Finite Group Databasebeta

← #102198

Subgroups — #102198

Order 512.

374 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 106 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#33256C211yesmaximal, characteristic
2#9891C211yesmaximal, characteristic
3#9686C211yesmaximal, characteristic
4#8274C211yesmaximal, characteristic
5#7842C211yesmaximal, characteristic
6#7683C211yesmaximal, characteristic
7#7680C211yesmaximal, characteristic
8#2859C2211yescharacteristic
9#2859C2211yescharacteristic
10#1941—21no—
11#1941—21no—
12#1941—21no—
13#1941—21no—
14#1844C411yescharacteristic
15#1844C411yescharacteristic
16#1834C411yescharacteristic
17#1834C411yescharacteristic
18#1767C2211yescharacteristic
19#1767C2211yescharacteristic
20#1744C2211yescharacteristic
21C4×#97C2211yescharacteristic
22C42×C8C2211yescharacteristic
23C2×#335—21no—
24C2×#330—22no—
25C2×#322—21no—
26C2×#322—21no—
27#1295—22no—
28#1277—21no—
29#1271—22no—
30#1271—22no—
31C2×#125—21no—
32C2×#125—21no—
33C2×#106—21no—
34C2×#106—21no—
35C2×#106—21no—
36C2×#106—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×#97—21no—
40C2×#97—21no—
41C2×#97—21no—
42C2×#97D412yes—
43C2×C4×C8—21no—
44C2×C4×C8—21no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51#389—21no—
52#389—21no—
53#389—21no—
54#389—21no—
55#389—21no—
56#389—21no—
57#389—21no—
58#389—21no—
59#384C2×C411yescharacteristic
60#384C2×C411yescharacteristic
61#384C2×C411yescharacteristic
62#384C2×C411yescharacteristic
63#382—21no—
64#382—21no—
65C43C2311yescharacteristic
66#340—21no—
67#340—21no—
68#340—21no—
69#340—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336D412yes—
74#335—24no—
75#330—44no—
76#322—24no—
77#322—24no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—41no—
81C22×C8—41no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86#125—42no—
87#125—42no—
88#125—82no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C42C22×C411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42#4111yescharacteristic
111C2×C42#4111yescharacteristic
112#106—42no—
113#106—42no—
114#106—42no—
115#106—42no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120#97—22no—
121#97—22no—
122#97—42no—
123#97—42no—
124#97—42no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8—22no—
131C4×C8—22no—
132C4×C8—22no—
133C4×C8—22no—
134C4×C8—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138#95—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C4×D411yescharacteristic
145C22×C4C4×D411yescharacteristic
146C22×C4#12111yescharacteristic
147C22×C4#12111yescharacteristic
148C22×C4#12411yescharacteristic
149C22×C4#12711yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—24no—
155C2×C8—24no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8#10112yes—
170C2×C8#10112yes—
171C2×C8#10412yes—
172C2×C8#10412yes—
173C2×C8#10412yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—81no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C4⋊C4—81no—
190C4⋊C4—82no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—22no—
202C42—22no—
203C42—41no—
204C42AGL(1,ℤ/8)12yes—
205C42#13712yes—
206C23#39011yescharacteristic
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4C2×#10111yescharacteristic
233C2×C4C2×#10111yescharacteristic
234C2×C4C2×#10411yescharacteristic
235C2×C4C2×#10411yescharacteristic
236C2×C4C2×#10411yescharacteristic
237C2×C4#42111yescharacteristic
238C2×C4#44012yes—
239C2×C4#44412yes—
240C2×C4#46812yes—
241C2×C4#47112yes—
242C2×C4#49611yescharacteristic
243C2×C4#49711yescharacteristic
244C8—22no—
245C8—22no—
246C8—44no—
247C8—44no—
248C8—41no—
249C8—82no—
250C8—82no—
251C8#33014yescomplemented
252C22#196211yescharacteristic
253C22#196611yescharacteristic
254C22#197311yescharacteristic
255C22#197511yescharacteristic
256C22#197611yescharacteristic
257C22#197711yescharacteristic
258C22#198011yescharacteristic
259C4—22no—
260C4—22no—
261C4—22no—
262C4—22no—
263C4—21no—
264C4—21no—
265C4—41no—
266C4—161no—
267C4—161no—
268C4—161no—
269C4—161no—
270C4C2×#33011yescharacteristic
271C4#152611yescharacteristic
272C4#152811yescharacteristic
273C4#153111yescharacteristic
274C2#968611yescharacteristic
275C2#972211yescharacteristic
276C2#972811yescharacteristic
277C2#976111yescharacteristic
278C2#979211yescharacteristic
279C2#980611yescharacteristic
280C2#985111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
251. C8#330semidirect

Maximal subgroups

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

subgroupindex
#332562
#98912
#96862
#82742
#78422
#76832
#76802