Finite Group Databasebeta

← #382464

Subgroups — #382464

Order 512.

4258 conjugacy classes of subgroups; only the 394 normal subgroups are shown.

normal subgroups only

This group has 4258 conjugacy classes of subgroups — too many to draw. Only its 394 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
2C2×#1819C214yesmaximal
3#14681C212yesmaximal
4#11381C218yesmaximal, complemented
5C24×Q8C411yescharacteristic, complemented
6C25×C4C2211yescharacteristic
7C22×C22⋊Q8—24no—
8C22×C22⋊Q8—24no—
9C22×C22⋊Q8—216no—
10C22×C22⋊Q8C414yescomplemented
11C22×C22⋊Q8C2214yes—
12C23×C4⋊C4C411yescharacteristic, complemented
13C23×C4⋊C4C412yescomplemented
14C23×C22⋊C4C2212yes—
15C2×#381—24no—
16C2×#379C2214yes—
17#1819—216no—
18C2×#342C22116yes—
19C22⋊#95—24no—
20C22⋊#95C2218yes—
21C26C2311yescharacteristic
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×C411yescharacteristic
31C24×C4C2×C412yes—
32C24×C4C2×C412yes—
33C24×C4C2×C412yes—
34C24×C4D414yes—
35C24×C4D414yes—
36C24×C4C2312yes—
37C2×C22⋊Q8—264no—
38C2×C22⋊Q8—264no—
39C2×C22⋊Q8—216no—
40C2×C22⋊Q8—216no—
41C2×C22⋊Q8—232no—
42C2×C22⋊Q8—232no—
43C22×C4⋊C4—28no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—24no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—28no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4C2×C418yes—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—24no—
54C22×C22⋊C4—28no—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—24no—
57C22×C22⋊C4C2×C414yes—
58C22×C22⋊C4D4116yes—
59C22×C22⋊C4C2318yes—
60C22×C22⋊C4C2314yes—
61#381—416no—
62#379—216no—
63C2×#95—216no—
64#342—232no—
65C25—28no—
66C25—44no—
67C25—44no—
68C25C22×C411yescharacteristic
69C25C22×C411yescharacteristic
70C25C2×D412yes—
71C25C2×D412yes—
72C25C2×D418yes—
73C25C2411yescharacteristic
74C22×Q8—216no—
75C22×Q8—216no—
76C22×Q8—216no—
77C22×Q8—28no—
78C22×Q8—416no—
79C22×Q8—464no—
80C22×Q8—48no—
81C22×Q8—416no—
82C22×Q8—48no—
83C23×C4—24no—
84C23×C4—28no—
85C23×C4—24no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—24no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—24no—
92C23×C4—28no—
93C23×C4—216no—
94C23×C4—216no—
95C23×C4—28no—
96C23×C4—216no—
97C23×C4—28no—
98C23×C4—48no—
99C23×C4—416no—
100C23×C4—48no—
101C23×C4—44no—
102C23×C4—48no—
103C23×C4—464no—
104C23×C4—48no—
105C23×C4—48no—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C412yes—
108C23×C4C22⋊C418yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C418yes—
111C23×C4C22⋊C418yes—
112C23×C4C22⋊C412yes—
113C23×C4C22×C412yes—
114C23×C4C22×C414yes—
115C23×C4C22×C414yes—
116C23×C4C22×C412yes—
117C23×C4C2×D418yes—
118C22⋊Q8—2128no—
119C22⋊Q8—2128no—
120C2×C4⋊C4—216no—
121C2×C4⋊C4—216no—
122C2×C4⋊C4—232no—
123C2×C4⋊C4—232no—
124C2×C4⋊C4—216no—
125C2×C4⋊C4—216no—
126C2×C4⋊C4—216no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4—216no—
129C2×C22⋊C4—28no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—232no—
132C2×C22⋊C4—232no—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—216no—
136C2×C22⋊C4—232no—
137C2×C22⋊C4—416no—
138C2×C22⋊C4—416no—
139C2×C22⋊C4—48no—
140C2×C22⋊C4—48no—
141C2×C22⋊C4C22⋊C4116yes—
142C2×C22⋊C4C2×D4116yes—
143#95—432no—
144C24—24no—
145C24—24no—
146C24—24no—
147C24—24no—
148C24—28no—
149C24—28no—
150C24—232no—
151C24—28no—
152C24—28no—
153C24—42no—
154C24—44no—
155C24—416no—
156C24—42no—
157C24—42no—
158C24—44no—
159C24—416no—
160C24—42no—
161C24—464no—
162C24C2×C22⋊C414yes—
163C24C2×C22⋊C412yes—
164C24C2×C22⋊C411yescharacteristic
165C24C2×C22⋊C411yescharacteristic
166C24C2×C22⋊C414yes—
167C24#12018yes—
168C24#12018yes—
169C24C23×C411yescharacteristic
170C24C22×D412yes—
171C24C22×D414yes—
172C242−1+418yes—
173C2×Q8—232no—
174C2×Q8—464no—
175C2×Q8—464no—
176C2×Q8—416no—
177C2×Q8—464no—
178C2×Q8—416no—
179C22×C4—232no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—28no—
185C22×C4—28no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—216no—
192C22×C4—28no—
193C22×C4—28no—
194C22×C4—28no—
195C22×C4—28no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—216no—
199C22×C4—28no—
200C22×C4—216no—
201C22×C4—216no—
202C22×C4—48no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—48no—
206C22×C4—464no—
207C22×C4—432no—
208C22×C4—464no—
209C22×C4—432no—
210C22×C4—48no—
211C22×C4—416no—
212C22×C4—416no—
213C22×C4—48no—
214C22×C4—416no—
215C22×C4—48no—
216C22×C4—48no—
217C22×C4—416no—
218C22×C4—416no—
219C22×C4—48no—
220C22×C4—48no—
221C22×C4—416no—
222C22×C4—416no—
223C22×C4—48no—
224C22×C4—416no—
225C22×C4#9918yes—
226C22×C4C2×C22⋊C414yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C418yes—
229C22×C4C2×C22⋊C418yes—
230C4⋊C4—232no—
231C4⋊C4—264no—
232C22⋊C4—232no—
233C22⋊C4—232no—
234C22⋊C4—432no—
235C22⋊C4—816no—
236C22⋊C4—816no—
237C23—232no—
238C23—28no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—24no—
244C23—22no—
245C23—22no—
246C23—24no—
247C23—28no—
248C23—28no—
249C23—28no—
250C23—24no—
251C23—216no—
252C23—216no—
253C23—24no—
254C23—24no—
255C23—216no—
256C23—216no—
257C23—44no—
258C23—48no—
259C23—44no—
260C23—44no—
261C23—48no—
262C23—44no—
263C23—416no—
264C23—432no—
265C23—432no—
266C23—4128no—
267C23—416no—
268C23#37914yes—
269C23#37912yes—
270C23#37912yes—
271C23C2×#9914yes—
272C23#41014yes—
273C23C22×C22⋊C412yes—
274C23C22×C22⋊C411yescharacteristic
275C23#51914yes—
276C23C2×#12014yes—
277C23#53618yes—
278C23#536116yes—
279Q8—464no—
280C2×C4—216no—
281C2×C4—216no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—216no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—24no—
289C2×C4—432no—
290C2×C4—432no—
291C2×C4—48no—
292C2×C4—416no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—416no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—416no—
301C2×C4—416no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—88no—
305C2×C4—88no—
306C2×C4—832no—
307C2×C4—88no—
308C2×C4C2×#9918yes—
309C22—232no—
310C22—22no—
311C22—22no—
312C22—216no—
313C22—28no—
314C22—28no—
315C22—28no—
316C22—22no—
317C22—22no—
318C22—24no—
319C22—22no—
320C22—22no—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—416no—
326C22—432no—
327C22—432no—
328C22—416no—
329C22—48no—
330C22#178114yes—
331C22#178214yes—
332C22C2×#37911yescharacteristic
333C22#231014yes—
334C22#231012yes—
335C22#261418yes—
336C22C22×#9912yes—
337C22C2×#41012yes—
338C4—28no—
339C4—44no—
340C4—48no—
341C4—48no—
342C4—168no—
343C2—28no—
344C2—24no—
345C2—24no—
346C2—24no—
347C2—44no—
348C2#1433611yescharacteristic
349C2C2×#178111yescharacteristic
350C2C2×#178211yescharacteristic
351C2#2182514yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11381C2semidirect
5. C24×Q8C4semidirect
10. C22×C22⋊Q8C4semidirect
12. C23×C4⋊C4C4semidirect
13. C23×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#18192
#146812
#113812