Finite Group Databasebeta

← #213513

Subgroups — #213513

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2323C211yesmaximal, characteristic
2#21818C212yesmaximal, complemented
3#9082C214yesmaximal, complemented
4#9080C214yesmaximal
5#8701C212yesmaximal, complemented
6#8300C212yesmaximal
7C22×C22⋊Q8C411yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#431C2214yes—
11#2323—28no—
12C2×#391C411yescharacteristic
13C2×#389C411yescharacteristic
14C2×#389C412yes—
15C2×#388C412yes—
16C2×#388C2212yes—
17C2×#387C2212yes—
18C2×#380C411yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1984—28no—
21#1817C2214yes—
22#1817C2214yes—
23C2×#324C2214yes—
24C2×#323C2214yes—
25#1280C2218yes—
26C23×Q8C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C2×C22⋊Q8—24no—
29C2×C22⋊Q8—24no—
30C2×C22⋊Q8C818yescomplemented
31C2×C4×Q8—44no—
32C2×C4×Q8D418yes—
33C2×#117—24no—
34C2×#117—44no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2311yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2311yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42C2×C412yes—
49C22×C42C2311yescharacteristic
50C22×C42C2312yes—
51#431—48no—
52C2×C4⋊C8—24no—
53C2×C22⋊C8C2318yes—
54C2×C4×C8—24no—
55#391—24no—
56#389—28no—
57#389—24no—
58#388—28no—
59#388—24no—
60#388—24no—
61#387—24no—
62#387—24no—
63#380—24no—
64C4×C4⋊C4—24no—
65C2×#95C2×C412yes—
66C2×#95C2×C412yes—
67C2×#95C2×C412yes—
68C2×#95C2×C412yes—
69C2×#95C2312yes—
70#336—28no—
71#336—28no—
72#324—44no—
73#323—44no—
74C25C22×C411yescharacteristic
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8C22⋊C414yes—
78C22×Q8C2×C814yes—
79C22×Q8M1614yes—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—42no—
83C23×C4—42no—
84C23×C4C2×C814yes—
85C23×C4C22×C411yescharacteristic
86C23×C4C22×C412yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C412yes—
89C23×C4C22×C411yescharacteristic
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C22×C412yes—
94C23×C4C2411yescharacteristic
95C22×C8—28no—
96C22⋊Q8—232no—
97C4×Q8—48no—
98C4×Q8—48no—
99#117—416no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—82no—
114C2×C4⋊C4C22⋊C414yes—
115C2×C4⋊C4C22⋊C418yes—
116C2×C4⋊C4C2×C818yes—
117C2×C4⋊C4C2×C814yes—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—82no—
126C2×C22⋊C4C2×C818yes—
127C2×C42—24no—
128C2×C42—28no—
129C2×C42—22no—
130C2×C42—24no—
131C2×C42—42no—
132C2×C42—82no—
133C2×C42C2×D414yes—
134C2×C42C2×D414yes—
135C2×C42C2×D414yes—
136C4⋊C8—48no—
137C22⋊C8—28no—
138C22⋊C8—28no—
139C4×C8—48no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—22no—
145#95—22no—
146C24—21no—
147C24—21no—
148C24—22no—
149C24—22no—
150C24—41no—
151C24—41no—
152C24—41no—
153C24—41no—
154C24C22×C812yes—
155C24C23×C411yescharacteristic
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—48no—
159C2×Q8—48no—
160C2×Q8C22⋊C8116yes—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—41no—
181C22×C4—48no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—48no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4C2×C22⋊C414yes—
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C414yes—
201C22×C4C2×C22⋊C412yes—
202C22×C4C22×C812yes—
203C22×C4C22×C814yes—
204C22×C4C22×C812yes—
205C22×C4C22×C814yes—
206C22×C4C2×M1614yes—
207C22×C4C2×M1612yes—
208C22×C4C22×D412yes—
209C22×C4H(5,𝔽2)14yes—
210C2×C8—48no—
211C2×C8—48no—
212C2×C8—48no—
213C4⋊C4—28no—
214C4⋊C4—216no—
215C4⋊C4—44no—
216C4⋊C4—48no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—84no—
221C22⋊C4—216no—
222C22⋊C4—44no—
223C22⋊C4—44no—
224C22⋊C4—84no—
225C42—24no—
226C42—24no—
227C42—28no—
228C42—28no—
229C42—44no—
230C42—84no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—24no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—42no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23—82no—
252C23C22×C22⋊C411yescharacteristic
253C23#51812yes—
254C23C23×C811yescharacteristic
255C23C22×M1611yescharacteristic
256C23#56812yes—
257Q8—416no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—28no—
266C2×C4—24no—
267C2×C4—28no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—44no—
278C2×C4—416no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4C2×C22⋊C818yes—
301C2×C4C2×C22⋊C814yes—
302C2×C4#53414yes—
303C8—88no—
304C22—22no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—84no—
316C22—84no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22#230811yescharacteristic
322C22C22×C22⋊C811yescharacteristic
323C22#287911yescharacteristic
324C22#288411yescharacteristic
325C22#288411yescharacteristic
326C22#288811yescharacteristic
327C22#288911yescharacteristic
328C22#297212yes—
329C22#297312yes—
330C4—24no—
331C4—28no—
332C4—24no—
333C4—44no—
334C4—42no—
335C4—42no—
336C4—44no—
337C4—82no—
338C4—81no—
339C4—81no—
340C2—22no—
341C2—22no—
342C2—82no—
343C2#1735211yescharacteristic
344C2#1735311yescharacteristic
345C2#1740911yescharacteristic
346C2#1741011yescharacteristic
347C2#2173811yescharacteristic
348C2#2175111yescharacteristic
349C2#2175711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21818C2semidirect
3. #9082C2semidirect
5. #8701C2semidirect
30. C2×C22⋊Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23232
#218182
#90822
#90802
#87012
#83002