Finite Group Databasebeta

← #186696

Subgroups — #186696

Order 512.

2450 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 2450 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
1C22×Q82C211yesmaximal, characteristic
2#21202C211yesmaximal, characteristic
3#20740C212yesmaximal
4C2×#1558C211yesmaximal, characteristic
5C2×#1491C212yesmaximal
6#8451C218yesmaximal, complemented
7C2×Q82—24no—
8C2×Q82—22no—
9C2×Q82—28no—
10C2×Q82—24no—
11C2×Q82C418yescomplemented
12C2×Q82C2214yes—
13C22×C4⋊Q8—22no—
14C22×C4⋊Q8—22no—
15C22×C4⋊Q8C2211yescharacteristic
16C22×C4×Q8—22no—
17C22×C4×Q8C2212yes—
18C2×#432C2212yes—
19C2×#419—22no—
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22#1960—22no—
23#1956C2214yes—
24#1823—22no—
25#1803C2218yes—
26#1787—22no—
27#1763—24no—
28#1558—24no—
29C2×#333C2214yes—
30#1491—28no—
31C2×Q8⋊C8C2218yes—
32C23×Q8—22no—
33C23×Q8D412yes—
34C23×Q8D412yes—
35C22×M16—21no—
36Q82—232no—
37Q82—216no—
38Q82—216no—
39Q82—28no—
40Q82—232no—
41Q82—216no—
42Q82C2×C4116yes—
43C2×C4⋊Q8—28no—
44C2×C4⋊Q8—216no—
45C2×C4⋊Q8—216no—
46C2×C4⋊Q8—28no—
47C2×C4⋊Q8—216no—
48C2×C4⋊Q8—28no—
49C2×C4⋊Q8—216no—
50C2×C4⋊Q8—28no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D412yes—
55C2×C4⋊Q8D414yes—
56C2×C4⋊Q8C2312yes—
57C2×C4×Q8—28no—
58C2×C4×Q8—216no—
59C2×C4×Q8—216no—
60C2×C4×Q8—28no—
61C2×C4×Q8—24no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C418yes—
66C2×C4×Q8D418yes—
67C2×C4×Q8D414yes—
68C2×C4×Q8C2314yes—
69C2×#117—21no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D412yes—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42C2311yescharacteristic
77#432—24no—
78#432—24no—
79C2×C4⋊C8C2314yes—
80#419—48no—
81C2×#103—28no—
82C2×#103—28no—
83C2×C22⋊C8—22no—
84C4×M16—24no—
85C2×#97—21no—
86C2×#97—22no—
87C2×C4×C8C2312yes—
88#384C2312yes—
89#382—21no—
90#380—22no—
91C4×C4⋊C4—22no—
92#340—28no—
93#333—28no—
94Q8⋊C8—216no—
95C22×Q8—216no—
96C22×Q8—28no—
97C22×Q8—216no—
98C22×Q8—22no—
99C22×Q8—24no—
100C22×Q8—22no—
101C22×Q8—24no—
102C22×Q8—44no—
103C22×Q8—44no—
104C22×Q8—84no—
105C22×Q8—84no—
106C22×Q8C22⋊C418yes—
107C22×Q8C22⋊C418yes—
108C22×Q8C2×D414yes—
109C22×Q8C2×D414yes—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C2×M16—24no—
116C2×M16—42no—
117C2×M16—42no—
118C22×C8—22no—
119C22×C8—24no—
120C4⋊Q8—232no—
121C4⋊Q8—264no—
122C4⋊Q8—232no—
123C4⋊Q8—264no—
124C4⋊Q8—232no—
125C4⋊Q8—232no—
126C4⋊Q8—28no—
127C4⋊Q8—28no—
128C4⋊Q8—24no—
129C4⋊Q8—24no—
130C4⋊Q8C22⋊C418yes—
131C4⋊Q8C22×C414yes—
132C4⋊Q8C2×D414yes—
133C4×Q8—232no—
134C4×Q8—264no—
135C4×Q8—232no—
136C4×Q8—28no—
137C4×Q8—28no—
138C4×Q8—28no—
139C4×Q8—28no—
140C4×Q8—28no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8C22⋊C4116yes—
144C4×Q8C22×C418yes—
145C4×Q8C2×D418yes—
146#117—44no—
147C2×C4⋊C4—216no—
148C2×C4⋊C4—232no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4C22⋊C418yes—
162C2×C4⋊C4C2×D414yes—
163C2×C22⋊C4—41no—
164C2×C42—28no—
165C2×C42—216no—
166C2×C42—28no—
167C2×C42—28no—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—41no—
171C2×C42C22×C412yes—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D414yes—
176C2×C42C2411yescharacteristic
177C4⋊C8—24no—
178C4⋊C8—24no—
179#103—416no—
180#103—416no—
181C22⋊C8—44no—
182#97—42no—
183#97—44no—
184C4×C8—24no—
185#95—42no—
186C24#12011yescharacteristic
187C2×Q8—232no—
188C2×Q8—216no—
189C2×Q8—232no—
190C2×Q8—232no—
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—28no—
194C2×Q8—28no—
195C2×Q8—28no—
196C2×Q8—44no—
197C2×Q8—416no—
198C2×Q8—44no—
199C2×Q8—88no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8—832no—
203C2×Q8—88no—
204C2×Q8—88no—
205C2×Q8C2×C22⋊C418yes—
206C2×Q8C2×C22⋊C418yes—
207C2×Q8#137116yes—
208C22×C4—216no—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—28no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4C2×C22⋊C414yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4#12014yes—
226C22×C4#12012yes—
227C22×C4#12012yes—
228C22×C4#12011yescharacteristic
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D412yes—
231M16—48no—
232C2×C8—24no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—44no—
236C2×C8—48no—
237C4⋊C4—264no—
238C4⋊C4—28no—
239C4⋊C4—28no—
240C4⋊C4—28no—
241C4⋊C4—28no—
242C4⋊C4—24no—
243C4⋊C4—216no—
244C4⋊C4—216no—
245C4⋊C4—28no—
246C4⋊C4—44no—
247C4⋊C4—48no—
248C4⋊C4C2×C22⋊C418yes—
249C22⋊C4—82no—
250C42—232no—
251C42—28no—
252C42—28no—
253C42—216no—
254C42—82no—
255C42C2×C22⋊C412yes—
256C42C2×C22⋊C414yes—
257C42#12012yes—
258C42#12014yes—
259C42C23×C411yescharacteristic
260C42C22×D412yes—
261C42C22×D411yescharacteristic
262C23—24no—
263C23#37912yes—
264C23#45312yes—
265C23#45512yes—
266C23C2×#12011yescharacteristic
267Q8—232no—
268Q8—28no—
269Q8—216no—
270Q8—48no—
271Q8—84no—
272Q8—88no—
273Q8—832no—
274Q8#419116yescomplemented
275C2×C4—232no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—216no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—416no—
290C2×C4—416no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—84no—
296C2×C4—82no—
297C2×C4#37914yes—
298C2×C4#37912yes—
299C2×C4C22×C22⋊C411yescharacteristic
300C2×C4C22×C22⋊C412yes—
301C2×C4C2×#12011yescharacteristic
302C2×C4C2×#12012yes—
303C2×C4C2×#13718yes—
304C8—44no—
305C8—84no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22#179214yes—
310C22C2×#37911yescharacteristic
311C22#300414yes—
312C22C2×#45311yescharacteristic
313C22C2×#45511yescharacteristic
314C4—216no—
315C4—24no—
316C4—21no—
317C4—21no—
318C4—22no—
319C4—48no—
320C4—48no—
321C4—162no—
322C4C2×#41914yes—
323C4#300414yes—
324C2—24no—
325C2#1854612yes—
326C2#1855112yes—
327C2#2059912yes—
328C2C2×#179211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8451C2semidirect
11. C2×Q82C4semidirect
274. Q8#419semidirect

Maximal subgroups

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

subgroupindex
C22×Q822
#212022
#207402
C2×#15582
C2×#14912
#84512