Finite Group Databasebeta

← #97019

Subgroups — #97019

Order 512.

1456 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1456 conjugacy classes of subgroups — too many to draw. Only its 142 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×#2310C211yesmaximal, characteristic
2#8300C212yesmaximal
3#7595C214yesmaximal, complemented
4C24×Q8C411yescharacteristic
5C22×#117C411yescharacteristic
6C22×#117C2211yescharacteristic
7#2310—28no—
8C2×#391—22no—
9C2×#391C412yes—
10C2×#391C2212yes—
11C2×#381—22no—
12#1817C2214yes—
13C2×#324—22no—
14C2×#323—22no—
15#1272—28no—
16C23×Q8—24no—
17C23×Q8—24no—
18C23×Q8—21no—
19C23×Q8—22no—
20C23×Q8—42no—
21C23×Q8—42no—
22C23×Q8—42no—
23C23×Q8—42no—
24C23×Q8C814yescomplemented
25C23×Q8C2×C412yes—
26C24×C4—21no—
27C24×C4C2×C411yescharacteristic
28C2×#117—24no—
29C2×#117—24no—
30C2×#117—42no—
31C2×#117—42no—
32C2×#117—42no—
33C2×#117—42no—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4D412yes—
36C22×C22⋊C4—21no—
37C22×C22⋊C4D412yes—
38C22×C42—21no—
39C22×C42C2×C411yescharacteristic
40C22×C42C2311yescharacteristic
41C2×C22⋊C8—24no—
42#391—216no—
43#391—24no—
44#391—24no—
45#391—24no—
46#391D418yes—
47#381—216no—
48C2×#95—24no—
49C2×#95—24no—
50#336—28no—
51#324—44no—
52#323—44no—
53C25C22⋊C411yescharacteristic
54C22×Q8—24no—
55C22×Q8—216no—
56C22×Q8—24no—
57C22×Q8—216no—
58C22×Q8—24no—
59C22×Q8—22no—
60C22×Q8—22no—
61C22×Q8—22no—
62C22×Q8—48no—
63C22×Q8—416no—
64C22×Q8—416no—
65C22×Q8—416no—
66C22×Q8—416no—
67C22×Q8—48no—
68C22×Q8—42no—
69C22×Q8—42no—
70C22×Q8—48no—
71C22×Q8—48no—
72C22×Q8—42no—
73C22×Q8—42no—
74C22×Q8—42no—
75C22×Q8—42no—
76C22×Q8—42no—
77C22×Q8—42no—
78C22×Q8C22⋊C414yes—
79C22×Q8C2×C814yes—
80C22×Q8M1614yes—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—24no—
88C23×C4—24no—
89C23×C4—24no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C412yes—
98C23×C4C2×C812yes—
99C23×C4M1612yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D412yes—
102C22×C8—44no—
103#117—416no—
104#117—416no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C22⋊C414yes—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C42—24no—
119C2×C42—22no—
120C2×C42—41no—
121C2×C42—41no—
122C2×C42—41no—
123C2×C42—41no—
124C2×C42—42no—
125C2×C42—42no—
126C2×C42C2×D414yes—
127C22⋊C8—44no—
128C22⋊C8—44no—
129#95—216no—
130#95—216no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—44no—
136C24C22⋊C812yes—
137C24#9912yes—
138C24#10012yes—
139C24C2×C22⋊C411yescharacteristic
140C2×Q8—216no—
141C2×Q8—24no—
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—464no—
145C2×Q8—464no—
146C2×Q8—48no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—48no—
150C2×Q8—48no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—48no—
155C2×Q8—48no—
156C2×Q8—44no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—44no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—48no—
164C2×Q8C22⋊C818yes—
165C22×C4—28no—
166C22×C4—28no—
167C22×C4—28no—
168C22×C4—28no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—216no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—44no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—48no—
199C22×C4—48no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—48no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4C22⋊C812yes—
209C22×C4#9912yes—
210C22×C4#10012yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4#12014yes—
213C22×C4C22×C811yescharacteristic
214C22×C4C2×M1612yes—
215C22×C4C2×M1611yescharacteristic
216C2×C8—84no—
217C2×C8—88no—
218C4⋊C4—48no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C22⋊C4—48no—
222C22⋊C4—48no—
223C42—44no—
224C42—44no—
225C42—48no—
226C23—24no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—44no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—44no—
239C23—44no—
240C23#32314yes—
241C23#37911yescharacteristic
242C23C2×C22⋊C811yescharacteristic
243C23#40711yescharacteristic
244C23C2×#9911yescharacteristic
245C23C2×#9911yescharacteristic
246C23C2×#10011yescharacteristic
247C23C2×#10011yescharacteristic
248Q8—48no—
249Q8—432no—
250Q8—48no—
251Q8—48no—
252Q8—48no—
253C2×C4—216no—
254C2×C4—216no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—24no—
272C2×C4—432no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—48no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4#32314yes—
290C2×C4C2×C22⋊C812yes—
291C2×C4#40712yes—
292C8—164no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—24no—
300C22—44no—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22C2×#32311yescharacteristic
308C22C2×#32311yescharacteristic
309C22#146211yescharacteristic
310C22#146211yescharacteristic
311C22#177911yescharacteristic
312C22#178111yescharacteristic
313C22#178311yescharacteristic
314C22#179614yes—
315C4—24no—
316C4—22no—
317C4—22no—
318C4—44no—
319C4—48no—
320C4—82no—
321C4—84no—
322C2—21no—
323C2—21no—
324C2—21no—
325C2—21no—
326C2—44no—
327C2#829811yescharacteristic
328C2#832312yes—
329C2#852012yes—
330C2#852412yes—
subgroupG/H
center, socleC23#379
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7595C2semidirect
24. C23×Q8C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23102
#83002
#75952