Finite Group Databasebeta

← #11774756

Subgroups — #11774756

Order 1152.

696 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 151 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
2#10592536C211yesmaximal, characteristic
3C4×C8×C3⋊D3C211yesmaximal, characteristic
4C32×Q8⋊C8C211yesmaximal, characteristic, complemented
5#10589401C211yesmaximal, characteristic, complemented
6#10589350C211yesmaximal, characteristic, complemented
7#10589322C211yesmaximal, characteristic, complemented
8#68240—34nomaximal
9C32×C4×Q8C2211yescharacteristic
10#64155C411yescharacteristic
11#64153C411yescharacteristic
12C2×#3743—21no—
13C2×C8×C3⋊D3—21no—
14#64106C411yescharacteristic
15#64105—21no—
16#64098—21no—
17#64097C411yescharacteristic
18#64081—21no—
19C42×C3⋊D3C2211yescharacteristic
20C4×#634—21no—
21C4×#634C2211yescharacteristic
22C32×C4⋊C8C2211yescharacteristic
23C32×C4×C8C2211yescharacteristic
24#63641—21no—
25C8×C3⋊Dic3—21no—
26C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
27C4×C3⋊(C3⋊C8)C2211yescharacteristic
28#5578—34no—
29#4837—34no—
30C4×D3×C8—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—
62#3743—42no—
63C8×C3⋊D3—24no—
64#1493—91noSylow
65C3×C4×Q8C2×D314yes—
66#946—34no—
67#944—34no—
68C2×ΓL(1,𝔽25)—64no—
69C2×D3×C8—64no—
70#897—34no—
71#896—64no—
72#889—64no—
73#888—34no—
74#872—64no—
75C42×D3—34no—
76C4×Dic6—34no—
77C4×Dic6—64no—
78C3×C4⋊C8C2×D314yes—
79C3×C4×C8C2×D314yes—
80Dic3⋊C8—64no—
81C8×Dic3—64no—
82C4⋊(C3⋊C8)—34no—
83C4×C3⋊C8—34no—
84C22×C3⋊D3C22⋊C411yescharacteristic
85C32×Q8—21no—
86C32×Q8—21no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4C22×C411yescharacteristic
90C2×C32×C4C2×D411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C2×C3⋊Dic3—21no—
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3C22⋊C411yescharacteristic
95C2×C3⋊Dic3C22⋊C411yescharacteristic
96C2×C3⋊Dic3C22⋊C411yescharacteristic
97C4×C3⋊D3—21no—
98C4×C3⋊D3—21no—
99C4×C3⋊D3—22no—
100C4×C3⋊D3SD1612yes—
101C4×C3⋊D3Q1612yes—
102#634—21no—
103#634—21no—
104#634—42no—
105C32×C8—22no—
106C32×C8—41no—
107C3⋊(C3⋊C8)—22no—
108C3⋊(C3⋊C8)—41no—
109#533—91no—
110#423—91no—
111C2×C4×C8—91no—
112Q8⋊C8—91no—
113Q8⋊C8—91no—
114Q8⋊C8—91no—
115Q8⋊C8C3⋊D311yescharacteristic, complemented
116C2×C3×Q8C4×D314yes—
117C2×C4×D3—34no—
118C2×C4×D3—34no—
119C2×C4×D3—34no—
120C2×Dic6—34no—
121C2×Dic6—64no—
122C2×C3×C8—24no—
123C2×C3×C8—24no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4C4×D314yes—
126C3×C42—24no—
127C3×C42C22×D314yes—
128(C2×D3)⋊C4—64no—
129(C2×D3)⋊C4—64no—
130C4⋊Dic3—34no—
131C4⋊Dic3—64no—
132Dic3⋊C4—64no—
133Dic3⋊C4—64no—
134Dic3⋊C4—64no—
135Dic3⋊C4—64no—
136Dic3⋊C4—64no—
137C4×Dic3—34no—
138C4×Dic3—34no—
139C4×Dic3—34no—
140C4×Dic3—64no—
141C2×C3⋊C8—64no—
142C2×C3⋊C8—64no—
143ΓL(1,𝔽25)—128no—
144D3×C8—616no—
145C22×C32C2×C22⋊C411yescharacteristic
146C2×C3⋊D3—21no—
147C2×C3⋊D3—21no—
148C2×C3⋊D3#10312yes—
149C32×C4—21no—
150C32×C4—41no—
151C32×C4—41no—
152C32×C4#13111yescharacteristic
153C32×C4#13111yescharacteristic
154C32×C4C2×SD1611yescharacteristic
155C32×C4C2×Q1611yescharacteristic
156C3⋊Dic3—21no—
157C3⋊Dic3—41no—
158C3⋊Dic3—41no—
159C3⋊Dic3#10312yes—
160C3⋊Dic3#10412yes—
161C2×M16—181no—
162C22×C8—181no—
163C4⋊Q8—91no—
164C4⋊Q8—91no—
165#125—181no—
166C22⋊Q8—91no—
167C22⋊Q8—91no—
168C22⋊Q8—181no—
169C4×Q8—91no—
170C4×Q8—181no—
171C4×Q8C2×C3⋊D311yescharacteristic
172#117—181no—
173C2×C42—91no—
174C4⋊C8—91no—
175C4⋊C8—181no—
176C4⋊C8C2×C3⋊D311yescharacteristic
177C4×C8—91no—
178C4×C8—181no—
179C4×C8C2×C3⋊D311yescharacteristic
180C22×D3—34no—
181C3×Q8—24no—
182C3×Q8—24no—
183C2×C3×C4—24no—
184C2×C3×C4—24no—
185C2×C3×C4C2×C4×D314yes—
186C2×C3×C4D3×D414yes—
187C2×C3×C4D3×D414yes—
188C2×Dic3—34no—
189C2×Dic3—34no—
190C2×Dic3—34no—
191C2×Dic3—64no—
192C2×Dic3—64no—
193C4×D3—38no—
194C4×D3—38no—
195C4×D3—68no—
196C4×D3—64no—
197C4×D3—64no—
198Dic6—64no—
199Dic6—64no—
200Dic6—128no—
201C3×C8—28no—
202C3×C8—44no—
203C3⋊C8—68no—
204C3⋊C8—124no—
205C2×C32#40811yescharacteristic
206C2×C32C2×#10311yescharacteristic
207C2×C32C2×#10411yescharacteristic
208C3⋊D3—22no—
209C2×Q8—91no—
210C2×Q8—181no—
211C2×Q8C4×C3⋊D311yescharacteristic
212C22×C4—91no—
213C22×C4—91no—
214C22×C4—91no—
215M16—362no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—184no—
219C2×C8—181no—
220C2×C8—181no—
221C4⋊C4—21no—
222C4⋊C4—91no—
223C4⋊C4—181no—
224C4⋊C4—181no—
225C4⋊C4—181no—
226C4⋊C4—181no—
227C4⋊C4—181no—
228C4⋊C4—181no—
229C4⋊C4C4×C3⋊D311yescharacteristic
230C22⋊C4—181no—
231C22⋊C4—181no—
232C42—21no—
233C42—91no—
234C42—91no—
235C42—91no—
236C42—181no—
237C42C22×C3⋊D311yescharacteristic
238C22×C3D3×C22⋊C414yes—
239C2×D3—38no—
240C2×D3—64no—
241C2×D3—64no—
242C3×C4—24no—
243C3×C4—44no—
244C3×C4—44no—
245C3×C4#90114yes—
246C3×C4#90714yes—
247C3×C4D3×SD1614yes—
248C3×C4D3×Q1614yes—
249Dic3—38no—
250Dic3—38no—
251Dic3—64no—
252Dic3—124no—
253Dic3—124no—
254C32#149311yescharacteristic, complemented, Sylow
255C23—91no—
256Q8—21no—
257Q8—21no—
258Q8—181no—
259Q8—181no—
260Q8—362no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—92no—
266C2×C4—92no—
267C2×C4—91no—
268C2×C4—182no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C4C2×C4×C3⋊D311yescharacteristic
274C2×C4D4×C3⋊D311yescharacteristic
275C2×C4D4×C3⋊D311yescharacteristic
276C8—22no—
277C8—41no—
278C8—182no—
279C8—361no—
280C2×C3#472714yes—
281C2×C3D3×#10314yes—
282C2×C3D3×#10414yes—
283D3—68no—
284C22—92no—
285C22—181no—
286C22—181no—
287C22C22⋊C4×C3⋊D311yescharacteristic
288C4—21no—
289C4—41no—
290C4—41no—
291C4—92no—
292C4—92no—
293C4—181no—
294C4—361no—
295C4—361no—
296C4#6411011yescharacteristic
297C4#6411611yescharacteristic
298C4SD16×C3⋊D311yescharacteristic
299C4Q16×C3⋊D311yescharacteristic
300C3#6824014yescomplemented
301C2—182no—
302C2#1059242611yescharacteristic
303C2C3⋊D3×#10311yescharacteristic
304C2C3⋊D3×#10411yescharacteristic
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#1493
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
115. Q8⋊C8C3⋊D3semidirect
254. C32#1493semidirect
300. C3#68240semidirect

Maximal subgroups

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

subgroupindex
#105960492
#105925362
C4×C8×C3⋊D32
C32×Q8⋊C82
#105894012
#105893502
#105893222
#682403