Finite Group Databasebeta

← #213895

Subgroups — #213895

Order 512.

909 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 909 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35672C211yesmaximal, characteristic
2#20730C212yesmaximal, complemented
3#9514C214yesmaximal, complemented
4#9512C214yesmaximal
5#8754C212yesmaximal, complemented
6#8550C212yesmaximal
7C22×C22⋊Q8C2211yescharacteristic
8C2×#519C2211yescharacteristic
9#2921C2214yes—
10#2484C411yescharacteristic
11#2484C411yescharacteristic
12#2484C2212yes—
13#2482C411yescharacteristic
14#2482C411yescharacteristic
15#2482C2212yes—
16#2423—22no—
17#2423—22no—
18#2421—22no—
19#2421—22no—
20#2317C412yes—
21#2310C2211yescharacteristic
22#2280C412yes—
23#1822C2214yes—
24#1801—22no—
25#1801—22no—
26#1784C2214yes—
27C4×#101—22no—
28C4×#101—22no—
29#1463C2214yes—
30#1462C2214yes—
31#1306C2218yes—
32C23×Q8C2311yescharacteristic
33C24×C4C2×C411yescharacteristic
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—44no—
37#519—44no—
38C2×C4×Q8D412yes—
39C2×C4×Q8D412yes—
40C2×C4×Q8D412yes—
41C2×C4×Q8D412yes—
42C2×#117C2×C412yes—
43C2×#117C2×C412yes—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2312yes—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2311yescharacteristic
52#433—48no—
53#423—24no—
54C2×#101—24no—
55C2×#101—24no—
56#407C2318yes—
57C4×M16—24no—
58#400—21no—
59#400—21no—
60#400—22no—
61#399—21no—
62#399—21no—
63#399—22no—
64#398—21no—
65#398—21no—
66#398—22no—
67#397—21no—
68#397—21no—
69#397—22no—
70#391—22no—
71#391—22no—
72#389—22no—
73#389—22no—
74#388—22no—
75#388—22no—
76#388—22no—
77#388—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#385C2×C412yes—
83#385C2×C412yes—
84#385C2×C412yes—
85#385C2×C412yes—
86#384—22no—
87#382—22no—
88#381C2×C411yescharacteristic
89#381C2×C411yescharacteristic
90#381C2×C412yes—
91#381C2×C412yes—
92#381C2312yes—
93#380—22no—
94#380—22no—
95#379C2×C412yes—
96#379C2×C412yes—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99#343—24no—
100#343—24no—
101#324—44no—
102#323—44no—
103C25C22×C411yescharacteristic
104C22×Q8—24no—
105C22×Q8—44no—
106C22×Q8C2×D412yes—
107C22×Q8C2×D412yes—
108C23×C4—22no—
109C23×C4—42no—
110C23×C4—44no—
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C2411yescharacteristic
117C2×M16—28no—
118C22⋊Q8—48no—
119C22⋊Q8—48no—
120C4×Q8—44no—
121C4×Q8—44no—
122C4×Q8—44no—
123C4×Q8—44no—
124#117—28no—
125#117—44no—
126#117—42no—
127#117—42no—
128#117—44no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22⋊C411yescharacteristic
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C2×D412yes—
148C2×C4⋊C4C2×D412yes—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4C22×C412yes—
159C2×C22⋊C4C22×C412yes—
160C2×C22⋊C4C22×C412yes—
161C2×C22⋊C4C22×C412yes—
162C2×C42—22no—
163C2×C42C22⋊C412yes—
164C2×C42C22⋊C412yes—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C4⋊C8—48no—
170#101—48no—
171#101—48no—
172C22⋊C8—28no—
173C22⋊C8—28no—
174#97—44no—
175C4×C8—44no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190C24—21no—
191C24—22no—
192C24—42no—
193C24—42no—
194C24—81no—
195C24C23×C411yescharacteristic
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—416no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—88no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C22×D412yes—
231C22×C4H(5,𝔽2)14yes—
232M16—48no—
233M16—48no—
234C2×C8—48no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—24no—
264C42—24no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—44no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—41no—
274C23—41no—
275C23—42no—
276C23—41no—
277C23—42no—
278C23—41no—
279C23—42no—
280C23—82no—
281C23—82no—
282C23—88no—
283C23C22×C22⋊C411yescharacteristic
284C23#51812yes—
285Q8—48no—
286Q8—48no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—44no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—88no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#41014yes—
324C2×C4#53414yes—
325C8—88no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—44no—
332C22—44no—
333C22—42no—
334C22—82no—
335C22—84no—
336C22—84no—
337C22—81no—
338C22—81no—
339C22#230811yescharacteristic
340C22C2×#41012yes—
341C4—24no—
342C4—24no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—82no—
350C4—82no—
351C4—81no—
352C4—81no—
353C4—81no—
354C2—22no—
355C2—42no—
356C2—82no—
357C2#2174711yescharacteristic
358C2#2178011yescharacteristic
359C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20730C2semidirect
3. #9514C2semidirect
5. #8754C2semidirect

Maximal subgroups

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

subgroupindex
#356722
#207302
#95142
#95122
#87542
#85502