Finite Group Databasebeta

← #11784619

Subgroups — #11784619

Order 1152.

1502 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 1502 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3828C211yesmaximal, characteristic
2C2×#63786C212yesmaximal, complemented
3#10593811C212yesmaximal, complemented
4C2×#63774C212yesmaximal, complemented
5C23×C32×C4C411yescharacteristic, complemented
6C22×#638C411yescharacteristic, complemented
7C22×#638C2211yescharacteristic
8C2×#3828—24no—
9C2×#3828—24no—
10C2×#3828—24no—
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×C4×C3⋊D3C411yescharacteristic, complemented
13C22×C4×C3⋊D3C2211yescharacteristic
14C22×#634C411yescharacteristic, complemented
15C2×C22⋊#157C2214yes—
16#63786—24no—
17#63781—24no—
18#63774—24no—
19C22×#235—62no—
20C24×C32C2×C411yescharacteristic
21C23×C3⋊D3C2×C411yescharacteristic
22C23×C3⋊D3C2311yescharacteristic
23C22×#157—24no—
24C22×C32×C4—22no—
25C22×C32×C4—22no—
26C22×C32×C4—22no—
27C22×C32×C4C2×C411yescharacteristic
28C22×C32×C4C2×C411yescharacteristic
29C2×#638—22no—
30C2×#638—22no—
31C2×#638—22no—
32C2×#638—22no—
33C2×#638—22no—
34C2×#638D414yes—
35C22×C3⋊Dic3C2×C411yescharacteristic
36C22×C3⋊Dic3C2×C411yescharacteristic
37#3828—232no—
38C2×C4⋊(C3⋊D3)—22no—
39C2×C4⋊(C3⋊D3)—22no—
40C2×C4⋊(C3⋊D3)D414yes—
41C2×C4×C3⋊D3—22no—
42C2×C4×C3⋊D3—22no—
43C2×C4×C3⋊D3—22no—
44C2×C4×C3⋊D3—22no—
45C2×C4×C3⋊D3—22no—
46C2×C4×C3⋊D3D414yes—
47C2×#634—22no—
48C2×#634—22no—
49C2×#634—22no—
50C22⋊#157—24no—
51C22⋊#157—24no—
52#1782—91nomaximal, Sylow
53C23×C3×C4—22no—
54C22×#67—62no—
55C22×#67—62no—
56C2×#235—616no—
57C2×#235—616no—
58C2×#235—616no—
59C22×D12—62no—
60C22×C4×D3—62no—
61C22×C4×D3—62no—
62C22×Dic6—62no—
63C23×C32—21no—
64C23×C32—21no—
65C23×C32—42no—
66C23×C32C22⋊C412yes—
67C23×C32C22×C411yescharacteristic
68C22×C3⋊D3—21no—
69C22×C3⋊D3—21no—
70C22×C3⋊D3—42no—
71C22×C3⋊D3—42no—
72C22×C3⋊D3C22⋊C412yes—
73C22×C3⋊D3C2×D412yes—
74C22×C3⋊D3C2×D412yes—
75C22×C3⋊D3C2×D412yes—
76C2×#157—48no—
77C2×#157—44no—
78C2×C32×C4—28no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4C22⋊C412yes—
84C2×C32×C4C22⋊C412yes—
85#638—28no—
86#638—28no—
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3C22⋊C412yes—
92C2×C3⋊Dic3C22⋊C412yes—
93C4⋊(C3⋊D3)—28no—
94C4×C3⋊D3—28no—
95C4×C3⋊D3—28no—
96#634—28no—
97C22×#41—91no—
98C2×#99—92no—
99C2×#99—92no—
100#381—92no—
101C24×C3—22no—
102C23×D3—62no—
103C23×D3—62no—
104C22×C3×C4—28no—
105C22×C3×C4—28no—
106C22×C3×C4—28no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C2×#67—68no—
110C2×#67—68no—
111C2×#67—68no—
112C2×#67—68no—
113C2×#67—68no—
114C2×#67—68no—
115C22×Dic3—62no—
116C22×Dic3—62no—
117#235—6128no—
118C2×D12—68no—
119C2×D12—68no—
120C2×D12—68no—
121C2×C4×D3—68no—
122C2×C4×D3—68no—
123C2×C4×D3—68no—
124C2×C4×D3—68no—
125C2×C4×D3—68no—
126C2×C4×D3—68no—
127C2×Dic6—68no—
128C2×Dic6—68no—
129C2×Dic6—68no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—42no—
133C22×C32—42no—
134C22×C32—42no—
135C22×C32C2×C22⋊C411yescharacteristic
136C22×C32C2×C22⋊C411yescharacteristic
137C22×C32C2×C22⋊C411yescharacteristic
138C2×C3⋊D3—22no—
139C2×C3⋊D3—24no—
140C2×C3⋊D3—42no—
141C2×C3⋊D3—42no—
142C2×C3⋊D3—42no—
143C2×C3⋊D3—42no—
144C2×C3⋊D3—42no—
145C2×C3⋊D3#12014yes—
146#157—84no—
147C32×C4—22no—
148C32×C4—22no—
149C3⋊Dic3—22no—
150C3⋊Dic3—22no—
151C2×#41—184no—
152C2×#41—184no—
153C2×#41—184no—
154C22×Q8—91no—
155C22×D4—91no—
156C22×D4—91no—
157C22×D4—91no—
158C23×C4—91no—
159C23×C4—91no—
160C23×C4#15711yescharacteristic, complemented
161C2×C22⋊C4—94no—
162#99—184no—
163#99—184no—
164#95—184no—
165C23×C3—24no—
166C23×C3—24no—
167C23×C3—22no—
168C23×C3—24no—
169C23×C3—48no—
170C22×D3—64no—
171C22×D3—64no—
172C22×D3—64no—
173C22×D3—64no—
174C22×D3—64no—
175C22×D3—64no—
176C22×D3—128no—
177C22×D3—128no—
178C2×C3×C4—24no—
179C2×C3×C4—24no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—232no—
183C2×C3×C4—24no—
184C2×C3×C4—24no—
185#67—632no—
186#67—632no—
187C2×Dic3—64no—
188C2×Dic3—64no—
189C2×Dic3—64no—
190C2×Dic3—64no—
191C2×Dic3—64no—
192C2×Dic3—64no—
193D12—632no—
194C4×D3—632no—
195C4×D3—632no—
196Dic6—632no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—42no—
200C2×C32#37911yescharacteristic
201C2×C32#41012yes—
202C3⋊D3—24no—
203C3⋊D3—42no—
204C24—91no—
205C24—91no—
206C24C2×#15711yescharacteristic
207#41—1832no—
208C2×Q8—182no—
209C2×Q8—182no—
210C2×Q8—182no—
211C2×D4—94no—
212C2×D4—94no—
213C2×D4—182no—
214C2×D4—182no—
215C2×D4—182no—
216C2×D4—182no—
217C2×D4—182no—
218C2×D4—182no—
219C2×D4—182no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—91no—
224C22×C4—91no—
225C22×C4—94no—
226C22×C4—182no—
227C22×C4—182no—
228C22×C4—182no—
229C22×C4—182no—
230C22×C4—182no—
231C22×C4—184no—
232C22×C4C2×#15711yescharacteristic
233C22×C4C2×#15711yescharacteristic
234C22⋊C4—184no—
235C22⋊C4—184no—
236C22×C3—28no—
237C22×C3—28no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—48no—
242C22×C3—48no—
243C22×C3—48no—
244C2×D3—68no—
245C2×D3—68no—
246C2×D3—128no—
247C2×D3—128no—
248C2×D3—128no—
249C2×D3—128no—
250C2×D3—128no—
251C2×D3—128no—
252C3×C4—28no—
253C3×C4—28no—
254Dic3—68no—
255Dic3—68no—
256C32#178211yescharacteristic, complemented, Sylow
257C23—21no—
258C23—21no—
259C23—42no—
260C23—92no—
261C23—92no—
262C23—92no—
263C23—92no—
264C23—181no—
265C23—181no—
266C23—362no—
267C23—362no—
268C23C22⋊#15712yes—
269C23C22×#15711yescharacteristic
270Q8—188no—
271D4—188no—
272D4—188no—
273D4—188no—
274C2×C4—28no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—92no—
280C2×C4—92no—
281C2×C4—188no—
282C2×C4—188no—
283C2×C4—181no—
284C2×C4—181no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4—364no—
288C2×C4—368no—
289C2×C4C22⋊#15712yes—
290C2×C4C22⋊#15712yes—
291C2×C3—24no—
292C2×C3—24no—
293C2×C3—24no—
294C2×C3—22no—
295C2×C3—48no—
296D3—128no—
297D3—128no—
298C22—22no—
299C22—22no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—94no—
304C22—184no—
305C22—182no—
306C22—362no—
307C22—362no—
308C22—362no—
309C22—362no—
310C22—362no—
311C22C2×C22⋊#15711yescharacteristic
312C22C2×C22⋊#15711yescharacteristic
313C22C2×C22⋊#15711yescharacteristic
314C4—22no—
315C4—22no—
316C4—182no—
317C4—182no—
318C4—724no—
319C3—22no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—184no—
324C2—362no—
325C2#1059379212yes—
326C2#1059384711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#157
commutatorC23×C32C22×C4
FrattiniC23C22×#157
FittingC23×C32×C4C4
socleC22×C32C2×C22⋊C4
prime radicalC22×#3828C2
cosocleC23×C3⋊D3C23
hypercenterC23×C4#157
hypocenterC32#1782
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C2×#63786C2semidirect
3. #10593811C2semidirect
4. C2×#63774C2semidirect
5. C23×C32×C4C4semidirect
6. C22×#638C4semidirect
12. C22×C4×C3⋊D3C4semidirect
14. C22×#634C4semidirect
160. C23×C4#157semidirect
256. C32#1782semidirect

Maximal subgroups

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

subgroupindex
C22×#38282
C2×#637862
#105938112
C2×#637742
#17829