Finite Group Databasebeta

← #142197

Subgroups — #142197

Order 512.

412 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 106 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#33208C211yesmaximal, characteristic
2#12783C211yesmaximal, characteristic
3#10820C211yesmaximal, characteristic
4#7258C212yesmaximal
5#7258C212yesmaximal
6C2×#499C411yescharacteristic, complemented
7C22×C4×C8C2211yescharacteristic
8#2851—21no—
9C2×C4⋊C16—21no—
10C2×C22⋊C16C2212yes—
11#1991—21no—
12#1942C411yescharacteristic, complemented
13#1942C411yescharacteristic, complemented
14#1941—21no—
15#1846C411yescharacteristic, complemented
16#1808C2211yescharacteristic
17#1808C2211yescharacteristic
18#1807—21no—
19C2×#340—21no—
20C2×#340C2212yes—
21C4⋊#95—21no—
22#1374—21no—
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#125—21no—
26C22×C4⋊C4—22no—
27C22×C42D411yescharacteristic
28C22×C16—22no—
29#499—42no—
30C2×#107C2×C411yescharacteristic
31C2×#107C2×C411yescharacteristic
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#106—21no—
35C2×#106—21no—
36C2×C4×C8—42no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8C2×C411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41#389—21no—
42#389—21no—
43#387—41no—
44#387—41no—
45#385—41no—
46#385—41no—
47#382—21no—
48#380—21no—
49#380—21no—
50C2×#95—41no—
51C2×#95—41no—
52C4⋊C16—42no—
53C22⋊C16—24no—
54#340—21no—
55#340—21no—
56#340—21no—
57#340—21no—
58#340—22no—
59#340—42no—
60#340Q814yes—
61C23×C4—42no—
62C23×C4D811yescharacteristic
63C23×C4D811yescharacteristic
64C23×C4C2×D411yescharacteristic
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—41no—
68C22×C8—41no—
69C22×C8—41no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22×C8C2×Q812yes—
75C22×C8#4112yes—
76#125—42no—
77#125—42no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—42no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—42no—
86C2×C4⋊C4—42no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—42no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C42—41no—
95C2×C42—41no—
96C2×C42C22⋊C411yescharacteristic
97C2×C42C22⋊C411yescharacteristic
98C2×C42SD1611yescharacteristic
99C2×C42SD1611yescharacteristic
100C2×C16—42no—
101C2×C16—42no—
102C2×C16—42no—
103#107—41no—
104#107—41no—
105#107—41no—
106#107—41no—
107#106—42no—
108#106—42no—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—41no—
112C4×C8—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—82no—
120#95—82no—
121C24C2×D811yescharacteristic
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—44no—
138C22×C4—82no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4—82no—
144C22×C4#10211yescharacteristic
145C22×C4#10211yescharacteristic
146C22×C4#10211yescharacteristic
147C22×C4#10211yescharacteristic
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C22⋊Q812yes—
150C22×C4C2×SD1611yescharacteristic
151C22×C4#13712yes—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8#9912yes—
165C2×C8C4×Q812yes—
166C2×C8#12411yescharacteristic
167C2×C8C4⋊Q811yescharacteristic
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C22⋊C4—82no—
185C22⋊C4—81no—
186C22⋊C4—81no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—41no—
192C42—41no—
193C16—82no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23C2×#10211yescharacteristic
199C23#47412yes—
200C23#50812yes—
201C23#50911yescharacteristic
202C23#51011yescharacteristic
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
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—21no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—84no—
224C2×C4—84no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—84no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4—162no—
239C2×C4—162no—
240C2×C4—162no—
241C2×C4—162no—
242C2×C4#36012yes—
243C2×C4#39111yescharacteristic
244C2×C4C2×#9911yescharacteristic
245C2×C4#41011yescharacteristic
246C2×C4#41911yescharacteristic
247C2×C4#42111yescharacteristic
248C2×C4#42111yescharacteristic
249C2×C4#47512yes—
250C8—21no—
251C8—22no—
252C8—21no—
253C8—22no—
254C8—41no—
255C22—22no—
256C22—21no—
257C22—21no—
258C22—22no—
259C22—21no—
260C22—21no—
261C22—22no—
262C22—22no—
263C22#151711yescharacteristic
264C22#151911yescharacteristic
265C22#198811yescharacteristic
266C22#213811yescharacteristic
267C22#213911yescharacteristic
268C22C2×#36011yescharacteristic
269C22#214511yescharacteristic
270C22#222812yes—
271C22#222912yes—
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—164no—
278C4—161no—
279C4—161no—
280C4—161no—
281C4—161no—
282C4#151111yescharacteristic
283C4#151511yescharacteristic
284C4#199511yescharacteristic
285C4#199611yescharacteristic
286C2—22no—
287C2—22no—
288C2#991211yescharacteristic
289C2#1185511yescharacteristic
290C2#1186011yescharacteristic
291C2#1188511yescharacteristic
292C2#1189011yescharacteristic
293C2#1327711yescharacteristic
294C2#1327811yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×#499C4semidirect
12. #1942C4semidirect
13. #1942C4semidirect
15. #1846C4semidirect

Maximal subgroups

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

subgroupindex
#332082
#127832
#108202
#72582