Finite Group Databasebeta

← #11774384

Subgroups — #11774384

Order 1152.

648 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 149 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#10596049C211yesmaximal, characteristic
2C3⋊D3×C4⋊C8C211yesmaximal, characteristic
3#10592384C211yesmaximal, characteristic
4C32×Q8⋊C8C211yesmaximal, characteristic, complemented
5#10589401C211yesmaximal, characteristic, complemented
6#10589350C211yesmaximal, characteristic, complemented
7#10589322C211yesmaximal, characteristic, complemented
8#68232—34nomaximal
9C32×C4×Q8C2211yescharacteristic
10#64155C411yescharacteristic
11#64153C411yescharacteristic
12C2×C8×C3⋊D3—21no—
13#64106C411yescharacteristic
14#64105—21no—
15#64098—21no—
16#64097C411yescharacteristic
17#64081—21no—
18C42×C3⋊D3C2211yescharacteristic
19C4×#634—21no—
20C4×#634C2211yescharacteristic
21C32×C4⋊C8C2211yescharacteristic
22C32×C4×C8C2211yescharacteristic
23#63647—21no—
24#63641—21no—
25#63641—21no—
26C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
27C4×C3⋊(C3⋊C8)C2211yescharacteristic
28#5578—34no—
29D3×C4⋊C8—34no—
30#4685—34no—
31C3×Q8⋊C8D314yescomplemented
32#4535—34no—
33#4484—34no—
34#4456—34no—
35C2×C32×Q8C2×C411yescharacteristic
36C2×C4×C3⋊D3C2×C411yescharacteristic
37C2×C4×C3⋊D3D411yescharacteristic
38C2×C4×C3⋊D3D411yescharacteristic
39C2×#634—21no—
40C2×#634C2×C411yescharacteristic
41C2×C32×C8—21no—
42C2×C32×C8—21no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4C2×C411yescharacteristic
45C32×C42—21no—
46C32×C42C2311yescharacteristic
47#3752—21no—
48#3752—21no—
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)C2×C411yescharacteristic
51#3750—21no—
52#3750—21no—
53#3750—21no—
54#3750—21no—
55#3750—21no—
56C4×C3⋊Dic3—21no—
57C4×C3⋊Dic3C2×C411yescharacteristic
58C4×C3⋊Dic3D411yescharacteristic
59C4×C3⋊Dic3D411yescharacteristic
60C2×C3⋊(C3⋊C8)—21no—
61C2×C3⋊(C3⋊C8)—21no—
62C8×C3⋊D3—42no—
63#1481—91noSylow
64C3×C4×Q8C2×D314yes—
65#946—34no—
66#944—34no—
67C2×D3×C8—64no—
68#897—34no—
69#896—64no—
70#889—64no—
71#888—34no—
72#872—64no—
73C42×D3—34no—
74C4×Dic6—34no—
75C4×Dic6—64no—
76C3×C4⋊C8C2×D314yes—
77C3×C4×C8C2×D314yes—
78(C2×D3)⋊C8—64no—
79Dic3⋊C8—64no—
80Dic3⋊C8—64no—
81C4⋊(C3⋊C8)—34no—
82C4×C3⋊C8—34no—
83C22×C3⋊D3C22⋊C411yescharacteristic
84C32×Q8—21no—
85C32×Q8—21no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4C22×C411yescharacteristic
89C2×C32×C4C2×D411yescharacteristic
90C2×C32×C4C2×D411yescharacteristic
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—21no—
93C2×C3⋊Dic3C22⋊C411yescharacteristic
94C2×C3⋊Dic3C22⋊C411yescharacteristic
95C2×C3⋊Dic3C22⋊C411yescharacteristic
96C4×C3⋊D3—21no—
97C4×C3⋊D3—21no—
98C4×C3⋊D3—41no—
99C4×C3⋊D3SD1612yes—
100C4×C3⋊D3Q1612yes—
101#634—21no—
102#634—21no—
103#634—42no—
104C32×C8—41no—
105C32×C8—41no—
106C3⋊(C3⋊C8)—41no—
107C3⋊(C3⋊C8)—41no—
108#533—91no—
109#431—91no—
110C2×C4⋊C8—91no—
111Q8⋊C8—91no—
112Q8⋊C8—91no—
113Q8⋊C8—91no—
114Q8⋊C8C3⋊D311yescharacteristic, complemented
115C2×C3×Q8C4×D314yes—
116C2×C4×D3—34no—
117C2×C4×D3—34no—
118C2×C4×D3—34no—
119C2×Dic6—34no—
120C2×Dic6—64no—
121C2×C3×C8—24no—
122C2×C3×C8—24no—
123C3×C4⋊C4—24no—
124C3×C4⋊C4C4×D314yes—
125C3×C42—24no—
126C3×C42C22×D314yes—
127(C2×D3)⋊C4—64no—
128(C2×D3)⋊C4—64no—
129C4⋊Dic3—34no—
130C4⋊Dic3—64no—
131Dic3⋊C4—64no—
132Dic3⋊C4—64no—
133Dic3⋊C4—64no—
134Dic3⋊C4—64no—
135Dic3⋊C4—64no—
136C4×Dic3—34no—
137C4×Dic3—34no—
138C4×Dic3—34no—
139C4×Dic3—64no—
140C2×C3⋊C8—64no—
141C2×C3⋊C8—64no—
142D3×C8—128no—
143C22×C32C2×C22⋊C411yescharacteristic
144C2×C3⋊D3—21no—
145C2×C3⋊D3—21no—
146C2×C3⋊D3#10312yes—
147C32×C4—21no—
148C32×C4—41no—
149C32×C4—41no—
150C32×C4#13111yescharacteristic
151C32×C4#13111yescharacteristic
152C32×C4C2×SD1611yescharacteristic
153C32×C4C2×Q1611yescharacteristic
154C3⋊Dic3—21no—
155C3⋊Dic3—21no—
156C3⋊Dic3—41no—
157C3⋊Dic3—41no—
158C3⋊Dic3#10312yes—
159C22×C8—181no—
160C4⋊Q8—91no—
161C4⋊Q8—91no—
162#125—181no—
163C22⋊Q8—91no—
164C22⋊Q8—91no—
165C22⋊Q8—181no—
166C4×Q8—91no—
167C4×Q8—181no—
168C4×Q8C2×C3⋊D311yescharacteristic
169#117—181no—
170C2×C42—91no—
171C4⋊C8—91no—
172C4⋊C8—181no—
173C4⋊C8—181no—
174C4⋊C8C2×C3⋊D311yescharacteristic
175C22⋊C8—181no—
176C4×C8—91no—
177C4×C8C2×C3⋊D311yescharacteristic
178C22×D3—34no—
179C3×Q8—24no—
180C3×Q8—24no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4C2×C4×D314yes—
184C2×C3×C4D3×D414yes—
185C2×C3×C4D3×D414yes—
186C2×Dic3—34no—
187C2×Dic3—34no—
188C2×Dic3—34no—
189C2×Dic3—64no—
190C2×Dic3—64no—
191C4×D3—38no—
192C4×D3—38no—
193C4×D3—64no—
194C4×D3—64no—
195C4×D3—124no—
196Dic6—64no—
197Dic6—64no—
198Dic6—128no—
199C3×C8—44no—
200C3×C8—44no—
201C3⋊C8—124no—
202C3⋊C8—124no—
203C2×C32#40811yescharacteristic
204C2×C32C2×#10311yescharacteristic
205C2×C32#42111yescharacteristic
206C3⋊D3—22no—
207C2×Q8—91no—
208C2×Q8—181no—
209C2×Q8C4×C3⋊D311yescharacteristic
210C22×C4—91no—
211C22×C4—91no—
212C22×C4—91no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—181no—
216C2×C8—181no—
217C2×C8—362no—
218C4⋊C4—21no—
219C4⋊C4—91no—
220C4⋊C4—181no—
221C4⋊C4—181no—
222C4⋊C4—181no—
223C4⋊C4—181no—
224C4⋊C4—181no—
225C4⋊C4—181no—
226C4⋊C4C4×C3⋊D311yescharacteristic
227C22⋊C4—181no—
228C22⋊C4—181no—
229C42—21no—
230C42—91no—
231C42—91no—
232C42—91no—
233C42—181no—
234C42C22×C3⋊D311yescharacteristic
235C22×C3D3×C22⋊C414yes—
236C2×D3—38no—
237C2×D3—64no—
238C2×D3—64no—
239C3×C4—24no—
240C3×C4—44no—
241C3×C4—44no—
242C3×C4#90114yes—
243C3×C4#90714yes—
244C3×C4D3×SD1614yes—
245C3×C4D3×Q1614yes—
246Dic3—38no—
247Dic3—64no—
248Dic3—64no—
249Dic3—124no—
250Dic3—124no—
251C32#148111yescharacteristic, complemented, Sylow
252C23—91no—
253Q8—21no—
254Q8—21no—
255Q8—181no—
256Q8—181no—
257Q8—362no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—91no—
261C2×C4—92no—
262C2×C4—92no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—361no—
270C2×C4C2×C4×C3⋊D311yescharacteristic
271C2×C4D4×C3⋊D311yescharacteristic
272C2×C4D4×C3⋊D311yescharacteristic
273C8—41no—
274C8—41no—
275C8—361no—
276C8—361no—
277C2×C3#472714yes—
278C2×C3D3×#10314yes—
279C2×C3#482114yes—
280D3—68no—
281C22—92no—
282C22—181no—
283C22—181no—
284C22C22⋊C4×C3⋊D311yescharacteristic
285C4—21no—
286C4—41no—
287C4—41no—
288C4—92no—
289C4—181no—
290C4—181no—
291C4—361no—
292C4—361no—
293C4#6411011yescharacteristic
294C4#6411611yescharacteristic
295C4SD16×C3⋊D311yescharacteristic
296C4Q16×C3⋊D311yescharacteristic
297C3#6823214yescomplemented
298C2—182no—
299C2#1059242611yescharacteristic
300C2C3⋊D3×#10311yescharacteristic
301C2#1059252011yescharacteristic
subgroupG/H
centerC2×C4D4×C3⋊D3
commutatorC2×C32×C4C22×C4
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C3⋊D3C22⋊C4
cosocleC32×C42C23
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1481
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. C32×Q8⋊C8C2semidirect
5. #10589401C2semidirect
6. #10589350C2semidirect
7. #10589322C2semidirect
31. C3×Q8⋊C8D3semidirect
114. Q8⋊C8C3⋊D3semidirect
251. C32#1481semidirect
297. C3#68232semidirect

Maximal subgroups

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

subgroupindex
#105960492
C3⋊D3×C4⋊C82
#105923842
C32×Q8⋊C82
#105894012
#105893502
#105893222
#682323