Finite Group Databasebeta

← #10713862

Subgroups — #10713862

Order 768.

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

normal subgroups only

This group has 500 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#71619C211yesmaximal, characteristic
3C4×#834C211yesmaximal, characteristic
4#67641C214yesmaximal
5#13299—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic, complemented
7C2×C3×C4×D4C2211yescharacteristic
8C3×#393—21no—
9C3×#393—21no—
10C3×#390C2212yes—
11C3×#382C412yescomplemented
12C3×#380—21no—
13C3×#380—21no—
14C3×C4×C4⋊C4C411yescharacteristic, complemented
15C2×#834C2212yes—
16C2×#834C2212yes—
17#5212—21no—
18C4×#217—21no—
19#4526—21no—
20#4526—21no—
21#2563D311yescharacteristic
22#1799—31no—
23C4×#99—31no—
24#1393—34no—
25C22×C3×Q8—21no—
26C22×C3×D4C2311yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C3×C4⋊Q8—41no—
30C3×C4⋊Q8—41no—
31C3×C4×D4—41no—
32C3×C4×D4—41no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42C2×C412yes—
43C2×#217—22no—
44C2×#217—22no—
45C2×C4⋊Dic3—41no—
46C2×C4×Dic3—41no—
47C3×#95—21no—
48C3×#95—22no—
49C3×#95—22no—
50#834—22no—
51#834—24no—
52#834Q814yescomplemented
53C3⋊#95—41no—
54C3⋊#95—41no—
55C2×C4⋊Q8Dic311yescharacteristic, complemented
56C2×C4×D4C2×D311yescharacteristic
57C2×#99—32no—
58C2×#99—32no—
59#393—21no—
60#393—21no—
61#390C2×D312yes—
62#382Dic312yescomplemented
63#380—21no—
64#380—21no—
65#380—61no—
66C4×C4⋊C4Dic311yescharacteristic, complemented
67C4×C22⋊C4—61no—
68#342—61no—
69#342—61no—
70C24×C3C2×D411yescharacteristic
71C24×C3C2×D411yescharacteristic
72C2×C3×Q8—42no—
73C2×C3×Q8—82no—
74C2×C3×D4—41no—
75C2×C3×D4—41no—
76C2×C3×D4C2×Q812yes—
77C2×C3×D4#4112yes—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22×C411yescharacteristic
91C22×Dic3—42no—
92C22×Dic3—42no—
93C3×C4⋊C4—41no—
94C3×C4⋊C4—42no—
95C3×C4⋊C4—42no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—82no—
101C3×C22⋊C4—82no—
102C3×C42—21no—
103C3×C42—21no—
104C3×C42—42no—
105#217—24no—
106#217—42no—
107#217—42no—
108#217—42no—
109C4⋊Dic3—82no—
110C4×Dic3—82no—
111C22×Q8—21no—
112C22×D4C22×D311yescharacteristic
113C23×C4#6711yescharacteristic
114C23×C4#6711yescharacteristic
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4×D4—41no—
118C4×D4—41no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—121no—
122C2×C4⋊C4C2×Dic312yes—
123C2×C4⋊C4C2×Dic311yescharacteristic
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4#6711yescharacteristic
129C2×C22⋊C4#6711yescharacteristic
130C2×C42—121no—
131C2×C42C2×Dic311yescharacteristic
132C2×C42C2×Dic312yes—
133#99—34no—
134#99—62no—
135#99—64no—
136#95—21no—
137#95—22no—
138#95—22no—
139#95—121no—
140#95—121no—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—41no—
146C23×C3—41no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3C2×C22⋊C411yescharacteristic
150C23×C3C22⋊Q812yes—
151C23×C3#12411yescharacteristic
152C23×C3C4⋊Q811yescharacteristic
153C3×Q8—84no—
154C3×D4—42no—
155C3×D4—42no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
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×C4#9911yescharacteristic
176C2×C3×C4#9911yescharacteristic
177C2×C3×C4C4×Q812yes—
178C2×Dic3—44no—
179C2×Dic3—82no—
180C2×Dic3—82no—
181C2×Dic3—82no—
182C2×Dic3—82no—
183C2×Dic3—82no—
184C24C2×#6711yescharacteristic
185C24C2×#6711yescharacteristic
186C2×Q8—42no—
187C2×Q8—82no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4D3×Q812yes—
191C2×D4#23912yes—
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—122no—
201C22×C4—122no—
202C22×C4#21711yescharacteristic
203C22×C4#21711yescharacteristic
204C22×C4#21711yescharacteristic
205C22×C4#21711yescharacteristic
206C22×C4C22×Dic311yescharacteristic
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—242no—
213C22⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—64no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C22⋊C4—122no—
219C22⋊C4—122no—
220C22⋊C4—122no—
221C42—21no—
222C42—21no—
223C42—42no—
224C42—242no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—42no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3—81no—
238C22×C3—81no—
239C22×C3#39111yescharacteristic
240C22×C3C2×#9911yescharacteristic
241C22×C3#41011yescharacteristic
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—41no—
245C3×C4—41no—
246C3×C4—41no—
247C3×C4—82no—
248C3×C4—82no—
249C3×C4#35412yes—
250Dic3—84no—
251Dic3—162no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#94411yescharacteristic
261C23#94612yes—
262C23#94711yescharacteristic
263C23C2×#21711yescharacteristic
264Q8—84no—
265D4—42no—
266D4—42no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—124no—
287C2×C4—242no—
288C2×C4—242no—
289C2×C4—242no—
290C2×C4—242no—
291C2×C4—242no—
292C2×C4#83411yescharacteristic
293C2×C4#83411yescharacteristic
294C2×C4Q8×Dic312yes—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—42no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3#199611yescharacteristic
301C2×C3C2×#35411yescharacteristic
302C2×C3#212811yescharacteristic
303C22—21no—
304C22—21no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—41no—
309C22—42no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#83411yescharacteristic
318C22#522911yescharacteristic
319C22#524011yescharacteristic
320C4—22no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C4—82no—
327C4—244no—
328C4—482no—
329C4#454112yes—
330C3#1329911yescharacteristic, complemented, Sylow
331C2—21no—
332C2—21no—
333C2—42no—
334C2—41no—
335C2—41no—
336C2#7170111yescharacteristic
337C2C2×#454111yescharacteristic
338C2#7186111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#2563C2
socleC22×C3C2×#99
prime radical, cosocleC22×C3×D4C23
hypercenter#2563D3
hypocenterC3#13299
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2563

Complements

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

NH ≅ G/Ntype
6. C2×C3×C4⋊Q8C4semidirect
11. C3×#382C4semidirect
14. C3×C4×C4⋊C4C4semidirect
52. #834Q8semidirect
55. C2×C4⋊Q8Dic3semidirect
62. #382Dic3semidirect
66. C4×C4⋊C4Dic3semidirect
330. C3#13299semidirect

Maximal subgroups

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

subgroupindex
C3×#25632
#716192
C4×#8342
#676412
#132993