Finite Group Databasebeta

← #11784795

Subgroups — #11784795

Order 1152.

1508 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1508 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3832C211yesmaximal, characteristic
2C2×#63785C214yesmaximal, complemented
3C4×C22⋊#157C211yesmaximal, characteristic, complemented
4#10593803C211yesmaximal, characteristic, complemented
5C22×C32×Q8C411yescharacteristic, complemented
6C2×#3832—28no—
7C2×#3832—21no—
8C2×#3832—21no—
9C2×#3832C412yescomplemented
10C2×#3832C2212yes—
11C22×C4⋊(C3⋊D3)—21no—
12C22×C4⋊(C3⋊D3)C411yescharacteristic, complemented
13C22×C4×C3⋊D3—21no—
14C22×C4×C3⋊D3C2211yescharacteristic
15C2×C22⋊#157—21no—
16C2×C4×#157C2212yescomplemented
17C2×#3788C2212yescomplemented
18C22⋊#622—21no—
19#63785—28no—
20#63781—21no—
21#63778—21no—
22C22×#239—62no—
23C23×C3⋊D3—21no—
24C23×C3⋊D3D411yescharacteristic
25C22×#157—22no—
26C2×C32×Q8—24no—
27C2×C32×Q8—21no—
28C2×C32×Q8C2×C412yes—
29C22×C32×C4—21no—
30C22×C32×C4C2×C411yescharacteristic
31C22×C3⋊Dic3D411yescharacteristic
32#3832—28no—
33#3832—28no—
34#3832—24no—
35#3832—24no—
36#3832—24no—
37#3832D418yes—
38C2×C4⋊(C3⋊D3)—28no—
39C2×C4⋊(C3⋊D3)—22no—
40C2×C4⋊(C3⋊D3)—22no—
41C2×C4⋊(C3⋊D3)—21no—
42C2×C4⋊(C3⋊D3)—24no—
43C2×C4⋊(C3⋊D3)C2×C412yes—
44C2×C4×C3⋊D3—28no—
45C2×C4×C3⋊D3—22no—
46C2×C4×C3⋊D3—22no—
47C2×C4×C3⋊D3—24no—
48C2×C4×C3⋊D3C2×C411yescharacteristic
49C2×C4×C3⋊D3D411yescharacteristic
50C2×C4×C3⋊D3D411yescharacteristic
51C2×C4×C3⋊D3C2311yescharacteristic
52C22⋊#157—24no—
53C2×#622—22no—
54C4×#157—24no—
55#3788—24no—
56#1789—91nomaximal, Sylow
57C22×C3×Q8—22no—
58C2×#239—64no—
59C2×#239—64no—
60C2×#239—64no—
61C2×#239—64no—
62C2×#239—632no—
63C22×D12—62no—
64C22×D12—64no—
65C22×C4×D3—62no—
66C22×C4×D3—64no—
67C23×C32C22⋊C411yescharacteristic
68C22×C3⋊D3—24no—
69C22×C3⋊D3—22no—
70C22×C3⋊D3—22no—
71C22×C3⋊D3—22no—
72C22×C3⋊D3—41no—
73C22×C3⋊D3—42no—
74C22×C3⋊D3—42no—
75C22×C3⋊D3C22⋊C411yescharacteristic
76C22×C3⋊D3C2×D411yescharacteristic
77C2×#157—24no—
78C2×#157—42no—
79C2×#157—42no—
80C32×Q8—24no—
81C32×Q8—22no—
82C32×Q8C22⋊C414yescomplemented
83C2×C32×C4—24no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4C22⋊C411yescharacteristic
87C2×C32×C4C22×C411yescharacteristic
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3C22⋊C411yescharacteristic
90C2×C3⋊Dic3C2×D411yescharacteristic
91C4⋊(C3⋊D3)—28no—
92C4⋊(C3⋊D3)—24no—
93C4⋊(C3⋊D3)—28no—
94C4⋊(C3⋊D3)—22no—
95C4⋊(C3⋊D3)C22⋊C414yescomplemented
96C4×C3⋊D3—28no—
97C4×C3⋊D3—22no—
98C4×C3⋊D3—22no—
99C4×C3⋊D3—28no—
100C4×C3⋊D3—22no—
101C4×C3⋊D3C2×D414yes—
102#622—42no—
103C22×#41—91no—
104C2×#104—94no—
105#407—91no—
106C4×C22⋊C4—91no—
107C23×D3—62no—
108C23×D3—64no—
109C2×C3×Q8—216no—
110C2×C3×Q8—24no—
111C2×C3×Q8—24no—
112C22×C3×C4—24no—
113C22×C3×C4—22no—
114C22×Dic3—62no—
115#239—616no—
116#239—616no—
117#239—616no—
118#239—632no—
119#239—632no—
120#239—616no—
121C2×D12—64no—
122C2×D12—616no—
123C2×D12—68no—
124C2×D12—68no—
125C2×D12—632no—
126C2×D12—64no—
127C2×C4×D3—62no—
128C2×C4×D3—62no—
129C2×C4×D3—62no—
130C2×C4×D3—62no—
131C2×C4×D3—616no—
132C2×C4×D3—68no—
133C2×C4×D3—68no—
134C2×C4×D3—632no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32#10412yescomplemented
138C22×C32C2×C22⋊C411yescharacteristic
139C2×C3⋊D3—21no—
140C2×C3⋊D3—24no—
141C2×C3⋊D3—21no—
142C2×C3⋊D3—21no—
143C2×C3⋊D3—21no—
144C2×C3⋊D3—21no—
145C2×C3⋊D3—44no—
146C2×C3⋊D3—48no—
147C2×C3⋊D3—42no—
148C2×C3⋊D3—42no—
149C2×C3⋊D3#12012yes—
150#157—44no—
151C32×C4—24no—
152C32×C4—21no—
153C32×C4C2×C22⋊C412yes—
154C3⋊Dic3—21no—
155C3⋊Dic3#12012yes—
156C2×#41—92no—
157C2×#41—92no—
158C2×#41—181no—
159C2×#41—181no—
160C2×#41—188no—
161C22×Q8#15711yescharacteristic, complemented
162C22×D4—91no—
163C22×D4—181no—
164C23×C4—91no—
165C23×C4—181no—
166C2×M16—92no—
167C2×C22⋊C4—181no—
168C2×C42—92no—
169#104—188no—
170C22⋊C8—181no—
171C22⋊C8—181no—
172#95—181no—
173C23×C3—22no—
174C22×D3—62no—
175C22×D3—62no—
176C22×D3—68no—
177C22×D3—616no—
178C22×D3—68no—
179C22×D3—124no—
180C22×D3—124no—
181C22×D3—128no—
182C22×D3—128no—
183C3×Q8—216no—
184C3×Q8—28no—
185C3×Q8—28no—
186C2×C3×C4—216no—
187C2×C3×C4—28no—
188C2×C3×C4—28no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—68no—
194D12—68no—
195D12—632no—
196D12—632no—
197D12—616no—
198D12—68no—
199C4×D3—68no—
200C4×D3—68no—
201C4×D3—632no—
202C4×D3—68no—
203C4×D3—632no—
204C4×D3—68no—
205C2×C32—22no—
206C2×C32#37911yescharacteristic
207C2×C32C2×#10411yescharacteristic
208C2×C32#42111yescharacteristic
209C3⋊D3—22no—
210C3⋊D3—41no—
211C3⋊D3—44no—
212C24—91no—
213C24—181no—
214#41—98no—
215#41—184no—
216#41—184no—
217#41—188no—
218#41—188no—
219#41—184no—
220C2×Q8—24no—
221C2×Q8—21no—
222C2×Q8C2×#15712yes—
223C2×D4—92no—
224C2×D4—184no—
225C2×D4—182no—
226C2×D4—182no—
227C2×D4—188no—
228C2×D4—181no—
229C22×C4—21no—
230C22×C4—91no—
231C22×C4—91no—
232C22×C4—91no—
233C22×C4—91no—
234C22×C4—91no—
235C22×C4—184no—
236C22×C4—182no—
237C22×C4—182no—
238C22×C4—188no—
239C22×C4—182no—
240C22×C4C2×#15711yescharacteristic
241M16—184no—
242C2×C8—182no—
243C22⋊C4—184no—
244C42—184no—
245C22×C3—24no—
246C22×C3—24no—
247C22×C3—22no—
248C22×C3—24no—
249C2×D3—64no—
250C2×D3—616no—
251C2×D3—64no—
252C2×D3—122no—
253C2×D3—122no—
254C2×D3—122no—
255C2×D3—122no—
256C2×D3—1216no—
257C2×D3—1232no—
258C2×D3—128no—
259C2×D3—128no—
260C3×C4—216no—
261C3×C4—24no—
262C3×C4—24no—
263Dic3—64no—
264Dic3—64no—
265C32#178911yescharacteristic, complemented, Sylow
266C23—91no—
267C23—91no—
268C23—182no—
269C23—184no—
270C23—182no—
271C23—182no—
272C23—361no—
273C23—362no—
274C23—362no—
275C23C22⋊#15711yescharacteristic
276Q8—24no—
277Q8—22no—
278Q8C22⋊#15714yescomplemented
279D4—94no—
280D4—188no—
281D4—188no—
282D4—184no—
283D4—182no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—91no—
288C2×C4—94no—
289C2×C4—91no—
290C2×C4—182no—
291C2×C4—188no—
292C2×C4—182no—
293C2×C4—188no—
294C2×C4—182no—
295C2×C4—182no—
296C2×C4—184no—
297C2×C4—362no—
298C2×C4—362no—
299C2×C4C22⋊#15711yescharacteristic
300C2×C4C22×#15711yescharacteristic
301C8—362no—
302C2×C3—28no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—22no—
306D3—124no—
307D3—124no—
308D3—1216no—
309C22—21no—
310C22—21no—
311C22—92no—
312C22—181no—
313C22—181no—
314C22—181no—
315C22—181no—
316C22—184no—
317C22—181no—
318C22—364no—
319C22—368no—
320C22—362no—
321C22—362no—
322C22#6378512yescomplemented
323C22C2×C22⋊#15711yescharacteristic
324C4—24no—
325C4—21no—
326C4—92no—
327C4—181no—
328C4—364no—
329C4C2×C22⋊#15712yes—
330C3—22no—
331C2—22no—
332C2—182no—
333C2—361no—
334C2—364no—
335C2C2×#6378511yescharacteristic
336C2#1059383111yescharacteristic
337C2#1059384711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#157
commutatorC2×C32×C4C22×C4
FrattiniC2×C4C22×#157
FittingC22×C32×Q8C4
socleC22×C32C2×C22⋊C4
prime radicalC22×#3832C2
cosocleC2×C4×C3⋊D3C23
hypercenterC22×Q8#157
hypocenterC32#1789
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×#63785C2semidirect
3. C4×C22⋊#157C2semidirect
4. #10593803C2semidirect
5. C22×C32×Q8C4semidirect
9. C2×#3832C4semidirect
12. C22×C4⋊(C3⋊D3)C4semidirect
16. C2×C4×#157C22semidirect
17. C2×#3788C22semidirect
82. C32×Q8C22⋊C4semidirect
95. C4⋊(C3⋊D3)C22⋊C4semidirect
137. C22×C32#104semidirect
161. C22×Q8#157semidirect
265. C32#1789semidirect
278. Q8C22⋊#157semidirect
322. C22#63785semidirect

Maximal subgroups

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

subgroupindex
C22×#38322
C2×#637852
C4×C22⋊#1572
#105938032
#17899