Finite Group Databasebeta

← #224712

Subgroups — #224712

Order 512.

883 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#22825C212yesmaximal, complemented
3#9707C212yesmaximal
4#9702C212yesmaximal, complemented
5#9699C212yesmaximal
6#8592C211yesmaximal, characteristic
7#8590C212yesmaximal, complemented
8#8235C212yesmaximal, complemented
9#8231C211yesmaximal, characteristic
10C2×#556C412yes—
11#2868C2211yescharacteristic
12#2858C2212yes—
13#2547C412yes—
14#2545C414yes—
15#2538—22no—
16#2512—22no—
17#2512—22no—
18#2354—22no—
19C4×C4⋊Q8C2212yes—
20C4×#127C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#1984—22no—
23#1917—22no—
24#1917—22no—
25#1842C2212yes—
26#1831C2212yes—
27#1831C2212yes—
28#1770—22no—
29#1767C2212yes—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32C2×#333C2212yes—
33C2×#333C2212yes—
34C2×#333C2212yes—
35C2×#333C2212yes—
36C2×#331C2212yes—
37C2×#331C2211yescharacteristic
38C2×#331C2211yescharacteristic
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#556—48no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2312yes—
46C2×#127C2311yescharacteristic
47C2×#126—24no—
48C2×#124C2×C414yes—
49C2×#124C2×C414yes—
50C2×C22⋊Q8—22no—
51C2×C4×Q8C2×C414yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×D4—22no—
55C2×C4×D4C2×C412yes—
56C2×#117—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×#97—22no—
64C2×#97—22no—
65C2×C4×C8—22no—
66#397—22no—
67#397—22no—
68#396—24no—
69#396—24no—
70#391C2×C414yes—
71#390—22no—
72#390—22no—
73#390—22no—
74#390—22no—
75#390—22no—
76#390C2×C412yes—
77#389—22no—
78#389—22no—
79#388—24no—
80#388—24no—
81#384—22no—
82#384—22no—
83#384C2311yescharacteristic
84#380—24no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4C2×C411yescharacteristic
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—42no—
104#333—42no—
105#331—24no—
106#331—41no—
107#331—41no—
108C22×Q8C22⋊C411yescharacteristic
109C22×Q8C22×C414yes—
110C22×D4—22no—
111C22×D4C22×C412yes—
112C23×C4—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—44no—
122C4⋊Q8—82no—
123#127—22no—
124#127—21no—
125#127—21no—
126#126—416no—
127#124—48no—
128#124—84no—
129C22⋊Q8—48no—
130C4×Q8—24no—
131C4×Q8—42no—
132C4×Q8—42no—
133C4×Q8—48no—
134C4×D4—44no—
135C4×D4—44no—
136C4×D4—44no—
137#117—48no—
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⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C412yes—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22×C412yes—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C42—24no—
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C412yes—
160C2×C42C22×C412yes—
161C2×C42C22×C411yescharacteristic
162C2×C42C2×D412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2411yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—44no—
177C4⋊C8—82no—
178C4⋊C8#4114yes—
179#97—82no—
180#97—82no—
181C4×C8—44no—
182#95—24no—
183#95—24no—
184#95C22⋊C412yes—
185#95C22⋊C412yes—
186#95C22⋊C411yescharacteristic
187#95C22⋊C411yescharacteristic
188C24—22no—
189C2×Q8—24no—
190C2×Q8—24no—
191C2×Q8—42no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×Q8#10014yes—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—48no—
208C22×C4—44no—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C23×C411yescharacteristic
217C22×C4C22×D412yes—
218C2×C8—24no—
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—
228C2×C8—42no—
229C2×C8—42no—
230C4⋊C4—22no—
231C4⋊C4—24no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—48no—
245C4⋊C4—84no—
246C22⋊C4—48no—
247C22⋊C4—48no—
248C22⋊C4—44no—
249C22⋊C4—48no—
250C22⋊C4—84no—
251C22⋊C4—84no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—24no—
266C42—42no—
267C42—41no—
268C42—48no—
269C42C2×#4112yes—
270C42H(5,𝔽2)11yescharacteristic
271C422−1+411yescharacteristic
272C23—42no—
273C23—42no—
274C23—42no—
275C23—44no—
276C23—44no—
277C23C22×C22⋊C411yescharacteristic
278Q8—44no—
279Q8—84no—
280Q8—84no—
281D4—82no—
282D4—82no—
283D4—82no—
284D4—82no—
285D4—82no—
286D4—82no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—88no—
312C2×C4C2×#10014yes—
313C2×C4C2×#10012yes—
314C2×C4C4×#4111yescharacteristic
315C2×C4#52011yescharacteristic
316C2×C4#54812yes—
317C2×C4#55012yes—
318C2×C4#57512yes—
319C2×C4#57512yes—
320C2×C4#57712yes—
321C2×C4#57812yes—
322C8—44no—
323C8—82no—
324C8—82no—
325C8—82no—
326C22—82no—
327C22—84no—
328C22—82no—
329C22—84no—
330C22—82no—
331C22#234111yescharacteristic
332C22C22×#10011yescharacteristic
333C22#288811yescharacteristic
334C22#289612yes—
335C22#289812yes—
336C4—21no—
337C4—21no—
338C4—22no—
339C4—24no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—41no—
345C4—84no—
346C4—82no—
347C4#337612yes—
348C4#337712yes—
349C4#340112yes—
350C4#340212yes—
351C2—162no—
352C2#1775411yescharacteristic
353C2#1776812yes—
354C2#1777311yescharacteristic
355C2#2253611yescharacteristic
356C2#2254812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
2. #22825C2semidirect
4. #9702C2semidirect
7. #8590C2semidirect
8. #8235C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#228252
#97072
#97022
#96992
#85922
#85902
#82352
#82312