Finite Group Databasebeta

← #10692861

Subgroups — #10692861

Order 768.

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

normal subgroups only

This group has 424 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
1C3×#2983C211yesmaximal, characteristic
2#82477C211yesmaximal, characteristic
3#71625C211yesmaximal, characteristic
4#70814C212yesmaximal
5#67749C212yesmaximal
6#9450—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#870C2211yescharacteristic
9C3×#446—24no—
10C3×#432—21no—
11C3×#432C2212yes—
12C3×#431—21no—
13#5229C411yescharacteristic
14#5229C411yescharacteristic
15#4982—21no—
16#4982—21no—
17#4977C411yescharacteristic
18#4975C411yescharacteristic
19C4×C4⋊Dic3C2211yescharacteristic
20#4933—21no—
21#4933—21no—
22#4926C2212yes—
23#4528—21no—
24#4528—21no—
25#4465—24no—
26#2983D311yescharacteristic
27#2565—31no—
28#1534—32no—
29#1467—31no—
30#1285—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C3×C4×Q8—22no—
33C3×C4×Q8—22no—
34C2×C3×C4⋊C4C2×C411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42C2311yescharacteristic
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×C4×Dic3C2×C411yescharacteristic
48C2×C4×Dic3C2×C411yescharacteristic
49#870—22no—
50#870—22no—
51C3×C4⋊C8—22no—
52C3×C4⋊C8—22no—
53C3×C4⋊C8—22no—
54C3×C22⋊C8—22no—
55C3×#97—22no—
56C3×#97D414yes—
57C3×C4×C8—22no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95—21no—
62C3⋊#95—21no—
63C3⋊#95—21no—
64C3⋊#95—21no—
65#830—22no—
66C4⋊(C3⋊C8)—22no—
67#803—22no—
68C2×#125—31no—
69C2×C4×Q8C2×D311yescharacteristic
70#446—24no—
71#432—21no—
72#432—32no—
73#432C2×D312yes—
74#431—21no—
75#398—61no—
76#398—61no—
77#391—31no—
78#391—31no—
79#387—61no—
80#387—61no—
81#384—31no—
82#382—31no—
83C4×C4⋊C4—31no—
84#330—64no—
85#324—61no—
86#324—61no—
87C2×C3×Q8—22no—
88C2×C3×Q8—22no—
89C2×C3×Q8—41no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C2×C3×C8—24no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42C2×D412yes—
110C3×C42#4112yes—
111C4⋊Dic3—22no—
112C4⋊Dic3—22no—
113C4⋊Dic3—42no—
114Dic3⋊C4—42no—
115Dic3⋊C4—42no—
116Dic3⋊C4—41no—
117Dic3⋊C4—42no—
118Dic3⋊C4—41no—
119C4×Dic3—41no—
120C4×Dic3—41no—
121C2×C3⋊C8—42no—
122C22×Q8C4×D311yescharacteristic
123#125—62no—
124#125—62no—
125C4×Q8—22no—
126C4×Q8—22no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—61no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4—61no—
134C2×C4⋊C4C4×D311yescharacteristic
135C2×C4⋊C4D1211yescharacteristic
136C2×C4⋊C4D1211yescharacteristic
137C2×C42—31no—
138C2×C42—31no—
139C2×C42#6711yescharacteristic
140C2×C42#6711yescharacteristic
141C2×C42C22×D311yescharacteristic
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—62no—
146C22⋊C8—22no—
147C22⋊C8—62no—
148#97—22no—
149#97—62no—
150#97#6714yes—
151C4×C8—22no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157#95—61no—
158#95—61no—
159C23×C3C2×C22⋊C411yescharacteristic
160C3×Q8—44no—
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#10112yes—
175C2×C3×C4C4×D412yes—
176C2×C3×C4#12011yescharacteristic
177C2×C3×C4#12112yes—
178C2×C3×C4#12311yescharacteristic
179C2×Dic3—22no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—42no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C3×C8—44no—
191C3⋊C8—82no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—41no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4(C2×D3)⋊C411yescharacteristic
200C22×C4(C2×D3)⋊C411yescharacteristic
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4(C2×D3)⋊C411yescharacteristic
203C22×C4C2×C4×D311yescharacteristic
204C22×C4C2×D1211yescharacteristic
205C22×C4C2×#6711yescharacteristic
206C2×C8—24no—
207C2×C8—122no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—122no—
214C4⋊C4—122no—
215C4⋊C4—122no—
216C4⋊C4—121no—
217C4⋊C4—122no—
218C4⋊C4—121no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—121no—
224C42—121no—
225C42#23512yes—
226C42C2×#6712yes—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3#38611yescharacteristic
230C22×C3C2×#10111yescharacteristic
231C22×C3#41311yescharacteristic
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—41no—
240Dic3—42no—
241Dic3—81no—
242Dic3—81no—
243Dic3—81no—
244C23C2×(C2×D3)⋊C411yescharacteristic
245Q8—44no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—62no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—121no—
263C2×C4—121no—
264C2×C4—121no—
265C2×C4—121no—
266C2×C4—122no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4#82412yes—
271C2×C4C4×#6712yes—
272C2×C4#92911yescharacteristic
273C2×C4#93012yes—
274C2×C4#95311yescharacteristic
275C8—44no—
276C8—242no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3#188411yescharacteristic
280C2×C3#190111yescharacteristic
281C2×C3#191211yescharacteristic
282C22—21no—
283C22—21no—
284C22#513211yescharacteristic
285C22C2×#82411yescharacteristic
286C22#521311yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—21no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—122no—
296C4—241no—
297C4—241no—
298C4—241no—
299C3#945011yescharacteristic, complemented, Sylow
300C2—21no—
301C2—21no—
302C2#7166311yescharacteristic
303C2#7200311yescharacteristic
304C2#7202111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2983C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2983D3
hypocenterC3#9450
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
299. C3#9450semidirect

Maximal subgroups

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

subgroupindex
C3×#29832
#824772
#716252
#708142
#677492
#94503