Finite Group Databasebeta

← #6376372

Subgroups — #6376372

Order 512.

1694 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic
3#34215C211yesmaximal, characteristic
4#22255C218yesmaximal, complemented
5#22249C218yesmaximal, complemented
6#21318C211yesmaximal, characteristic
7C2×#1991C211yesmaximal, characteristic
8#21036C214yesmaximal
9#21035C214yesmaximal
10#20765C211yesmaximal, characteristic
11#20366C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#125C2211yescharacteristic
14#3166C4116yescomplemented
15C2×#501C2214yes—
16C2×#499C2214yes—
17C2×#428C2214yes—
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2370—24no—
21#2368—24no—
22#2355—24no—
23#2345—24no—
24#2331C2214yes—
25#2330C2214yes—
26#2317C2212yes—
27#2316C2212yes—
28C4×#125C2214yes—
29C4×#125C2214yes—
30#2285C2211yescharacteristic
31C2×#382C2211yescharacteristic
32#2274C2212yes—
33#2274C2212yes—
34C4×#117C2211yescharacteristic
35#1991C2214yes—
36#1991C2214yes—
37#1991C2214yes—
38#1990C2214yes—
39#1921C2218yes—
40#1921C2218yes—
41#1920C2218yes—
42#1920C2218yes—
43#1919C2218yes—
44#1919C2218yes—
45#1918C2218yes—
46#1918C2218yes—
47#1836C2214yes—
48#1835C2214yes—
49#1746C2214yes—
50#1744C2214yes—
51#1736C2214yes—
52#1735C2214yes—
53C2×#340C2214yes—
54#1729C2212yes—
55#1729C2212yes—
56C22×M16C2312yes—
57C23×C8C2312yes—
58#533—24no—
59#533C2×C418yes—
60C2×C4⋊Q8C2312yes—
61C2×#125—22no—
62C2×#125—24no—
63C2×#125C2×C418yes—
64C2×#125C2312yes—
65C2×#125C2312yes—
66C2×#125C2312yes—
67C2×C22⋊Q8—22no—
68C2×C4×Q8—22no—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C42C2311yescharacteristic
77#501C2×C4132yescomplemented
78#499C2×C4132yescomplemented
79#428C2×C4132yes—
80C2×#107C2318yes—
81C2×#106C2318yes—
82C4×M16C2×C418yes—
83C2×#97C2312yes—
84C2×C4×C8—24no—
85C2×C4×C8C2312yes—
86C2×C4×C8C2312yes—
87C2×C4×C8C2312yes—
88#391—28no—
89#389—24no—
90#389—28no—
91#389C2318yes—
92#389C2318yes—
93#387—28no—
94#387—28no—
95#387—28no—
96#387—28no—
97#385—24no—
98#384—22no—
99#384C2312yes—
100#382—22no—
101#382—22no—
102#382C2312yes—
103#382C2312yes—
104#382C2312yes—
105#380—22no—
106#380—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—28no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4D414yes—
113C4×C4⋊C4C2318yes—
114C4×C4⋊C4C2318yes—
115C4×C22⋊C4—24no—
116C4×C22⋊C4—24no—
117C4×C22⋊C4D414yes—
118#376C2312yes—
119C2×#95—22no—
120C43C2312yes—
121#340C2318yes—
122#340C2318yes—
123#340C2314yes—
124#340C2318yes—
125#340C2314yes—
126#340C2314yes—
127#340C2318yes—
128#340C2314yes—
129#336C2318yes—
130#336C2318yes—
131C22×Q8—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C2×D412yes—
135C23×C4C2411yescharacteristic
136C2×M16—24no—
137C2×M16—24no—
138C2×M16C22×C418yes—
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8C2414yes—
143C22×C8C2414yes—
144C4⋊Q8—24no—
145C4⋊Q8C22×C418yes—
146#125—216no—
147#125—24no—
148#125—24no—
149#125—24no—
150#125C22×C418yes—
151#125C22×C418yes—
152#125C22×C418yes—
153C22⋊Q8—216no—
154C4×Q8—216no—
155#117—24no—
156#117—24no—
157#117—24no—
158#117—28no—
159#117C22×C418yes—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4C22×C418yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2414yes—
176C2×C4⋊C4C2414yes—
177C2×C4⋊C4C2414yes—
178C2×C4⋊C4C2414yes—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×D412yes—
182C2×C22⋊C4C2×D412yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—28no—
186C2×C42—28no—
187C2×C42C22×C414yes—
188C2×C42C2×D418yes—
189C2×C42C2×D412yes—
190C2×C42#4114yes—
191C2×C42C2412yes—
192C2×C42C2412yes—
193C2×C42C2411yescharacteristic
194C2×C42C2411yescharacteristic
195#107C22×C4132yes—
196#106C22×C4132yes—
197#97C22×C418yes—
198C4×C8—24no—
199C4×C8—24no—
200C4×C8C22×C418yes—
201#95—24no—
202#95—24no—
203#95—28no—
204#95—28no—
205#95—24no—
206#95—28no—
207#95—28no—
208#95C2×D418yes—
209C24C22×D411yescharacteristic
210C2×Q8—28no—
211C2×Q8—84no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—84no—
220C22×C4—84no—
221C22×C4C4×D418yes—
222C22×C4C23×C412yes—
223C22×C4C22×D414yes—
224C22×C4C22×D414yes—
225C22×C4C22×D412yes—
226C22×C4C22×D412yes—
227C22×C4C22×D411yescharacteristic
228C22×C4C2×#4112yes—
229C22×C42−1+414yes—
230C22×C42−1+414yes—
231C22×C4C2511yescharacteristic
232M16—216no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—24no—
237C2×C8—24no—
238C2×C8—48no—
239C2×C8C23×C418yes—
240C4⋊C4—216no—
241C4⋊C4—216no—
242C4⋊C4—28no—
243C4⋊C4—216no—
244C4⋊C4—28no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—44no—
252C4⋊C4—84no—
253C4⋊C4—88no—
254C4⋊C4—88no—
255C4⋊C4C23×C418yes—
256C4⋊C4C23×C418yes—
257C22⋊C4—28no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—44no—
261C42—28no—
262C42—24no—
263C42—24no—
264C42—48no—
265C42—44no—
266C42—88no—
267C42—88no—
268C42C4×D418yes—
269C42C23×C412yes—
270C42C23×C412yes—
271C42C2×#4112yes—
272C42C2×#4112yes—
273C23—24no—
274C23C2×C4×D412yes—
275C23#53612yes—
276C23#53612yes—
277C23C23×D411yescharacteristic
278Q8—88no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—48no—
288C2×C4—44no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—88no—
297C2×C4—164no—
298C2×C4C2×C4×D414yes—
299C2×C4C2×C4×D414yes—
300C2×C4C2×C4×D412yes—
301C2×C4#51918yes—
302C2×C4#53612yes—
303C2×C4#53612yes—
304C2×C4#54118yes—
305C2×C4C24×C411yescharacteristic
306C2×C4C22×#4111yescharacteristic
307C2×C4C2×2−1+412yes—
308C2×C4C2×2−1+412yes—
309C8—28no—
310C8—44no—
311C22—24no—
312C22—24no—
313C22—24no—
314C22#232714yes—
315C22C22×C4×D411yescharacteristic
316C22C2×#53611yescharacteristic
317C22C2×#53611yescharacteristic
318C22#358414yes—
319C4—24no—
320C4—24no—
321C4—48no—
322C4—48no—
323C4—44no—
324C4—44no—
325C4—84no—
326C4—84no—
327C4#232714yes—
328C4C2×#51912yes—
329C4C2×#54112yes—
330C2—24no—
331C2C2×#232711yescharacteristic
332C2#6016112yes—
333C2#6087912yes—
334C2#6088212yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22255C2semidirect
5. #22249C2semidirect
14. #3166C4semidirect
77. #501C2×C4semidirect
78. #499C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
#342172
#342152
#222552
#222492
#213182
C2×#19912
#210362
#210352
#207652
#203662