Finite Group Databasebeta

← #190110

Subgroups — #190110

Order 512.

1510 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 1510 conjugacy classes of subgroups — too many to draw. Only its 238 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
1#34176C211yesmaximal, characteristic
2C2×#1819C212yesmaximal
3#8750C218yesmaximal, complemented
4#8101C212yesmaximal
5C2×#1304C212yesmaximal
6C23×C22⋊C4C2211yescharacteristic
7#2342—24no—
8#2342—24no—
9#2315C412yescomplemented
10#2312C2214yes—
11#2280C414yescomplemented
12C2×#379C2214yes—
13C2×#379C2211yescharacteristic
14C2×#379C2211yescharacteristic
15#2273C412yescomplemented
16#1819—24no—
17#1819C2218yes—
18C2×#342—24no—
19#1304C2218yes—
20#1304C2218yes—
21C26C2311yescharacteristic
22C24×C4C2×C412yes—
23C2×#117—24no—
24C22×C22⋊C4—22no—
25C22×C22⋊C4—24no—
26C22×C22⋊C4—22no—
27C22×C22⋊C4—22no—
28C22×C22⋊C4—22no—
29C22×C22⋊C4—24no—
30C22×C22⋊C4—24no—
31C22×C22⋊C4—24no—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4C2×C411yescharacteristic
34C22×C22⋊C4C2×C412yes—
35C22×C22⋊C4C2×C412yes—
36C22×C22⋊C4D414yes—
37C22×C22⋊C4D414yes—
38C22×C22⋊C4C2312yes—
39#388—216no—
40#385—24no—
41#385—24no—
42#381—24no—
43#380—24no—
44#380C2×C418yescomplemented
45#379—22no—
46#379—22no—
47#379—24no—
48#379—24no—
49#379—24no—
50#379C2×C412yes—
51#379C2×C412yes—
52#379C2318yes—
53#379C2312yes—
54#379C2312yes—
55C4×C22⋊C4—24no—
56C4×C22⋊C4—24no—
57C4×C22⋊C4—24no—
58C4×C22⋊C4C2×C418yescomplemented
59#342—216no—
60C25—44no—
61C25—84no—
62C25C22×C411yescharacteristic
63C25C22×C411yescharacteristic
64C25C2×D412yes—
65C25C2×D412yes—
66C25#4112yes—
67C25#4114yes—
68C25#4112yes—
69C25C2411yescharacteristic
70C23×C4—22no—
71C23×C4—22no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—44no—
77C23×C4—84no—
78C23×C4C22⋊C414yes—
79C23×C4C22⋊C414yes—
80C23×C4C22×C412yes—
81C23×C4C22×C412yes—
82#117—88no—
83C2×C4⋊C4—28no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—28no—
89C2×C22⋊C4—28no—
90C2×C22⋊C4—28no—
91C2×C22⋊C4—28no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—24no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—28no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—28no—
102C2×C22⋊C4—28no—
103C2×C22⋊C4—44no—
104C2×C22⋊C4—44no—
105C2×C22⋊C4—416no—
106C2×C22⋊C4—44no—
107C2×C22⋊C4—44no—
108C2×C22⋊C4—44no—
109C2×C22⋊C4—84no—
110C2×C22⋊C4—84no—
111C2×C22⋊C4—88no—
112C2×C22⋊C4—88no—
113C2×C22⋊C4C22⋊C412yes—
114C2×C22⋊C4C22⋊C412yes—
115C2×C22⋊C4C22⋊C412yes—
116C2×C22⋊C4C22⋊C412yes—
117C2×C22⋊C4C22×C414yes—
118C2×C22⋊C4C22×C414yes—
119C2×C22⋊C4C2×D418yes—
120C2×C42—28no—
121#95—28no—
122#95—28no—
123C24—24no—
124C24—28no—
125C24—44no—
126C24—42no—
127C24—44no—
128C24—44no—
129C24—48no—
130C24—48no—
131C24—44no—
132C24—44no—
133C24—48no—
134C24—44no—
135C24—42no—
136C24—82no—
137C24—82no—
138C24—82no—
139C24—84no—
140C24—84no—
141C24—82no—
142C24—832no—
143C24C2×C22⋊C412yes—
144C24C2×C22⋊C411yescharacteristic
145C24C2×C22⋊C411yescharacteristic
146C24#11711yescharacteristic
147C24#11712yes—
148C24#11712yes—
149C24#11712yes—
150C24#11711yescharacteristic
151C24#12314yes—
152C24#12314yes—
153C24#12314yes—
154C24#12314yes—
155C24C23×C411yescharacteristic
156C24C22×D412yes—
157C24C2×#4112yes—
158C24C2×#4112yes—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—48no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—48no—
173C22×C4—84no—
174C22×C4—84no—
175C22×C4—84no—
176C22×C4—84no—
177C22×C4—84no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—84no—
181C22×C4—84no—
182C22×C4—88no—
183C22×C4—88no—
184C22×C4—88no—
185C22×C4C2×C22⋊C414yes—
186C22×C4C2×C22⋊C414yes—
187C4⋊C4—88no—
188C22⋊C4—216no—
189C22⋊C4—24no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—44no—
194C22⋊C4—48no—
195C22⋊C4—48no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—416no—
200C22⋊C4—84no—
201C22⋊C4—84no—
202C22⋊C4—88no—
203C22⋊C4—88no—
204C22⋊C4—88no—
205C22⋊C4—88no—
206C22⋊C4—88no—
207C42—88no—
208C23—28no—
209C23—24no—
210C23—24no—
211C23—24no—
212C23—28no—
213C23—22no—
214C23—22no—
215C23—24no—
216C23—28no—
217C23—24no—
218C23—24no—
219C23—28no—
220C23—24no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—44no—
227C23—42no—
228C23—41no—
229C23—42no—
230C23—41no—
231C23—44no—
232C23—42no—
233C23—42no—
234C23—44no—
235C23—44no—
236C23—44no—
237C23—44no—
238C23—42no—
239C23—42no—
240C23—44no—
241C23—44no—
242C23—44no—
243C23—44no—
244C23—44no—
245C23—44no—
246C23—416no—
247C23—816no—
248C23—88no—
249C23—816no—
250C23—832no—
251C23—816no—
252C23—88no—
253C23—816no—
254C23#38112yes—
255C23#38112yes—
256C23#38112yes—
257C23#38112yes—
258C23#41014yes—
259C23C22×C22⋊C411yescharacteristic
260C23C2×#11711yescharacteristic
261C23C2×#11711yescharacteristic
262C23C2×#12314yes—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—48no—
266C2×C4—48no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—84no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4—88no—
280C2×C4—88no—
281C2×C4—88no—
282C2×C4—164no—
283C2×C4—164no—
284C2×C4—1616no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—24no—
289C22—24no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—24no—
298C22—24no—
299C22—22no—
300C22—24no—
301C22—44no—
302C22—416no—
303C22—48no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22—48no—
308C22—44no—
309C22—84no—
310C22—88no—
311C22—88no—
312C22—84no—
313C22—84no—
314C22—84no—
315C22—84no—
316C22—88no—
317C22#182014yes—
318C22#182814yes—
319C22C2×#38111yescharacteristic
320C22C2×#41012yes—
321C4—44no—
322C4—44no—
323C4—84no—
324C4—84no—
325C4—168no—
326C2—22no—
327C2—24no—
328C2—24no—
329C2—44no—
330C2—41no—
331C2—41no—
332C2—42no—
333C2—42no—
334C2C2×#182011yescharacteristic
335C2C2×#182812yes—
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C26 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8750C2semidirect
9. #2315C4semidirect
11. #2280C4semidirect
15. #2273C4semidirect
44. #380C2×C4semidirect
58. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341762
C2×#18192
#87502
#81012
C2×#13042