal, characteristic
3#71977C211yesmaximal, characteristic
4#69923C212yesmaximal
5#69445C212yesmaximal
6#9814—31nomaximal, Sylow
7C23×C3×C8C2211yescharacteristic
8C2×#924C2212yes—
9C22×Dic12C411yescharacteristic, complemented
10C2×C3×C22⋊C8C2211yescharacteristic
11C2×C3×C4×C8—21no—
12C3×C4×C22⋊C4C2211yescharacteristic
13#5207C2212yes—
14#5116—21no—
15#5116—21no—
16#5116—21no—
17#5116—21no—
18C2×#819—21no—
19C2×#818C411yescharacteristic, complemented
20C2×#816C412yescomplemented
21C3×#336—21no—
22#4552—21no—
23#4469—24no—
24#4452—24no—
25C2×#467—31no—
26#1810—31no—
27C8×C22⋊C4D311yescharacteristic
28#1508—32no—
29#1472—32no—
30C23×C3×C4C2311yescharacteristic
31C22×Dic6C2×C412yes—
32C22×C3×C8—21no—
33C22×C3×C8—42no—
34C22×C3×C8C2×C411yescharacteristic
35C22×C3×C8C2×C411yescharacteristic
36C22×C3×C8D411yescharacteristic
37C22×C3×C8D411yescharacteristic
38C22×C3×C8D411yescharacteristic
39C22×C3×C8D411yescharacteristic
40C2×C3×C22⋊C4—21no—
41C2×C3×C42—21no—
42C2×#217—22no—
43C2×C4⋊Dic3C2×C412yes—
44#924—22no—
45#924—22no—
46#924—22no—
47#924—22no—
48C2×Dic3⋊C4—22no—
49C2×#208—22no—
50C2×Dic12—21no—
51C2×Dic12—21no—
52C2×Dic12—42no—
53C3×C22⋊C8—21no—
54C3×C22⋊C8D412yes—
55C3×C4×C8—42no—
56C3×#95D412yes—
57#830—22no—
58#830—22no—
59#819—42no—
60#818—21no—
61#818—21no—
62#816—22no—
63#816—22no—
64C22×Q16—31no—
65C23×C8C2×D311yescharacteristic
66C2×C22⋊Q8—32no—
67#467—61no—
68#467—61no—
69#467—61no—
70#467—61no—
71C2×#108—61no—
72C2×#107—31no—
73C2×#103—32no—
74#407—32no—
75C2×C22⋊C8C2×D311yescharacteristic
76C2×C4×C8—21no—
77C4×C22⋊C4C2×D311yescharacteristic
78#341—61no—
79#336—21no—
80#328—64no—
81#324—64no—
82C24×C3#4111yescharacteristic
83C22×C3×C4—22no—
84C22×C3×C4—41no—
85C22×C3×C4—41no—
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×Dic3—22no—
93C22×Dic3—22no—
94C2×Dic6—22no—
95C2×Dic6—22no—
96C2×Dic6—42no—
97C2×Dic6—42no—
98C2×C3×C8—22no—
99C2×C3×C8—22no—
100C2×C3×C8—22no—
101C2×C3×C8—22no—
102C2×C3×C8—41no—
103C2×C3×C8—41no—
104C2×C3×C8—41no—
105C2×C3×C8—41no—
106C2×C3×C8—41no—
107C2×C3×C8C22⋊C411yescharacteristic
108C2×C3×C8C22⋊C411yescharacteristic
109C2×C3×C8C22⋊C411yescharacteristic
110C2×C3×C8C22⋊C411yescharacteristic
111C3×C22⋊C4—24no—
112C3×C42—41no—
113C3×C42—41no—
114#217—42no—
115#217—42no—
116C4⋊Dic3—22no—
117C4⋊Dic3—22no—
118Dic3⋊C4—42no—
119Dic3⋊C4—42no—
120#208—42no—
121#208—42no—
122C2×C3⋊C8—42no—
123Dic12—42no—
124Dic12—42no—
125C22×Q8—32no—
126C23×C4C22×D311yescharacteristic
127C2×Q16—61no—
128C2×Q16—61no—
129C2×Q16—122no—
130C2×M16—62no—
131C22×C8—21no—
132C22×C8—42no—
133C22×C8C4×D311yescharacteristic
134C22×C8C4×D311yescharacteristic
135C22×C8D1211yescharacteristic
136C22×C8D1211yescharacteristic
137C22×C8#6711yescharacteristic
138C22×C8#6711yescharacteristic
139C22⋊Q8—62no—
140C22⋊Q8—62no—
141C22⋊Q8—62no—
142C22⋊Q8—62no—
143C2×C4⋊C4—32no—
144C2×C4⋊C4—62no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—62no—
147C2×C42—21no—
148#108—122no—
149#107—61no—
150#107—61no—
151#103—62no—
152#103—62no—
153C22⋊C8—21no—
154C22⋊C8—62no—
155C22⋊C8—62no—
156C22⋊C8D1212yes—
157C4×C8—42no—
158#95D1212yes—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—41no—
162C23×C3C4×D411yescharacteristic
163C23×C3#12111yescharacteristic
164C23×C3#12411yescharacteristic
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—24no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4C2×C22⋊C411yescharacteristic
178C2×C3×C4C4×D411yescharacteristic
179C2×C3×C4#12111yescharacteristic
180C2×C3×C4#12711yescharacteristic
181C2×Dic3—42no—
182C2×Dic3—42no—
183C2×Dic3—44no—
184C2×Dic3—82no—
185Dic6—42no—
186Dic6—42no—
187Dic6—82no—
188C3×C8—21no—
189C3×C8—21no—
190C3×C8—21no—
191C3×C8—42no—
192C3×C8#9912yes—
193C3⋊C8—82no—
194C24#23511yescharacteristic
195C2×Q8—62no—
196C2×Q8—62no—
197C2×Q8—122no—
198C2×Q8—122no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—62no—
203C22×C4—62no—
204C22×C4C2×C4×D311yescharacteristic
205C22×C4C2×D1211yescharacteristic
206C22×C4C2×D1211yescharacteristic
207C22×C4C2×D1211yescharacteristic
208C22×C4#23511yescharacteristic
209C22×C4C2×#6711yescharacteristic
210Q16—122no—
211Q16—122no—
212M16—122no—
213M16—122no—
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—122no—
224C2×C8(C2×D3)⋊C411yescharacteristic
225C2×C8(C2×D3)⋊C411yescharacteristic
226C2×C8(C2×D3)⋊C411yescharacteristic
227C2×C8(C2×D3)⋊C411yescharacteristic
228C4⋊C4—62no—
229C4⋊C4—62no—
230C4⋊C4—122no—
231C4⋊C4—122no—
232C22⋊C4—24no—
233C22⋊C4—122no—
234C22⋊C4—122no—
235C42—41no—
236C42—41no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—44no—
245C22×C3#39011yescharacteristic
246C22×C3#44312yes—
247C22×C3#47312yes—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—44no—
252C3×C4C2×#9911yescharacteristic
253C3×C4#41011yescharacteristic
254Dic3—82no—
255Dic3—82no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23C4×D1211yescharacteristic
260C23#87511yescharacteristic
261C23#93011yescharacteristic
262Q8—122no—
263Q8—122no—
264Q8—242no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—124no—
280C2×C4—242no—
281C2×C4C4×D1211yescharacteristic
282C2×C4#87411yescharacteristic
283C2×C4C2×(C2×D3)⋊C411yescharacteristic
284C2×C4#93011yescharacteristic
285C8—21no—
286C8—21no—
287C8—21no—
288C8—42no—
289C8—242no—
290C8#80612yes—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—41no—
294C2×C3#196411yescharacteristic
295C2×C3#197411yescharacteristic
296C2×C3#197911yescharacteristic
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—44no—
305C22#470012yes—
306C22#493911yescharacteristic
307C22#514312yes—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—44no—
312C4—242no—
313C4—242no—
314C4C2×#80611yescharacteristic
315C4#500611yescharacteristic
316C3#981411yescharacteristic, complemented, Sylow
317C2—22no—
318C2—22no—
319C2—41no—
320C2#6945611yescharacteristic
321C2#6979511yescharacteristic
322C2#6990411yescharacteristic
subgroupG/H
centerC22#4939
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×C8×C22⋊C4C2
socleC22×C3#390
prime radicalC24×C3#41
cosocleC23×C3×C4C23
hypercenterC8×C22⋊C4D3
hypocenterC3#9814
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
9. C22×Dic12C4semidirect
19. C2×#818C4semidirect
20. C2×#816C4semidirect
316. C3#9814semidirect

Maximal subgroups

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

subgroupindex
C2×#51162
C3×C8×C22⋊C42
#719772
#699232
#694452
#98143