Finite Group Databasebeta

← #11784733

Subgroups — #11784733

Order 1152.

1442 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 1442 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C22×Q8×C3⋊D3C211yesmaximal, characteristic
2C2×#63784C214yesmaximal, complemented
3#10593812C211yesmaximal, characteristic
4#10593800C211yesmaximal, characteristic
5C22×C32×Q8C411yescharacteristic, complemented
6C2×Q8×C3⋊D3—28no—
7C2×Q8×C3⋊D3—21no—
8C2×Q8×C3⋊D3—21no—
9C2×Q8×C3⋊D3C412yescomplemented
10C2×Q8×C3⋊D3C2212yes—
11C22×C4×C3⋊D3—21no—
12C22×C4×C3⋊D3C2211yescharacteristic
13C22×#634—21no—
14C22×#634C411yescharacteristic, complemented
15C2×C22⋊#157—21no—
16C2×C4⋊#157C2212yes—
17C2×#3787C2212yes—
18C22⋊#622—21no—
19#63784—28no—
20#63781—21no—
21#63778—21no—
22C22×D3×Q8—62no—
23C23×C3⋊D3D411yescharacteristic
24C22×#157—22no—
25C2×C32×Q8—24no—
26C2×C32×Q8—21no—
27C2×C32×Q8C2×C412yes—
28C22×C32×C4—21no—
29C22×C32×C4C2×C411yescharacteristic
30C22×C3⋊Dic3—21no—
31C22×C3⋊Dic3D411yescharacteristic
32Q8×C3⋊D3—28no—
33Q8×C3⋊D3—28no—
34Q8×C3⋊D3—24no—
35Q8×C3⋊D3—24no—
36Q8×C3⋊D3—24no—
37Q8×C3⋊D3D418yes—
38C2×C4×C3⋊D3—28no—
39C2×C4×C3⋊D3—22no—
40C2×C4×C3⋊D3—22no—
41C2×C4×C3⋊D3—24no—
42C2×C4×C3⋊D3C2×C411yescharacteristic
43C2×C4×C3⋊D3D411yescharacteristic
44C2×C4×C3⋊D3D411yescharacteristic
45C2×C4×C3⋊D3C2311yescharacteristic
46C2×#634—28no—
47C2×#634—21no—
48C2×#634—22no—
49C2×#634—22no—
50C2×#634—24no—
51C2×#634C2×C412yes—
52C22⋊#157—42no—
53C2×#622—22no—
54C4⋊#157—24no—
55#3787—24no—
56#1787—91nomaximal, Sylow
57C22×C3×Q8—22no—
58C2×D3×Q8—64no—
59C2×D3×Q8—64no—
60C2×D3×Q8—64no—
61C2×D3×Q8—64no—
62C2×D3×Q8—632no—
63C22×C4×D3—62no—
64C22×C4×D3—64no—
65C22×Dic6—62no—
66C22×Dic6—64no—
67C23×C32C22⋊C411yescharacteristic
68C22×C3⋊D3—22no—
69C22×C3⋊D3—22no—
70C22×C3⋊D3C22⋊C411yescharacteristic
71C22×C3⋊D3SD1612yes—
72C22×C3⋊D3Q1612yes—
73C22×C3⋊D3C2×D411yescharacteristic
74C2×#157—42no—
75C2×#157—42no—
76C2×#157—42no—
77C32×Q8—24no—
78C32×Q8—22no—
79C32×Q8C22⋊C414yescomplemented
80C2×C32×C4—24no—
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4C22⋊C411yescharacteristic
84C2×C32×C4C22×C411yescharacteristic
85C2×C3⋊Dic3—24no—
86C2×C3⋊Dic3—22no—
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3C22⋊C411yescharacteristic
89C2×C3⋊Dic3C2×D411yescharacteristic
90C4×C3⋊D3—28no—
91C4×C3⋊D3—28no—
92C4×C3⋊D3—22no—
93C4×C3⋊D3—22no—
94C4×C3⋊D3—22no—
95C4×C3⋊D3C2×D414yes—
96#634—28no—
97#634—24no—
98#634—22no—
99#634—28no—
100#634C22⋊C414yescomplemented
101#622—42no—
102C23×Q8—91no—
103C2×#103—94no—
104C2×C22⋊C8—91no—
105#380—91no—
106C23×D3—62no—
107C2×C3×Q8—216no—
108C2×C3×Q8—24no—
109C2×C3×Q8—24no—
110C22×C3×C4—24no—
111C22×C3×C4—22no—
112C22×Dic3—62no—
113C22×Dic3—64no—
114D3×Q8—616no—
115D3×Q8—616no—
116D3×Q8—616no—
117D3×Q8—632no—
118D3×Q8—632no—
119D3×Q8—616no—
120C2×C4×D3—62no—
121C2×C4×D3—62no—
122C2×C4×D3—62no—
123C2×C4×D3—62no—
124C2×C4×D3—616no—
125C2×C4×D3—68no—
126C2×C4×D3—68no—
127C2×C4×D3—632no—
128C2×Dic6—64no—
129C2×Dic6—68no—
130C2×Dic6—68no—
131C2×Dic6—616no—
132C2×Dic6—632no—
133C2×Dic6—64no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32#10312yescomplemented
137C22×C32C2×C22⋊C411yescharacteristic
138C2×C3⋊D3—21no—
139C2×C3⋊D3—21no—
140C2×C3⋊D3—24no—
141C2×C3⋊D3—24no—
142C2×C3⋊D3#10312yescomplemented
143C2×C3⋊D3#12012yes—
144C2×C3⋊D3C2×SD1611yescharacteristic
145C2×C3⋊D3C2×Q1611yescharacteristic
146C2×C3⋊D3#13712yes—
147#157—82no—
148C32×C4—24no—
149C32×C4—21no—
150C32×C4C2×C22⋊C412yes—
151C3⋊Dic3—21no—
152C3⋊Dic3—24no—
153C3⋊Dic3#12012yes—
154C22×Q8—92no—
155C22×Q8—92no—
156C22×Q8—91no—
157C22×Q8—181no—
158C22×Q8—181no—
159C22×Q8—181no—
160C22×Q8—188no—
161C22×Q8#15711yescharacteristic, complemented
162C23×C4—91no—
163C23×C4—181no—
164C22×C8—92no—
165C2×C4⋊C4—92no—
166C2×C22⋊C4—181no—
167#103—188no—
168C22⋊C8—181no—
169C22⋊C8—181no—
170#95—181no—
171C23×C3—22no—
172C22×D3—62no—
173C22×D3—64no—
174C22×D3—64no—
175C22×D3—62no—
176C22×D3—68no—
177C22×D3—68no—
178C3×Q8—216no—
179C3×Q8—28no—
180C3×Q8—28no—
181C2×C3×C4—216no—
182C2×C3×C4—28no—
183C2×C3×C4—28no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×Dic3—62no—
187C2×Dic3—62no—
188C2×Dic3—68no—
189C2×Dic3—616no—
190C2×Dic3—68no—
191C4×D3—68no—
192C4×D3—68no—
193C4×D3—68no—
194C4×D3—632no—
195C4×D3—632no—
196C4×D3—68no—
197Dic6—68no—
198Dic6—632no—
199Dic6—632no—
200Dic6—616no—
201Dic6—68no—
202C2×C32—22no—
203C2×C32#37911yescharacteristic
204C2×C32C2×#10311yescharacteristic
205C2×C32#41911yescharacteristic
206C3⋊D3—22no—
207C3⋊D3#44812yes—
208C3⋊D3#45112yes—
209C24—91no—
210C2×Q8—24no—
211C2×Q8—21no—
212C2×Q8—98no—
213C2×Q8—92no—
214C2×Q8—182no—
215C2×Q8—182no—
216C2×Q8—184no—
217C2×Q8—188no—
218C2×Q8—184no—
219C2×Q8—184no—
220C2×Q8—188no—
221C2×Q8—188no—
222C2×Q8—184no—
223C2×Q8—181no—
224C2×Q8C2×#15712yes—
225C22×C4—21no—
226C22×C4—91no—
227C22×C4—91no—
228C22×C4—91no—
229C22×C4—91no—
230C22×C4—91no—
231C22×C4—184no—
232C22×C4—182no—
233C22×C4—182no—
234C22×C4—188no—
235C22×C4—181no—
236C22×C4—182no—
237C22×C4C2×#15711yescharacteristic
238C2×C8—182no—
239C2×C8—184no—
240C4⋊C4—184no—
241C22⋊C4—362no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3—22no—
245C22×C3—24no—
246C2×D3—64no—
247C2×D3—64no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—64no—
251C2×D3—64no—
252C2×D3—616no—
253C2×D3—616no—
254C2×D3—64no—
255C3×C4—216no—
256C3×C4—24no—
257C3×C4—24no—
258Dic3—64no—
259Dic3—64no—
260Dic3—616no—
261C32#178711yescharacteristic, complemented, Sylow
262C23—91no—
263C23—91no—
264C23—92no—
265C23—92no—
266C23—182no—
267C23—182no—
268C23C22⋊#15711yescharacteristic
269Q8—24no—
270Q8—22no—
271Q8—94no—
272Q8—188no—
273Q8—188no—
274Q8—184no—
275Q8—182no—
276Q8C22⋊#15714yescomplemented
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—91no—
281C2×C4—94no—
282C2×C4—91no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4—188no—
286C2×C4—188no—
287C2×C4—182no—
288C2×C4—184no—
289C2×C4—182no—
290C2×C4—182no—
291C2×C4—362no—
292C2×C4—362no—
293C2×C4—362no—
294C2×C4C22⋊#15711yescharacteristic
295C2×C4C22×#15711yescharacteristic
296C8—362no—
297C2×C3—28no—
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—22no—
301D3—64no—
302D3—64no—
303D3—68no—
304C22—21no—
305C22—21no—
306C22—91no—
307C22—91no—
308C22—92no—
309C22—92no—
310C22—92no—
311C22—184no—
312C22—184no—
313C22—181no—
314C22—181no—
315C22#6378412yescomplemented
316C22C2×C22⋊#15711yescharacteristic
317C4—24no—
318C4—21no—
319C4—92no—
320C4—181no—
321C4—184no—
322C4—722no—
323C4C2×C22⋊#15712yes—
324C3—22no—
325C2—22no—
326C2—92no—
327C2—92no—
328C2—182no—
329C2C2×#6378411yescharacteristic
330C2#1059382911yescharacteristic
331C2#1059384711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#157
commutatorC2×C32×C4C22×C4
FrattiniC2×C4C22×#157
FittingC22×C32×Q8C4
socleC22×C32C2×C22⋊C4
prime radicalC23×C3⋊D3D4
cosocleC2×C4×C3⋊D3C23
hypercenterC22×Q8#157
hypocenterC32#1787
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#63784C2semidirect
5. C22×C32×Q8C4semidirect
9. C2×Q8×C3⋊D3C4semidirect
14. C22×#634C4semidirect
79. C32×Q8C22⋊C4semidirect
100. #634C22⋊C4semidirect
136. C22×C32#103semidirect
142. C2×C3⋊D3#103semidirect
161. C22×Q8#157semidirect
261. C32#1787semidirect
276. Q8C22⋊#157semidirect
315. C22#63784semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×C3⋊D32
C2×#637842
#105938122
#105938002
#17879