Finite Group Databasebeta

← #11776146

Subgroups — #11776146

Order 1152.

667 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 667 conjugacy classes of subgroups — too many to draw. Only its 108 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
1#10596196C211yesmaximal, characteristic
2#10590331C211yesmaximal, characteristic, complemented
3#10590256C211yesmaximal, characteristic, complemented
4C3×#4547C212yesmaximal, complemented
5#10589091C212yesmaximal, complemented
6#68417—32nomaximal
7C2×#3832—21no—
8C32×C4⋊Q8C2211yescharacteristic
9#64156—21no—
10#64104—21no—
11#64104C412yes—
12#64102—21no—
13#64102C412yes—
14#64083C2211yescharacteristic
15C42×C3⋊D3C2211yescharacteristic
16C2×#3710—21no—
17C3×C4⋊(C3⋊C8)C2212yes—
18C3×C4⋊(C3⋊C8)C2212yes—
19#63572—21no—
20#63557—21no—
21#63553—21no—
22#5725—32no—
23#5725—61no—
24#4833—32no—
25D3×C4⋊C8—32no—
26C3×#332—32no—
27#4547D312yescomplemented
28#4485—32no—
29#4485—32no—
30C2×C32×Q8—21no—
31#3832—42no—
32#3832—42no—
33C2×C4⋊(C3⋊D3)—22no—
34C2×C4⋊(C3⋊D3)—21no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×C4⋊(C3⋊D3)C2×C412yes—
37C2×C4×C3⋊D3—22no—
38C2×C4×C3⋊D3C2×C411yescharacteristic
39C2×C4×C3⋊D3D411yescharacteristic
40C2×C4×C3⋊D3D411yescharacteristic
41C32×C4⋊C4—21no—
42C32×C4⋊C4C2×C412yes—
43C32×C42C2311yescharacteristic
44#3752—22no—
45#3752—22no—
46C4×C3⋊Dic3C2×C411yescharacteristic
47C4×C3⋊Dic3D411yescharacteristic
48C4×C3⋊Dic3D411yescharacteristic
49C2×C3×C3⋊C8—22no—
50C2×C3×C3⋊C8—22no—
51#3710—42no—
52#1540—91noSylow
53C2×#239—62no—
54C2×#239—62no—
55C3×C4⋊Q8—21no—
56C3×C4⋊Q8C2×D312yes—
57#947—62no—
58#947—62no—
59C2×D3×C8—62no—
60#895—34no—
61#895—62no—
62#895—62no—
63#895—62no—
64#893—34no—
65#893—62no—
66#893—62no—
67#893—62no—
68#874—32no—
69#874—61no—
70C42×D3—32no—
71C42×D3—61no—
72C3×C4⋊C8—32no—
73C3×C4⋊C8—32no—
74(C2×D3)⋊C8—62no—
75Dic3⋊C8—62no—
76C8×Dic3—62no—
77C4⋊(C3⋊C8)C2×D312yes—
78C4⋊(C3⋊C8)C2×D312yes—
79C22×C3⋊D3—22no—
80C22×C3⋊D3C22⋊C411yescharacteristic
81C32×Q8—41no—
82C32×Q8—41no—
83C2×C32×C4—22no—
84C2×C32×C4C22×C411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86C2×C32×C4C2×D411yescharacteristic
87C2×C3⋊Dic3C22⋊C411yescharacteristic
88C2×C3⋊Dic3C22⋊C411yescharacteristic
89C2×C3⋊Dic3C22⋊C411yescharacteristic
90C4⋊(C3⋊D3)—42no—
91C4⋊(C3⋊D3)—44no—
92C4⋊(C3⋊D3)—41no—
93C4⋊(C3⋊D3)—41no—
94C4⋊(C3⋊D3)—42no—
95C4×C3⋊D3—22no—
96C4×C3⋊D3—41no—
97C4×C3⋊D3—44no—
98C4×C3⋊D3D812yes—
99C4×C3⋊D3SD1612yes—
100C3×C3⋊C8—42no—
101C3×C3⋊C8—42no—
102#532—91no—
103#431—91no—
104C2×C4⋊C8—91no—
105#332—32no—
106#331—92no—
107C2×C3×Q8—22no—
108C2×C3×Q8—22no—
109#239—124no—
110#239—124no—
111#239—124no—
112#239—124no—
113C2×D12—34no—
114C2×D12—62no—
115C2×D12—64no—
116C2×D12—62no—
117C2×D12—62no—
118C2×D12—64no—
119C2×D12—62no—
120C2×D12—62no—
121C2×C4×D3—32no—
122C2×C4×D3—32no—
123C2×C4×D3—32no—
124C2×C4×D3—61no—
125C2×C4×D3—61no—
126C2×C4×D3—61no—
127C2×C4×D3—64no—
128C2×C4×D3—64no—
129C2×C3×C8—62no—
130C2×C3×C8—62no—
131C3×C4⋊C4—22no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4C4×D314yes—
135C3×C42—21no—
136C3×C42C22×D312yes—
137(C2×D3)⋊C4—64no—
138(C2×D3)⋊C4—64no—
139(C2×D3)⋊C4—64no—
140(C2×D3)⋊C4—64no—
141C4×Dic3—32no—
142C4×Dic3—32no—
143C4×Dic3—32no—
144C4×Dic3—61no—
145C4×Dic3—61no—
146C4×Dic3—61no—
147C2×C3⋊C8—22no—
148C2×C3⋊C8—22no—
149D3×C8—124no—
150C22×C32C2×C22⋊C411yescharacteristic
151C2×C3⋊D3—21no—
152C2×C3⋊D3—21no—
153C2×C3⋊D3—42no—
154C2×C3⋊D3—42no—
155C2×C3⋊D3—42no—
156C2×C3⋊D3#10212yes—
157C32×C4—21no—
158C32×C4—42no—
159C32×C4C2×D811yescharacteristic
160C32×C4C2×SD1611yescharacteristic
161C32×C4#13512yes—
162C3⋊Dic3—21no—
163C3⋊Dic3—21no—
164C3⋊Dic3#10212yes—
165C2×#41—181no—
166C22×C8—181no—
167C4⋊Q8D3211yescharacteristic
168#127—91no—
169#124—181no—
170#121—92no—
171#121—181no—
172C4×D4—92no—
173C4×D4—181no—
174C2×C42—91no—
175C4⋊C8—32no—
176C4⋊C8—32no—
177C4⋊C8—181no—
178C22⋊C8—181no—
179C4×C8—181no—
180C22×D3—32no—
181C22×D3—61no—
182C22×D3—64no—
183C22×D3—64no—
184C3×Q8—42no—
185C3×Q8—42no—
186C3×Q8—42no—
187C3×Q8—42no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4C2×C4×D312yes—
194C2×C3×C4D3×D412yes—
195C2×C3×C4D3×D412yes—
196C2×Dic3—32no—
197C2×Dic3—32no—
198C2×Dic3—32no—
199C2×Dic3—61no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202D12—124no—
203D12—124no—
204D12—124no—
205D12—122no—
206D12—122no—
207D12—128no—
208D12—128no—
209D12—122no—
210D12—122no—
211D12—122no—
212D12—122no—
213C4×D3—34no—
214C4×D3—34no—
215C4×D3—62no—
216C4×D3—62no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—122no—
220C4×D3—128no—
221C4×D3—122no—
222C4×D3—128no—
223C4×D3—122no—
224C3×C8—122no—
225C3×C8—122no—
226C3⋊C8—42no—
227C3⋊C8—42no—
228C2×C32#41311yescharacteristic
229C2×C32C2×#10211yescharacteristic
230C2×C32#41611yescharacteristic
231C3⋊D3—22no—
232C3⋊D3—82no—
233#41—362no—
234#41—362no—
235C2×Q8—21no—
236C2×D4—92no—
237C2×D4—182no—
238C2×D4—181no—
239C2×D4—181no—
240C22×C4—91no—
241C22×C4—91no—
242C22×C4—91no—
243C22×C4—182no—
244C2×C8—62no—
245C2×C8—62no—
246C2×C8—362no—
247C4⋊C4—21no—
248C4⋊C4#62412yes—
249C22⋊C4—182no—
250C22⋊C4—182no—
251C42—91no—
252C42—91no—
253C42—91no—
254C42C2×D3211yescharacteristic
255C22×C3—21no—
256C22×C3D3×C22⋊C412yes—
257C2×D3—34no—
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—121no—
262C2×D3—121no—
263C2×D3—124no—
264C2×D3—124no—
265C2×D3—124no—
266C2×D3—124no—
267C2×D3—124no—
268C2×D3—124no—
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—22no—
274C3×C4—44no—
275C3×C4—44no—
276C3×C4D3×D812yes—
277C3×C4D3×SD1612yes—
278C3×C4#91612yes—
279C3×C4#91912yes—
280Dic3—34no—
281Dic3—62no—
282Dic3—62no—
283Dic3—62no—
284Dic3—121no—
285Dic3—121no—
286C32#154011yescharacteristic, complemented, Sylow
287C23—91no—
288C23—182no—
289Q8—41no—
290Q8—41no—
291D4—362no—
292D4—364no—
293D4—361no—
294D4—361no—
295D4—362no—
296C2×C4—22no—
297C2×C4—91no—
298C2×C4—92no—
299C2×C4—92no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—182no—
303C2×C4—361no—
304C2×C4—364no—
305C2×C4C2×#62411yescharacteristic
306C2×C4#381111yescharacteristic
307C2×C4#381111yescharacteristic
308C8—122no—
309C8—122no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3#475112yes—
314C2×C3D3×#10212yes—
315C2×C3#480412yes—
316D3—64no—
317D3—122no—
318D3—244no—
319D3—244no—
320C22—92no—
321C22—181no—
322C22—181no—
323C22—362no—
324C22—362no—
325C22—362no—
326C22#6397411yescharacteristic
327C4—21no—
328C4—42no—
329C4—92no—
330C4—181no—
331C4—181no—
332C4#6392611yescharacteristic
333C4#6393011yescharacteristic
334C4#6395012yes—
335C3—21no—
336C3#6841712yescomplemented
337C2—182no—
338C2—722no—
339C2#1059141611yescharacteristic
340C2#1059156011yescharacteristic
341C2#1059164011yescharacteristic
subgroupG/H
centerC22#63974
commutatorC2×C32×C4C22×C4
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32C2×C22⋊C4
prime radical#10596196C2
cosocleC32×C42C23
hypercenterC4⋊Q8D32
hypocenterC32#1540
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10590331C2semidirect
3. #10590256C2semidirect
4. C3×#4547C2semidirect
5. #10589091C2semidirect
27. #4547D3semidirect
286. C32#1540semidirect
336. C3#68417semidirect

Maximal subgroups

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

subgroupindex
#105961962
#105903312
#105902562
C3×#45472
#105890912
#684173