Finite Group Databasebeta

← #480701

Subgroups — #480701

Order 512.

816 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35674C211yesmaximal, characteristic
2#17371C212yesmaximal
3#11548C214yesmaximal
4#11544C214yesmaximal
5#11518C212yesmaximal
6#11504C212yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8C2×#519C2211yescharacteristic
9#2983—22no—
10#2484C412yes—
11#2484C412yes—
12#2484C2212yes—
13#2484C2212yes—
14#2446—22no—
15#2446—22no—
16#2446—22no—
17#2446—22no—
18#2310C411yescharacteristic
19#2310C411yescharacteristic
20#2310C2211yescharacteristic
21#2296C411yescharacteristic
22#2296C2211yescharacteristic
23#1467—22no—
24#1467—22no—
25#1464C2214yes—
26#1464C2214yes—
27#1461—22no—
28#1458C2214yes—
29#1326C2218yes—
30#1317C2214yes—
31#1317C2214yes—
32C23×Q8C2×C411yescharacteristic
33C23×Q8C2×C411yescharacteristic
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—44no—
38#519—44no—
39C2×C4×Q8D412yes—
40C2×C4×Q8D412yes—
41C2×C4×Q8D412yes—
42C2×C4×Q8D412yes—
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C412yes—
47C2×#117C2311yescharacteristic
48C2×#117C2312yes—
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C412yes—
51C22×C42C2×C411yescharacteristic
52#446—48no—
53#432—24no—
54#432—24no—
55#431—24no—
56#400—22no—
57#400—22no—
58#400—22no—
59#400—22no—
60#398—22no—
61#398—22no—
62#398—22no—
63#398—22no—
64#391—22no—
65#391—21no—
66#391—21no—
67#391—22no—
68#391—22no—
69#391—21no—
70#391—21no—
71#391—22no—
72#389—22no—
73#389—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#385C2×C412yes—
83#385C2×C412yes—
84#385C2×C412yes—
85#385C2312yes—
86#381C2×C412yes—
87#381C2×C412yes—
88#381C2×C412yes—
89#381C2×C412yes—
90#381C2×C411yescharacteristic
91#381C2×C411yescharacteristic
92#381C2311yescharacteristic
93#381C2311yescharacteristic
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—22no—
97#324—24no—
98#324—24no—
99#324—24no—
100#323C2318yes—
101C22×Q8—24no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8C22⋊C412yes—
110C22×Q8C22⋊C411yescharacteristic
111C22×Q8C22⋊C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C412yes—
117C4⋊Q8—48no—
118C4⋊Q8—44no—
119C4⋊Q8—44no—
120C4×Q8—28no—
121C4×Q8—44no—
122C4×Q8—44no—
123C4×Q8—44no—
124#117—44no—
125#117—42no—
126#117—42no—
127#117—42no—
128#117—44no—
129#117—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C22⋊C4C22×C412yes—
149C2×C22⋊C4C22×C411yescharacteristic
150C2×C22⋊C4C22×C411yescharacteristic
151C2×C22⋊C4C22×C412yes—
152C2×C22⋊C4C22×C411yescharacteristic
153C2×C22⋊C4C2411yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—42no—
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C412yes—
162C2×C42C22⋊C412yes—
163C2×C42C2×D412yes—
164C2×C42C2×D412yes—
165C2×C42C2×D412yes—
166C4⋊C8—44no—
167C4⋊C8—44no—
168C4⋊C8—44no—
169C22⋊C8—28no—
170#97—44no—
171#97—44no—
172C4×C8—44no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189C24C23×C411yescharacteristic
190C2×Q8—24no—
191C2×Q8—22no—
192C2×Q8—24no—
193C2×Q8—41no—
194C2×Q8—42no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—416no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—88no—
206C2×Q8#9914yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C22×D412yes—
235C2×C8—48no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C22⋊C4—24no—
253C22⋊C4—42no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42—24no—
259C42—24no—
260C42—24no—
261C42—41no—
262C42—41no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23C22×C22⋊C411yescharacteristic
273C23#56812yes—
274Q8—44no—
275Q8—48no—
276Q8—44no—
277Q8—84no—
278Q8—84no—
279C2×C4—28no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—48no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4C2×#9912yes—
319C2×C4C2×#9914yes—
320C2×C4#41312yes—
321C8—88no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22#287911yescharacteristic
328C22C22×#9911yescharacteristic
329C22C2×#41311yescharacteristic
330C4—22no—
331C4—22no—
332C4—24no—
333C4—42no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—44no—
338C4—42no—
339C4—82no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C2—21no—
345C2—21no—
346C2—42no—
347C2#1739211yescharacteristic
348C2#3334912yes—
subgroupG/H
center, socleC22C22×#99
commutator, prime radicalC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#356742
#173712
#115482
#115442
#115182
#115042