Finite Group Databasebeta

← #10692877

Subgroups — #10692877

Order 768.

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

normal subgroups only

This group has 498 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×#2563C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#71776C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic
5#67621C212yesmaximal, complemented
6#9453—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#393—21no—
10C3×#393—21no—
11C3×#390C412yes—
12C3×#382C412yes—
13C3×#380—21no—
14C3×#380—21no—
15C3×C4×C4⋊C4C2211yescharacteristic
16#5207—21no—
17C2×#830—21no—
18#5011—24no—
19#4926C2212yes—
20#4926C2212yes—
21#4547—24no—
22#4525—21no—
23#4525—21no—
24#2973—31no—
25#2563D311yescharacteristic
26#1556—32no—
27#1466—31no—
28#1285—32no—
29C22×C3×Q8—21no—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C3×C4⋊Q8—22no—
34C3×C4⋊Q8—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C22⋊C4D411yescharacteristic
45C2×C3×C42C2×C412yes—
46C2×C3×C42C2311yescharacteristic
47C2×#208—22no—
48C22×C3⋊C8—22no—
49C3×#95—22no—
50C3×#95—22no—
51C3×#95—21no—
52#830—22no—
53#830—22no—
54#830—22no—
55#830—22no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)D414yes—
58#803—22no—
59#803—22no—
60C2×C4⋊Q8C2×D311yescharacteristic
61C2×C4×D4C2×D311yescharacteristic
62#435—64no—
63#432—32no—
64#432—32no—
65#407—61no—
66C2×C22⋊C8—61no—
67#393—21no—
68#393—21no—
69#390Dic312yes—
70#382Dic312yes—
71#380—21no—
72#380—21no—
73C4×C4⋊C4C2×D311yescharacteristic
74#332—64no—
75#323—61no—
76#323—61no—
77C24×C3C22⋊C411yescharacteristic
78C24×C3C22⋊C411yescharacteristic
79C2×C3×Q8—42no—
80C2×C3×Q8—82no—
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—41no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C3×C4⋊C4—22no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C22⋊C4—22no—
103C3×C22⋊C4—22no—
104C3×C22⋊C4—82no—
105C3×C22⋊C4—82no—
106C3×C42—42no—
107C3×C42C2×D412yes—
108C3×C42#4112yes—
109#208—82no—
110C2×C3⋊C8—24no—
111C2×C3⋊C8—42no—
112C2×C3⋊C8—82no—
113C22×Q8—21no—
114C22×D4C2×Dic311yescharacteristic
115C23×C4#6711yescharacteristic
116C23×C4#6711yescharacteristic
117C2×M16—62no—
118C22×C8—62no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4C2×Dic312yes—
126C2×C4⋊C4C2×Dic311yescharacteristic
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4#6711yescharacteristic
130C2×C22⋊C4#6711yescharacteristic
131C2×C42C2×Dic312yes—
132C2×C42C22×D311yescharacteristic
133C4⋊C8—34no—
134C4⋊C8—62no—
135C22⋊C8—62no—
136C22⋊C8—62no—
137C22⋊C8—62no—
138C22⋊C8—62no—
139#97—62no—
140#97—62no—
141#95—22no—
142#95—22no—
143#95—21no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—41no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3#10112yes—
154C23×C3C2×C22⋊C411yescharacteristic
155C3×Q8—84no—
156C3×D4—44no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—81no—
174C2×C3×C4—81no—
175C2×C3×C4C4×D412yes—
176C2×C3×C4#12011yescharacteristic
177C2×C3×C4#12112yes—
178C2×C3×C4#12311yescharacteristic
179C3⋊C8—44no—
180C3⋊C8—82no—
181C24#21711yescharacteristic
182C24#21711yescharacteristic
183C2×Q8—42no—
184C2×Q8—82no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—41no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4#21711yescharacteristic
197C22×C4#21711yescharacteristic
198C22×C4C22×Dic311yescharacteristic
199C22×C4C2×#6711yescharacteristic
200C22×C4C2×#6711yescharacteristic
201M16—242no—
202C2×C8—64no—
203C2×C8—122no—
204C2×C8—242no—
205C4⋊C4—22no—
206C4⋊C4—24no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—82no—
213C22⋊C4—82no—
214C42—42no—
215C42D3×D412yes—
216C42#23712yes—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—22no—
220C22×C3—42no—
221C22×C3—42no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3#38611yescharacteristic
230C22×C3C2×#10111yescharacteristic
231C22×C3#41311yescharacteristic
232C3×C4—21no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4—82no—
238C3×C4—82no—
239C3×C4#45512yes—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#83612yes—
250C23C2×#21711yescharacteristic
251Q8—84no—
252D4—44no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4D4×Dic312yes—
272C2×C4#93511yescharacteristic
273C2×C4#93711yescharacteristic
274C2×C4#93912yes—
275C8—124no—
276C8—242no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—42no—
280C2×C3—81no—
281C2×C3#188511yescharacteristic
282C2×C3#190111yescharacteristic
283C2×C3#191011yescharacteristic
284C22—21no—
285C22—21no—
286C22—22no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22#522011yescharacteristic
297C22C2×#83611yescharacteristic
298C22#523711yescharacteristic
299C4—22no—
300C4—21no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—82no—
306C4#506112yes—
307C3#945311yescharacteristic, complemented, Sylow
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#7169811yescharacteristic
313C2#7172311yescharacteristic
314C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2563C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2563D3
hypocenterC3#9453
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2563

Complements

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

NH ≅ G/Ntype
3. #71776C2semidirect
5. #67621C2semidirect
307. C3#9453semidirect

Maximal subgroups

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

subgroupindex
C3×#25632
#823022
#717762
#716032
#676212
#94533