Finite Group Databasebeta

← #122808

Subgroups — #122808

Order 512.

339 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 64 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#28552C211yesmaximal, characteristic
2#11487C211yesmaximal, characteristic, complemented
3#8611C211yesmaximal, characteristic, complemented
4#7085C212yesmaximal, complemented
5#7084C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2414—21no—
9#2162—21no—
10#2087C411yescharacteristic
11#2076C2211yescharacteristic
12#2054C411yescharacteristic
13#1999C2211yescharacteristic
14#1934—21no—
15#1893—21no—
16#1875—21no—
17#1810—21no—
18#1735C411yescharacteristic
19#1465—21no—
20#1314—21no—
21#1314C2212yes—
22#1293C2212yes—
23C22×D8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#121—21no—
27C2×#121—21no—
28C2×C4×D4D411yescharacteristic
29C2×#117—21no—
30#466—41no—
31#466—41no—
32C8×D4—41no—
33C8×D4—41no—
34C2×#108—41no—
35C2×#107C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×#102—21no—
38C2×#102—21no—
39C2×#102—21no—
40#407—22no—
41C2×C22⋊C8D411yescharacteristic
42C2×C22⋊C8C2311yescharacteristic
43C2×C4×C8D411yescharacteristic
44#394—41no—
45#392—41no—
46#390—21no—
47#390—41no—
48#389—21no—
49#388—41no—
50#380—21no—
51C4⋊C16—41no—
52C22⋊C16—22no—
53C4×C16—41no—
54#341—41no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340C2×C411yescharacteristic
59#340C2×C411yescharacteristic
60#324—41no—
61#323—41no—
62C22×D4—21no—
63C22×D4—41no—
64C22×D4—41no—
65C22×D4SD1611yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C2×D411yescharacteristic
68C2×D8—41no—
69C2×D8—81no—
70C2×M16—42no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—81no—
76C22×C8—81no—
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D411yescharacteristic
81#121—42no—
82#121—42no—
83#121—42no—
84#121—81no—
85C4×D4—41no—
86C4×D4—41no—
87#117—42no—
88#117—81no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4D811yescharacteristic
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4D811yescharacteristic
101C2×C22⋊C4SD1611yescharacteristic
102C2×C42—41no—
103C2×C42C22⋊C411yescharacteristic
104C2×C16—42no—
105#108—82no—
106#107—41no—
107C4⋊C8—41no—
108#102—42no—
109#102—41no—
110#102—41no—
111#102—81no—
112C22⋊C8—41no—
113C22⋊C8—41no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116C4×C8—21no—
117C4×C8—21no—
118#95—41no—
119#95—41no—
120C24—41no—
121C24#10211yescharacteristic
122C24C2×SD1611yescharacteristic
123C2×D4—22no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—82no—
133C2×D4—82no—
134C2×D4—82no—
135C2×D4—161no—
136C2×D4—161no—
137C2×D4—161no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×D811yescharacteristic
154D8—82no—
155M16—82no—
156M16—82no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—42no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8—81no—
168C2×C8—82no—
169C2×C8#9911yescharacteristic
170C2×C8#9911yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—161no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—82no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C22⋊C4—161no—
186C22⋊C4—161no—
187C42—21no—
188C42—21no—
189C42—82no—
190C16—82no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—81no—
202C23—81no—
203C23—81no—
204C23—81no—
205C23—81no—
206C23—81no—
207C23—81no—
208C23C2×#10211yescharacteristic
209D4—44no—
210D4—81no—
211D4—81no—
212D4—81no—
213D4—81no—
214D4—164no—
215D4—164no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4#35912yes—
242C2×C4C2×#9911yescharacteristic
243C2×C4#42111yescharacteristic
244C8—21no—
245C8—21no—
246C8—41no—
247C8—41no—
248C8—41no—
249C8—162no—
250C22—21no—
251C22—21no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—161no—
263C22—161no—
264C22—161no—
265C22—161no—
266C22—161no—
267C22—161no—
268C22—161no—
269C22#151911yescharacteristic
270C22C2×#35911yescharacteristic
271C22#214811yescharacteristic
272C4—21no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—161no—
279C4—161no—
280C4#212111yescharacteristic
281C4#212611yescharacteristic
282C2—21no—
283C2—21no—
284C2—81no—
285C2—81no—
286C2—321no—
287C2#1187011yescharacteristic
288C2#1187611yescharacteristic
289C2#1189311yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C22⋊C8C23
prime radical#28552C2
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11487C2semidirect
3. #8611C2semidirect
4. #7085C2semidirect
5. #7084C2semidirect

Maximal subgroups

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

subgroupindex
#285522
#114872
#86112
#70852
#70842