Finite Group Databasebeta

← #189701

Subgroups — #189701

Order 512.

2422 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 2422 conjugacy classes of subgroups — too many to draw. Only its 310 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×#2280C211yesmaximal, characteristic
2C2×#1779C212yesmaximal
3#8704C214yesmaximal, complemented
4#8702C214yesmaximal, complemented
5#7907C214yesmaximal, complemented
6C25×C4C2211yescharacteristic
7C22×#117C412yes—
8C22×C22⋊C8—24no—
9C22×C22⋊C8C2214yes—
10#2280—24no—
11#2280—24no—
12C2×#381C2212yes—
13C2×#380C412yes—
14C2×#380C2212yes—
15C2×#379C412yes—
16C2×C4×C22⋊C4C412yes—
17C2×C4×C22⋊C4C2212yes—
18#1779—24no—
19#1779C2214yes—
20#1779C2214yes—
21#1280C22116yescomplemented
22C26C2×C411yescharacteristic
23C24×C4—28no—
24C24×C4—48no—
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C24×C4D414yes—
29C24×C4D414yes—
30C24×C4C2311yescharacteristic
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33C23×C8—24no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—48no—
39C22×C4⋊C4C2×C414yes—
40C22×C22⋊C4C2×C414yes—
41C22×C22⋊C4C2×C414yes—
42C22×C22⋊C4C2×C414yes—
43C22×C42C2×C414yes—
44C2×C22⋊C8—216no—
45C2×C22⋊C8—28no—
46C2×C22⋊C8—28no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—416no—
49C2×C22⋊C8C2318yes—
50#381—28no—
51#380—24no—
52#380—24no—
53#380—24no—
54#380—24no—
55#379—28no—
56C4×C22⋊C4—24no—
57C4×C22⋊C4—24no—
58C4×C22⋊C4—24no—
59C4×C22⋊C4—24no—
60C2×#95C2×C414yes—
61C2×#95C2314yes—
62C25—24no—
63C25—28no—
64C25—44no—
65C25—44no—
66C25C22⋊C414yes—
67C25C22×C411yescharacteristic
68C25C22×C411yescharacteristic
69C25C22×C411yescharacteristic
70C23×C4—28no—
71C23×C4—24no—
72C23×C4—24no—
73C23×C4—216no—
74C23×C4—24no—
75C23×C4—216no—
76C23×C4—216no—
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—44no—
80C23×C4—44no—
81C23×C4—48no—
82C23×C4—48no—
83C23×C4—44no—
84C23×C4—44no—
85C23×C4—48no—
86C23×C4—44no—
87C23×C4—464no—
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22⋊C414yes—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C22×C418yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C2×D414yes—
98C23×C4C2×D414yes—
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4#4118yes—
102C23×C4C2411yescharacteristic
103C22×C8—28no—
104C22×C8—416no—
105C22×C8—48no—
106#117—48no—
107#117—48no—
108#117—48no—
109#117—48no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—48no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—216no—
120C2×C22⋊C4—48no—
121C2×C22⋊C4—48no—
122C2×C42—24no—
123C2×C42—24no—
124C2×C42—48no—
125C22⋊C8—216no—
126C22⋊C8—416no—
127C22⋊C8—416no—
128C22⋊C8—48no—
129#95—28no—
130#95—28no—
131C24—22no—
132C24—22no—
133C24—28no—
134C24—24no—
135C24—28no—
136C24—24no—
137C24—28no—
138C24—24no—
139C24—24no—
140C24—28no—
141C24—28no—
142C24—44no—
143C24—42no—
144C24—42no—
145C24—44no—
146C24—42no—
147C24—42no—
148C24—416no—
149C24—416no—
150C24—464no—
151C24—416no—
152C24C2×C22⋊C412yes—
153C24C2×C22⋊C412yes—
154C24C2×C22⋊C412yes—
155C24#11714yes—
156C24#13118yes—
157C24C23×C411yescharacteristic
158C22×C4—232no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—28no—
168C22×C4—22no—
169C22×C4—28no—
170C22×C4—28no—
171C22×C4—28no—
172C22×C4—28no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—28no—
176C22×C4—28no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—216no—
180C22×C4—416no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—42no—
188C22×C4—464no—
189C22×C4—464no—
190C22×C4—416no—
191C22×C4—416no—
192C22×C4—416no—
193C22×C4—416no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4#9918yes—
197C22×C4C2×C22⋊C412yes—
198C22×C4C2×C22⋊C411yescharacteristic
199C22×C4C2×C22⋊C412yes—
200C22×C4C2×C22⋊C411yescharacteristic
201C22×C4#11714yes—
202C22×C4#12318yes—
203C22×C4#12318yes—
204C22×C4C22×D412yes—
205C22×C4C2×#4114yes—
206C2×C8—416no—
207C2×C8—48no—
208C2×C8—816no—
209C4⋊C4—48no—
210C4⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—48no—
213C22⋊C4—48no—
214C22⋊C4—48no—
215C22⋊C4—48no—
216C22⋊C4—48no—
217C42—48no—
218C42—48no—
219C23—22no—
220C23—24no—
221C23—22no—
222C23—24no—
223C23—28no—
224C23—28no—
225C23—232no—
226C23—24no—
227C23—24no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—24no—
232C23—22no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—416no—
237C23—416no—
238C23—416no—
239C23—416no—
240C23—416no—
241C23—416no—
242C23—416no—
243C23—44no—
244C23—44no—
245C23—416no—
246C23—4128no—
247C23—416no—
248C23—44no—
249C23—44no—
250C23#38114yes—
251C23#40814yes—
252C23#40814yes—
253C23C2×#9914yes—
254C23#41314yes—
255C23#433116yes—
256C23C22×C22⋊C411yescharacteristic
257C23C2×#11712yes—
258C23C2×#13114yes—
259C2×C4—216no—
260C2×C4—28no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—216no—
266C2×C4—216no—
267C2×C4—28no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—416no—
273C2×C4—416no—
274C2×C4—416no—
275C2×C4—48no—
276C2×C4—44no—
277C2×C4—48no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—816no—
282C2×C4#38112yes—
283C2×C4#38112yes—
284C2×C4C2×#9914yes—
285C2×C4C2×#9914yes—
286C2×C4C2×#12314yes—
287C8—88no—
288C22—22no—
289C22—24no—
290C22—22no—
291C22—24no—
292C22—28no—
293C22—28no—
294C22—28no—
295C22—24no—
296C22—24no—
297C22—28no—
298C22—28no—
299C22—416no—
300C22—464no—
301C22—416no—
302C22—416no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—44no—
308C22—44no—
309C22#181818yes—
310C22C2×#38111yescharacteristic
311C22C2×#40812yes—
312C22C22×#9912yes—
313C22C2×#41312yes—
314C22C2×#43314yes—
315C4—24no—
316C4—22no—
317C4—24no—
318C4—42no—
319C4—88no—
320C4#181914yes—
321C2—28no—
322C2—24no—
323C2—44no—
324C2—44no—
325C2#1738114yes—
326C2C2×#181811yescharacteristic
327C2C2×#181911yescharacteristic
328C2#2069411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8704C2semidirect
4. #8702C2semidirect
5. #7907C2semidirect
21. #1280C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22802
C2×#17792
#87042
#87022
#79072