Finite Group Databasebeta

← #224185

Subgroups — #224185

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36565C211yesmaximal, characteristic
2#21345C212yesmaximal
3#9895C212yesmaximal
4#9894C212yesmaximal
5#9890C212yesmaximal
6#8584C211yesmaximal, characteristic
7#8582C212yesmaximal
8#8227C212yesmaximal
9#8225C211yesmaximal, characteristic
10#2862C2212yes—
11#2859C2211yescharacteristic
12#2586—22no—
13#2586—22no—
14#2551C412yes—
15#2550C412yes—
16#2550C412yes—
17#2451—22no—
18#2369—22no—
19#2355C412yes—
20C4×#125C2211yescharacteristic
21C4×#125C2212yes—
22C42×Q8C2211yescharacteristic
23#1984—22no—
24#1939—22no—
25#1917—22no—
26#1917—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#1295C2212yes—
40#1295C2212yes—
41#1295C2211yescharacteristic
42#1295C2211yescharacteristic
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×#125C2×C411yescharacteristic
47C2×#125C2×C411yescharacteristic
48C2×#125C2311yescharacteristic
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8D412yes—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—22no—
54C2×#97—22no—
55C2×#97C2312yes—
56C2×#97C2312yes—
57C2×#97C2312yes—
58C2×#97C2312yes—
59C2×C4×C8—22no—
60#398—22no—
61#398—22no—
62#395—22no—
63#395—22no—
64#395—22no—
65#395—22no—
66#391—22no—
67#391—22no—
68#391—22no—
69#391—22no—
70#391—22no—
71#391—22no—
72#391—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389C2×C412yes—
79#389C2×C412yes—
80#389C2×C412yes—
81#387—22no—
82#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#384C2×C412yes—
87#382C2×C412yes—
88#382C2×C412yes—
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×C412yes—
99C4×C4⋊C4C2×C411yescharacteristic
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×Q8—22no—
116C22×Q8—22no—
117C22×Q8C22⋊C411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C4⋊Q8—48no—
123C4⋊Q8—82no—
124C4⋊Q8—82no—
125#125—22no—
126#125—81no—
127#125—81no—
128#125C2×Q812yes—
129#125#4112yes—
130C4×Q8—24no—
131C4×Q8—44no—
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⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
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×C4⋊C4C22×C412yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C412yes—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C22×C412yes—
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2411yescharacteristic
165C4⋊C8—44no—
166C4⋊C8—44no—
167C4⋊C8—44no—
168#97—22no—
169#97—22no—
170#97—22no—
171#97—22no—
172#97—22no—
173#97—22no—
174#97—22no—
175#97—44no—
176#97C2×Q814yes—
177C4×C8—44no—
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#95—22no—
189#95—22no—
190#95—22no—
191#95C22⋊C412yes—
192#95C22⋊C412yes—
193#95C22⋊C411yescharacteristic
194#95C22⋊C411yescharacteristic
195C2×Q8—42no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8#10114yes—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C412yes—
214C22×C4C23×C411yescharacteristic
215C22×C4C22×D412yes—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C4⋊C4—22no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4C4×Q814yes—
249C42—24no—
250C42—22no—
251C42—24no—
252C42—42no—
253C42—42no—
254C42—42no—
255C42—41no—
256C42—42no—
257C42—42no—
258C42—42no—
259C42—44no—
260C42—82no—
261C42—82no—
262C42C22×Q811yescharacteristic
263C42C2×#4111yescharacteristic
264C422−1+412yes—
265C23C22×C22⋊C411yescharacteristic
266Q8—44no—
267Q8—164no—
268Q8—164no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4C2×#10114yes—
302C2×C4C2×#10112yes—
303C2×C4#42114yes—
304C2×C4C2×C4×Q811yescharacteristic
305C2×C4#51911yescharacteristic
306C2×C4#54812yes—
307C2×C4D4×Q812yes—
308C8—44no—
309C8—44no—
310C8—82no—
311C8—82no—
312C22Q8×C22⋊C411yescharacteristic
313C22C22×#10111yescharacteristic
314C22#288811yescharacteristic
315C22C2×#42112yes—
316C22#290112yes—
317C4—22no—
318C4—22no—
319C4—24no—
320C4—22no—
321C4—24no—
322C4—21no—
323C4—21no—
324C4—44no—
325C4—42no—
326C4—42no—
327C4—42no—
328C4—41no—
329C4—82no—
330C4—82no—
331C4—82no—
332C2#1767611yescharacteristic
333C2#1767912yes—
334C2#1769211yescharacteristic
335C2Q8×#10111yescharacteristic
336C2#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
#365652
#213452
#98952
#98942
#98902
#85842
#85822
#82272
#82252