Finite Group Databasebeta

← #10704586

Subgroups — #10704586

Order 768.

578 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 95 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×#4975C211yesmaximal, characteristic
2C3×#1463C211yesmaximal, characteristic
3#71552C211yesmaximal, characteristic
4#67453C214yesmaximal, complemented
5#11512—31nomaximal, Sylow
6C22×C3×C4⋊C4C2211yescharacteristic
7C22×C4⋊Dic3C411yescharacteristic
8C22×Dic3⋊C4—21no—
9C22×Dic3⋊C4—21no—
10C22×C4×Dic3C411yescharacteristic
11C3×#407—21no—
12#5207—21no—
13#4975—21no—
14#4975—21no—
15#4975C412yes—
16#4975C2212yes—
17C3×#324—21no—
18C3×#324C2212yes—
19#4528—21no—
20#4528C2212yes—
21C2×#384—31no—
22#1463—31no—
23#1463D311yescharacteristic
24#1325—34no—
25C23×C3×C4C2×C411yescharacteristic
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×Dic3—21no—
29C23×Dic3—21no—
30C2×C3×M16—41no—
31C2×C3×C4⋊C4—41no—
32C2×C3×C4⋊C4—41no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C4⋊C4C2311yescharacteristic
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—41no—
39C2×C4⋊Dic3—41no—
40C2×C4⋊Dic3C2×C412yes—
41C2×Dic3⋊C4—22no—
42C2×Dic3⋊C4—22no—
43C2×Dic3⋊C4—22no—
44C2×Dic3⋊C4—22no—
45C2×Dic3⋊C4—44no—
46C2×Dic3⋊C4—44no—
47C2×C4×Dic3—21no—
48C2×C4×Dic3—24no—
49C2×C4×Dic3C2×C412yes—
50C2×#208—41no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8—41no—
53#830—22no—
54#830—41no—
55C22×C4⋊C4—31no—
56C22×C4⋊C4—61no—
57C22×C4⋊C4—61no—
58C22×C4⋊C4C2×D311yescharacteristic
59C22×C42—31no—
60#407—21no—
61#407—61no—
62#384—32no—
63#384—32no—
64#384—61no—
65#384—61no—
66#324—21no—
67#324—32no—
68#324—61no—
69#324C2×D312yes—
70C24×C3C22⋊C411yescharacteristic
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—42no—
80C22×C3×C4—42no—
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×Dic3—24no—
88C22×Dic3—24no—
89C22×Dic3—24no—
90C22×Dic3—22no—
91C22×Dic3—22no—
92C22×Dic3—42no—
93C22×Dic3—42no—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C3×M16—82no—
97C2×C3×C8—42no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C4⋊Dic3—42no—
103C4⋊Dic3—42no—
104Dic3⋊C4—44no—
105Dic3⋊C4—44no—
106Dic3⋊C4—44no—
107Dic3⋊C4—44no—
108C4×Dic3—24no—
109C4×Dic3—22no—
110C4×Dic3M1614yescomplemented
111#208—82no—
112C2×C3⋊C8—42no—
113C23×C4—61no—
114C23×C4—61no—
115C23×C4C4×D311yescharacteristic
116C23×C4D1211yescharacteristic
117C23×C4#6711yescharacteristic
118C2×M16—41no—
119C2×M16—121no—
120C2×C4⋊C4—32no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—62no—
125C2×C4⋊C4—62no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—121no—
129C2×C4⋊C4—121no—
130C2×C4⋊C4—124no—
131C2×C4⋊C4—124no—
132C2×C4⋊C4C4×D311yescharacteristic
133C2×C4⋊C4D1211yescharacteristic
134C2×C4⋊C4#6711yescharacteristic
135C2×C4⋊C4C22×D311yescharacteristic
136C2×C42—32no—
137C2×C42—64no—
138C2×C42—61no—
139C22⋊C8—22no—
140C22⋊C8—41no—
141C22⋊C8—62no—
142C22⋊C8—121no—
143C23×C3—22no—
144C23×C3—22no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3#9911yescharacteristic
148C23×C3#9911yescharacteristic
149C23×C3C2×C22⋊C411yescharacteristic
150C2×C3×C4—24no—
151C2×C3×C4—24no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—81no—
167C2×C3×C4—81no—
168C2×C3×C4#10112yes—
169C2×C3×C4C2×M1612yes—
170C2×Dic3—24no—
171C2×Dic3—216no—
172C2×Dic3—24no—
173C2×Dic3—24no—
174C2×Dic3—42no—
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—82no—
178C2×Dic3—82no—
179C3×C8—82no—
180C3⋊C8—82no—
181C24(C2×D3)⋊C411yescharacteristic
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—64no—
193C22×C4—64no—
194C22×C4—64no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—122no—
198C22×C4—122no—
199C22×C4—121no—
200C22×C4—121no—
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4(C2×D3)⋊C411yescharacteristic
203C22×C4(C2×D3)⋊C411yescharacteristic
204C22×C4C2×C4×D311yescharacteristic
205C22×C4C2×D1211yescharacteristic
206C22×C4C2×#6711yescharacteristic
207M16—82no—
208M16—242no—
209C2×C8—42no—
210C2×C8—122no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—122no—
216C4⋊C4—122no—
217C4⋊C4—124no—
218C4⋊C4—124no—
219C4⋊C4—124no—
220C4⋊C4—124no—
221C42—34no—
222C42—64no—
223C42—62no—
224C22×C3—24no—
225C22×C3—24no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3#35312yes—
233C22×C3#35612yes—
234C22×C3#40711yescharacteristic
235C22×C3C2×#9911yescharacteristic
236C22×C3C2×#10111yescharacteristic
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243Dic3—28no—
244Dic3—82no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23#80611yescharacteristic
250C23#80611yescharacteristic
251C23C2×(C2×D3)⋊C411yescharacteristic
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—64no—
269C2×C4—616no—
270C2×C4—64no—
271C2×C4—64no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—242no—
278C2×C4—242no—
279C2×C4#82412yes—
280C2×C4D3×M1612yes—
281C8—82no—
282C8—242no—
283C2×C3—21no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—21no—
287C2×C3#146311yescharacteristic
288C2×C3C2×#35311yescharacteristic
289C2×C3C2×#35611yescharacteristic
290C22—24no—
291C22—24no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22#447412yes—
299C22#447812yes—
300C22C2×#80611yescharacteristic
301C22#512611yescharacteristic
302C22C2×#82411yescharacteristic
303C4—22no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—68no—
310C4—242no—
311C3#1151211yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2—22no—
315C2—22no—
316C2#6997111yescharacteristic
317C2C2×#447411yescharacteristic
318C2C2×#447811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1463C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1463D3
hypocenterC3#11512
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
4. #67453C2semidirect
110. C4×Dic3M16semidirect
311. C3#11512semidirect

Maximal subgroups

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

subgroupindex
C2×#49752
C3×#14632
#715522
#674532
#115123