Finite Group Databasebeta

← #115307

Subgroups — #115307

Order 512.

1148 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 150 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×#2502C211yesmaximal, characteristic
2#21825C211yesmaximal, characteristic
3#21753C211yesmaximal, characteristic
4#20700C212yesmaximal
5#8051C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9#2502—24no—
10#2502—24no—
11C2×#398C414yescomplemented
12C2×#391—21no—
13C2×#385—22no—
14C2×#384—21no—
15#2280C2212yes—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1996—24no—
18#1901—24no—
19#1824—24no—
20#1823—24no—
21#1821—21no—
22#1819—22no—
23#1784—21no—
24#1736—24no—
25#1297—24no—
26C23×Q8D411yescharacteristic
27C24×C4—21no—
28C24×C4C2311yescharacteristic
29C22×M16D414yes—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×#117—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—24no—
44C22×C4⋊C4C2×C412yes—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D414yes—
48C22×C22⋊C4—21no—
49C22×C22⋊C4D412yes—
50C22×C42—21no—
51C2×#101—24no—
52#407—22no—
53#407—22no—
54#407—22no—
55#407—22no—
56C2×C22⋊C8—22no—
57C2×C22⋊C8—22no—
58#398—28no—
59#398—28no—
60#391—28no—
61#385—216no—
62#384—28no—
63#381—22no—
64#380—22no—
65#380D414yes—
66#379—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—22no—
70C4×C22⋊C4D414yes—
71C2×#95—24no—
72C2×#95C2×C412yes—
73#343—24no—
74#342—28no—
75#340—216no—
76C25C2×D411yescharacteristic
77C22×Q8—22no—
78C22×Q8—21no—
79C22×Q8—22no—
80C22×Q8—21no—
81C22×Q8—44no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—24no—
85C23×C4—44no—
86C23×C4—41no—
87C23×C4—42no—
88C23×C4—41no—
89C23×C4—84no—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4#4112yes—
99C23×C4#4112yes—
100C2×M16—24no—
101C2×M16—24no—
102C2×M16—48no—
103C22×C8—24no—
104C22⋊Q8—42no—
105C22⋊Q8—42no—
106C22⋊Q8—42no—
107C22⋊Q8—42no—
108C22⋊Q8—42no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111C22⋊Q8—42no—
112#117—48no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—216no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—88no—
139C2×C4⋊C4C22⋊C414yes—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C42—24no—
150C2×C42—44no—
151C2×C42—42no—
152#101—84no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156C22⋊C8—42no—
157#95—216no—
158#95—22no—
159#95—24no—
160#95—42no—
161#95C22⋊C414yes—
162C24—24no—
163C24—42no—
164C24—42no—
165C24C2×C22⋊C411yescharacteristic
166C24#12012yes—
167C24#12011yescharacteristic
168C24#12112yes—
169C24#12311yescharacteristic
170C2×Q8—48no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—44no—
174C2×Q8—82no—
175C2×Q8—84no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—48no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—88no—
204C22×C4—84no—
205C22×C4—88no—
206C22×C4—84no—
207C22×C4C2×C22⋊C411yescharacteristic
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C4×D412yes—
210C22×C4C4×D412yes—
211C22×C4#12012yes—
212C22×C4#12012yes—
213C22×C4#12011yescharacteristic
214C22×C4#12112yes—
215C22×C4#12112yes—
216C22×C4#12112yes—
217C22×C4#12112yes—
218C22×C4#12112yes—
219C22×C4#12311yescharacteristic
220C22×C4#12312yes—
221C22×C4#12412yes—
222C22×C4#12712yes—
223M16—48no—
224M16—84no—
225C2×C8—48no—
226C2×C8—44no—
227C4⋊C4—28no—
228C4⋊C4—48no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—44no—
235C4⋊C4—42no—
236C4⋊C4—816no—
237C22⋊C4—216no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—48no—
241C22⋊C4—84no—
242C22⋊C4—84no—
243C42—48no—
244C42—84no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—24no—
249C23—22no—
250C23—24no—
251C23—24no—
252C23—24no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—42no—
258C23—41no—
259C23—416no—
260C23#37911yescharacteristic
261C23#38611yescharacteristic
262C23#38611yescharacteristic
263C23#39212yes—
264C23#39412yes—
265C23#45614yes—
266Q8—84no—
267Q8—84no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—28no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—416no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—84no—
295C2×C4—84no—
296C2×C4—88no—
297C2×C4—88no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4#38612yes—
305C2×C4#39012yes—
306C2×C4#39212yes—
307C2×C4#39412yes—
308C2×C4#45814yes—
309C8—84no—
310C22—28no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—44no—
323C22—48no—
324C22—48no—
325C22—44no—
326C22#143811yescharacteristic
327C22#179512yes—
328C22#188612yes—
329C22#189012yes—
330C22#200514yes—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—48no—
335C4—82no—
336C4—84no—
337C4—82no—
338C4—164no—
339C4#202214yes—
340C2—22no—
341C2—24no—
342C2—22no—
343C2—22no—
344C2—42no—
345C2#1097811yescharacteristic
346C2#1098411yescharacteristic
347C2#1099211yescharacteristic
subgroupG/H
center, socleC22#1438
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
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
11. C2×#398C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
#218252
#217532
#207002
#80512