Finite Group Databasebeta

← #108256

Subgroups — #108256

Order 512.

934 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 186 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#34179C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3#20683C211yesmaximal, characteristic
4#7943C212yesmaximal
5#7943C212yesmaximal
6C22×#117C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2211yescharacteristic
9#2347—24no—
10#2346—24no—
11#2316C412yescomplemented
12#2312—22no—
13#2283—22no—
14#2280C2212yes—
15C2×#381C2212yes—
16#2274C412yescomplemented
17#1820—22no—
18#1820C414yescomplemented
19#1818—22no—
20#1818C414yescomplemented
21C2×#342—22no—
22#1728—22no—
23#1284—24no—
24#1284—24no—
25C24×C4C2311yescharacteristic
26C22×M16—21no—
27C22×M16—21no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117C2×C414yes—
33C2×#117C2×C412yes—
34C2×#117C2×C412yes—
35C2×#117D412yes—
36C2×#117Q812yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C22⋊C4—22no—
40C22×C22⋊C4D412yes—
41C22×C42D412yes—
42#407—22no—
43#407—22no—
44#407—22no—
45#407—22no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8C2×C412yes—
49C2×C22⋊C8C2×C412yes—
50#389—28no—
51#388—216no—
52#383—24no—
53#382—24no—
54#381—22no—
55#381—22no—
56#381—22no—
57#381—22no—
58#381C2×C414yes—
59#380—24no—
60#380—22no—
61#380—22no—
62#379—22no—
63C4×C4⋊C4—24no—
64C4×C4⋊C4—28no—
65C4×C22⋊C4—24no—
66C4×C22⋊C4D414yes—
67C4×C22⋊C4D414yes—
68#376—24no—
69C2×#95—22no—
70C2×#95—22no—
71#342—28no—
72#336—28no—
73C25C2×D411yescharacteristic
74C23×C4—44no—
75C23×C4—42no—
76C23×C4C22⋊C412yes—
77C23×C4C22⋊C412yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C23×C4C2×D412yes—
85C23×C4C2×Q811yescharacteristic
86C23×C4#4112yes—
87C23×C4#4112yes—
88C2×M16—44no—
89C2×M16—44no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C22×C8—22no—
95C22×C8—22no—
96#117—24no—
97#117—24no—
98#117—24no—
99#117—24no—
100#117—24no—
101#117—24no—
102#117—24no—
103#117—24no—
104#117—48no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4C4212yes—
112C2×C4⋊C4C4212yes—
113C2×C4⋊C4C4⋊C412yes—
114C2×C4⋊C4C4⋊C412yes—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—48no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4C22⋊C414yes—
124C2×C22⋊C4C4⋊C414yes—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—28no—
130C2×C42C22⋊C414yes—
131C2×C42C22⋊C418yes—
132C2×C42C4⋊C414yes—
133C22⋊C8—24no—
134C22⋊C8—24no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137#95—28no—
138#95—28no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—42no—
144#95—42no—
145C24—24no—
146C24—81no—
147C24—81no—
148C24C2×C22⋊C411yescharacteristic
149C24C2×C22⋊C411yescharacteristic
150C24C2×C4⋊C411yescharacteristic
151C24#12012yes—
152C24#12312yes—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—28no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4—44no—
165C22×C4—42no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—44no—
169C22×C4—82no—
170C22×C4—82no—
171C22×C4—84no—
172C22×C4—84no—
173C22×C4—84no—
174C22×C4#9514yes—
175C22×C4#9514yes—
176C22×C4C2×C4211yescharacteristic
177C22×C4C2×C22⋊C412yes—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C2×C4⋊C411yescharacteristic
180C22×C4C2×C4⋊C411yescharacteristic
181C22×C4#11712yes—
182C22×C4C4×D412yes—
183C22×C4C4×D412yes—
184C22×C4C4×D412yes—
185C22×C4C4×D412yes—
186C22×C4#12011yescharacteristic
187C22×C4#12011yescharacteristic
188C22×C4#12112yes—
189C22×C4#12112yes—
190C22×C4C22⋊Q812yes—
191C22×C4C22⋊Q812yes—
192C22×C4#12311yescharacteristic
193C22×C4#12311yescharacteristic
194M16—82no—
195M16—82no—
196M16—82no—
197M16—82no—
198C2×C8—44no—
199C2×C8—44no—
200C2×C8—42no—
201C2×C8—42no—
202C4⋊C4—24no—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—48no—
207C4⋊C4—88no—
208C22⋊C4—28no—
209C22⋊C4—28no—
210C22⋊C4—48no—
211C22⋊C4—48no—
212C22⋊C4—84no—
213C22⋊C4—84no—
214C22⋊C4—84no—
215C42—24no—
216C42—24no—
217C42—24no—
218C42—24no—
219C42—48no—
220C42—88no—
221C23—24no—
222C23—22no—
223C23—22no—
224C23—24no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—88no—
236C23C2×#9511yescharacteristic
237C23#37911yescharacteristic
238C23#38111yescharacteristic
239C23#38512yes—
240C23#38612yes—
241C23#41014yes—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—48no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—168no—
275C2×C4C4×C22⋊C412yes—
276C2×C4#38012yes—
277C2×C4#38511yescharacteristic
278C2×C4#38511yescharacteristic
279C2×C4#38611yescharacteristic
280C2×C4#38611yescharacteristic
281C8—82no—
282C8—82no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—44no—
296C22—48no—
297C22—84no—
298C22—82no—
299C22—82no—
300C22C22⋊#9511yescharacteristic
301C22#174012yes—
302C22#178212yes—
303C22#182012yes—
304C4—21no—
305C4—21no—
306C4—22no—
307C4—22no—
308C4—21no—
309C4—21no—
310C4—44no—
311C4—44no—
312C4—88no—
313C4—164no—
314C2—22no—
315C2—22no—
316C2—22no—
317C2—42no—
318C2—42no—
319C2#1072611yescharacteristic
320C2#1074311yescharacteristic
321C2#1074311yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2316C4semidirect
16. #2274C4semidirect
18. #1820C4semidirect
20. #1818C4semidirect

Maximal subgroups

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

subgroupindex
#341792
C2×#18202
#206832
#79432