Finite Group Databasebeta

← #213981

Subgroups — #213981

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35639C211yesmaximal, characteristic
2#17370C212yesmaximal
3#9518C214yesmaximal, complemented
4#9510C214yesmaximal, complemented
5#8757C212yesmaximal, complemented
6#8554C212yesmaximal, complemented
7C22×#125C2211yescharacteristic
8C2×#520C2211yescharacteristic
9#2975C2214yes—
10#2579C412yes—
11#2579C2212yes—
12#2487C412yes—
13#2487C2212yes—
14#2454—22no—
15#2453—22no—
16#2442—22no—
17#2441—22no—
18#2317C411yescharacteristic
19#2316C2211yescharacteristic
20#2311C411yescharacteristic
21#2297C411yescharacteristic
22#2296C411yescharacteristic
23#1821C2214yes—
24#1465—22no—
25#1465C2214yes—
26#1463—22no—
27#1463C2214yes—
28#1463C2214yes—
29#1461—22no—
30#1458—22no—
31#1305C2218yes—
32C22×#41C2×C411yescharacteristic
33C2×#125—22no—
34C2×#125—22no—
35C2×#125—44no—
36#520—44no—
37C2×C4×Q8D412yes—
38C2×C4×D4C2×C412yes—
39C2×C4×D4D412yes—
40C2×C4×D4C2312yes—
41C2×#117C2×C411yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117C2×C411yescharacteristic
44C2×#117D412yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C42C2×C411yescharacteristic
53#436—48no—
54#431—24no—
55#407—24no—
56#407C2318yes—
57#400—22no—
58#400—22no—
59#399—22no—
60#399—22no—
61#398—22no—
62#398—22no—
63#396—22no—
64#396—22no—
65#391—21no—
66#391—21no—
67#390—21no—
68#390—21no—
69#389—21no—
70#389—21no—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#387—22no—
77#387—22no—
78#386C2×C412yes—
79#386C2×C412yes—
80#386C2×C412yes—
81#386C2×C412yes—
82#385—22no—
83#385—22no—
84#385C2×C412yes—
85#385C2×C412yes—
86#385C2×C412yes—
87#385C2×C412yes—
88#384—21no—
89#382—21no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380C2×C412yes—
94#380C2312yes—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—22no—
97#343—44no—
98#324—24no—
99#324—24no—
100#324—24no—
101#324—44no—
102#323—24no—
103C2×#41—24no—
104C2×#41—42no—
105C2×#41—42no—
106C22×Q8C22⋊C411yescharacteristic
107C22×D4C22⋊C411yescharacteristic
108C22×D4C22×C412yes—
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C412yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2411yescharacteristic
119C2×M16—28no—
120C2×M16—44no—
121#125—48no—
122#125—48no—
123C4×Q8—44no—
124C4×D4—28no—
125C4×D4—44no—
126C4×D4—44no—
127#117—44no—
128#117—42no—
129#117—42no—
130#117—42no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22×C412yes—
164C2×C22⋊C4C22×C412yes—
165C2×C22⋊C4C22×C412yes—
166C2×C22⋊C4C2×D412yes—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42C22⋊C412yes—
173C2×C42C22⋊C412yes—
174C2×C42C2×D412yes—
175C4⋊C8—44no—
176C22⋊C8—28no—
177C22⋊C8—28no—
178C22⋊C8—44no—
179C4×C8—44no—
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—
189#95—22no—
190C24C2×C22⋊C412yes—
191C24C23×C411yescharacteristic
192#41—416no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8#10114yes—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—24no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—42no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C22×D412yes—
236C22×C4H(5,𝔽2)14yes—
237M16—88no—
238M16—164no—
239C2×C8—48no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C42—24no—
270C42—42no—
271C42—42no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—42no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—41no—
282C23—41no—
283C23—42no—
284C23—42no—
285C23C22×C22⋊C411yescharacteristic
286C23#51812yes—
287Q8—44no—
288D4—48no—
289D4—44no—
290C2×C4—28no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—48no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—84no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—162no—
329C2×C4C2×#10112yes—
330C2×C4C2×#10114yes—
331C2×C4#41312yes—
332C2×C4#53414yes—
333C8—88no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—42no—
340C22—42no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—82no—
346C22#230811yescharacteristic
347C22C22×#10111yescharacteristic
348C22C2×#41311yescharacteristic
349C4—24no—
350C4—22no—
351C4—22no—
352C4—24no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—82no—
357C4—82no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—82no—
366C2#2175311yescharacteristic
367C2#2177711yescharacteristic
368C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C2×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
3. #9518C2semidirect
4. #9510C2semidirect
5. #8757C2semidirect
6. #8554C2semidirect

Maximal subgroups

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

subgroupindex
#356392
#173702
#95182
#95102
#87572
#85542