Finite Group Databasebeta

← #117632

Subgroups — #117632

Order 512.

706 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 130 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#21943C211yesmaximal, characteristic
2#20705C212yesmaximal
3#20694C211yesmaximal, characteristic
4#8115C212yesmaximal
5#8055C211yesmaximal, characteristic
6C22×#117C2211yescharacteristic
7C2×#407C2211yescharacteristic
8C2×#407C2212yes—
9C2×#407C2211yescharacteristic
10#2280C2212yes—
11#1854—24no—
12#1826—24no—
13#1826—24no—
14#1822—22no—
15#1821C414yescomplemented
16#1809—22no—
17#1742—22no—
18#1733—24no—
19#1306—24no—
20#1298—22no—
21C24×C4C2311yescharacteristic
22C22×M16—22no—
23C22×M16—21no—
24C22×M16D412yes—
25C2×#117—21no—
26C2×#117—21no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117D414yes—
32C2×#117D412yes—
33C2×#117Q812yes—
34C22×C4⋊C4C2×C412yes—
35C22×C22⋊C4—21no—
36C22×C42—21no—
37C2×#107—22no—
38C2×#106—22no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407—22no—
43#407—21no—
44#407—22no—
45#407C2×C412yes—
46#407C2×C412yes—
47#407D414yes—
48#407D412yes—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8—22no—
53#381—22no—
54#380—22no—
55#380—22no—
56#379—22no—
57C4×C22⋊C4—22no—
58C4×C22⋊C4—22no—
59#343—28no—
60#340—24no—
61#339—216no—
62C25#4111yescharacteristic
63C23×C4—22no—
64C23×C4—42no—
65C23×C4—42no—
66C23×C4C22×C411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C23×C4C2×D412yes—
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D412yes—
72C23×C4C2×Q811yescharacteristic
73C23×C4#4112yes—
74C23×C4#4112yes—
75C23×C4#4111yescharacteristic
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—24no—
79C2×M16—42no—
80C2×M16—41no—
81C2×M16—42no—
82C2×M16—42no—
83C2×M16—44no—
84C2×M16—44no—
85C2×M16—42no—
86C2×M16—42no—
87C2×M16C22⋊C414yes—
88C22×C8—22no—
89C22×C8—42no—
90C22×C8—41no—
91#117—24no—
92#117—24no—
93#117—24no—
94#117—24no—
95#117—41no—
96#117—42no—
97#117—42no—
98#117—41no—
99#117—41no—
100#117—41no—
101#117—44no—
102#117—44no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4C4⋊C414yes—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—24no—
120C2×C42—42no—
121C2×C42—42no—
122#107—82no—
123#106—82no—
124C22⋊C8—24no—
125C22⋊C8—24no—
126C22⋊C8—24no—
127C22⋊C8—24no—
128C22⋊C8—24no—
129C22⋊C8—42no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132#95—42no—
133#95—42no—
134C24—42no—
135C24—82no—
136C24#11711yescharacteristic
137C24#12111yescharacteristic
138C24#12112yes—
139C24C22⋊Q811yescharacteristic
140C24#12612yes—
141C22×C4—24no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—44no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—84no—
161C22×C4C2×C22⋊C411yescharacteristic
162C22×C4C2×C4⋊C411yescharacteristic
163C22×C4C4×D412yes—
164C22×C4C4×D412yes—
165C22×C4#12012yes—
166C22×C4#12011yescharacteristic
167C22×C4#12112yes—
168C22×C4#12112yes—
169C22×C4#12111yescharacteristic
170C22×C4C22⋊Q812yes—
171C22×C4C22⋊Q811yescharacteristic
172C22×C4C22⋊Q812yes—
173C22×C4#12311yescharacteristic
174C22×C4#12312yes—
175C22×C4#12312yes—
176C22×C4#12411yescharacteristic
177C22×C4#12412yes—
178C22×C4#12711yescharacteristic
179M16—44no—
180M16—48no—
181M16—44no—
182M16—82no—
183M16—82no—
184M16—82no—
185M16—82no—
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—44no—
191C2×C8—84no—
192C2×C8—81no—
193C2×C8—81no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C22⋊C4—42no—
201C22⋊C4—44no—
202C22⋊C4—42no—
203C22⋊C4—84no—
204C22⋊C4—84no—
205C22⋊C4—84no—
206C42—48no—
207C42—42no—
208C42—44no—
209C42—42no—
210C23—24no—
211C23—22no—
212C23—41no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—41no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—88no—
223C23#38011yescharacteristic
224C23#38812yes—
225C23#39211yescharacteristic
226C23#39612yes—
227C23#39711yescharacteristic
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—84no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4#38512yes—
257C2×C4#38611yescharacteristic
258C2×C4#39011yescharacteristic
259C2×C4#39311yescharacteristic
260C2×C4#39412yes—
261C2×C4#39911yescharacteristic
262C2×C4#45812yes—
263C2×C4#45812yes—
264C8—44no—
265C8—82no—
266C8—84no—
267C22—22no—
268C22—21no—
269C22—22no—
270C22—22no—
271C22—21no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—44no—
279C22—84no—
280C22—82no—
281C22—82no—
282C22—84no—
283C22#143911yescharacteristic
284C22#202212yes—
285C4—21no—
286C4—22no—
287C4—22no—
288C4—21no—
289C4—84no—
290C4—164no—
291C4#189012yes—
292C4#203011yescharacteristic
293C4#203011yescharacteristic
294C2—22no—
295C2—41no—
296C2—42no—
297C2—42no—
298C2—41no—
299C2#1107011yescharacteristic
300C2#1114112yes—
subgroupG/H
center, socleC22#1439
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25#41
LCSG ▷ C23×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
15. #1821C4semidirect

Maximal subgroups

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

subgroupindex
#219432
#207052
#206942
#81152
#80552