Finite Group Databasebeta

← #115653

Subgroups — #115653

Order 512.

1174 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 154 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
1C2×#2547C211yesmaximal, characteristic
2#21829C211yesmaximal, characteristic, complemented
3#21829C211yesmaximal, characteristic, complemented
4#17448C212yesmaximal
5#7950C212yesmaximal, complemented
6C22×C4⋊Q8C2211yescharacteristic
7C22×#124C412yes—
8C22×C4×D4C2211yescharacteristic
9#2920C2212yes—
10#2920C2212yes—
11#2547—24no—
12#2547—24no—
13C2×#397—22no—
14C2×#391C412yes—
15C2×#380—22no—
16C2×C4×C4⋊C4C2211yescharacteristic
17#1997—24no—
18#1997—24no—
19#1821—22no—
20#1821—22no—
21#1490—24no—
22#1486—24no—
23#1478—24no—
24#1285—24no—
25C23×Q8C2×C412yes—
26C23×D4C2×C411yescharacteristic
27C24×C4—21no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8—22no—
34C2×#124—24no—
35C2×#124—24no—
36C2×#124—48no—
37C2×C4×D4—22no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4D414yes—
43C22×C4⋊C4—21no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—21no—
49C22×C42C2×C412yes—
50C22×C42C2311yescharacteristic
51#432—22no—
52#432—22no—
53#432—22no—
54#432—22no—
55#431—22no—
56#431—22no—
57#431—22no—
58#431—22no—
59#423—22no—
60#423D414yes—
61#407—22no—
62#407—22no—
63C4×M16—22no—
64C4×M16D414yes—
65#397—216no—
66#391—24no—
67#391—24no—
68#380—216no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4D414yes—
71C2×#95—24no—
72C2×#95—21no—
73#343—28no—
74#343—28no—
75D4⋊C8—216no—
76C25—21no—
77C22×Q8—22no—
78C22×Q8—22no—
79C22×Q8—24no—
80C22×Q8—24no—
81C22×Q8—44no—
82C22×Q8—44no—
83C22×D4—24no—
84C22×D4—21no—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4—42no—
88C22×D4C22⋊C412yes—
89C23×C4—22no—
90C23×C4—21no—
91C23×C4—24no—
92C23×C4—48no—
93C23×C4—41no—
94C23×C4—42no—
95C23×C4—41no—
96C23×C4C22⋊C414yes—
97C23×C4C22×C411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C2×M16—24no—
101C2×M16—24no—
102C4⋊Q8—42no—
103C4⋊Q8—42no—
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110#124—416no—
111#124—48no—
112#124—48no—
113C4×D4—24no—
114C4×D4—24no—
115C4×D4—24no—
116C4×D4—24no—
117C4×D4—42no—
118C4×D4—42no—
119C4×D4—42no—
120C4×D4—42no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4C22⋊C412yes—
135C2×C22⋊C4—216no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—416no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—41no—
141C2×C42—22no—
142C2×C42—44no—
143C2×C42C22⋊C414yes—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42#4112yes—
149C2×C42#4112yes—
150C4⋊C8—24no—
151C4⋊C8—24no—
152C4⋊C8—42no—
153C4⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157C22⋊C8—42no—
158#97—24no—
159#97—42no—
160C4×C8—24no—
161C4×C8—42no—
162#95—216no—
163#95—24no—
164C24—24no—
165C24—22no—
166C24—44no—
167C24—42no—
168C24—44no—
169C24—41no—
170C24—41no—
171C24C2×C22⋊C411yescharacteristic
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—88no—
179C2×Q8—88no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—48no—
185C2×D4—48no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4#10014yes—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—416no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—84no—
210C22×C4—88no—
211C22×C4—84no—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C4×D412yes—
215C22×C4C4×D412yes—
216C22×C4#12011yescharacteristic
217C22×C4#12012yes—
218C22×C4#12012yes—
219C22×C4#12011yescharacteristic
220C22×C4#12112yes—
221C22×C4#12112yes—
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224M16—48no—
225M16—48no—
226C2×C8—44no—
227C2×C8—44no—
228C4⋊C4—24no—
229C4⋊C4—48no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C22⋊C4—432no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C42—44no—
239C42—44no—
240C42#12012yes—
241C42#12112yes—
242C42#12112yes—
243C42#12112yes—
244C42#12112yes—
245C42#12312yes—
246C42#12412yes—
247C42#12712yes—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—24no—
253C23—44no—
254C23—44no—
255C23—44no—
256C23—416no—
257C23—44no—
258C23—42no—
259C23—42no—
260C23—82no—
261C23—88no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23#37911yescharacteristic
266C23#38611yescharacteristic
267C23#38611yescharacteristic
268Q8—88no—
269Q8—88no—
270D4—28no—
271D4—48no—
272D4—44no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—88no—
293C2×C4—88no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—88no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4#38612yes—
300C2×C4#39012yes—
301C2×C4#39212yes—
302C2×C4#39212yes—
303C2×C4#39412yes—
304C2×C4#39412yes—
305C2×C4C2×#10012yes—
306C2×C4#41312yes—
307C2×C4#45414yes—
308C8—84no—
309C8—84no—
310C22—24no—
311C22—24no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—48no—
319C22—44no—
320C22—88no—
321C22—88no—
322C22—84no—
323C22—82no—
324C22—82no—
325C22#143811yescharacteristic
326C22#178311yescharacteristic
327C22#178511yescharacteristic
328C22#188812yes—
329C22#191212yes—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—84no—
335C4—84no—
336C4—82no—
337C4#190014yes—
338C4#200314yes—
339C2—21no—
340C2—22no—
341C2—22no—
342C2—21no—
343C2—84no—
344C2#1096711yescharacteristic
345C2#1097611yescharacteristic
346C2#1099011yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21829C2semidirect
3. #21829C2semidirect
5. #7950C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
#218292
#174482
#79502