Finite Group Databasebeta

← #217063

Subgroups — #217063

Order 512.

735 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20815C211yesmaximal, characteristic
4#9907C212yesmaximal, complemented
5#9705C212yesmaximal, complemented
6#9138C212yesmaximal
7#9130C212yesmaximal
8#8591C212yesmaximal, complemented
9#8586C212yesmaximal, complemented
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857C2212yes—
15#2647—21no—
16#2629—21no—
17#2507C414yes—
18#2497C414yes—
19#2446—22no—
20#2429—22no—
21#2330—21no—
22#2314—21no—
23C4×C4⋊Q8C2212yes—
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1917—22no—
27#1842C2212yes—
28#1842C2211yescharacteristic
29#1831C2212yes—
30#1831C2211yescharacteristic
31#1798—21no—
32#1798—21no—
33#1797C2212yes—
34#1768—21no—
35#1768—21no—
36#1767C2212yes—
37C2×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8C2312yes—
47C2×#125—21no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×#117D411yescharacteristic
51C2×#117D411yescharacteristic
52C22×C42C2311yescharacteristic
53#432—41no—
54#432—41no—
55#431—41no—
56#431—41no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×#97—21no—
68C2×#97—22no—
69C2×C4×C8—21no—
70#400—24no—
71#398—24no—
72#398—24no—
73#396—24no—
74#391—22no—
75#391—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—24no—
80#389C2×C414yes—
81#389C2×C414yes—
82#389C2×C414yes—
83#388—24no—
84#387—24no—
85#387—24no—
86#384—21no—
87#384—21no—
88#384C2×C414yes—
89#384C2312yes—
90#382—21no—
91#380—22no—
92#380C2×C414yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C414yes—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4D412yes—
97C4×C22⋊C4C2×C412yes—
98C4×C22⋊C4D412yes—
99#376D411yescharacteristic
100#376D411yescharacteristic
101C43C2311yescharacteristic
102C43C2311yescharacteristic
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#332—42no—
109#332—42no—
110#332—42no—
111#332—42no—
112C22×Q8—22no—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—82no—
125C4⋊Q8—82no—
126#125—44no—
127C22⋊Q8—82no—
128C22⋊Q8—82no—
129C4×Q8—82no—
130C4×Q8—82no—
131#117—24no—
132#117—42no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22×C414yes—
144C2×C4⋊C4C22×C414yes—
145C2×C22⋊C4—24no—
146C2×C22⋊C4C22⋊C412yes—
147C2×C22⋊C4C22⋊C412yes—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—24no—
153C2×C42C22×C411yescharacteristic
154C2×C42C22×C412yes—
155C2×C42C22×C412yes—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2411yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—44no—
174C4⋊C8—44no—
175C4⋊C8—82no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—82no—
179#97—42no—
180#97—82no—
181C4×C8—42no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95C22⋊C412yes—
188#95C22⋊C412yes—
189#95C22⋊C412yes—
190#95C22⋊C412yes—
191C24C2×C22⋊C411yescharacteristic
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C23×C411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C22×D411yescharacteristic
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—48no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—84no—
245C4⋊C4—84no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C22⋊C4—24no—
249C22⋊C4—42no—
250C22⋊C4—84no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—42no—
265C42—42no—
266C42—84no—
267C42#13714yes—
268C42H(5,𝔽2)11yescharacteristic
269C42H(5,𝔽2)11yescharacteristic
270C422−1+411yescharacteristic
271C422−1+411yescharacteristic
272C23—21no—
273C23—22no—
274C23—21no—
275C23—21no—
276C23—22no—
277C23C22×C22⋊C411yescharacteristic
278Q8—82no—
279Q8—82no—
280Q8—82no—
281Q8—82no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4#41914yes—
315C2×C4#52011yescharacteristic
316C2×C4#52011yescharacteristic
317C2×C4#53511yescharacteristic
318C2×C4#53611yescharacteristic
319C2×C4#53711yescharacteristic
320C2×C4#53711yescharacteristic
321C2×C4C2×#13712yes—
322C2×C4#57512yes—
323C2×C4#57612yes—
324C2×C4#57712yes—
325C8—82no—
326C8—82no—
327C8—82no—
328C8—82no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22#231111yescharacteristic
335C22#288912yes—
336C22C2×#41911yescharacteristic
337C22#289611yescharacteristic
338C22#289712yes—
339C4—24no—
340C4—21no—
341C4—21no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—84no—
349C4—84no—
350C4#321612yes—
351C4#323712yes—
352C4#325712yes—
353C4#325911yescharacteristic
354C4#326011yescharacteristic
355C2—81no—
356C2#1776112yes—
357C2#1776912yes—
358C2#2187011yescharacteristic
359C2#2188311yescharacteristic
360C2#2189711yescharacteristic
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. #9907C2semidirect
5. #9705C2semidirect
8. #8591C2semidirect
9. #8586C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#219262
#208152
#99072
#97052
#91382
#91302
#85912
#85862