Finite Group Databasebeta

← #125509

Subgroups — #125509

Order 512.

375 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 86 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#33129C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic, complemented
3#8596C211yesmaximal, characteristic, complemented
4#7121C212yesmaximal, complemented
5#7112C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2854—21no—
9C2×C4⋊C16—21no—
10C2×C22⋊C16—21no—
11#1970C411yescharacteristic
12#1970C2211yescharacteristic
13#1942C411yescharacteristic
14#1942C2211yescharacteristic
15#1934—21no—
16#1934—21no—
17#1893—21no—
18#1808C411yescharacteristic
19#1797—21no—
20C2×#335—21no—
21#1374C2212yes—
22#1277C2212yes—
23C22×D8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#123—41no—
27C2×#121—21no—
28C22×C42D411yescharacteristic
29C22×C16—22no—
30#466—42no—
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×C4⋊C8—22no—
35C2×#102—21no—
36C2×#102—21no—
37C2×#102—21no—
38C2×C22⋊C8—41no—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8C2311yescharacteristic
44#390—21no—
45#390—21no—
46#389—21no—
47#389—21no—
48#388—41no—
49#388—41no—
50#386—41no—
51#380—21no—
52C4⋊C16—42no—
53C22⋊C16—42no—
54#340—21no—
55#340—21no—
56#340—21no—
57#336—41no—
58#335—42no—
59C22×D4—21no—
60C22×D4—41no—
61C23×C4C22⋊C411yescharacteristic
62C23×C4D811yescharacteristic
63C23×C4SD1611yescharacteristic
64C2×D8—22no—
65C2×D8—41no—
66C2×D8—41no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—21no—
70C22×C8—21no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8—41no—
74C22×C8—42no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79#123—84no—
80#121—42no—
81#121—42no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C42—41no—
94C2×C42—41no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42D811yescharacteristic
97C2×C42SD1611yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C16—42no—
100C2×C16—42no—
101C2×C16—42no—
102#107—22no—
103#107—41no—
104#107—41no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107#102—42no—
108#102—42no—
109#102—41no—
110#102—41no—
111C22⋊C8—82no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—41no—
115C4×C8—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119C24—41no—
120C24#10211yescharacteristic
121C2×D4—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—82no—
128C2×D4—82no—
129C2×D4—82no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4#10211yescharacteristic
147C22×C4#10211yescharacteristic
148C22×C4#10211yescharacteristic
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C2×D811yescharacteristic
151C22×C4C2×SD1611yescharacteristic
152D8—82no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—44no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—82no—
166C2×C8—82no—
167C2×C8—82no—
168C2×C8#9912yes—
169C2×C8#13112yes—
170C2×C8AGL(1,ℤ/8)11yescharacteristic
171C2×C8#13711yescharacteristic
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C22⋊C4—82no—
181C22⋊C4—82no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—41no—
192C42—41no—
193C16—82no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—81no—
201C23—81no—
202C23—81no—
203C23—81no—
204C23—81no—
205C23—81no—
206C23—81no—
207C23#35912yes—
208C23C2×#10211yescharacteristic
209D4—81no—
210D4—81no—
211D4—81no—
212D4—81no—
213D4—164no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—44no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4#36112yes—
244C2×C4#40811yescharacteristic
245C2×C4C2×#9911yescharacteristic
246C2×C4#41311yescharacteristic
247C2×C4#41711yescharacteristic
248C2×C4#42111yescharacteristic
249C2×C4#42111yescharacteristic
250C2×C4#50911yescharacteristic
251C2×C4#51011yescharacteristic
252C8—22no—
253C8—22no—
254C8—21no—
255C8—21no—
256C8—41no—
257C8—162no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—161no—
266C22—161no—
267C22—161no—
268C22—161no—
269C22—161no—
270C22—161no—
271C22—161no—
272C22#148011yescharacteristic
273C22#151911yescharacteristic
274C22#154711yescharacteristic
275C22#213911yescharacteristic
276C22C2×#35911yescharacteristic
277C22#214511yescharacteristic
278C22C2×#36111yescharacteristic
279C4—21no—
280C4—21no—
281C4—22no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—161no—
286C4—161no—
287C4—161no—
288C4#146811yescharacteristic
289C4#149511yescharacteristic
290C4#151311yescharacteristic
291C4#153511yescharacteristic
292C2—42no—
293C2—321no—
294C2#771411yescharacteristic
295C2#1164811yescharacteristic
296C2#1165211yescharacteristic
297C2#1189011yescharacteristic
298C2#1189111yescharacteristic
299C2#1197811yescharacteristic
300C2#1198611yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#466C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12783C2semidirect
3. #8596C2semidirect
4. #7121C2semidirect
5. #7112C2semidirect

Maximal subgroups

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

subgroupindex
#331292
#127832
#85962
#71212
#71122