Finite Group Databasebeta

← #10705773

Subgroups — #10705773

Order 768.

464 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 87 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#82373C211yesmaximal, characteristic
2C3×C4×#100C211yesmaximal, characteristic
3#71602C211yesmaximal, characteristic
4#67495C214yesmaximal, complemented
5#11780—31nomaximal, Sylow
6C2×C3×C4×D4C2211yescharacteristic
7C2×#935—21no—
8C22×C4⋊Dic3C411yescharacteristic
9C2×C3×#100—21no—
10C3×C4×M16—21no—
11#5212—21no—
12#5212C411yescharacteristic
13#5207—21no—
14#4981C2212yes—
15#4953C412yes—
16#4943—21no—
17#4928—21no—
18C3×#343C2212yes—
19#4528C2212yes—
20#4525—21no—
21#2486—31no—
22C4×#100D311yescharacteristic
23#1465—31no—
24#1346—34no—
25C22×C3×D4D411yescharacteristic
26C23×C3×C4C2×C411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×Dic3—21no—
29C2×C3×M16—22no—
30C3×C4×D4—41no—
31C3×C4×D4—41no—
32C2×C3×C4⋊C4C2311yescharacteristic
33C2×C3×C22⋊C4C2×C411yescharacteristic
34C2×C3×C22⋊C4D411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×#217—21no—
37C2×#217—22no—
38#935—44no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—41no—
42C2×C4⋊Dic3—41no—
43C2×C4⋊Dic3C2×C412yes—
44C2×Dic3⋊C4—22no—
45C2×#208—41no—
46C3×#100—24no—
47C3×#97—41no—
48C3×C4×C8—41no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95C2×C412yes—
52#830—22no—
53#830—41no—
54C2×#123—61no—
55C2×C4×D4C2×D311yescharacteristic
56C22×C4⋊C4—31no—
57C2×#100—21no—
58#407—61no—
59C4×M16—21no—
60#398—32no—
61#397—32no—
62#389—61no—
63#381—61no—
64#380—31no—
65#380—61no—
66#343C2×D312yes—
67#324—32no—
68#323—61no—
69C24×C3C22⋊C411yescharacteristic
70C24×C3C22⋊C411yescharacteristic
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4—41no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—42no—
79C22×C3×C4—42no—
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—42no—
88C22×Dic3—42no—
89C22×Dic3—42no—
90C3×M16—24no—
91C3×M16—42no—
92C2×C3×C8—42no—
93C3×C4⋊C4—41no—
94C3×C22⋊C4—41no—
95C3×C22⋊C4—41no—
96C3×C42—21no—
97C3×C42—21no—
98#217—42no—
99#217—44no—
100C4⋊Dic3—42no—
101C4⋊Dic3—41no—
102C4⋊Dic3—41no—
103C4⋊Dic3—42no—
104Dic3⋊C4—44no—
105#208—82no—
106C2×C3⋊C8—42no—
107C22×D4#6711yescharacteristic
108C23×C4—61no—
109C23×C4C4×D311yescharacteristic
110C23×C4D1211yescharacteristic
111C2×M16—22no—
112C2×M16—121no—
113#123—124no—
114C4×D4—41no—
115C4×D4—41no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—121no—
121C2×C4⋊C4—121no—
122C2×C4⋊C4C22×D311yescharacteristic
123C2×C22⋊C4—61no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4C4×D311yescharacteristic
126C2×C22⋊C4D1211yescharacteristic
127C2×C42#6711yescharacteristic
128#100—24no—
129C22⋊C8—62no—
130C22⋊C8—121no—
131#97—41no—
132C4×C8—41no—
133#95—32no—
134#95—62no—
135#95—62no—
136C23×C3—21no—
137C23×C3—21no—
138C23×C3—21no—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3—41no—
145C23×C3#10412yes—
146C23×C3C2×C22⋊C411yescharacteristic
147C3×D4—44no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—81no—
161C2×C3×C4#9911yescharacteristic
162C2×C3×C4#9911yescharacteristic
163C2×C3×C4AGL(1,ℤ/8)12yes—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—84no—
171C3×C8—44no—
172C3⋊C8—82no—
173C24(C2×D3)⋊C411yescharacteristic
174C24(C2×D3)⋊C411yescharacteristic
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—41no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4—122no—
187C22×C4—122no—
188C22×C4—122no—
189C22×C4(C2×D3)⋊C411yescharacteristic
190C22×C4(C2×D3)⋊C411yescharacteristic
191C22×C4C2×C4×D311yescharacteristic
192C22×C4C2×D1211yescharacteristic
193C22×C4C2×#6711yescharacteristic
194M16—24no—
195M16—42no—
196M16—242no—
197C2×C8—42no—
198C2×C8—122no—
199C4⋊C4—41no—
200C4⋊C4—122no—
201C4⋊C4—122no—
202C4⋊C4—121no—
203C4⋊C4—121no—
204C4⋊C4—124no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—122no—
208C22⋊C4—124no—
209C42—21no—
210C42—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—81no—
223C22×C3C2×#9911yescharacteristic
224C22×C3#41811yescharacteristic
225C22×C3C2×#10411yescharacteristic
226C3×C4—21no—
227C3×C4—22no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4#35212yes—
232Dic3—82no—
233Dic3—82no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#80512yes—
244C23C2×(C2×D3)⋊C411yescharacteristic
245D4—44no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—81no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—244no—
266C2×C4#80611yescharacteristic
267C2×C4#80611yescharacteristic
268C2×C4#94912yes—
269C8—44no—
270C8—242no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—42no—
274C2×C3—81no—
275C2×C3#150711yescharacteristic
276C2×C3C2×#35211yescharacteristic
277C2×C3#213511yescharacteristic
278C22—21no—
279C22—21no—
280C22—22no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—81no—
290C22C2×#80511yescharacteristic
291C22C2×#80611yescharacteristic
292C22#500111yescharacteristic
293C4—22no—
294C4—21no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—242no—
299C4—242no—
300C4#447612yes—
301C3#1178011yescharacteristic, complemented, Sylow
302C2—21no—
303C2—21no—
304C2—42no—
305C2—81no—
306C2#6944411yescharacteristic
307C2C2×#447611yescharacteristic
308C2#7015011yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×C4×#100C2
socleC22×C3C2×#99
prime radicalC22×C3×D4D4
cosocleC2×C3×C4⋊C4C23
hypercenterC4×#100D3
hypocenterC3#11780
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
4. #67495C2semidirect
301. C3#11780semidirect

Maximal subgroups

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

subgroupindex
#823732
C3×C4×#1002
#716022
#674952
#117803