Finite Group Databasebeta

← #108241

Subgroups — #108241

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#17364C211yesmaximal, characteristic
3#14345C211yesmaximal, characteristic
4#7975C212yesmaximal, complemented
5#7959C212yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7#2975C412yescomplemented
8#2974—24no—
9#2973—22no—
10#2972C412yescomplemented
11D4×M16—24no—
12#2920C2212yes—
13#2920C2212yes—
14C2×#407—22no—
15C2×#383C2211yescharacteristic
16C2×#382C2211yescharacteristic
17#1466C412yescomplemented
18#1466C412yescomplemented
19#1462—22no—
20#1460—22no—
21#1445C412yes—
22#1439—22no—
23#1438C412yes—
24C22⋊#95—22no—
25#1288—24no—
26#1286—24no—
27C23×D4C2×C411yescharacteristic
28C24×C4D412yes—
29C22×M16—24no—
30C2×C4×D4—22no—
31C2×C4×D4—22no—
32C2×C4×D4—22no—
33C2×C4×D4—22no—
34C2×C4×D4C2×C411yescharacteristic
35C2×C4×D4C2×C411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4D412yes—
39C2×C4×D4Q812yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C22⋊C4—24no—
43C22×C22⋊C4D412yes—
44C22×C42C2311yescharacteristic
45#436—28no—
46#435—216no—
47C8×D4—28no—
48#432—22no—
49#432—22no—
50#432—22no—
51#432C2×C414yes—
52#431—22no—
53#431—22no—
54#431C2×C414yes—
55#431D414yes—
56#423—22no—
57#423—22no—
58#407—24no—
59#407—24no—
60#407—24no—
61#407—22no—
62#407—22no—
63C2×C22⋊C8—24no—
64C2×C22⋊C8—24no—
65C4×M16—22no—
66C4×M16D414yes—
67#383—22no—
68#383—22no—
69#382—22no—
70#382—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—24no—
75C2×#95C2×C412yes—
76C2×#95C2×C412yes—
77#323—24no—
78#323—24no—
79C25C22⋊C412yes—
80C22×D4—22no—
81C22×D4—22no—
82C22×D4—44no—
83C22×D4C4211yescharacteristic
84C22×D4C4211yescharacteristic
85C22×D4C4⋊C412yes—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—84no—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C412yes—
95C23×C4C22⋊C412yes—
96C23×C4C4⋊C412yes—
97C23×C4C4⋊C412yes—
98C23×C4C22×C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—24no—
102C2×M16—28no—
103C2×M16—42no—
104C2×M16—88no—
105C22×C8—28no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—22no—
109C4×D4—22no—
110C4×D4—22no—
111C4×D4—22no—
112C4×D4—22no—
113C4×D4—22no—
114C4×D4—24no—
115C4×D4—24no—
116C4×D4—24no—
117C4×D4—24no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—82no—
122C2×C4⋊C4C4211yescharacteristic
123C2×C4⋊C4C4211yescharacteristic
124C2×C4⋊C4C4⋊C412yes—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—48no—
128C2×C22⋊C4—84no—
129C2×C22⋊C4—84no—
130C2×C22⋊C4C22⋊C412yes—
131C2×C22⋊C4C22⋊C412yes—
132C2×C22⋊C4C4⋊C412yes—
133C2×C22⋊C4C4⋊C412yes—
134C2×C42C22×C411yescharacteristic
135C2×C42C22×C411yescharacteristic
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×Q811yescharacteristic
140C2×C42#4112yes—
141C2×C42#4112yes—
142C4⋊C8—24no—
143C4⋊C8—24no—
144C4⋊C8—24no—
145C4⋊C8—42no—
146C22⋊C8—28no—
147C22⋊C8—28no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C22⋊C8—24no—
151C22⋊C8—42no—
152#97—24no—
153#97—24no—
154C4×C8—42no—
155C4×C8C22⋊C418yescomplemented
156#95—24no—
157#95—24no—
158#95—41no—
159#95—41no—
160#95—42no—
161#95—41no—
162#95—41no—
163#95—84no—
164C24—24no—
165C24—22no—
166C24—22no—
167C24—42no—
168C24—42no—
169C24—44no—
170C24—42no—
171C24#9512yes—
172C24#9512yes—
173C24C2×C22⋊C411yescharacteristic
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—24no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—48no—
182C2×D4—48no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4—48no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—44no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—84no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4#9512yes—
217C22×C4#9512yes—
218C22×C4C2×C4211yescharacteristic
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C4⋊C411yescharacteristic
222C22×C4C2×C4⋊C411yescharacteristic
223C22×C4C2×C4⋊C411yescharacteristic
224C22×C4C4×D412yes—
225C22×C4C4×D412yes—
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12112yes—
229C22×C4#12112yes—
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232M16—48no—
233M16—88no—
234M16—82no—
235M16—82no—
236C2×C8—28no—
237C2×C8—44no—
238C2×C8—88no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—24no—
242C4⋊C4—84no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—816no—
248C42C2×C22⋊C412yes—
249C42#11712yes—
250C42C4×D412yes—
251C42C4×D412yes—
252C42#12012yes—
253C42C22⋊Q812yes—
254C42C22⋊Q812yes—
255C42#12312yes—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—44no—
266C23—44no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—48no—
272C23—48no—
273C23—42no—
274C23—42no—
275C23—44no—
276C23—82no—
277C23—82no—
278C23—84no—
279C23—84no—
280C23#34414yes—
281C23C2×#9511yescharacteristic
282C23#38611yescharacteristic
283C23#38611yescharacteristic
284D4—28no—
285D4—48no—
286D4—44no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C4×C22⋊C412yes—
327C2×C4#37911yescharacteristic
328C2×C4#38012yes—
329C2×C4#38111yescharacteristic
330C2×C4#38511yescharacteristic
331C2×C4#38511yescharacteristic
332C2×C4#38512yes—
333C2×C4#38612yes—
334C8—48no—
335C8—84no—
336C22—24no—
337C22—24no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—44no—
346C22—44no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—82no—
353C22—84no—
354C22—88no—
355C22—84no—
356C22—82no—
357C22C22⋊#9511yescharacteristic
358C22C2×#34412yes—
359C22#190911yescharacteristic
360C22#191011yescharacteristic
361C22#191112yes—
362C4—22no—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—22no—
369C4—22no—
370C4—42no—
371C4—42no—
372C4—162no—
373C4—162no—
374C2—21no—
375C2—22no—
376C2—22no—
377C2—21no—
378C2—44no—
379C2—82no—
380C2#1072911yescharacteristic
381C2#1074011yescharacteristic
382C2#1074211yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7975C2semidirect
5. #7959C2semidirect
7. #2975C4semidirect
10. #2972C4semidirect
17. #1466C4semidirect
18. #1466C4semidirect
155. C4×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#602302
#173642
#143452
#79752
#79592