Finite Group Databasebeta

← #105828

Subgroups — #105828

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1464C212yesmaximal
3#8040C212yesmaximal, complemented
4#8030C212yesmaximal, complemented
5C22×#117C411yescharacteristic
6C22×#117C2211yescharacteristic
7C23×C4⋊C4C411yescharacteristic
8C2×#432C2214yes—
9#2317—24no—
10#2317—24no—
11C2×#389—22no—
12C2×#385—22no—
13C2×#384C411yescharacteristic
14C2×#384C2211yescharacteristic
15C2×#382C411yescharacteristic
16C2×#382C2211yescharacteristic
17#1464—24no—
18#1464C418yescomplemented
19C2×#324—22no—
20C2×#323—22no—
21#1296—24no—
22#1295—24no—
23C24×C4—21no—
24C24×C4C2×C411yescharacteristic
25C2×#117—22no—
26C2×#117—22no—
27C2×#117—22no—
28C2×#117—22no—
29C2×#117—22no—
30C2×#117—22no—
31C2×#117—42no—
32C2×#117—42no—
33C2×#117C2×C414yes—
34C2×#117D412yes—
35C2×#117Q812yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—42no—
42C22×C4⋊C4—42no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4D412yes—
48C22×C22⋊C4—21no—
49C22×C22⋊C4D412yes—
50C22×C42—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2311yescharacteristic
53#432—24no—
54#432—24no—
55#432—24no—
56#432C2×C418yescomplemented
57C2×C4⋊C8—24no—
58C2×C22⋊C8—24no—
59C2×#97—24no—
60#389—216no—
61#385—216no—
62#384—22no—
63#384—22no—
64#384—22no—
65#384D414yes—
66#382—22no—
67#382—22no—
68#382—22no—
69#382D414yes—
70C2×#95—22no—
71C2×#95—22no—
72#324—28no—
73#323—28no—
74C25C22⋊C411yescharacteristic
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—24no—
88C23×C4—42no—
89C23×C4—42no—
90C23×C4—84no—
91C23×C4C4212yes—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C412yes—
94C23×C4C4⋊C412yes—
95C23×C4C22×C411yescharacteristic
96C23×C4C2×D412yes—
97C22×C8—44no—
98#117—28no—
99#117—28no—
100#117—24no—
101#117—24no—
102#117—24no—
103#117—24no—
104#117—48no—
105#117—48no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—48no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—84no—
128C2×C4⋊C4—84no—
129C2×C4⋊C4C22⋊C414yes—
130C2×C4⋊C4C22⋊C412yes—
131C2×C4⋊C4C22⋊C412yes—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22⋊C412yes—
134C2×C4⋊C4C4⋊C414yes—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4C22⋊C414yes—
141C2×C22⋊C4C4⋊C414yes—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—24no—
147C2×C42—42no—
148C2×C42—42no—
149C2×C42C4212yes—
150C2×C42C4⋊C412yes—
151C2×C42C22×C412yes—
152C2×C42C2×D412yes—
153C2×C42C2×D412yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×Q811yescharacteristic
156C2×C42#4112yes—
157C2×C42#4112yes—
158C4⋊C8—28no—
159C4⋊C8—44no—
160C22⋊C8—28no—
161C22⋊C8—44no—
162#97—28no—
163#97—44no—
164#95—28no—
165#95—28no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—42no—
171C24—42no—
172C24#9512yes—
173C24#10012yes—
174C24#10112yes—
175C24C2×C22⋊C411yescharacteristic
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—48no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—88no—
224C22×C4—88no—
225C22×C4—88no—
226C22×C4#9512yes—
227C22×C4#9514yes—
228C22×C4#10012yes—
229C22×C4#10112yes—
230C22×C4C2×C4211yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C4⋊C411yescharacteristic
235C22×C4C2×C4⋊C412yes—
236C22×C4#11711yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4#12014yes—
239C22×C4#12314yes—
240C2×C8—48no—
241C2×C8—84no—
242C2×C8—84no—
243C4⋊C4—28no—
244C4⋊C4—28no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—48no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—88no—
255C4⋊C4—88no—
256C22⋊C4—28no—
257C22⋊C4—28no—
258C22⋊C4—48no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—48no—
264C42C4×D414yes—
265C42C4×D414yes—
266C42#12112yes—
267C42#12112yes—
268C42C22⋊Q812yes—
269C42C22⋊Q812yes—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—44no—
280C23—42no—
281C23—44no—
282C23—42no—
283C23—44no—
284C23—42no—
285C23—44no—
286C23—42no—
287C23—42no—
288C23#34314yes—
289C23C2×#9511yescharacteristic
290C23#37911yescharacteristic
291C23#38111yescharacteristic
292C23C2×#10011yescharacteristic
293C23C2×#10011yescharacteristic
294C23C2×#10111yescharacteristic
295C23C2×#10111yescharacteristic
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—24no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—816no—
345C2×C4#34314yes—
346C2×C4C4×C22⋊C411yescharacteristic
347C2×C4C4×C22⋊C411yescharacteristic
348C2×C4#38011yescharacteristic
349C2×C4#38011yescharacteristic
350C2×C4#38514yes—
351C2×C4#38614yes—
352C8—88no—
353C22—22no—
354C22—22no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—24no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22—44no—
370C22C22⋊#9511yescharacteristic
371C22C2×#34311yescharacteristic
372C22C2×#34311yescharacteristic
373C22#178311yescharacteristic
374C22#178411yescharacteristic
375C22#179512yes—
376C22#179612yes—
377C22#182111yescharacteristic
378C22#182111yescharacteristic
379C4—24no—
380C4—24no—
381C4—24no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—44no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4—82no—
391C2—21no—
392C2—21no—
393C2—21no—
394C2—21no—
395C2—44no—
396C2#852211yescharacteristic
397C2#852411yescharacteristic
398C2#852512yes—
399C2#1072711yescharacteristic
400C2#1076512yes—
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8040C2semidirect
4. #8030C2semidirect
18. #1464C4semidirect
56. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#14642
#80402
#80302