Finite Group Databasebeta

← #224720

Subgroups — #224720

Order 512.

841 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#36593C211yesmaximal, characteristic
2#22823C212yesmaximal
3#9702C212yesmaximal
4#9702C212yesmaximal
5#9691C212yesmaximal
6#8594C211yesmaximal, characteristic
7#8590C212yesmaximal
8#8235C212yesmaximal
9#8232C211yesmaximal, characteristic
10C2×#557C412yes—
11#2868C2212yes—
12#2868C2211yescharacteristic
13#2552—22no—
14#2552—22no—
15#2552—22no—
16#2546C414yes—
17#2546C412yes—
18Q8×C4⋊C4—22no—
19C4×C4⋊Q8C2211yescharacteristic
20C4×C4⋊Q8C2212yes—
21C42×Q8C2211yescharacteristic
22#1985—22no—
23#1917—22no—
24#1916—22no—
25#1842C2212yes—
26#1831C2212yes—
27#1831C2212yes—
28#1771—22no—
29#1767C2212yes—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32C2×#333C2212yes—
33C2×#333C2212yes—
34C2×#333C2212yes—
35C2×#333C2212yes—
36C2×#333C2212yes—
37C2×#333C2211yescharacteristic
38C2×#333C2211yescharacteristic
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#557—48no—
43C2×C4⋊Q8C2×C414yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C414yes—
46C2×C4⋊Q8C2×C412yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2312yes—
49C2×#125—24no—
50C2×#125—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8C2×C414yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×#97—22no—
63C2×#97—22no—
64C2×C4×C8—22no—
65#398—22no—
66#398—24no—
67#398—24no—
68#398—22no—
69#391C2×C412yes—
70#391C2×C414yes—
71#389—24no—
72#389—24no—
73#389—24no—
74#389—22no—
75#389—22no—
76#389—22no—
77#384—22no—
78#384—22no—
79#384—22no—
80#384—22no—
81#384—22no—
82#384—22no—
83#384C2311yescharacteristic
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C411yescharacteristic
88C4×C4⋊C4C2×C412yes—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92C43C2311yescharacteristic
93C43C2312yes—
94#336—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#336—22no—
99#336—22no—
100#336—22no—
101#333—24no—
102#333—24no—
103#333—24no—
104#333—42no—
105#333—42no—
106#333—41no—
107#333—41no—
108C22×Q8C22⋊C411yescharacteristic
109C22×Q8C22×C414yes—
110C22×Q8C22×C412yes—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—22no—
120C4⋊Q8—21no—
121C4⋊Q8—22no—
122C4⋊Q8—48no—
123C4⋊Q8—44no—
124C4⋊Q8—84no—
125C4⋊Q8—82no—
126#125—416no—
127#125—48no—
128C4×Q8—24no—
129C4×Q8—24no—
130C4×Q8—24no—
131C4×Q8—48no—
132C4×Q8—44no—
133C4×Q8—48no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22×C412yes—
153C2×C42—24no—
154C2×C42—22no—
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C412yes—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2411yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—44no—
174C4⋊C8—82no—
175C4⋊C8#4114yes—
176#97—44no—
177#97—82no—
178C4×C8—82no—
179#95—24no—
180#95—24no—
181#95—24no—
182#95C22⋊C411yescharacteristic
183#95C22⋊C412yes—
184#95C22⋊C412yes—
185#95C22⋊C411yescharacteristic
186C2×Q8—22no—
187C2×Q8—24no—
188C2×Q8—24no—
189C2×Q8—22no—
190C2×Q8—24no—
191C2×Q8—22no—
192C2×Q8—44no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—44no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C412yes—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C411yescharacteristic
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C23×C411yescharacteristic
207C22×C4C22×D412yes—
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—
219C2×C8—42no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—48no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—48no—
238C4⋊C4—48no—
239C4⋊C4—84no—
240C4⋊C4—84no—
241C4⋊C4—84no—
242C4⋊C4#10014yes—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—24no—
254C42—22no—
255C42—42no—
256C42—41no—
257C42—48no—
258C42—44no—
259C42#13714yes—
260C42#13714yes—
261C42C2×#4112yes—
262C42H(5,𝔽2)11yescharacteristic
263C422−1+411yescharacteristic
264C23C22×C22⋊C411yescharacteristic
265Q8—28no—
266Q8—84no—
267Q8—82no—
268Q8—84no—
269Q8—82no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4C2×#10012yes—
296C2×C4C2×#10014yes—
297C2×C4#41914yes—
298C2×C4#41914yes—
299C2×C4C4×#4111yescharacteristic
300C2×C4#52011yescharacteristic
301C2×C4#54812yes—
302C2×C4#55012yes—
303C2×C4C2×#13712yes—
304C2×C4C2×#13712yes—
305C2×C4#57712yes—
306C2×C4#57812yes—
307C8—44no—
308C8—82no—
309C8—82no—
310C8—82no—
311C22#234111yescharacteristic
312C22C22×#10011yescharacteristic
313C22#288811yescharacteristic
314C22C2×#41912yes—
315C22#289812yes—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—24no—
320C4—24no—
321C4—24no—
322C4—42no—
323C4—42no—
324C4—41no—
325C4—84no—
326C4—82no—
327C4—82no—
328C4#337212yes—
329C4#337412yes—
330C4#339812yes—
331C4#340012yes—
332C2#1774611yescharacteristic
333C2#1776012yes—
334C2#1777911yescharacteristic
335C2#2253611yescharacteristic
336C2#2254712yes—
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

Maximal subgroups

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

subgroupindex
#365932
#228232
#97022
#96912
#85942
#85902
#82352
#82322