Finite Group Databasebeta

← #10692867

Subgroups — #10692867

Order 768.

518 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2547C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#71777C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic
5#67621C212yesmaximal, complemented
6#9451—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×#124C412yes—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#397—21no—
11C3×#397—21no—
12C3×#391C412yes—
13C3×#380—21no—
14C3×#380—21no—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5207—21no—
17C2×#830—21no—
18#5011—24no—
19#4926C2212yes—
20#4926C2212yes—
21#4546—24no—
22#4525—21no—
23#4525—21no—
24#2973—31no—
25#2547D311yescharacteristic
26#1558—32no—
27#1466—31no—
28#1285—32no—
29C22×C3×Q8C2×C412yes—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8—22no—
35C3×#124—44no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42C2×C412yes—
47C2×C3×C42C2311yescharacteristic
48C2×#208—22no—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95—22no—
52C3×#95—21no—
53#830—22no—
54#830—22no—
55#830—22no—
56#830—22no—
57C4⋊(C3⋊C8)—22no—
58C4⋊(C3⋊C8)D414yes—
59#803—22no—
60#803—22no—
61C2×C4⋊Q8C2×D311yescharacteristic
62C2×#124Dic312yes—
63C2×C4×D4C2×D311yescharacteristic
64#435—64no—
65#432—32no—
66#432—32no—
67#407—61no—
68C2×C22⋊C8—61no—
69#397—21no—
70#397—21no—
71#391Dic312yes—
72#380—21no—
73#380—21no—
74C4×C4⋊C4C2×D311yescharacteristic
75#333—64no—
76#323—61no—
77#323—61no—
78C24×C3C22⋊C411yescharacteristic
79C24×C3C22⋊C411yescharacteristic
80C2×C3×Q8—24no—
81C2×C3×Q8—42no—
82C2×C3×Q8—42no—
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—41no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4C2×D411yescharacteristic
99C3×C4⋊C4—22no—
100C3×C4⋊C4—42no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—42no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—22no—
106C3×C22⋊C4—44no—
107C3×C22⋊C4—44no—
108C3×C42—42no—
109C3×C42C2×D412yes—
110C3×C42#4112yes—
111#208—82no—
112C2×C3⋊C8—24no—
113C2×C3⋊C8—42no—
114C2×C3⋊C8—82no—
115C22×Q8C2×Dic312yes—
116C22×D4C2×Dic311yescharacteristic
117C23×C4#6711yescharacteristic
118C23×C4#6711yescharacteristic
119C2×M16—62no—
120C22×C8—62no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123#124—44no—
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4C2×Dic311yescharacteristic
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4#6711yescharacteristic
133C2×C22⋊C4#6711yescharacteristic
134C2×C42C2×Dic312yes—
135C2×C42C22×D311yescharacteristic
136C4⋊C8—34no—
137C4⋊C8—62no—
138C22⋊C8—62no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141C22⋊C8—62no—
142#97—62no—
143#97—62no—
144#95—22no—
145#95—22no—
146#95—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3#10012yes—
157C23×C3C2×C22⋊C411yescharacteristic
158C3×Q8—84no—
159C3×D4—44no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—22no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—41no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—42no—
176C2×C3×C4—81no—
177C2×C3×C4—81no—
178C2×C3×C4C4×D412yes—
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12112yes—
181C2×C3×C4#12311yescharacteristic
182C3⋊C8—44no—
183C3⋊C8—82no—
184C24#21711yescharacteristic
185C24#21711yescharacteristic
186C2×Q8—24no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—41no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4#21711yescharacteristic
201C22×C4#21711yescharacteristic
202C22×C4C22×Dic311yescharacteristic
203C22×C4C2×#6711yescharacteristic
204C22×C4C2×#6711yescharacteristic
205M16—242no—
206C2×C8—64no—
207C2×C8—122no—
208C2×C8—242no—
209C4⋊C4—22no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C42—42no—
219C42D3×D412yes—
220C42#23712yes—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—22no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—81no—
232C22×C3—81no—
233C22×C3#38611yescharacteristic
234C22×C3C2×#10011yescharacteristic
235C22×C3#41311yescharacteristic
236C3×C4—21no—
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—82no—
242C3×C4—82no—
243C3×C4#45512yes—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#83312yes—
254C23C2×#21711yescharacteristic
255Q8—84no—
256D4—44no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4D4×Dic312yes—
276C2×C4#93511yescharacteristic
277C2×C4#93711yescharacteristic
278C2×C4#93912yes—
279C8—124no—
280C8—242no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—42no—
284C2×C3—81no—
285C2×C3#188511yescharacteristic
286C2×C3#190011yescharacteristic
287C2×C3#190911yescharacteristic
288C22—21no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22—81no—
300C22C2×#83311yescharacteristic
301C22#522011yescharacteristic
302C22#523711yescharacteristic
303C4—22no—
304C4—21no—
305C4—22no—
306C4—41no—
307C4—41no—
308C4—82no—
309C4—82no—
310C4#506112yes—
311C3#945111yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#7168311yescharacteristic
317C2#7172311yescharacteristic
318C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2547C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2547D3
hypocenterC3#9451
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #71777C2semidirect
5. #67621C2semidirect
311. C3#9451semidirect

Maximal subgroups

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

subgroupindex
C3×#25472
#823022
#717772
#716032
#676212
#94513