Finite Group Databasebeta

← #10706244

Subgroups — #10706244

Order 768.

408 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#82475C211yesmaximal, characteristic
2C3×#1801C211yesmaximal, characteristic
3#71624C211yesmaximal, characteristic
4#67495C214yesmaximal
5#11883—31nomaximal, Sylow
6C2×C3×C4×Q8C2211yescharacteristic
7C2×#944—21no—
8C2×#869C411yescharacteristic
9C3×#423—21no—
10C2×C3×#101—21no—
11#4981C412yes—
12#4981C2212yes—
13#4973—21no—
14C4×C4⋊Dic3C411yescharacteristic
15#4928—21no—
16#4928—21no—
17#4925—21no—
18C3×#343C2212yes—
19#4528—21no—
20#4528C2212yes—
21#2546—31no—
22#1801D311yescharacteristic
23#1461—31no—
24#1346—34no—
25C22×C3×Q8D411yescharacteristic
26C22×Dic6—21no—
27C2×C3×M16—22no—
28C3×C4×Q8—41no—
29C3×C4×Q8—41no—
30C2×C3×C4⋊C4C2×C411yescharacteristic
31C2×C3×C4⋊C4D411yescharacteristic
32C2×C3×C4⋊C4C2311yescharacteristic
33C2×C3×C42C2×C411yescharacteristic
34C2×C3×C42D411yescharacteristic
35C2×C3×C42D411yescharacteristic
36#944—42no—
37#944—42no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3C2×C412yes—
41C2×Dic3⋊C4—22no—
42C2×Dic3⋊C4—22no—
43C2×C4×Dic3—21no—
44#869—41no—
45#869—41no—
46C3×C4⋊C8—41no—
47C3×C4⋊C8—41no—
48C3×#101—42no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95C2×C412yes—
52#830—22no—
53C4⋊(C3⋊C8)—41no—
54C4×C3⋊C8—41no—
55C2×C4⋊Q8—31no—
56C2×C4⋊Q8—61no—
57C2×C4×Q8C2×D311yescharacteristic
58#431—61no—
59#423—21no—
60C2×#101—21no—
61#398—32no—
62#398—32no—
63#391—61no—
64#389—61no—
65#389—61no—
66C4×C4⋊C4—31no—
67#343C2×D312yes—
68#324—32no—
69#324—61no—
70C2×C3×Q8—41no—
71C2×C3×Q8—41no—
72C2×C3×Q8D812yes—
73C2×C3×Q8SD1612yes—
74C22×C3×C4C22⋊C411yescharacteristic
75C22×C3×C4C22⋊C411yescharacteristic
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4C22×C411yescharacteristic
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×Dic3—22no—
82C22×Dic3—22no—
83C2×Dic6—42no—
84C2×Dic6—42no—
85C2×Dic6—42no—
86C3×M16—42no—
87C3×M16—42no—
88C2×C3×C8—42no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—41no—
92C3×C42—21no—
93C3×C42—21no—
94C3×C42—21no—
95C3×C42—21no—
96C3×C42—21no—
97C3×C42—21no—
98C4⋊Dic3—24no—
99C4⋊Dic3—41no—
100C4⋊Dic3—41no—
101C4⋊Dic3—42no—
102Dic3⋊C4—44no—
103Dic3⋊C4—42no—
104Dic3⋊C4—42no—
105C4×Dic3—42no—
106C2×C3⋊C8—42no—
107C22×Q8—61no—
108C22×Q8#6711yescharacteristic
109C2×M16—22no—
110C4⋊Q8—121no—
111C4⋊Q8—121no—
112C4⋊Q8—122no—
113C4⋊Q8—122no—
114C4×Q8—41no—
115C4×Q8—41no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—61no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4C4×D311yescharacteristic
122C2×C4⋊C4D1211yescharacteristic
123C2×C4⋊C4C22×D311yescharacteristic
124C2×C42—61no—
125C2×C42C4×D311yescharacteristic
126C2×C42D1211yescharacteristic
127C2×C42#6711yescharacteristic
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—121no—
131#101—42no—
132C22⋊C8—62no—
133C4×C8—121no—
134#95—32no—
135#95—62no—
136#95—62no—
137C23×C3C2×C22⋊C411yescharacteristic
138C3×Q8—42no—
139C3×Q8—42no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4#9911yescharacteristic
154C2×C3×C4#9911yescharacteristic
155C2×C3×C4#10212yes—
156C2×C3×C4#10212yes—
157C2×C3×C4C2×D811yescharacteristic
158C2×C3×C4C2×SD1611yescharacteristic
159C2×Dic3—24no—
160C2×Dic3—42no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165Dic6—84no—
166C3×C8—82no—
167C3⋊C8—82no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—122no—
171C2×Q8—122no—
172C2×Q8—122no—
173C2×Q8#21312yes—
174C2×Q8#21512yes—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4(C2×D3)⋊C411yescharacteristic
178C22×C4(C2×D3)⋊C411yescharacteristic
179C22×C4(C2×D3)⋊C411yescharacteristic
180C22×C4(C2×D3)⋊C411yescharacteristic
181C22×C4C2×C4×D311yescharacteristic
182C22×C4C2×D1211yescharacteristic
183C22×C4C2×#6711yescharacteristic
184M16—42no—
185M16—42no—
186C2×C8—42no—
187C2×C8—122no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—64no—
192C4⋊C4—122no—
193C4⋊C4—121no—
194C4⋊C4—121no—
195C4⋊C4—124no—
196C4⋊C4—122no—
197C4⋊C4—122no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—122no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3C2×#9911yescharacteristic
208C22×C3C2×#10211yescharacteristic
209C22×C3#42111yescharacteristic
210C3×C4—21no—
211C3×C4—21no—
212C3×C4—22no—
213C3×C4—22no—
214C3×C4—21no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4#35612yes—
219Dic3—44no—
220Dic3—82no—
221C23C2×(C2×D3)⋊C411yescharacteristic
222Q8—42no—
223Q8—42no—
224Q8—244no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—64no—
239C2×C4—122no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4—122no—
243C2×C4—122no—
244C2×C4#80611yescharacteristic
245C2×C4#80611yescharacteristic
246C2×C4#80912yes—
247C2×C4#80912yes—
248C2×C4C2×#21311yescharacteristic
249C2×C4C2×#21511yescharacteristic
250C8—82no—
251C8—242no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3#151911yescharacteristic
255C2×C3#212311yescharacteristic
256C2×C3C2×#35611yescharacteristic
257C22—21no—
258C22—21no—
259C22C2×#80611yescharacteristic
260C22C2×#80911yescharacteristic
261C22#500511yescharacteristic
262C4—21no—
263C4—22no—
264C4—22no—
265C4—21no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—124no—
271C4—242no—
272C4#447812yes—
273C3#1188311yescharacteristic, complemented, Sylow
274C2—21no—
275C2—21no—
276C2#6950911yescharacteristic
277C2C2×#447811yescharacteristic
278C2#7014711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1801C2
socleC22×C3C2×#99
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1801D3
hypocenterC3#11883
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
273. C3#11883semidirect

Maximal subgroups

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

subgroupindex
#824752
C3×#18012
#716242
#674952
#118833