Finite Group Databasebeta

← #115645

Subgroups — #115645

Order 512.

1094 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1094 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×#2503C211yesmaximal, characteristic
2#21825C211yesmaximal, characteristic
3#20700C212yesmaximal
4#17397C211yesmaximal, characteristic
5#7942C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C23×C4⋊C4C2211yescharacteristic
8C2×#407C2212yes—
9#2503—24no—
10#2503—24no—
11C2×#400—22no—
12C2×#391C412yescomplemented
13C2×#382C412yescomplemented
14#2280C2212yes—
15C2×#381—22no—
16C2×C4×C22⋊C4C2211yescharacteristic
17#1996—24no—
18#1824—24no—
19#1823—24no—
20#1819—22no—
21#1736—24no—
22#1472—24no—
23#1463—22no—
24#1284—24no—
25C23×Q8C2×C411yescharacteristic
26C24×C4—21no—
27C24×C4C2311yescharacteristic
28C22×M16—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—22no—
32C2×C22⋊Q8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×#117—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D414yes—
46C22×C22⋊C4—21no—
47C22×C22⋊C4D412yes—
48C22×C42C2×C412yes—
49#407—22no—
50#407—22no—
51#407—22no—
52#407—22no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8D414yes—
55#400—216no—
56#391—24no—
57#391—24no—
58#382—24no—
59#382—24no—
60#381—216no—
61#381—22no—
62#380—22no—
63#380D414yes—
64#379—22no—
65C4×C22⋊C4—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4D414yes—
69C2×#95—24no—
70C2×#95—24no—
71C2×#95—21no—
72#342—28no—
73#340—216no—
74#324—28no—
75C25C2×D411yescharacteristic
76C22×Q8—21no—
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—22no—
80C22×Q8—44no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—24no—
84C23×C4—44no—
85C23×C4—48no—
86C23×C4—41no—
87C23×C4—42no—
88C23×C4—41no—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C23×C4#4112yes—
98C23×C4#4112yes—
99C2×M16—48no—
100C2×M16—42no—
101C2×M16—42no—
102C22×C8—24no—
103C22⋊Q8—42no—
104C22⋊Q8—42no—
105C22⋊Q8—42no—
106C22⋊Q8—42no—
107C22⋊Q8—42no—
108C22⋊Q8—42no—
109C22⋊Q8—42no—
110C22⋊Q8—42no—
111#117—48no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C22⋊C414yes—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C42—22no—
147C2×C42—44no—
148C2×C42—42no—
149C2×C42C22⋊C414yes—
150C22⋊C8—24no—
151C22⋊C8—24no—
152C22⋊C8—42no—
153C22⋊C8—42no—
154#95—216no—
155#95—216no—
156#95—24no—
157#95—24no—
158#95—42no—
159C24—24no—
160C24—42no—
161C24—42no—
162C24C2×C22⋊C411yescharacteristic
163C24#12012yes—
164C24#12011yescharacteristic
165C24#12112yes—
166C24#12311yescharacteristic
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—84no—
172C2×Q8—84no—
173C22×C4—24no—
174C22×C4—28no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—48no—
194C22×C4—416no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—84no—
202C22×C4—88no—
203C22×C4—84no—
204C22×C4C2×C22⋊C411yescharacteristic
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C4×D412yes—
207C22×C4C4×D412yes—
208C22×C4#12012yes—
209C22×C4#12012yes—
210C22×C4#12011yescharacteristic
211C22×C4#12112yes—
212C22×C4#12112yes—
213C22×C4#12112yes—
214C22×C4#12112yes—
215C22×C4#12112yes—
216C22×C4#12312yes—
217C22×C4#12311yescharacteristic
218C22×C4#12412yes—
219C22×C4#12712yes—
220M16—84no—
221M16—84no—
222C2×C8—48no—
223C2×C8—44no—
224C4⋊C4—28no—
225C4⋊C4—48no—
226C4⋊C4—42no—
227C4⋊C4—44no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—44no—
233C22⋊C4—48no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—48no—
237C22⋊C4—84no—
238C22⋊C4—84no—
239C42—44no—
240C42—44no—
241C42—84no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—24no—
247C23—24no—
248C23—24no—
249C23—24no—
250C23—41no—
251C23—42no—
252C23—41no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23—416no—
257C23#37911yescharacteristic
258C23#38611yescharacteristic
259C23#38611yescharacteristic
260C23#39212yes—
261C23#39412yes—
262C23#45414yes—
263Q8—84no—
264Q8—84no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—88no—
291C2×C4—84no—
292C2×C4—88no—
293C2×C4—84no—
294C2×C4—88no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4#38612yes—
302C2×C4#39012yes—
303C2×C4#39212yes—
304C2×C4#39412yes—
305C2×C4#41014yes—
306C8—84no—
307C22—28no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—44no—
320C22—48no—
321C22—48no—
322C22—44no—
323C22#143811yescharacteristic
324C22#178212yes—
325C22#188812yes—
326C22#190912yes—
327C22#200314yes—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—84no—
332C4—82no—
333C4—84no—
334C4—82no—
335C4—164no—
336C4#189914yes—
337C2—22no—
338C2—24no—
339C2—22no—
340C2—22no—
341C2—42no—
342C2#1096611yescharacteristic
343C2#1097611yescharacteristic
344C2#1098811yescharacteristic
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
12. C2×#391C4semidirect
13. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
#218252
#207002
#173972
#79422