Finite Group Databasebeta

← #122850

Subgroups — #122850

Order 512.

345 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 66 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#11465C211yesmaximal, characteristic, complemented
3#8614C211yesmaximal, characteristic, complemented
4#7089C212yesmaximal, complemented
5#7088C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8#2414—21no—
9C2×C22⋊C16—21no—
10#2087C2211yescharacteristic
11#2076C411yescharacteristic
12#2054C2211yescharacteristic
13#1999C411yescharacteristic
14#1934—21no—
15#1893—21no—
16#1875—21no—
17#1800—21no—
18#1735C411yescharacteristic
19#1566—21no—
20#1322C2212yes—
21#1314—21no—
22#1292C2212yes—
23C22×D8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#121—21no—
27C2×#121—21no—
28C2×C4×D4D411yescharacteristic
29C2×#117—21no—
30C22×C16—41no—
31#466—41no—
32#466—41no—
33C8×D4—41no—
34C8×D4—41no—
35C2×#107C2×C411yescharacteristic
36#423—22no—
37C2×C4⋊C8C2311yescharacteristic
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×#100—41no—
42C2×C22⋊C8C2×C411yescharacteristic
43C2×C22⋊C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#394—41no—
46#392—41no—
47#390—21no—
48#390—41no—
49#389—21no—
50#388—41no—
51#380—21no—
52C22⋊C16—22no—
53C22⋊C16—41no—
54#343—41no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340C2×C411yescharacteristic
59#340C2×C411yescharacteristic
60#334—41no—
61#334—41no—
62C22×D4—21no—
63C22×D4—41no—
64C22×D4—41no—
65C22×D4D811yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4C22⋊C411yescharacteristic
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⋊C4SD1611yescharacteristic
101C2×C22⋊C4SD1611yescharacteristic
102C2×C42—41no—
103C2×C42C2×D411yescharacteristic
104C2×C16—42no—
105C2×C16—82no—
106#107—41no—
107C4⋊C8—41no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110#102—42no—
111#102—41no—
112#102—41no—
113#102—81no—
114#100—82no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117C4×C8—21no—
118C4×C8—21no—
119#95—41no—
120#95—41no—
121C24—41no—
122C24#10211yescharacteristic
123C24#10211yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—82no—
134C2×D4—82no—
135C2×D4—82no—
136C2×D4—161no—
137C2×D4—161no—
138C2×D4—161no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4#10211yescharacteristic
151C22×C4#10211yescharacteristic
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×D811yescharacteristic
154C22×C4C2×SD1611yescharacteristic
155D8—82no—
156M16—82no—
157M16—82no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—41no—
162C2×C8—42no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—81no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8—82no—
170C2×C8#9911yescharacteristic
171C2×C8#9911yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—161no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C22⋊C4—82no—
186C22⋊C4—161no—
187C22⋊C4—161no—
188C42—21no—
189C42—21no—
190C42—82no—
191C16—82no—
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—
208C23#36012yes—
209C23C2×#10211yescharacteristic
210D4—44no—
211D4—81no—
212D4—81no—
213D4—81no—
214D4—81no—
215D4—164no—
216D4—164no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
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—81no—
242C2×C4C2×#9911yescharacteristic
243C2×C4#42111yescharacteristic
244C2×C4#50812yes—
245C8—21no—
246C8—21no—
247C8—41no—
248C8—41no—
249C8—41no—
250C8—162no—
251C22—21no—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—161no—
264C22—161no—
265C22—161no—
266C22—161no—
267C22—161no—
268C22—161no—
269C22—161no—
270C22#151911yescharacteristic
271C22#213811yescharacteristic
272C22C2×#36011yescharacteristic
273C4—21no—
274C4—22no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—82no—
279C4—161no—
280C4—161no—
281C4#212111yescharacteristic
282C4#213211yescharacteristic
283C2—21no—
284C2—21no—
285C2—42no—
286C2—81no—
287C2—321no—
288C2#1187011yescharacteristic
289C2#1188411yescharacteristic
290C2#1188511yescharacteristic
subgroupG/H
center, socleC22#1519
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radical#28552C2
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #11465C2semidirect
3. #8614C2semidirect
4. #7089C2semidirect
5. #7088C2semidirect

Maximal subgroups

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

subgroupindex
#285522
#114652
#86142
#70892
#70882