Finite Group Databasebeta

← #215722

Subgroups — #215722

Order 512.

828 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic
2#21924C212yesmaximal
3#9906C212yesmaximal, complemented
4#9905C212yesmaximal, complemented
5#9901C212yesmaximal, complemented
6#9896C212yesmaximal, complemented
7#8728C211yesmaximal, characteristic
8#8728C211yesmaximal, characteristic
9#8724C212yesmaximal
10C2×#544C412yes—
11#2868C2212yes—
12#2859C2212yes—
13#2660C414yes—
14#2552—22no—
15#2551—22no—
16#2440—22no—
17#2440—22no—
18#2317C412yes—
19#2285C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#1924—22no—
23#1923—22no—
24#1917—22no—
25#1916—22no—
26#1842C2212yes—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32#1296C2212yes—
33#1296C2212yes—
34#1296C2212yes—
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1296C2212yes—
38#1296C2211yescharacteristic
39#1296C2212yes—
40#1296C2211yescharacteristic
41#1296C2211yescharacteristic
42#544—48no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2×C414yes—
47C2×#125—22no—
48C2×#125C2×C414yes—
49C2×C22⋊Q8—24no—
50C2×#117C2×C412yes—
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2×C412yes—
55C22×C42C2311yescharacteristic
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×#97—22no—
65C2×C4×C8—22no—
66#398—24no—
67#398—24no—
68#395—24no—
69#395—24no—
70#389—24no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#385—22no—
76#385—22no—
77#384—22no—
78#384—22no—
79#384—22no—
80#384C2×C414yes—
81#384C2×C412yes—
82#384C2311yescharacteristic
83#384C2312yes—
84#384C2311yescharacteristic
85#382C2×C414yes—
86#382C2×C412yes—
87#380—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C22⋊C4D412yes—
92C4×C22⋊C4D412yes—
93#376C2×C412yes—
94C43C2312yes—
95#336—22no—
96#336—22no—
97#336—22no—
98#336—22no—
99#336—22no—
100#336—22no—
101C22×Q8—24no—
102C23×C4—22no—
103C23×C4C22×C411yescharacteristic
104C23×C4C2×D412yes—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C4⋊Q8—48no—
110C4⋊Q8—48no—
111C4⋊Q8—82no—
112C4⋊Q8—82no—
113C4⋊Q8—82no—
114C4⋊Q8—82no—
115#125—48no—
116#125—48no—
117C22⋊Q8—416no—
118#117—24no—
119#117—24no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4C22⋊C411yescharacteristic
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C4⋊C4C22×C414yes—
139C2×C4⋊C4C22×C412yes—
140C2×C4⋊C4C22×C412yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4C22⋊C412yes—
143C2×C22⋊C4C22⋊C411yescharacteristic
144C2×C22⋊C4C22⋊C411yescharacteristic
145C2×C42—21no—
146C2×C42—22no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42C22×C412yes—
150C2×C42C22×C412yes—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C2×C42C2×D412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2411yescharacteristic
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—82no—
167C4⋊C8—82no—
168C22⋊C8—82no—
169C22⋊C8—82no—
170#97—82no—
171C4×C8—82no—
172#95—24no—
173#95—24no—
174#95—24no—
175#95C22⋊C412yes—
176#95C22⋊C412yes—
177#95C22⋊C412yes—
178#95C22⋊C412yes—
179C24C2×C22⋊C411yescharacteristic
180C2×Q8—48no—
181C2×Q8—84no—
182C2×Q8—84no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—44no—
197C22×C4—84no—
198C22×C4C2×C22⋊C412yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×C22⋊C412yes—
204C22×C4C23×C411yescharacteristic
205C22×C4C22×D412yes—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
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—
218C4⋊C4—22no—
219C4⋊C4—24no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—48no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—48no—
234C4⋊C4—48no—
235C4⋊C4—84no—
236C4⋊C4—84no—
237C4⋊C4—84no—
238C4⋊C4#10014yes—
239C22⋊C4—22no—
240C22⋊C4—24no—
241C22⋊C4—44no—
242C22⋊C4#10114yes—
243C42—21no—
244C42—21no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—24no—
256C42—24no—
257C42—44no—
258C422−1+411yescharacteristic
259C422−1+412yes—
260C422−1+411yescharacteristic
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23C2×#10112yes—
266C23C22×C22⋊C411yescharacteristic
267Q8—168no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
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—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—44no—
298C2×C4—164no—
299C2×C4C2×#10014yes—
300C2×C4C2×#10012yes—
301C2×C4C2×#10114yes—
302C2×C4#51911yescharacteristic
303C2×C4#51911yescharacteristic
304C2×C4#53612yes—
305C2×C4#53612yes—
306C2×C4#57612yes—
307C2×C4#57712yes—
308C2×C4#57712yes—
309C2×C4#57812yes—
310C8—82no—
311C8—82no—
312C8—82no—
313C8—82no—
314C22—22no—
315C22—24no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22#231011yescharacteristic
320C22C22×#10011yescharacteristic
321C22C22×#10111yescharacteristic
322C22#289712yes—
323C22#289812yes—
324C4—21no—
325C4—21no—
326C4—22no—
327C4—24no—
328C4—24no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—82no—
333C4—84no—
334C4—82no—
335C4#324112yes—
336C4#324212yes—
337C4#324312yes—
338C4#324412yes—
339C2—42no—
340C2#1777412yes—
341C2#1777911yescharacteristic
342C2#1778211yescharacteristic
343C2#2182911yescharacteristic
344C2#2184312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9906C2semidirect
4. #9905C2semidirect
5. #9901C2semidirect
6. #9896C2semidirect

Maximal subgroups

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

subgroupindex
#352752
#219242
#99062
#99052
#99012
#98962
#87282
#87242