Finite Group Databasebeta

← #129303

Subgroups — #129303

Order 512.

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

normal subgroups only

This group has 660 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
1C2×#2921C211yesmaximal, characteristic
2#12849C212yesmaximal, complemented
3C2×#1327C212yesmaximal, complemented
4C2×#1320C212yesmaximal
5C22×#117C411yescharacteristic
6#2921—24no—
7#2921—24no—
8C2×#433—22no—
9C2×#433—22no—
10C2×#423C411yescharacteristic
11C2×#423C2211yescharacteristic
12C2×#407C411yescharacteristic
13C2×#407C2211yescharacteristic
14C2×C4×M16C411yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C2×#349C2214yes—
17#1376—22no—
18#1375—22no—
19#1327—24no—
20#1320—24no—
21C24×C4C2×C411yescharacteristic
22C22×M16C2×C411yescharacteristic
23C22×M16C2×C411yescharacteristic
24C22×M16C2×C411yescharacteristic
25C22×M16C2311yescharacteristic
26C2×#117—22no—
27C2×#117—21no—
28C2×#117—22no—
29C2×#117—21no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117C814yes—
33C22×C4⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C42C2×C411yescharacteristic
36C22×C42C2×C411yescharacteristic
37C2×M32—24no—
38#433—216no—
39#433—216no—
40#423—22no—
41#423—22no—
42#423—22no—
43#423D414yes—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46#407—22no—
47#407—22no—
48#407—22no—
49#407D414yes—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—22no—
52C4×M16—22no—
53C4×M16—22no—
54C4×M16—22no—
55C4×M16—22no—
56C2×#97—21no—
57C2×#97D412yes—
58C2×C4×C8—21no—
59C2×C4×C8D412yes—
60#349—24no—
61#349—24no—
62C25M1611yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4C22⋊C412yes—
74C23×C4C2×C812yes—
75C23×C4M1611yescharacteristic
76C23×C4C22×C411yescharacteristic
77C2×M16—21no—
78C2×M16—21no—
79C2×M16—21no—
80C2×M16—21no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16C22⋊C412yes—
90C2×M16C22⋊C412yes—
91C2×M16C2×D412yes—
92C2×M16C2×D412yes—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8C22⋊C412yes—
96C22×C8C2×D412yes—
97#117—28no—
98#117—24no—
99#117—24no—
100#117—24no—
101#117—28no—
102#117—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4—22no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—21no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42C22⋊C411yescharacteristic
118C2×C42C22⋊C412yes—
119C2×C42C22⋊C411yescharacteristic
120C2×C42C2×C812yes—
121C2×C42C2×C812yes—
122C2×C42M1612yes—
123C2×C42M1612yes—
124M32—44no—
125M32—44no—
126C2×C16—44no—
127C4⋊C8—28no—
128C4⋊C8—28no—
129C22⋊C8—28no—
130C22⋊C8—28no—
131#97—24no—
132#97—24no—
133C4×C8—24no—
134C4×C8—24no—
135C24—21no—
136C24—21no—
137C24—41no—
138C24—41no—
139C24—82no—
140C24C22⋊C812yes—
141C24C2×M1611yescharacteristic
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4C22⋊C812yes—
178C22×C4C2×C22⋊C411yescharacteristic
179C22×C4C2×C22⋊C411yescharacteristic
180C22×C4C2×C22⋊C411yescharacteristic
181C22×C4C22×C811yescharacteristic
182C22×C4C2×M1611yescharacteristic
183C22×C4C2×M1611yescharacteristic
184M16—42no—
185M16—42no—
186M16—42no—
187M16—42no—
188M16—42no—
189M16—42no—
190M16—42no—
191M16—42no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8#12014yes—
204C4⋊C4—28no—
205C4⋊C4—28no—
206C22⋊C4—28no—
207C22⋊C4—28no—
208C42—21no—
209C42—21no—
210C42—22no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—22no—
215C42C22⋊C814yes—
216C42C22⋊C812yes—
217C42C22⋊C812yes—
218C16—84no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—42no—
223C23—42no—
224C23—44no—
225C23—42no—
226C23—41no—
227C23—41no—
228C23—82no—
229C23—82no—
230C23—84no—
231C23—84no—
232C23C2×C22⋊C811yescharacteristic
233C23#40711yescharacteristic
234C23#40711yescharacteristic
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#37911yescharacteristic
267C2×C4C2×C22⋊C811yescharacteristic
268C2×C4C2×C22⋊C811yescharacteristic
269C2×C4#40711yescharacteristic
270C2×C4#41011yescharacteristic
271C2×C4#41011yescharacteristic
272C2×C4#41311yescharacteristic
273C2×C4#41311yescharacteristic
274C8—24no—
275C8—42no—
276C8—42no—
277C8—42no—
278C22—22no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—42no—
284C22—44no—
285C22—42no—
286C22—42no—
287C22—82no—
288C22—84no—
289C22—84no—
290C22—82no—
291C22#146411yescharacteristic
292C22#146411yescharacteristic
293C22#177911yescharacteristic
294C4—21no—
295C4—21no—
296C4—42no—
297C4—42no—
298C4—44no—
299C4—42no—
300C4#145811yescharacteristic
301C4#145811yescharacteristic
302C4#178211yescharacteristic
303C4#178511yescharacteristic
304C2—21no—
305C2—21no—
306C2—41no—
307C2—41no—
308C2—82no—
309C2#830011yescharacteristic
310C2#1152212yes—
subgroupG/H
centerC2×C4#410
commutatorC22×C4C22×C8
Frattini, cosocleC22×M16C23
socleC22#1464
prime radicalC25M16
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12849C2semidirect
3. C2×#1327C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29212
#128492
C2×#13272
C2×#13202