Finite Group Databasebeta

← #6372235

Subgroups — #6372235

Order 512.

3294 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3504C211yesmaximal, characteristic
2#60099C211yesmaximal, characteristic
3C4×#568C211yesmaximal, characteristic
4#21338C2112yesmaximal, complemented
5#21337C216yesmaximal, complemented
6C2×#1998C216yesmaximal, complemented
7#20358C214yesmaximal
8C2×#568C2212yes—
9C2×#568C2212yes—
10#3504—24no—
11#3504—24no—
12#3504—28no—
13C2×#557C414yes—
14C2×#554C4112yes—
15C2×#544C2216yes—
16C2×#533C2213yes—
17C22×#125C2213yes—
18C2×#520C2212yes—
19C2×C4×#41C2211yescharacteristic
20#2906—28no—
21#2905C2216yes—
22C2×#423C2216yes—
23#2875—24no—
24C4×#131—24no—
25#2873C2216yes—
26C2×C4×M16C2216yes—
27#1998C22112yes—
28#1998C22124yes—
29#1998C2214yes—
30#1997C22124yes—
31#1742C22124yes—
32C2×#343C22124yes—
33C22×#41C2311yescharacteristic
34#568—28no—
35#568—216no—
36#568Q8116yescomplemented
37C2×#131—28no—
38C2×#131—28no—
39C22×M16C23112yes—
40C22×M16C23112yes—
41#557—216no—
42#557—216no—
43#554—248no—
44#554—248no—
45#544—224no—
46#544—224no—
47#533—26no—
48#533—26no—
49#533—212no—
50C2×C4⋊Q8C2×C4112yes—
51C2×C4⋊Q8C2316yes—
52C2×#125—26no—
53C2×#125—26no—
54C2×#125—212no—
55C2×#125C2×C4112yes—
56C2×#125C2×C4112yes—
57C2×#125C2313yes—
58C2×#125C2316yes—
59C2×#125C2311yescharacteristic
60C2×C22⋊Q8C2×C4112yes—
61C2×C22⋊Q8C2×C4124yes—
62#520—28no—
63#520—28no—
64C4×#41D414yes—
65C4×#41D414yes—
66C4×#41D418yes—
67C2×C4×Q8C2×C414yes—
68C2×C4×Q8C2×C4112yes—
69C2×C4×Q8C2×C412yes—
70C2×C4×D4C2×C4112yes—
71C2×C4×D4C2×C416yes—
72C2×#117C2316yes—
73C2×#117C2313yes—
74C22×C4⋊C4C2×C4112yes—
75C22×C4⋊C4C2316yes—
76C22×C42C2313yes—
77#424—224no—
78#423—212no—
79#423C2314yes—
80#423C23112yes—
81C2×C4⋊C8—212no—
82#405—224no—
83C4×M16—26no—
84C4×M16—26no—
85C4×M16C2314yes—
86C4×M16C23112yes—
87C2×#97—26no—
88C2×C4×C8—26no—
89#343C23148yes—
90#343C23116yes—
91C2×#41C2×D414yes—
92C2×#41C2×D414yes—
93C2×#41C2×Q818yes—
94C2×#41#4118yes—
95C22×Q8C22×C4112yes—
96C22×Q8C22×C411yescharacteristic
97C22×D4C22×C413yes—
98C23×C4C22×C4112yes—
99C23×C4C22×C416yes—
100C23×C4C2413yes—
101#131—264no—
102#131—432no—
103C2×M16—212no—
104C2×M16—224no—
105C2×M16—212no—
106C2×M16—248no—
107C2×M16C2×Q8148yes—
108C2×M16C2418yes—
109C2×M16C2418yes—
110C22×C8—212no—
111C22×C8—212no—
112C4⋊Q8—248no—
113C4⋊Q8—212no—
114C4⋊Q8—212no—
115#125—248no—
116#125—224no—
117#125—224no—
118#125—23no—
119#125—23no—
120#125—26no—
121#125—212no—
122#125—212no—
123#125—21no—
124#125—21no—
125#125—22no—
126C22⋊Q8—224no—
127C22⋊Q8—224no—
128C22⋊Q8—296no—
129C4×Q8—216no—
130C4×Q8—224no—
131C4×Q8—224no—
132C4×Q8C22⋊C418yes—
133C4×Q8C22⋊C418yes—
134C4×D4—248no—
135C4×D4C22⋊C4124yes—
136C4×D4C22⋊C4124yes—
137#117—212no—
138#117—212no—
139#117C2×D4112yes—
140#117C2×D4112yes—
141C2×C4⋊C4—224no—
142C2×C4⋊C4—224no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4—212no—
145C2×C4⋊C4C22×C4112yes—
146C2×C4⋊C4C22×C4124yes—
147C2×C4⋊C4C22×C4112yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C4112yes—
150C2×C4⋊C4C22×C416yes—
151C2×C4⋊C4C2416yes—
152C2×C4⋊C4C2412yes—
153C2×C22⋊C4C22×C4112yes—
154C2×C22⋊C4C22×C416yes—
155C2×C42C22×C4112yes—
156C2×C42C22×C416yes—
157C2×C42C2×D416yes—
158C2×C42C2×D416yes—
159C2×C42C2×D4112yes—
160C2×C42C2413yes—
161C2×C42C2411yescharacteristic
162C4⋊C8—28no—
163C4⋊C8—224no—
164#97—24no—
165#97—212no—
166C4×C8—24no—
167C4×C8—212no—
168C24C23×C413yes—
169#41C22⋊Q8132yes—
170#41#124116yes—
171#41C4⋊Q8116yes—
172C2×Q8—224no—
173C2×Q8—812no—
174C2×Q8C2×C22⋊C414yes—
175C2×Q8C4×Q818yes—
176C2×D4C2×C22⋊C4112yes—
177C2×D4C4×Q8124yes—
178C22×C4—224no—
179C22×C4—86no—
180C22×C4—812no—
181C22×C4C2×C22⋊C4112yes—
182C22×C4C2×C22⋊C4112yes—
183C22×C4C23×C414yes—
184C22×C4C23×C4112yes—
185C22×C4C23×C412yes—
186C22×C4C23×C416yes—
187C22×C4C23×C413yes—
188C22×C4C22×D416yes—
189C22×C4C22×D416yes—
190C22×C4C22×Q8112yes—
191C22×C4C2×#41112yes—
192C22×C4C2511yescharacteristic
193M16—248no—
194M16—28no—
195M16—216no—
196M16—424no—
197M16C22×Q8116yes—
198C2×C8—248no—
199C2×C8—28no—
200C2×C8—28no—
201C2×C8—424no—
202C4⋊C4—212no—
203C4⋊C4—212no—
204C4⋊C4—224no—
205C4⋊C4—248no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—224no—
209C4⋊C4—26no—
210C4⋊C4—26no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4C2×C22⋊C4124yes—
214C22⋊C4—224no—
215C22⋊C4C2×C22⋊C4124yes—
216C42—224no—
217C42C2×C22⋊C4112yes—
218C42C2×C22⋊C4112yes—
219C42C22×D411yescharacteristic
220C42C22×D416yes—
221C42C22×D411yescharacteristic
222C42C22×D412yes—
223C42C22×D416yes—
224C23—46no—
225C23C22×C22⋊C416yes—
226C23C2×C4×Q8112yes—
227C23C24×C411yescharacteristic
228Q8—824no—
229Q8#391116yes—
230D4#391148yes—
231C2×C4—24no—
232C2×C4—212no—
233C2×C4—42no—
234C2×C4—412no—
235C2×C4—412no—
236C2×C4—46no—
237C2×C4—412no—
238C2×C4—44no—
239C2×C4—824no—
240C2×C4C22×C22⋊C412yes—
241C2×C4C22×C22⋊C412yes—
242C2×C4C22×C22⋊C4112yes—
243C2×C4C22×C22⋊C416yes—
244C2×C4C2×C4×Q8112yes—
245C2×C4C2×C22⋊Q8124yes—
246C2×C4C2×#124112yes—
247C2×C4C2×C4⋊Q8112yes—
248C2×C4C23×D411yescharacteristic
249C2×C4C23×D411yescharacteristic
250C2×C4C23×Q812yes—
251C2×C4C22×#4112yes—
252C8—216no—
253C8—48no—
254C22—22no—
255C22—212no—
256C22—412no—
257C22C2×#391112yes—
258C22C23×C22⋊C411yescharacteristic
259C22C22×C4×Q812yes—
260C4—22no—
261C4—22no—
262C4—212no—
263C4—412no—
264C4—44no—
265C4C2×#391112yes—
266C4C22×C22⋊Q812yes—
267C4C22×#12411yescharacteristic
268C4C22×C4⋊Q811yescharacteristic
269C2—22no—
270C2—212no—
271C2C22×#39111yescharacteristic
272C2#6007612yes—
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21338C2semidirect
5. #21337C2semidirect
6. C2×#1998C2semidirect
36. #568Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#35042
#600992
C4×#5682
#213382
#213372
C2×#19982
#203582