Finite Group Databasebeta

← #10713879

Subgroups — #10713879

Order 768.

396 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 103 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#82477C211yesmaximal, characteristic
2C3×C4×#101C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#67699C212yesmaximal
5#67699C212yesmaximal
6#13303—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#870C411yescharacteristic
9C2×C3×#101C2212yes—
10C3×C4×M16—21no—
11#5229C2211yescharacteristic
12#5229C2211yescharacteristic
13C2×#836C2212yes—
14#4982—21no—
15#4982—21no—
16#4977C411yescharacteristic
17#4975C411yescharacteristic
18C4×C4⋊Dic3C411yescharacteristic
19#4933—21no—
20#4933—21no—
21#4924—21no—
22C3×#343—21no—
23#4530—21no—
24#2565—31no—
25#1801—31no—
26C4×#101D311yescharacteristic
27#1394—32no—
28#1394—32no—
29C22×C3×Q8C2311yescharacteristic
30C2×C3×M16—22no—
31C3×C4×Q8—41no—
32C3×C4×Q8—41no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C42C2×C411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3C2×C411yescharacteristic
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×Dic3⋊C4—21no—
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×C4×Dic3C2×C411yescharacteristic
47C2×C4×Dic3C2×C411yescharacteristic
48C2×#208—22no—
49#870—41no—
50#870—41no—
51C3×#101—22no—
52C3×#101Q814yes—
53C3×#97—41no—
54C3×C4×C8—41no—
55#836—24no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C4⋊(C3⋊C8)—41no—
64C4⋊(C3⋊C8)—41no—
65C2×#125—31no—
66C2×C4×Q8C2×D311yescharacteristic
67#423—61no—
68C2×#101—32no—
69C2×#101C2×D312yes—
70C4×M16—21no—
71#398—61no—
72#398—61no—
73#391—31no—
74#391—31no—
75#387—61no—
76#387—61no—
77#384—31no—
78#382—31no—
79C4×C4⋊C4—31no—
80#343—21no—
81#343—61no—
82C2×C3×Q8—41no—
83C2×C3×Q8—41no—
84C2×C3×Q8C2×Q812yes—
85C2×C3×Q8#4112yes—
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—21no—
94C22×Dic3—21no—
95C22×Dic3—21no—
96C22×Dic3—21no—
97C3×M16—24no—
98C3×M16—42no—
99C2×C3×C8—42no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C42—21no—
104C3×C42—21no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109C4⋊Dic3—42no—
110C4⋊Dic3—41no—
111C4⋊Dic3—41no—
112Dic3⋊C4—42no—
113Dic3⋊C4—42no—
114Dic3⋊C4—42no—
115Dic3⋊C4—41no—
116Dic3⋊C4—41no—
117C4×Dic3—41no—
118C4×Dic3—41no—
119#208—42no—
120#208—42no—
121C2×C3⋊C8—42no—
122C22×Q8C22×D311yescharacteristic
123C2×M16—22no—
124C2×M16—62no—
125#125—121no—
126#125—121no—
127C4×Q8—41no—
128C4×Q8—41no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4—61no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4C4×D311yescharacteristic
137C2×C4⋊C4D1211yescharacteristic
138C2×C4⋊C4#6711yescharacteristic
139C2×C42—31no—
140C2×C42—31no—
141C2×C42C4×D311yescharacteristic
142C2×C42D1211yescharacteristic
143C2×C42#6711yescharacteristic
144C4⋊C8—121no—
145C4⋊C8—121no—
146#101—22no—
147#101—64no—
148#101Dic614yes—
149#97—41no—
150C4×C8—41no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158C23×C3C2×C22⋊C411yescharacteristic
159C3×Q8—42no—
160C3×Q8—42no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4#9911yescharacteristic
175C2×C3×C4#9911yescharacteristic
176C2×C3×C4C4×Q812yes—
177C2×C3×C4C22⋊Q812yes—
178C2×C3×C4#12411yescharacteristic
179C2×C3×C4C4⋊Q811yescharacteristic
180C2×Dic3—22no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—42no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C3×C8—44no—
192C3⋊C8—82no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8C2×Dic612yes—
196C2×Q8#23512yes—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4—61no—
200C22×C4—61no—
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4(C2×D3)⋊C411yescharacteristic
203C22×C4(C2×D3)⋊C411yescharacteristic
204C22×C4(C2×D3)⋊C411yescharacteristic
205C22×C4C2×C4×D311yescharacteristic
206C22×C4C2×D1211yescharacteristic
207C22×C4C2×#6711yescharacteristic
208M16—24no—
209M16—42no—
210M16—122no—
211M16—122no—
212C2×C8—42no—
213C2×C8—122no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—121no—
218C4⋊C4—121no—
219C4⋊C4—122no—
220C4⋊C4—122no—
221C4⋊C4—122no—
222C4⋊C4—122no—
223C4⋊C4—121no—
224C4⋊C4—121no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—121no—
232C42—121no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3#39111yescharacteristic
236C22×C3C2×#9911yescharacteristic
237C22×C3#41011yescharacteristic
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—21no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—41no—
247Dic3—42no—
248Dic3—81no—
249Dic3—81no—
250Dic3—81no—
251C23C2×(C2×D3)⋊C411yescharacteristic
252Q8—42no—
253Q8—42no—
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—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—62no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—122no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4#80611yescharacteristic
279C2×C4#80611yescharacteristic
280C2×C4C4×Dic612yes—
281C2×C4#86911yescharacteristic
282C2×C4#87511yescharacteristic
283C2×C4#92412yes—
284C8—44no—
285C8—242no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3#199611yescharacteristic
289C2×C3#213211yescharacteristic
290C2×C3#213311yescharacteristic
291C22—21no—
292C22—21no—
293C22#492911yescharacteristic
294C22C2×#80611yescharacteristic
295C22#500611yescharacteristic
296C4—22no—
297C4—21no—
298C4—22no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—122no—
306C4—241no—
307C4—241no—
308C4—241no—
309C3#1330311yescharacteristic, complemented, Sylow
310C2—21no—
311C2—21no—
312C2#6945711yescharacteristic
313C2#7012811yescharacteristic
314C2#7014811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC22×Q8C22×D3
FittingC3×C4×#101C2
socleC22×C3C2×#99
prime radicalC23×C3C2×C22⋊C4
cosocleC22×C3×Q8C23
hypercenterC4×#101D3
hypocenterC3#13303
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
309. C3#13303semidirect

Maximal subgroups

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

subgroupindex
#824772
C3×C4×#1012
#716372
#676992
#133033