Finite Group Databasebeta

← #224201

Subgroups — #224201

Order 512.

683 conjugacy classes of subgroups.

The subgroup lattice has 329 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36567C211yesmaximal, characteristic
2#21346C212yesmaximal
3#9894C212yesmaximal
4#9893C212yesmaximal
5#9892C212yesmaximal
6#8580C211yesmaximal, characteristic
7#8580C212yesmaximal
8#8227C212yesmaximal
9#8225C211yesmaximal, characteristic
10#2864C2212yes—
11#2859C2211yescharacteristic
12#2590—22no—
13#2585—22no—
14#2584C412yes—
15#2507C412yes—
16#2506C412yes—
17#2462—22no—
18#2370—22no—
19#2370C412yes—
20C4×#125C2211yescharacteristic
21C4×#125C2212yes—
22C42×Q8C2211yescharacteristic
23#1985—22no—
24#1940—22no—
25#1916—22no—
26#1916—22no—
27#1767C2212yes—
28#1767C2212yes—
29#1767C2212yes—
30#1745C2212yes—
31#1745C2212yes—
32C4×#97C2212yes—
33C2×#330C2211yescharacteristic
34C2×#330C2212yes—
35C2×#330C2211yescharacteristic
36#1295C2212yes—
37#1295C2212yes—
38#1295C2212yes—
39#1295C2211yescharacteristic
40#1295C2212yes—
41#1295C2211yescharacteristic
42#1295C2212yes—
43C2×#125—22no—
44C2×#125C2×C412yes—
45C2×#125C2311yescharacteristic
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8D412yes—
48C2×C4⋊C8—22no—
49C2×C4⋊C8—22no—
50C2×C4⋊C8—22no—
51C2×#97—22no—
52C2×#97C2312yes—
53C2×#97C2312yes—
54C2×#97C2312yes—
55C2×#97C2312yes—
56C2×C4×C8—22no—
57#400—22no—
58#398—22no—
59#398—22no—
60#398—22no—
61#398—22no—
62#395—22no—
63#389—22no—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389C2×C412yes—
70#389C2×C412yes—
71#389C2×C412yes—
72#389C2×C412yes—
73#389C2×C412yes—
74#389C2×C412yes—
75#389C2×C412yes—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#384C2×C411yescharacteristic
88#384C2×C411yescharacteristic
89#382C2312yes—
90#382C2311yescharacteristic
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4C2×C411yescharacteristic
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4D412yes—
103C43C2311yescharacteristic
104C43C2312yes—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—22no—
111#336—22no—
112#330—22no—
113#330—24no—
114#330Q814yescomplemented
115C22×Q8C22⋊C411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120#125—22no—
121#125—44no—
122#125—48no—
123#125C2×Q812yes—
124#125#4112yes—
125C4×Q8—42no—
126C4×Q8—44no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4C22⋊C412yes—
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22×C412yes—
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42C22×C411yescharacteristic
152C2×C42C22×C412yes—
153C2×C42C22×C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2411yescharacteristic
162C4⋊C8—44no—
163C4⋊C8—44no—
164C4⋊C8—44no—
165#97—22no—
166#97—22no—
167#97—22no—
168#97—22no—
169#97—22no—
170#97—22no—
171#97—22no—
172#97—44no—
173#97C2×Q814yes—
174C4×C8—44no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95C22⋊C411yescharacteristic
189#95C22⋊C411yescharacteristic
190#95C22⋊C412yes—
191#95C22⋊C412yes—
192C2×Q8—41no—
193C2×Q8—42no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C22⋊C412yes—
205C22×C4C23×C411yescharacteristic
206C22×C4C22×D412yes—
207C2×C8—24no—
208C2×C8—24no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C4⋊C4—22no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4C4×Q814yes—
242C42—22no—
243C42—24no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—42no—
248C42—41no—
249C42—42no—
250C42—42no—
251C42—42no—
252C42—44no—
253C42—82no—
254C42—82no—
255C42C22×Q811yescharacteristic
256C42C2×#4111yescharacteristic
257C422−1+412yes—
258C23C22×C22⋊C411yescharacteristic
259Q8—44no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4#41312yes—
294C2×C4#42114yes—
295C2×C4C2×C4×Q811yescharacteristic
296C2×C4#51911yescharacteristic
297C2×C4#54812yes—
298C2×C4D4×Q812yes—
299C8—44no—
300C8—44no—
301C8—82no—
302C8—82no—
303C22Q8×C22⋊C411yescharacteristic
304C22C2×#41311yescharacteristic
305C22#288911yescharacteristic
306C22C2×#42112yes—
307C22#290112yes—
308C4—22no—
309C4—22no—
310C4—24no—
311C4—22no—
312C4—21no—
313C4—21no—
314C4—44no—
315C4—41no—
316C4—42no—
317C4—42no—
318C4—42no—
319C4—42no—
320C4—82no—
321C4—82no—
322C4—82no—
323C2#1767311yescharacteristic
324C2#1768012yes—
325C2#1769011yescharacteristic
326C2#2251711yescharacteristic
327C2#2252912yes—
subgroupG/H
center, socle, prime radicalC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
114. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#365672
#213462
#98942
#98932
#98922
#85802
#82272
#82252