Finite Group Databasebeta

← #125338

Subgroups — #125338

Order 512.

410 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 114 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#33209C211yesmaximal, characteristic
2C4×C22⋊C16C211yesmaximal, characteristic
3#7874C211yesmaximal, characteristic
4#7121C214yesmaximal, complemented
5C2×#500C411yescharacteristic
6C2×#423—21no—
7C22×C4×C8C2211yescharacteristic
8#2857—21no—
9C2×C22⋊C16—21no—
10C2×C4×C16—21no—
11#1990—21no—
12#1942—21no—
13#1942C2212yes—
14#1845C411yescharacteristic
15#1808—21no—
16#1808C412yes—
17#1374C2212yes—
18#1277—21no—
19#1277C2212yes—
20C22×M16—41no—
21C23×C8C2×C411yescharacteristic
22C23×C8D411yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C22⋊Q8—41no—
25C22×C42D411yescharacteristic
26C22×C16—22no—
27#500—42no—
28C2×#107—21no—
29C2×#107—21no—
30C2×#107C2×C412yes—
31C2×#107C2×C412yes—
32#423—24no—
33#423—42no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—41no—
36C2×C4×C8—42no—
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8C2311yescharacteristic
41#391—41no—
42#389—22no—
43#389—41no—
44#384—21no—
45#380—22no—
46#380—41no—
47C22⋊C16—24no—
48C4×C16—42no—
49#340—22no—
50#340—22no—
51C22×Q8—41no—
52C23×C4C22⋊C411yescharacteristic
53C23×C4D811yescharacteristic
54C23×C4SD1611yescharacteristic
55C2×M16—44no—
56C2×M16—82no—
57C2×M16—82no—
58C22×C8—22no—
59C22×C8—22no—
60C22×C8—22no—
61C22×C8—22no—
62C22×C8—41no—
63C22×C8—41no—
64C22×C8—42no—
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22⋊C411yescharacteristic
67C22×C8C22×C411yescharacteristic
68C22×C8C2×D411yescharacteristic
69C4⋊Q8—42no—
70C4⋊Q8—42no—
71C22⋊Q8—84no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42C22⋊C411yescharacteristic
82C2×C42D811yescharacteristic
83C2×C42SD1611yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C16—24no—
86C2×C16—42no—
87C2×C16—42no—
88#107—24no—
89#107—42no—
90#107—42no—
91#107—42no—
92C4⋊C8—24no—
93C4⋊C8—42no—
94C4⋊C8—44no—
95C4×C8—22no—
96C4×C8—22no—
97C4×C8—41no—
98C4×C8—41no—
99#95—42no—
100#95—42no—
101C24#10211yescharacteristic
102C2×Q8—82no—
103C2×Q8—82no—
104C2×Q8—82no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4#10211yescharacteristic
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4SD3212yes—
122C22×C4Q3212yes—
123C22×C4C2×C22⋊C411yescharacteristic
124C22×C4C2×D811yescharacteristic
125C22×C4C2×SD1611yescharacteristic
126M16—88no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—44no—
136C2×C8—44no—
137C2×C8—44no—
138C2×C8—82no—
139C2×C8—82no—
140C2×C8#10112yes—
141C2×C8#10412yes—
142C2×C8#13112yes—
143C2×C8#13712yes—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—84no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C4⋊C4—82no—
152C22⋊C4—84no—
153C42—22no—
154C42—22no—
155C42—21no—
156C42—21no—
157C42#9912yes—
158C42#10412yes—
159C16—44no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23#35812yes—
167C23C2×#10211yescharacteristic
168Q8—164no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—44no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—82no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4#32712yes—
191C2×C4#32712yes—
192C2×C4#35812yes—
193C2×C4#35912yes—
194C2×C4#36012yes—
195C2×C4#36112yes—
196C2×C4#40811yescharacteristic
197C2×C4C2×#9911yescharacteristic
198C2×C4C2×#10111yescharacteristic
199C2×C4#41911yescharacteristic
200C2×C4C2×#10411yescharacteristic
201C2×C4C2×#10411yescharacteristic
202C2×C4C2×SD3211yescharacteristic
203C2×C4C2×Q3211yescharacteristic
204C8—22no—
205C8—22no—
206C8—22no—
207C8—22no—
208C8—84no—
209C22—22no—
210C22—22no—
211C22—22no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22#149211yescharacteristic
217C22C2×#32711yescharacteristic
218C22#154611yescharacteristic
219C22C2×#35811yescharacteristic
220C22C2×#35911yescharacteristic
221C22C2×#36011yescharacteristic
222C22C2×#36111yescharacteristic
223C4—21no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—162no—
228C4—162no—
229C4#134814yes—
230C4#134914yes—
231C4#147211yescharacteristic
232C4#148511yescharacteristic
233C4#150811yescharacteristic
234C4#152611yescharacteristic
235C2—42no—
236C2#768511yescharacteristic
237C2#1167211yescharacteristic
238C2#1168011yescharacteristic
239C2C2×#134811yescharacteristic
240C2C2×#134911yescharacteristic
241C2#1196811yescharacteristic
242C2#1197211yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7121C2semidirect

Maximal subgroups

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

subgroupindex
#332092
C4×C22⋊C162
#78742
#71212