Finite Group Databasebeta

← #213511

Subgroups — #213511

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2#21786C211yesmaximal, characteristic
3#21738C211yesmaximal, characteristic
4#9082C214yesmaximal, complemented
5#9081C214yesmaximal, complemented
6#8700C212yesmaximal, complemented
7#8299C212yesmaximal, complemented
8C22×#123C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#431C2212yes—
12C22×C22⋊C8C2212yes—
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#1869—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
36C23×C8—22no—
37C2×#123—24no—
38C2×#123—24no—
39C2×#123C818yescomplemented
40C2×C4×D4—44no—
41C2×C4×D4D414yes—
42C2×C4×D4D414yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—44no—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2×C412yes—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C412yes—
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C42C2×C412yes—
58C22×C42C2311yescharacteristic
59#431—44no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8—44no—
66C2×C22⋊C8C2314yes—
67C2×C22⋊C8C2314yes—
68C2×C4×C8—22no—
69#389—28no—
70#388—28no—
71#387—24no—
72#387—24no—
73#386—24no—
74#385—24no—
75#385—24no—
76#385—24no—
77#381—24no—
78C4×C22⋊C4—22no—
79C4×C22⋊C4—22no—
80C2×#95C2×C412yes—
81C2×#95C2×C412yes—
82C2×#95C2×C412yes—
83C2×#95C2×C412yes—
84C2×#95C2312yes—
85#336—24no—
86#324—48no—
87C25C22×C411yescharacteristic
88C25C22×C411yescharacteristic
89C22×D4—42no—
90C22×D4—42no—
91C22×D4C22⋊C414yes—
92C22×D4C2×C814yes—
93C22×D4M1614yes—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—22no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—82no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C2×C814yescomplemented
106C23×C4C22×C412yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C22×C412yes—
109C23×C4C22×C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2411yescharacteristic
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—44no—
122C22×C8—84no—
123#123—232no—
124C4×D4—44no—
125C4×D4—44no—
126C4×D4—44no—
127C4×D4—44no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—48no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—82no—
140C2×C4⋊C4C22⋊C414yes—
141C2×C4⋊C4C2×C818yescomplemented
142C2×C22⋊C4—22no—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—82no—
152C2×C22⋊C4C22⋊C414yes—
153C2×C22⋊C4C22⋊C414yes—
154C2×C22⋊C4C2×C818yes—
155C2×C22⋊C4C2×C814yescomplemented
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—82no—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C4⋊C8—44no—
162C22⋊C8—28no—
163C22⋊C8—28no—
164C22⋊C8—44no—
165C22⋊C8—84no—
166C22⋊C8—84no—
167C22⋊C8—84no—
168C4×C8—44no—
169#95—24no—
170#95—24no—
171#95—24no—
172#95—24no—
173#95—22no—
174#95—22no—
175C24—22no—
176C24—22no—
177C24—22no—
178C24—22no—
179C24—42no—
180C24—42no—
181C24—42no—
182C24—42no—
183C24C2×C22⋊C412yes—
184C24C2×C22⋊C412yes—
185C24C22×C812yes—
186C24C22×C812yes—
187C24C2×M1612yes—
188C24C2×M1612yes—
189C24C23×C411yescharacteristic
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—48no—
193C2×D4—48no—
194C2×D4C22⋊C8116yes—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—28no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—48no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—88no—
236C22×C4—84no—
237C22×C4—88no—
238C22×C4—88no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C22×C814yes—
244C22×C4C22×C812yes—
245C22×C4C22×C814yes—
246C22×C4C2×M1612yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4H(5,𝔽2)14yes—
250C2×C8—48no—
251C2×C8—48no—
252C2×C8—44no—
253C2×C8—44no—
254C2×C8—88no—
255C4⋊C4—216no—
256C4⋊C4—44no—
257C4⋊C4—48no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—84no—
261C22⋊C4—216no—
262C22⋊C4—28no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—84no—
267C42—22no—
268C42—22no—
269C42—24no—
270C42—84no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—24no—
278C23—24no—
279C23—24no—
280C23—24no—
281C23—24no—
282C23—24no—
283C23—24no—
284C23—24no—
285C23—44no—
286C23—44no—
287C23—44no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—48no—
292C23—82no—
293C23—82no—
294C23—84no—
295C23C2×C22⋊C814yes—
296C23C2×C22⋊C814yes—
297C23C22×C22⋊C411yescharacteristic
298C23#51812yes—
299C23C23×C811yescharacteristic
300C23C22×M1611yescharacteristic
301C23#56812yes—
302D4—416no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—416no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—88no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4C2×C22⋊C814yes—
345C2×C4#53412yes—
346C2×C4#53412yes—
347C8—84no—
348C8—84no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—24no—
354C22—28no—
355C22—28no—
356C22—24no—
357C22—24no—
358C22—48no—
359C22—48no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—44no—
365C22—44no—
366C22—84no—
367C22—84no—
368C22#230811yescharacteristic
369C22C22×C22⋊C811yescharacteristic
370C22#287911yescharacteristic
371C22#288312yes—
372C22#288912yes—
373C22#297212yes—
374C22#297312yes—
375C4—24no—
376C4—22no—
377C4—22no—
378C4—44no—
379C4—44no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—82no—
384C4—82no—
385C2—22no—
386C2—22no—
387C2—44no—
388C2—44no—
389C2#1735112yes—
390C2#1740612yes—
391C2#2173811yescharacteristic
392C2#2174911yescharacteristic
393C2#2175611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ 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. #9082C2semidirect
5. #9081C2semidirect
6. #8700C2semidirect
7. #8299C2semidirect
39. C2×#123C8semidirect
105. C23×C4C2×C8semidirect
141. C2×C4⋊C4C2×C8semidirect
155. C2×C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
#217862
#217382
#90822
#90812
#87002
#82992