Finite Group Databasebeta

← #99020

Subgroups — #99020

Order 512.

369 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 94 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#21631C211yesmaximal, characteristic
2#8273C211yesmaximal, characteristic, complemented
3#8273C211yesmaximal, characteristic, complemented
4#7635C212yesmaximal
5#7042C212yesmaximal, complemented
6C2×#446C411yescharacteristic
7C2×#436C411yescharacteristic
8C4×#124C2211yescharacteristic
9#1984C411yescharacteristic
10#1954C411yescharacteristic
11#1923—21no—
12#1923—21no—
13#1831C2211yescharacteristic
14#1767C2212yes—
15#1767C2212yes—
16#1753—21no—
17#1753—21no—
18C42×C8C2211yescharacteristic
19C2×#334—21no—
20C2×#334—21no—
21#1277—21no—
22#1277—21no—
23#1272—22no—
24#1270—22no—
25C22×M16—21no—
26C2×#124Q811yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×C4×D4C2×C411yescharacteristic
29#446—42no—
30#436—42no—
31C2×C4⋊C8—22no—
32C2×C4⋊C8—22no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C22⋊C8—21no—
37C2×C22⋊C8—21no—
38C2×C22⋊C8—21no—
39C2×#97—21no—
40C2×#97—22no—
41C2×#97—22no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#391D411yescharacteristic
49#390D411yescharacteristic
50#388—21no—
51#388—21no—
52#382Q811yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C43C2311yescharacteristic
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—22no—
61#336—22no—
62#334—24no—
63#334—24no—
64C22×Q8C4⋊C411yescharacteristic
65C22×D4C4⋊C411yescharacteristic
66C23×C4—21no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C22×C8—21no—
71C22×C8—21no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78#124—42no—
79C4×Q8—41no—
80C4×Q8—41no—
81C4×D4—41no—
82C4×D4—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4C4⋊C411yescharacteristic
85C2×C4⋊C4C4⋊C411yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C42—21no—
92C2×C42C22×C411yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×Q811yescharacteristic
97C2×C42#4111yescharacteristic
98C2×C42#4111yescharacteristic
99C4⋊C8—24no—
100C4⋊C8—24no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—42no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C22⋊C8—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#97—42no—
110#97—82no—
111#97—82no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—41no—
116C4×C8—41no—
117C4×C8—41no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122C24—21no—
123C2×Q8—22no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×D4—22no—
127C2×D4—41no—
128C2×D4—41no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C2×C4⋊C411yescharacteristic
140C22×C4#11711yescharacteristic
141C22×C4#12111yescharacteristic
142C22×C4#12111yescharacteristic
143C22×C4C22⋊Q811yescharacteristic
144C22×C4C22⋊Q811yescharacteristic
145M16—84no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—44no—
156C2×C8—44no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8—82no—
166C2×C8—82no—
167C2×C8#10012yes—
168C2×C8#10412yes—
169C2×C8#10412yes—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—22no—
185C42—22no—
186C42—22no—
187C42—41no—
188C42—41no—
189C42—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23#38011yescharacteristic
198Q8—82no—
199D4—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
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—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4C2×#10011yescharacteristic
229C2×C4#41311yescharacteristic
230C2×C4C2×#10411yescharacteristic
231C2×C4C2×#10411yescharacteristic
232C2×C4#42111yescharacteristic
233C2×C4#42111yescharacteristic
234C2×C4#43012yes—
235C2×C4#47112yes—
236C2×C4#47112yes—
237C8—22no—
238C8—22no—
239C8—22no—
240C8—41no—
241C8—84no—
242C8—84no—
243C8—81no—
244C8—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22#180011yescharacteristic
253C22#180611yescharacteristic
254C22#180611yescharacteristic
255C22#181111yescharacteristic
256C22#181211yescharacteristic
257C22#181511yescharacteristic
258C22#181511yescharacteristic
259C4—22no—
260C4—22no—
261C4—22no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4#152411yescharacteristic
269C4#152711yescharacteristic
270C4#153011yescharacteristic
271C4#153511yescharacteristic
272C2—161no—
273C2#858111yescharacteristic
274C2#862411yescharacteristic
275C2#863611yescharacteristic
276C2#867211yescharacteristic
277C2#867211yescharacteristic
278C2#869411yescharacteristic
279C2#869611yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C4⋊C4
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
2. #8273C2semidirect
3. #8273C2semidirect
5. #7042C2semidirect

Maximal subgroups

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

subgroupindex
#216312
#82732
#76352
#70422