Finite Group Databasebeta

← #184797

Subgroups — #184797

Order 512.

4062 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 4062 conjugacy classes of subgroups — too many to draw. Only its 370 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
1C23×C22⋊Q8C211yesmaximal, characteristic
2#20579C212yesmaximal
3C2×#1463C214yesmaximal, complemented
4#8308C218yesmaximal, complemented
5C24×Q8C411yescharacteristic
6C25×C4C2211yescharacteristic
7C22×C22⋊Q8—24no—
8C22×C22⋊Q8—24no—
9C22×C22⋊Q8—216no—
10C22×C22⋊Q8C414yes—
11C22×C22⋊Q8C2214yes—
12C23×C4⋊C4C411yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C22⋊C4C412yes—
15C2×#407—24no—
16C2×#407C2214yes—
17#1779—24no—
18#1779C2218yes—
19#1463—216no—
20C2×#324C22116yescomplemented
21C26C2×C411yescharacteristic
22C23×Q8—28no—
23C23×Q8—22no—
24C23×Q8—22no—
25C23×Q8—48no—
26C23×Q8C2×C414yes—
27C24×C4—28no—
28C24×C4—48no—
29C24×C4C2×C411yescharacteristic
30C24×C4C2×C412yes—
31C24×C4C2×C412yes—
32C24×C4C2×C412yes—
33C24×C4D414yes—
34C24×C4D414yes—
35C24×C4C2312yes—
36C24×C4C2311yescharacteristic
37C22×M16—24no—
38C2×C22⋊Q8—264no—
39C2×C22⋊Q8—264no—
40C2×C22⋊Q8—216no—
41C2×C22⋊Q8—216no—
42C2×C22⋊Q8—232no—
43C2×C22⋊Q8—232no—
44C22×C4⋊C4—28no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—24no—
49C22×C4⋊C4—28no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4C2×C414yes—
52C22×C4⋊C4D4116yes—
53C22×C4⋊C4C2314yes—
54C22×C22⋊C4—24no—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—28no—
57C22×C22⋊C4—28no—
58C22×C22⋊C4C2×C418yes—
59#407—216no—
60#407—416no—
61C2×C22⋊C8—216no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8C2318yes—
64#324—232no—
65C25—28no—
66C25—24no—
67C25—44no—
68C25—44no—
69C25C22⋊C412yes—
70C25C22⋊C412yes—
71C25C22×C411yescharacteristic
72C25C22×C412yes—
73C22×Q8—216no—
74C22×Q8—216no—
75C22×Q8—216no—
76C22×Q8—28no—
77C22×Q8—416no—
78C22×Q8—464no—
79C22×Q8—48no—
80C22×Q8—416no—
81C22×Q8—48no—
82C23×C4—24no—
83C23×C4—28no—
84C23×C4—24no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—24no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—216no—
91C23×C4—28no—
92C23×C4—216no—
93C23×C4—216no—
94C23×C4—28no—
95C23×C4—48no—
96C23×C4—416no—
97C23×C4—48no—
98C23×C4—48no—
99C23×C4—44no—
100C23×C4—48no—
101C23×C4—464no—
102C23×C4—48no—
103C23×C4C22⋊C412yes—
104C23×C4C22⋊C418yes—
105C23×C4C22⋊C412yes—
106C23×C4C22⋊C418yes—
107C23×C4C22⋊C418yes—
108C23×C4C22×C414yes—
109C23×C4C22×C412yes—
110C23×C4C22×C414yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D418yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D418yes—
116C23×C4C2411yescharacteristic
117C2×M16—416no—
118C2×M16—48no—
119C22×C8—28no—
120C22⋊Q8—2128no—
121C22⋊Q8—2128no—
122C2×C4⋊C4—216no—
123C2×C4⋊C4—216no—
124C2×C4⋊C4—216no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—232no—
129C2×C4⋊C4—232no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4C22⋊C4116yes—
133C2×C4⋊C4C2×D4116yes—
134C2×C22⋊C4—232no—
135C2×C22⋊C4—232no—
136C2×C22⋊C4—216no—
137C2×C22⋊C4—216no—
138C2×C22⋊C4—216no—
139C22⋊C8—216no—
140C22⋊C8—432no—
141C22⋊C8—48no—
142C24—24no—
143C24—24no—
144C24—24no—
145C24—232no—
146C24—24no—
147C24—24no—
148C24—24no—
149C24—28no—
150C24—24no—
151C24—28no—
152C24—28no—
153C24—28no—
154C24—44no—
155C24—44no—
156C24—44no—
157C24—44no—
158C24—416no—
159C24—464no—
160C24—416no—
161C24#10118yes—
162C24C2×C22⋊C414yes—
163C24C2×C22⋊C414yes—
164C24C2×C22⋊C411yescharacteristic
165C24C2×C22⋊C411yescharacteristic
166C24C23×C411yescharacteristic
167C2×Q8—232no—
168C2×Q8—464no—
169C2×Q8—464no—
170C2×Q8—416no—
171C2×Q8—464no—
172C2×Q8—416no—
173C22×C4—232no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—28no—
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—28no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—216no—
186C22×C4—28no—
187C22×C4—28no—
188C22×C4—28no—
189C22×C4—28no—
190C22×C4—216no—
191C22×C4—216no—
192C22×C4—216no—
193C22×C4—216no—
194C22×C4—48no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—48no—
198C22×C4—416no—
199C22×C4—42no—
200C22×C4—416no—
201C22×C4—464no—
202C22×C4—432no—
203C22×C4—464no—
204C22×C4—432no—
205C22×C4—416no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4—48no—
209C22×C4—416no—
210C22×C4—48no—
211C22×C4—416no—
212C22×C4—48no—
213C22×C4#9918yes—
214C22×C4C2×C22⋊C418yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C2×C22⋊C414yes—
217C22×C4C2×C22⋊C418yes—
218C22×C4C2×C22⋊C414yes—
219C22×C4#12018yes—
220C22×C4#12018yes—
221C22×C4C22×D412yes—
222C22×C4C22×D414yes—
223M16—816no—
224C2×C8—416no—
225C2×C8—48no—
226C4⋊C4—232no—
227C4⋊C4—232no—
228C4⋊C4—232no—
229C22⋊C4—264no—
230C23—232no—
231C23—28no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—28no—
239C23—28no—
240C23—216no—
241C23—216no—
242C23—28no—
243C23—28no—
244C23—24no—
245C23—24no—
246C23—216no—
247C23—416no—
248C23—432no—
249C23—432no—
250C23—48no—
251C23—48no—
252C23—4128no—
253C23—48no—
254C23—48no—
255C23—416no—
256C23—48no—
257C23—48no—
258C23#37912yes—
259C23#37912yes—
260C23C2×#9914yes—
261C23C2×#10118yes—
262C23C2×#10114yes—
263C23C22×C22⋊C412yes—
264C23C22×C22⋊C411yescharacteristic
265C23#56814yes—
266Q8—464no—
267C2×C4—216no—
268C2×C4—216no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—216no—
274C2×C4—216no—
275C2×C4—216no—
276C2×C4—28no—
277C2×C4—28no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—432no—
282C2×C4—432no—
283C2×C4—416no—
284C2×C4—416no—
285C2×C4—48no—
286C2×C4—416no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—416no—
291C2×C4—48no—
292C2×C4#37914yes—
293C2×C4C2×#9918yes—
294C2×C4C2×#12014yes—
295C8—88no—
296C22—232no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—28no—
304C22—28no—
305C22—28no—
306C22—28no—
307C22—216no—
308C22—432no—
309C22—432no—
310C22—432no—
311C22—44no—
312C22—44no—
313C22—48no—
314C22#178114yes—
315C22#178414yes—
316C22C2×#37911yescharacteristic
317C22#287914yes—
318C22#287912yes—
319C22C22×#9912yes—
320C22C22×#10112yes—
321C4—24no—
322C4—28no—
323C4—24no—
324C4—48no—
325C4—48no—
326C2—24no—
327C2—24no—
328C2—24no—
329C2—48no—
330C2#1739714yes—
331C2#2058011yescharacteristic
332C2C2×#178111yescharacteristic
333C2C2×#178411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1463C2semidirect
4. #8308C2semidirect
20. C2×#324C22semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
#205792
C2×#14632
#83082