Finite Group Databasebeta

← #125111

Subgroups — #125111

Order 512.

329 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 74 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#33132C211yesmaximal, characteristic
2#12602C211yesmaximal, characteristic, complemented
3#8602C211yesmaximal, characteristic, complemented
4#7127C212yesmaximal, complemented
5#7110C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×#405C2211yescharacteristic
8#2624—21no—
9C2×#349—21no—
10C4×M32—21no—
11#1972C411yescharacteristic
12#1972C2211yescharacteristic
13#1944C411yescharacteristic
14#1944C2211yescharacteristic
15#1934—21no—
16#1934—21no—
17#1895—21no—
18#1807C411yescharacteristic
19#1564—21no—
20#1458—21no—
21#1377C2212yes—
22#1275C2212yes—
23C22×D8C2×C411yescharacteristic
24C22×M16D411yescharacteristic
25C23×C8C2×C411yescharacteristic
26C2×#121—21no—
27C2×#117D411yescharacteristic
28C2×M32—22no—
29#466—41no—
30#466—41no—
31#431—22no—
32C2×#107C2×C411yescharacteristic
33C2×#106C2×C411yescharacteristic
34C2×#106C2×C411yescharacteristic
35C2×#102—21no—
36C2×#102—21no—
37C2×#102—21no—
38#405—41no—
39#405—41no—
40C2×#97C2×C411yescharacteristic
41C2×#97C2311yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#399—41no—
45#397—41no—
46#394—41no—
47#394—41no—
48#390—21no—
49#390—21no—
50#389—21no—
51#389—21no—
52#380—21no—
53#349—24no—
54#346—41no—
55C4×C16—41no—
56#340—21no—
57#340—21no—
58#340—21no—
59#335—41no—
60#334—41no—
61#324—41no—
62#323—41no—
63C22×D4—21no—
64C22×D4—41no—
65C23×C4C22⋊C411yescharacteristic
66C2×D8—22no—
67C2×D8—41no—
68C2×D8—41no—
69C2×M16—22no—
70C2×M16—22no—
71C2×M16—41no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22×C411yescharacteristic
79C22×C8C2×D411yescharacteristic
80#121—42no—
81#121—42no—
82#117—41no—
83#117—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4D811yescharacteristic
90C2×C4⋊C4SD1611yescharacteristic
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4D811yescharacteristic
98C2×C22⋊C4SD1611yescharacteristic
99C2×C42C22⋊C411yescharacteristic
100C2×C42C2×D411yescharacteristic
101M32—24no—
102M32—42no—
103C2×C16—42no—
104#107—41no—
105#106—22no—
106#106—41no—
107C4⋊C8—42no—
108#102—41no—
109#102—41no—
110#102—42no—
111#102—42no—
112C22⋊C8—42no—
113#97—41no—
114#97—41no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—42no—
120#95—41no—
121#95—41no—
122#95—41no—
123C24—41no—
124C24#10211yescharacteristic
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—82no—
132C2×D4—161no—
133C2×D4—161no—
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#10211yescharacteristic
144C22×C4#10211yescharacteristic
145C22×C4#10211yescharacteristic
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C2×D811yescharacteristic
148C22×C4C2×SD1611yescharacteristic
149D8—82no—
150M16—44no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—82no—
162C2×C8#9911yescharacteristic
163C2×C8#9911yescharacteristic
164C2×C8#10412yes—
165C2×C8#13112yes—
166C2×C8#13512yes—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—161no—
176C4⋊C4—161no—
177C22⋊C4—41no—
178C22⋊C4—41no—
179C22⋊C4—82no—
180C22⋊C4—161no—
181C22⋊C4—161no—
182C22⋊C4—161no—
183C22⋊C4—161no—
184C22⋊C4—161no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C16—44no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—41no—
194C23—41no—
195C23—81no—
196C23—81no—
197C23—81no—
198C23—81no—
199C23—81no—
200C23—81no—
201C23—81no—
202C23C2×#10211yescharacteristic
203D4—81no—
204D4—81no—
205D4—81no—
206D4—81no—
207D4—164no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
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×C4#40811yescharacteristic
233C2×C4C2×#9911yescharacteristic
234C2×C4#41311yescharacteristic
235C2×C4#41611yescharacteristic
236C2×C4C2×#10411yescharacteristic
237C2×C4#42111yescharacteristic
238C8—21no—
239C8—21no—
240C8—22no—
241C8—22no—
242C8—22no—
243C8—162no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—81no—
250C22—81no—
251C22—161no—
252C22—161no—
253C22—161no—
254C22—161no—
255C22—161no—
256C22—161no—
257C22—161no—
258C22#149211yescharacteristic
259C22#151911yescharacteristic
260C22#154011yescharacteristic
261C4—41no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—161no—
267C4—161no—
268C4—161no—
269C4#146811yescharacteristic
270C4#148211yescharacteristic
271C4#150111yescharacteristic
272C4#152911yescharacteristic
273C2—21no—
274C2—21no—
275C2—81no—
276C2—321no—
277C2#765811yescharacteristic
278C2#1186811yescharacteristic
279C2#1187211yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC2×#466C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12602C2semidirect
3. #8602C2semidirect
4. #7127C2semidirect
5. #7110C2semidirect

Maximal subgroups

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

subgroupindex
#331322
#126022
#86022
#71272
#71102