Finite Group Databasebeta

← #124652

Subgroups — #124652

Order 512.

515 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#1971C211yesmaximal, characteristic
2#12846C211yesmaximal, characteristic
3#7861C211yesmaximal, characteristic
4#7111C214yesmaximal, complemented
5C23×Q16C411yescharacteristic
6#2920—21no—
7C22×#106C411yescharacteristic
8C22×#103—21no—
9C22×#97C2211yescharacteristic
10C2×#391—21no—
11#2112—21no—
12#1971—21no—
13#1971—21no—
14#1971C412yes—
15#1971C2212yes—
16#1377—21no—
17#1377C2212yes—
18#1275—21no—
19#1275C2212yes—
20C23×Q8—21no—
21C22×Q16—24no—
22C22×Q16—21no—
23C22×Q16—41no—
24C22×Q16—41no—
25C22×Q16—42no—
26C22×Q16C2×C412yes—
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C22×C4⋊C4—21no—
30C22×C42D411yescharacteristic
31C2×M32—22no—
32#432—41no—
33#432—41no—
34#431—22no—
35#431—22no—
36C2×#106—21no—
37C2×#106—42no—
38C2×#106C2×C412yes—
39#423—41no—
40C2×#103—22no—
41C2×#103—22no—
42C2×#103—44no—
43#407—41no—
44C4×M16—22no—
45C2×#97—41no—
46C2×#97—41no—
47C2×#97C2×C411yescharacteristic
48C2×#97D411yescharacteristic
49C2×#97D411yescharacteristic
50C2×#97C2311yescharacteristic
51#391—22no—
52#391—22no—
53#391—41no—
54#391—41no—
55C2×#95—41no—
56C2×#95—41no—
57C22⋊C16—41no—
58C22⋊C16—41no—
59C22×Q8—24no—
60C22×Q8—22no—
61C22×Q8—42no—
62C22×Q8—42no—
63C22×Q8—42no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—41no—
69C23×C4—41no—
70C23×C4—41no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4D811yescharacteristic
73C23×C4SD1611yescharacteristic
74C2×Q16—24no—
75C2×Q16—22no—
76C2×Q16—42no—
77C2×Q16—42no—
78C2×Q16—48no—
79C2×Q16—48no—
80C2×Q16—42no—
81C2×Q16—42no—
82C2×Q16M1614yescomplemented
83C2×M16—42no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—22no—
91C22×C8—41no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22×C411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C2×C4⋊C4—22no—
97C2×C4⋊C4—42no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—41no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42D811yescharacteristic
106C2×C42SD1611yescharacteristic
107C2×C42C2×D411yescharacteristic
108M32—42no—
109M32—42no—
110C2×C16—42no—
111#106—42no—
112#106—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115#103—44no—
116#103—44no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#97—41no—
120#97—41no—
121#97—41no—
122#97—41no—
123#97—42no—
124C4×C8—24no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129C24#10211yescharacteristic
130C2×Q8—24no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×Q8—44no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—82no—
142C2×Q8—82no—
143C2×Q8—82no—
144C2×Q8—82no—
145C2×Q8—82no—
146C2×Q8—81no—
147C2×Q8—81no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—44no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—82no—
162C22×C4—81no—
163C22×C4—81no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4#9911yescharacteristic
168C22×C4#9911yescharacteristic
169C22×C4#10211yescharacteristic
170C22×C4#10211yescharacteristic
171C22×C4#10211yescharacteristic
172C22×C4#10411yescharacteristic
173C22×C4#10411yescharacteristic
174C22×C4C2×C22⋊C411yescharacteristic
175C22×C4C2×D811yescharacteristic
176C22×C4C2×SD1611yescharacteristic
177Q16—44no—
178Q16—48no—
179Q16—44no—
180M16—84no—
181C2×C8—24no—
182C2×C8—24no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—44no—
195C2×C8#10012yes—
196C2×C8C2×M1612yes—
197C2×C8AGL(1,ℤ/8)12yes—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—82no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C16—82no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—21no—
214C23#32711yescharacteristic
215C23#32711yescharacteristic
216C23C2×#10211yescharacteristic
217Q8—44no—
218Q8—44no—
219Q8—84no—
220Q8—84no—
221Q8—82no—
222Q8—82no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—44no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—84no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—164no—
254C2×C4#32711yescharacteristic
255C2×C4#32711yescharacteristic
256C2×C4#40711yescharacteristic
257C2×C4C2×#9911yescharacteristic
258C2×C4C2×#10011yescharacteristic
259C2×C4#41811yescharacteristic
260C2×C4C2×#10411yescharacteristic
261C2×C4#42111yescharacteristic
262C8—22no—
263C8—22no—
264C8—22no—
265C8—41no—
266C8—84no—
267C22—24no—
268C22—24no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22#134012yes—
276C22#134212yes—
277C22#148611yescharacteristic
278C22C2×#32711yescharacteristic
279C22#154511yescharacteristic
280C4—21no—
281C4—21no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—41no—
286C4—84no—
287C4—162no—
288C4#146211yescharacteristic
289C4#149011yescharacteristic
290C4#151011yescharacteristic
291C4#152711yescharacteristic
292C2—21no—
293C2—22no—
294C2—22no—
295C2—21no—
296C2#760611yescharacteristic
297C2C2×#134011yescharacteristic
298C2C2×#134211yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7111C2semidirect
82. C2×Q16M16semidirect

Maximal subgroups

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

subgroupindex
C2×#19712
#128462
#78612
#71112