Finite Group Databasebeta

← #214085

Subgroups — #214085

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2#21786C211yesmaximal, characteristic
3#21742C211yesmaximal, characteristic
4#9085C214yesmaximal, complemented
5#9082C214yesmaximal, complemented
6#8700C212yesmaximal, complemented
7#8299C212yesmaximal, complemented
8C22×#123C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#431C2212yes—
12C2×#407C2212yes—
13#2322—24no—
14#2322—24no—
15C2×#389C412yes—
16C2×#388C412yes—
17C2×#387C2212yes—
18C2×#386C411yescharacteristic
19C2×#385C411yescharacteristic
20C2×#385C2212yes—
21C2×#381C411yescharacteristic
22C2×C4×C22⋊C4C2211yescharacteristic
23#1954—24no—
24#1870—24no—
25#1817C2212yes—
26#1779C2212yes—
27#1779C2212yes—
28#1779C2212yes—
29C2×#324C2218yes—
30#1280C2218yes—
31C23×D4C2×C411yescharacteristic
32C24×C4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2311yescharacteristic
35C24×C4C2311yescharacteristic
36C22×M16—22no—
37C2×#123—24no—
38C2×#123—28no—
39C2×C4×D4—44no—
40C2×C4×D4D414yes—
41C2×C4×D4D414yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—44no—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C412yes—
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2311yescharacteristic
58#431—44no—
59C2×C4⋊C8—22no—
60#407—44no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8C2314yes—
66C2×C22⋊C8C2314yes—
67C2×C4×C8—22no—
68#389—28no—
69#388—28no—
70#387—24no—
71#387—24no—
72#386—24no—
73#385—24no—
74#385—24no—
75#385—24no—
76#381—24no—
77C4×C22⋊C4—22no—
78C4×C22⋊C4—22no—
79C2×#95C2×C412yes—
80C2×#95C2×C412yes—
81C2×#95C2×C412yes—
82C2×#95C2×C412yes—
83C2×#95C2312yes—
84#336—24no—
85#324—48no—
86C25C22×C411yescharacteristic
87C25C22×C411yescharacteristic
88C22×D4—24no—
89C22×D4—24no—
90C22×D4—42no—
91C22×D4C22⋊C414yes—
92C23×C4—22no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—82no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4C22×C412yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C412yes—
106C23×C4C22×C412yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C22×C411yescharacteristic
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C412yes—
111C23×C4C2×D412yes—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2411yescharacteristic
116C2×M16—44no—
117C2×M16—84no—
118C22×C8—24no—
119C22×C8—24no—
120#123—232no—
121C4×D4—44no—
122C4×D4—44no—
123C4×D4—44no—
124C4×D4—44no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—48no—
135C2×C4⋊C4—48no—
136C2×C4⋊C4—82no—
137C2×C4⋊C4C22⋊C414yes—
138C2×C4⋊C4M1618yescomplemented
139C2×C22⋊C4—22no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—82no—
150C2×C22⋊C4C22⋊C414yes—
151C2×C22⋊C4C22⋊C414yes—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—82no—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C4⋊C8—44no—
158C22⋊C8—28no—
159C22⋊C8—28no—
160C22⋊C8—44no—
161C22⋊C8—84no—
162C22⋊C8—84no—
163C22⋊C8—84no—
164C4×C8—44no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—22no—
170#95—22no—
171C24—22no—
172C24—22no—
173C24—22no—
174C24—22no—
175C24—22no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24C2×C22⋊C412yes—
181C24C2×C22⋊C412yes—
182C24#13114yes—
183C24C23×C411yescharacteristic
184C2×D4—28no—
185C2×D4—28no—
186C2×D4—44no—
187C2×D4—416no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—28no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—416no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—88no—
228C22×C4—84no—
229C22×C4—88no—
230C22×C4—88no—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×M1614yes—
236C22×C4C2×M1614yes—
237C22×C4C2×M1614yes—
238C22×C4C22×D411yescharacteristic
239C22×C4C22×D411yescharacteristic
240C22×C4H(5,𝔽2)14yes—
241M16—88no—
242C2×C8—48no—
243C2×C8—48no—
244C2×C8—44no—
245C2×C8—44no—
246C4⋊C4—216no—
247C4⋊C4—44no—
248C4⋊C4—48no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—84no—
252C22⋊C4—28no—
253C22⋊C4—216no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—84no—
258C42—22no—
259C42—22no—
260C42—24no—
261C42—84no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—24no—
267C23—24no—
268C23—28no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—24no—
274C23—48no—
275C23—44no—
276C23—44no—
277C23—44no—
278C23—44no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—82no—
283C23—84no—
284C23—82no—
285C23#40814yes—
286C23C22×C22⋊C411yescharacteristic
287C23#51812yes—
288C23C22×M1612yes—
289C23C2×#13112yes—
290D4—416no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—28no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—28no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—416no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—88no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4#40714yes—
332C2×C4#53412yes—
333C2×C4#53412yes—
334C8—84no—
335C8—84no—
336C22—22no—
337C22—22no—
338C22—24no—
339C22—22no—
340C22—22no—
341C22—28no—
342C22—24no—
343C22—416no—
344C22—44no—
345C22—42no—
346C22—42no—
347C22—44no—
348C22—44no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—88no—
353C22#230811yescharacteristic
354C22C2×#40711yescharacteristic
355C22C2×#40811yescharacteristic
356C22#288312yes—
357C22#288912yes—
358C22#297414yes—
359C4—24no—
360C4—22no—
361C4—22no—
362C4—44no—
363C4—44no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—82no—
368C4—82no—
369C2—24no—
370C2—44no—
371C2—82no—
372C2#1737012yes—
373C2#1738712yes—
374C2#2174011yescharacteristic
375C2#2174911yescharacteristic
376C2#2175611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9085C2semidirect
5. #9082C2semidirect
6. #8700C2semidirect
7. #8299C2semidirect
138. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
#217862
#217422
#90852
#90822
#87002
#82992