Finite Group Databasebeta

← #106663

Subgroups — #106663

Order 512.

1076 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2C2×#1821C211yesmaximal, characteristic
3C2×#1821C211yesmaximal, characteristic
4#8107C212yesmaximal
5#7929C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2212yes—
8C2×#407C2212yes—
9#2345—24no—
10#2345—24no—
11C2×#389—21no—
12C2×#387—22no—
13C2×#380C412yes—
14C2×#380C2211yescharacteristic
15C2×C4×C4⋊C4—21no—
16C2×C4×C22⋊C4C412yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18#1821—22no—
19#1821—21no—
20#1821—22no—
21#1821—21no—
22#1821C414yescomplemented
23C2×#343—22no—
24C2×#343C414yescomplemented
25#1305—24no—
26#1282—24no—
27C24×C4C2×C412yes—
28C24×C4C2311yescharacteristic
29C22×M16—22no—
30C22×M16C2×C414yes—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C412yes—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D412yes—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4Q812yes—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C42—21no—
48C22×C42—22no—
49#407—22no—
50#407—22no—
51#407—22no—
52#407—22no—
53#407—22no—
54#407—22no—
55#407—22no—
56#407C2×C414yes—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61#389—28no—
62#387—216no—
63#380—22no—
64#380—24no—
65#380—24no—
66#380D414yes—
67C4×C4⋊C4—28no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—24no—
70C4×C22⋊C4—24no—
71C4×C22⋊C4D414yes—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—21no—
75C2×#95—21no—
76#343—24no—
77#343—24no—
78#343—28no—
79C25C22×C411yescharacteristic
80C23×C4—21no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—44no—
88C23×C4—84no—
89C23×C4C4214yes—
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C412yes—
92C23×C4C4⋊C414yes—
93C23×C4C22×C411yescharacteristic
94C23×C4C22×C411yescharacteristic
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×Q811yescharacteristic
101C23×C4#4112yes—
102C23×C4#4112yes—
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—42no—
107C2×M16—48no—
108C2×M16—48no—
109C22×C8—24no—
110C22×C8—42no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—84no—
129C2×C4⋊C4C22⋊C414yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C4⋊C412yes—
133C2×C4⋊C4C4⋊C412yes—
134C2×C4⋊C4C4⋊C414yes—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—44no—
141C2×C22⋊C4C22⋊C412yes—
142C2×C22⋊C4C22⋊C412yes—
143C2×C22⋊C4C22⋊C414yes—
144C2×C42—24no—
145C2×C42—28no—
146C2×C42—44no—
147C2×C42—84no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153C22⋊C8—42no—
154C22⋊C8—42no—
155C22⋊C8—42no—
156#95—28no—
157#95—28no—
158#95—24no—
159#95—24no—
160C24—22no—
161C24—22no—
162C24—42no—
163C24—42no—
164C24C2×C4211yescharacteristic
165C24C2×C22⋊C411yescharacteristic
166C24C2×C22⋊C411yescharacteristic
167C24C2×C22⋊C411yescharacteristic
168C24C2×C4⋊C411yescharacteristic
169C24#11711yescharacteristic
170C24#11711yescharacteristic
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—84no—
209C22×C4#9514yes—
210C22×C4#9514yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C4⋊C411yescharacteristic
214C22×C4C2×C4⋊C411yescharacteristic
215C22×C4C4×D412yes—
216C22×C4C4×D412yes—
217C22×C4C4×D412yes—
218C22×C4C4×D412yes—
219C22×C4#12012yes—
220C22×C4#12012yes—
221C22×C4#12112yes—
222C22×C4#12112yes—
223C22×C4C22⋊Q812yes—
224C22×C4C22⋊Q812yes—
225C22×C4#12312yes—
226C22×C4#12312yes—
227M16—48no—
228M16—48no—
229M16—84no—
230M16—84no—
231C2×C8—44no—
232C2×C8—44no—
233C2×C8—84no—
234C2×C8—84no—
235C4⋊C4—24no—
236C4⋊C4—28no—
237C4⋊C4—24no—
238C4⋊C4—28no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—48no—
242C4⋊C4—88no—
243C22⋊C4—24no—
244C22⋊C4—22no—
245C22⋊C4—24no—
246C22⋊C4—42no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4#10114yes—
250C42—48no—
251C42—88no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—48no—
275C23—48no—
276C23C2×#9511yescharacteristic
277C23C4×C22⋊C411yescharacteristic
278C23C4×C22⋊C411yescharacteristic
279C23#37911yescharacteristic
280C23#38011yescharacteristic
281C23#38011yescharacteristic
282C23#38111yescharacteristic
283C23C2×#10112yes—
284C23#41312yes—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—48no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—88no—
324C2×C4#38512yes—
325C2×C4#38512yes—
326C2×C4#38612yes—
327C2×C4#38612yes—
328C2×C4#41014yes—
329C8—84no—
330C8—84no—
331C22—24no—
332C22—24no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—42no—
350C22—42no—
351C22—44no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22C22⋊#9511yescharacteristic
358C22#174012yes—
359C22C4×#10112yes—
360C22#178411yescharacteristic
361C22#178511yescharacteristic
362C22#180112yes—
363C22#182912yes—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—44no—
368C4—44no—
369C4—44no—
370C4—82no—
371C4—84no—
372C4#189914yes—
373C2—21no—
374C2—21no—
375C2—22no—
376C2—22no—
377C2—22no—
378C2—22no—
379C2—41no—
380C2—41no—
381C2#1073611yescharacteristic
382C2#1076311yescharacteristic
383C2#1076911yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
22. #1821C4semidirect
24. C2×#343C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
C2×#18212
#81072
#79292