Finite Group Databasebeta

← #99008

Subgroups — #99008

Order 512.

388 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C8×#124C211yesmaximal, characteristic
2#8274C211yesmaximal, characteristic, complemented
3#8273C211yesmaximal, characteristic, complemented
4#7635C212yesmaximal
5#7042C212yesmaximal, complemented
6C2×C8×Q8C411yescharacteristic
7C2×C8×D4C411yescharacteristic
8C4×#124C2211yescharacteristic
9#1984C411yescharacteristic
10#1954C411yescharacteristic
11#1923—21no—
12#1923—21no—
13#1831C2211yescharacteristic
14#1767C2212yes—
15#1767C2212yes—
16C8×C22⋊C4—21no—
17C8×C22⋊C4—21no—
18C42×C8C2211yescharacteristic
19C2×#335—21no—
20C2×#334—21no—
21#1277—21no—
22#1277—21no—
23#1272—22no—
24#1270—22no—
25C23×C8—21no—
26C2×#124Q811yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×C4×D4C2×C411yescharacteristic
29C8×Q8—42no—
30C8×D4—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—22no—
40C2×#97—22no—
41C2×C4×C8—21no—
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#335—24no—
63#334—24no—
64C22×Q8C4⋊C411yescharacteristic
65C22×D4C4⋊C411yescharacteristic
66C23×C4—21no—
67C22×C8—21no—
68C22×C8—21no—
69C22×C8—42no—
70C22×C8—42no—
71C22×C8—42no—
72C22×C8—42no—
73C22×C8—42no—
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78#124—22no—
79#124—22no—
80C4×Q8—41no—
81C4×Q8—41no—
82C4×D4—41no—
83C4×D4—41no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C4⋊C411yescharacteristic
86C2×C4⋊C4C4⋊C411yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C42—21no—
93C2×C42C22×C411yescharacteristic
94C2×C42C2×D411yescharacteristic
95C2×C42C2×D411yescharacteristic
96C2×C42C2×D411yescharacteristic
97C2×C42C2×Q811yescharacteristic
98C2×C42#4111yescharacteristic
99C2×C42#4111yescharacteristic
100C4⋊C8—24no—
101C4⋊C8—24no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—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—
118C4×C8—42no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123C24—21no—
124C2×Q8—21no—
125C2×Q8—41no—
126C2×Q8—41no—
127C2×Q8#10812yes—
128C2×D4—21no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4#10812yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4C2×C22⋊C411yescharacteristic
142C22×C4C2×C4⋊C411yescharacteristic
143C22×C4#11711yescharacteristic
144C22×C4#12111yescharacteristic
145C22×C4#12111yescharacteristic
146C22×C4C22⋊Q811yescharacteristic
147C22×C4C22⋊Q811yescharacteristic
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—84no—
166C2×C8—82no—
167C2×C8—82no—
168C2×C8—82no—
169C2×C8—82no—
170C2×C8#10012yes—
171C2×C8#10412yes—
172C2×C8#10412yes—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—41no—
181C22⋊C4—41no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—22no—
191C42—21no—
192C42—21no—
193C42—41no—
194C42—41no—
195C42—41no—
196C23—22no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23#38011yescharacteristic
204Q8—42no—
205Q8—42no—
206D4—42no—
207D4—42no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4C2×#10011yescharacteristic
237C2×C4#41311yescharacteristic
238C2×C4C2×#10411yescharacteristic
239C2×C4C2×#10411yescharacteristic
240C2×C4#42111yescharacteristic
241C2×C4#42111yescharacteristic
242C2×C4C2×#10811yescharacteristic
243C2×C4C2×#10811yescharacteristic
244C2×C4#47112yes—
245C2×C4#47211yescharacteristic
246C2×C4#47311yescharacteristic
247C8—22no—
248C8—22no—
249C8—22no—
250C8—41no—
251C8—84no—
252C8—84no—
253C8—81no—
254C8—81no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22#180011yescharacteristic
263C22#180611yescharacteristic
264C22#180611yescharacteristic
265C22#181011yescharacteristic
266C22#181211yescharacteristic
267C22#181511yescharacteristic
268C22#181611yescharacteristic
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—22no—
274C4—42no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—81no—
279C4—81no—
280C4#152411yescharacteristic
281C4#152711yescharacteristic
282C4#153011yescharacteristic
283C4#153511yescharacteristic
284C2—82no—
285C2#858111yescharacteristic
286C2#862011yescharacteristic
287C2#864011yescharacteristic
288C2#867011yescharacteristic
289C2#867111yescharacteristic
290C2#869511yescharacteristic
291C2#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. #8274C2semidirect
3. #8273C2semidirect
5. #7042C2semidirect

Maximal subgroups

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

subgroupindex
C8×#1242
#82742
#82732
#76352
#70422