Finite Group Databasebeta

← #117634

Subgroups — #117634

Order 512.

1386 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21949C211yesmaximal, characteristic
2#20599C212yesmaximal
3C2×#1784C211yesmaximal, characteristic
4#8108C212yesmaximal
5#8049C211yesmaximal, characteristic
6C24×Q8C2211yescharacteristic
7C2×#419—24no—
8C22×#101—22no—
9C2×#407C2211yescharacteristic
10C2×#407C2212yes—
11C2×#407C2211yescharacteristic
12#2280C2212yes—
13#1855—24no—
14#1811—22no—
15#1787—24no—
16#1787—24no—
17#1784—22no—
18#1784C414yescomplemented
19#1305—24no—
20#1297—22no—
21C23×Q8—21no—
22C23×Q8—21no—
23C23×Q8—21no—
24C23×Q8—22no—
25C23×Q8—22no—
26C23×Q8—22no—
27C23×Q8—22no—
28C23×Q8—48no—
29C23×Q8C2×C412yes—
30C23×Q8D412yes—
31C23×Q8Q812yes—
32C24×C4—21no—
33C24×C4C2311yescharacteristic
34C22×M16—21no—
35C22×M16D414yes—
36C22×M16D412yes—
37C2×#117—22no—
38C2×#108—24no—
39#419—416no—
40C2×#103—216no—
41C2×#101—28no—
42C2×#101—48no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407—22no—
47#407—22no—
48#407—21no—
49#407—22no—
50#407C2×C412yes—
51#407C2×C412yes—
52#407D412yes—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C22⋊C8—22no—
57#381—22no—
58#380—22no—
59#380D414yes—
60#379—22no—
61C4×C22⋊C4—22no—
62C4×C22⋊C4—22no—
63#341—24no—
64C25#4111yescharacteristic
65C22×Q8—28no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—22no—
69C22×Q8—216no—
70C22×Q8—24no—
71C22×Q8—24no—
72C22×Q8—24no—
73C22×Q8—24no—
74C22×Q8—24no—
75C22×Q8—24no—
76C22×Q8—48no—
77C22×Q8—48no—
78C22×Q8—44no—
79C22×Q8—48no—
80C22×Q8—48no—
81C22×Q8—48no—
82C22×Q8—416no—
83C22×Q8—48no—
84C22×Q8—416no—
85C22×Q8—416no—
86C22×Q8—41no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8—42no—
90C22×Q8—41no—
91C22×Q8—41no—
92C22×Q8—42no—
93C22×Q8—41no—
94C22×Q8C4⋊C414yes—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—44no—
99C23×C4—48no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4C22×C411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×Q811yescharacteristic
109C23×C4#4112yes—
110C23×C4#4112yes—
111C23×C4#4111yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—48no—
117C2×M16—41no—
118C2×M16—44no—
119C2×M16—44no—
120C2×M16C22⋊C414yes—
121C22×C8—22no—
122C22×C8—24no—
123C22×C8—41no—
124#117—48no—
125C2×C4⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C42—42no—
130#108—84no—
131#103—432no—
132#101—416no—
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—24no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139C22⋊C8—42no—
140C22⋊C8—42no—
141#95—24no—
142#95—42no—
143C24—42no—
144C24—44no—
145C24#11711yescharacteristic
146C24#12111yescharacteristic
147C24C22⋊Q811yescharacteristic
148C24#12312yes—
149C24#12412yes—
150C2×Q8—28no—
151C2×Q8—28no—
152C2×Q8—216no—
153C2×Q8—216no—
154C2×Q8—216no—
155C2×Q8—28no—
156C2×Q8—48no—
157C2×Q8—48no—
158C2×Q8—48no—
159C2×Q8—416no—
160C2×Q8—44no—
161C2×Q8—48no—
162C2×Q8—48no—
163C2×Q8—48no—
164C2×Q8—416no—
165C2×Q8—48no—
166C2×Q8—44no—
167C2×Q8—48no—
168C2×Q8—416no—
169C2×Q8—416no—
170C2×Q8—416no—
171C2×Q8—416no—
172C2×Q8—416no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—42no—
176C2×Q8—44no—
177C2×Q8—42no—
178C2×Q8—48no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—28no—
182C22×C4—24no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—44no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—44no—
194C22×C4—416no—
195C22×C4—416no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C2×C4⋊C411yescharacteristic
203C22×C4C4×D412yes—
204C22×C4C4×D412yes—
205C22×C4#12012yes—
206C22×C4#12011yescharacteristic
207C22×C4#12112yes—
208C22×C4#12112yes—
209C22×C4#12111yescharacteristic
210C22×C4#12112yes—
211C22×C4C22⋊Q812yes—
212C22×C4C22⋊Q812yes—
213C22×C4C22⋊Q811yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12312yes—
216C22×C4#12411yescharacteristic
217C22×C4#12612yes—
218C22×C4#12711yescharacteristic
219M16—44no—
220M16—44no—
221M16—48no—
222M16—82no—
223M16—82no—
224M16—84no—
225C2×C8—24no—
226C2×C8—44no—
227C2×C8—48no—
228C2×C8—42no—
229C2×C8—44no—
230C2×C8—82no—
231C4⋊C4—48no—
232C22⋊C4—84no—
233C22⋊C4—84no—
234C22⋊C4—84no—
235C42—84no—
236C23—24no—
237C23—24no—
238C23—24no—
239C23—24no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—416no—
249C23#38011yescharacteristic
250C23#38812yes—
251C23#39311yescharacteristic
252C23#39412yes—
253C23#39911yescharacteristic
254Q8—216no—
255Q8—48no—
256Q8—48no—
257Q8—416no—
258Q8—48no—
259Q8—48no—
260Q8—48no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—28no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—416no—
282C2×C4—416no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4#38512yes—
289C2×C4#38611yescharacteristic
290C2×C4#39011yescharacteristic
291C2×C4#39211yescharacteristic
292C2×C4#39612yes—
293C2×C4#39711yescharacteristic
294C2×C4B(4,𝔽2)14yes—
295C8—44no—
296C8—82no—
297C8—84no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—44no—
309C22—48no—
310C22—48no—
311C22—44no—
312C22—44no—
313C22#143911yescharacteristic
314C22#202912yes—
315C22#204914yes—
316C4—21no—
317C4—22no—
318C4—22no—
319C4—21no—
320C4—48no—
321C4—164no—
322C4#188912yes—
323C4#202112yes—
324C2—24no—
325C2—22no—
326C2—41no—
327C2—42no—
328C2—41no—
329C2#1106911yescharacteristic
330C2#1114312yes—
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #1784C4semidirect

Maximal subgroups

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

subgroupindex
#219492
#205992
C2×#17842
#81082
#80492