Finite Group Databasebeta

← #137659

Subgroups — #137659

Order 512.

413 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 98 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#33419C211yesmaximal, characteristic, complemented
2#21536C211yesmaximal, characteristic
3#11593C212yesmaximal, complemented
4#8602C211yesmaximal, characteristic, complemented
5#7174C212yesmaximal, complemented
6C2×#469C411yescharacteristic
7C2×C4×D8C2211yescharacteristic
8C2×#405C2211yescharacteristic
9#2324—21no—
10#2164—24no—
11#2150—21no—
12#2112—21no—
13C4×M32C2212yes—
14#1972C411yescharacteristic
15#1944C411yescharacteristic
16#1934—21no—
17#1893—21no—
18#1809C411yescharacteristic
19C4×#106C2211yescharacteristic
20#1762—21no—
21#1564C2212yes—
22#1458—21no—
23#1331—24no—
24#1275—21no—
25C22×D8C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#121—21no—
29C2×C4×D4—21no—
30C2×#117D411yescharacteristic
31C2×M32—22no—
32#469—22no—
33#469—22no—
34C4×D8—22no—
35C4×D8—22no—
36#431—22no—
37C2×#107C2×C411yescharacteristic
38C2×#106C2×C411yescharacteristic
39C2×#106C2×C411yescharacteristic
40C2×#102—21no—
41C2×#102—21no—
42C2×#102—21no—
43#405—22no—
44#405—22no—
45C2×#97C2×C411yescharacteristic
46C2×#97D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8C2311yescharacteristic
49#390—21no—
50#390—41no—
51#389—21no—
52#388—41no—
53#386—41no—
54#385—41no—
55#380—21no—
56C4×C4⋊C4—21no—
57C4⋊C16—24no—
58C22⋊C16—24no—
59#346—22no—
60C4×C16D414yescomplemented
61#340—21no—
62#340—21no—
63#340—21no—
64#335—22no—
65#334—22no—
66#324—41no—
67#323—41no—
68C22×D4—21no—
69C22×D4—41no—
70C23×C4—41no—
71C23×C4C22⋊C411yescharacteristic
72C2×D8—22no—
73C2×D8—21no—
74C2×D8—22no—
75C2×D8M1612yes—
76C2×M16—22no—
77C2×M16—21no—
78C2×M16—21no—
79C2×M16—22no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86C22×C8C22×C411yescharacteristic
87C22×C8C2×D411yescharacteristic
88#121—24no—
89#121—42no—
90C4×D4—42no—
91C4×D4—42no—
92#117—22no—
93#117—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4D811yescharacteristic
99C2×C4⋊C4SD1611yescharacteristic
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—41no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4D811yescharacteristic
106C2×C22⋊C4SD1611yescharacteristic
107C2×C42—41no—
108C2×C42C22⋊C411yescharacteristic
109C2×C42C2×D411yescharacteristic
110M32—42no—
111M32—42no—
112C2×C16—24no—
113#107—22no—
114#106—21no—
115#106—22no—
116#106M1612yes—
117C4⋊C8—42no—
118#102—24no—
119#102—42no—
120#102—41no—
121#102—41no—
122C22⋊C8—42no—
123#97—22no—
124#97—22no—
125C4×C8—21no—
126C4×C8—21no—
127C4×C8—42no—
128C4×C8C2×D412yes—
129C4×C8#4112yes—
130#95—41no—
131#95—41no—
132#95—41no—
133C24—41no—
134C24#10211yescharacteristic
135C2×D4—22no—
136C2×D4—44no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—41no—
141C2×D4—41no—
142C2×D4—161no—
143C2×D4—161no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—82no—
154C22×C4—161no—
155C22×C4—161no—
156C22×C4#10211yescharacteristic
157C22×C4#10211yescharacteristic
158C22×C4#10211yescharacteristic
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4C2×D811yescharacteristic
161C22×C4C2×SD1611yescharacteristic
162D8—24no—
163D8—42no—
164M16—42no—
165M16—42no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—82no—
177C2×C8C4×D412yes—
178C2×C8#12011yescharacteristic
179C2×C8#12311yescharacteristic
180C2×C8C2×M1611yescharacteristic
181C2×C8C2×M1611yescharacteristic
182C2×C8#13112yes—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—161no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—44no—
195C22⋊C4—82no—
196C22⋊C4—161no—
197C22⋊C4—161no—
198C22⋊C4—161no—
199C42—21no—
200C42—21no—
201C42—161no—
202C42#12112yes—
203C42#13512yes—
204C16—44no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—41no—
209C23—41no—
210C23—42no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23—81no—
217C23C2×#10211yescharacteristic
218D4—42no—
219D4—42no—
220D4—81no—
221D4—81no—
222D4—164no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—164no—
246C2×C4#38611yescharacteristic
247C2×C4#40711yescharacteristic
248C2×C4#40811yescharacteristic
249C2×C4#41611yescharacteristic
250C2×C4#42111yescharacteristic
251C2×C4#42111yescharacteristic
252C2×C4#46512yes—
253C2×C4#46612yes—
254C8—21no—
255C8—21no—
256C8—41no—
257C8—41no—
258C8—162no—
259C8#43512yes—
260C8#43612yes—
261C22—21no—
262C22—21no—
263C22—42no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—161no—
272C22—161no—
273C22—161no—
274C22—161no—
275C22#148011yescharacteristic
276C22#148611yescharacteristic
277C22#189311yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—161no—
285C4—161no—
286C4#148211yescharacteristic
287C4#148811yescharacteristic
288C4#187011yescharacteristic
289C4#189711yescharacteristic
290C2—21no—
291C2—21no—
292C2—81no—
293C2—162no—
294C2#907611yescharacteristic
295C2#1307411yescharacteristic
296C2#1307611yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#469C4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33419C2semidirect
3. #11593C2semidirect
4. #8602C2semidirect
5. #7174C2semidirect
60. C4×C16D4semidirect

Maximal subgroups

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

subgroupindex
#334192
#215362
#115932
#86022
#71742