Finite Group Databasebeta

← #217102

Subgroups — #217102

Order 512.

763 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9904C211yesmaximal, characteristic, complemented
5#9899C211yesmaximal, characteristic, complemented
6#9702C211yesmaximal, characteristic, complemented
7#9691C211yesmaximal, characteristic, complemented
8#9136C212yesmaximal
9#9135C212yesmaximal
10#8590C212yesmaximal, complemented
11#8587C212yesmaximal, complemented
12C2×#433—21no—
13C2×#432C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2857C2212yes—
18#2647—21no—
19#2629—21no—
20#2507—22no—
21#2497—22no—
22#2446C414yes—
23#2429C414yes—
24#2330—21no—
25#2314—21no—
26C4×C4⋊Q8C2212yes—
27C4×#117C2211yescharacteristic
28#1940—22no—
29#1851—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1832—21no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1798—21no—
37#1797C2212yes—
38#1768—21no—
39#1767C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C22×M16—21no—
50C2×C4⋊Q8C2×C414yes—
51C2×C4⋊Q8C2312yes—
52C2×#125—21no—
53C2×C22⋊Q8C2×C414yes—
54C2×C4×Q8C2×C414yes—
55C2×#117D411yescharacteristic
56C2×#117D411yescharacteristic
57C22×C42C2311yescharacteristic
58#433—44no—
59#432—42no—
60#423—42no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×C4⋊C8C2311yescharacteristic
71C2×C4⋊C8C2311yescharacteristic
72C2×C22⋊C8—22no—
73C2×C22⋊C8—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77#400—24no—
78#398—24no—
79#398—24no—
80#396—24no—
81#391C2×C414yes—
82#391C2×C414yes—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#389—22no—
88#389—24no—
89#389C2×C414yes—
90#388—24no—
91#387—24no—
92#387—24no—
93#384—21no—
94#384—21no—
95#384—22no—
96#384C2312yes—
97#382—21no—
98#380—22no—
99#380—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C412yes—
103C4×C4⋊C4D412yes—
104C4×C22⋊C4C2×C412yes—
105C4×C22⋊C4D412yes—
106#376D411yescharacteristic
107#376D411yescharacteristic
108C43C2311yescharacteristic
109C43C2311yescharacteristic
110#336—22no—
111#336—21no—
112#336—21no—
113#336—22no—
114#336—22no—
115#333—42no—
116#333—42no—
117#333—42no—
118#333—42no—
119C22×Q8C22×C414yes—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—42no—
124C2×M16—82no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—21no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—84no—
136#125—42no—
137#125—42no—
138C22⋊Q8—84no—
139C4×Q8—84no—
140#117—22no—
141#117—22no—
142#117—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C414yes—
155C2×C22⋊C4—24no—
156C2×C22⋊C4C22⋊C412yes—
157C2×C22⋊C4C22⋊C412yes—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C412yes—
163C2×C42C22×C412yes—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2411yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—42no—
182C4⋊C8—44no—
183C4⋊C8—44no—
184C4⋊C8—82no—
185C4⋊C8—82no—
186C22⋊C8—42no—
187C22⋊C8—44no—
188#97—42no—
189#97—42no—
190C4×C8—42no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95C22⋊C412yes—
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199#95C22⋊C412yes—
200C24C2×C22⋊C411yescharacteristic
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—44no—
204C2×Q8—44no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—42no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D411yescharacteristic
227M16—84no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—41no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—48no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—84no—
261C42—22no—
262C42—21no—
263C42—21no—
264C42—22no—
265C42—21no—
266C42—22no—
267C42—21no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—42no—
273C42—42no—
274C42—84no—
275C42AGL(1,ℤ/8)12yes—
276C42#13712yes—
277C42#13714yes—
278C42H(5,𝔽2)12yes—
279C422−1+412yes—
280C23—21no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23C22×C22⋊C411yescharacteristic
286Q8—84no—
287Q8—84no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—82no—
318C2×C4#41714yes—
319C2×C4#41914yes—
320C2×C4#52012yes—
321C2×C4#53511yescharacteristic
322C2×C4#53611yescharacteristic
323C2×C4#53712yes—
324C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
325C2×C4C2×#13712yes—
326C2×C4C2×#13711yescharacteristic
327C2×C4#57712yes—
328C2×C4#57812yes—
329C8—81no—
330C8—81no—
331C8—81no—
332C8—81no—
333C8—82no—
334C8—82no—
335C22—42no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#231111yescharacteristic
341C22#288812yes—
342C22C2×#41911yescharacteristic
343C22C2×#41711yescharacteristic
344C22#289812yes—
345C4—24no—
346C4—24no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—84no—
354C4—84no—
355C4#322211yescharacteristic
356C4#322311yescharacteristic
357C4#324212yes—
358C4#325412yes—
359C4#325512yes—
360C2—81no—
361C2#1776012yes—
362C2#1776412yes—
363C2#2187311yescharacteristic
364C2#2188211yescharacteristic
365C2#2189411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9904C2semidirect
5. #9899C2semidirect
6. #9702C2semidirect
7. #9691C2semidirect
10. #8590C2semidirect
11. #8587C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#219782
#208122
#99042
#98992
#97022
#96912
#91362
#91352
#85902
#85872