Finite Group Databasebeta

← #119916

Subgroups — #119916

Order 512.

730 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 118 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#21825C211yesmaximal, characteristic
2#17397C211yesmaximal, characteristic
3#14660C211yesmaximal, characteristic
4#7983C212yesmaximal
5#7943C212yesmaximal
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8#2280C2212yes—
9C2×#381C2211yescharacteristic
10C2×#380C2211yescharacteristic
11#1996—24no—
12#1819—22no—
13#1472—24no—
14#1463—22no—
15#1446C412yescomplemented
16#1445C412yescomplemented
17#1441—22no—
18#1440—22no—
19#1289—24no—
20#1284—24no—
21C23×Q8C2×C411yescharacteristic
22C24×C4C2311yescharacteristic
23C22×M16—22no—
24C2×C22⋊Q8—22no—
25C2×C22⋊Q8—22no—
26C2×C22⋊Q8—22no—
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×#117—22no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4D412yes—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4D412yes—
37#407—22no—
38#407—22no—
39#407—22no—
40#407—22no—
41C2×C22⋊C8—22no—
42C2×C22⋊C8D414yes—
43#381—22no—
44#381—22no—
45#381—22no—
46#380—22no—
47#380—22no—
48#380—22no—
49#380—22no—
50#379—22no—
51C4×C22⋊C4—22no—
52C4×C22⋊C4D414yes—
53C2×#95—21no—
54C2×#95—24no—
55C2×#95—24no—
56C2×#95C2×C412yes—
57C2×#95C2×C412yes—
58#342—28no—
59#324—28no—
60C25C2×D411yescharacteristic
61C22×Q8—21no—
62C22×Q8—21no—
63C22×Q8—22no—
64C22×Q8—22no—
65C22×Q8—82no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—82no—
69C23×C4C22⋊C412yes—
70C23×C4C22⋊C412yes—
71C23×C4C22×C411yescharacteristic
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D412yes—
74C23×C4C2×D412yes—
75C23×C4#4112yes—
76C23×C4#4112yes—
77C23×C4#4112yes—
78C23×C4#4112yes—
79C2×M16—42no—
80C2×M16—42no—
81C2×M16—84no—
82C22×C8—24no—
83C22⋊Q8—42no—
84C22⋊Q8—42no—
85C22⋊Q8—42no—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C22⋊Q8—42no—
89C22⋊Q8—42no—
90C22⋊Q8—42no—
91#117—84no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—82no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—24no—
106C2×C22⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—42no—
114C2×C22⋊C4—82no—
115C2×C22⋊C4—82no—
116C2×C42—24no—
117C22⋊C8—24no—
118C22⋊C8—24no—
119C22⋊C8—42no—
120C22⋊C8—42no—
121#95—24no—
122#95—24no—
123#95—24no—
124#95—42no—
125#95—42no—
126#95—84no—
127#95—84no—
128C24—41no—
129C24—42no—
130C24—41no—
131C24—42no—
132C24C2×C22⋊C411yescharacteristic
133C24#12011yescharacteristic
134C24#12112yes—
135C24#12312yes—
136C24#12311yescharacteristic
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—82no—
140C2×Q8—82no—
141C2×Q8—84no—
142C2×Q8—84no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—44no—
151C22×C4—44no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—44no—
166C22×C4—82no—
167C22×C4—82no—
168C22×C4—82no—
169C22×C4—82no—
170C22×C4#11711yescharacteristic
171C22×C4#11711yescharacteristic
172C22×C4C4×D412yes—
173C22×C4C4×D412yes—
174C22×C4#12011yescharacteristic
175C22×C4#12112yes—
176C22×C4#12112yes—
177C22×C4#12112yes—
178C22×C4#12312yes—
179C22×C4#12312yes—
180C22×C4#12312yes—
181C22×C4#12311yescharacteristic
182C22×C4#12412yes—
183C22×C4#12412yes—
184C22×C4#12612yes—
185C22×C4#12612yes—
186M16—84no—
187M16—84no—
188C2×C8—44no—
189C2×C8—84no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—82no—
195C4⋊C4—82no—
196C4⋊C4—84no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—48no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C22⋊C4—84no—
203C22⋊C4—84no—
204C22⋊C4—82no—
205C22⋊C4—82no—
206C42—84no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—24no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23—41no—
217C23—41no—
218C23—42no—
219C23—41no—
220C23—42no—
221C23—42no—
222C23—41no—
223C23—42no—
224C23—84no—
225C23—84no—
226C23#38111yescharacteristic
227C23#38611yescharacteristic
228C23#38611yescharacteristic
229C23#39412yes—
230C23#39712yes—
231Q8—84no—
232Q8—84no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4—82no—
261C2×C4—84no—
262C2×C4—82no—
263C2×C4—84no—
264C2×C4—88no—
265C2×C4—84no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4#38812yes—
271C2×C4#38812yes—
272C2×C4#39412yes—
273C2×C4#39712yes—
274C8—84no—
275C22—24no—
276C22—24no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—84no—
293C22—82no—
294C22#144011yescharacteristic
295C22#182812yes—
296C22#190911yescharacteristic
297C22#191011yescharacteristic
298C22#191212yes—
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—82no—
304C4—82no—
305C4—162no—
306C4—162no—
307C4—164no—
308C2—21no—
309C2—22no—
310C2—22no—
311C2—21no—
312C2—42no—
313C2—42no—
314C2#1115411yescharacteristic
315C2#1115611yescharacteristic
316C2#1116611yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
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. #1446C4semidirect
16. #1445C4semidirect

Maximal subgroups

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

subgroupindex
#218252
#173972
#146602
#79832
#79432