Finite Group Databasebeta

← #213957

Subgroups — #213957

Order 512.

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

normal subgroups only

This group has 889 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#35454C211yesmaximal, characteristic
2#17370C212yesmaximal, complemented
3#9516C214yesmaximal, complemented
4#9510C214yesmaximal, complemented
5#8758C212yesmaximal, complemented
6#8556C212yesmaximal, complemented
7C22×#126C2211yescharacteristic
8C2×#520C2211yescharacteristic
9#2975C2214yes—
10#2534C2212yes—
11#2487C2212yes—
12#2484C411yescharacteristic
13#2482C411yescharacteristic
14#2481—22no—
15#2479—22no—
16#2469—22no—
17#2467—22no—
18#2406C411yescharacteristic
19#2404C411yescharacteristic
20#2321C412yes—
21#2313C2211yescharacteristic
22C4×#123C412yes—
23#1821C2214yes—
24#1465—22no—
25#1465C2214yes—
26#1463—22no—
27#1463C2214yes—
28#1462C2214yes—
29#1461—22no—
30#1458—22no—
31#1305C2218yes—
32C22×#41C2×C411yescharacteristic
33C2×#126—22no—
34C2×#126—22no—
35C2×#126—44no—
36C2×#123C2×C412yes—
37C2×#120C2×C412yes—
38#520—44no—
39C2×C4×Q8D412yes—
40C2×C4×D4C2×C412yes—
41C2×C4×D4D412yes—
42C2×C4×D4C2312yes—
43C2×#117C2×C411yescharacteristic
44C2×#117D412yes—
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4D412yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2311yescharacteristic
52C22×C42C2×C411yescharacteristic
53#436—48no—
54#431—24no—
55#407—24no—
56#407C2318yes—
57#400—21no—
58#400—22no—
59#399—21no—
60#399—22no—
61#398—21no—
62#397—21no—
63#396—22no—
64#395—21no—
65#394—21no—
66#394—22no—
67#393—21no—
68#392—21no—
69#390—22no—
70#389—22no—
71#388—22no—
72#388—22no—
73#388—22no—
74#388—22no—
75#388—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#386—22no—
81#386—22no—
82#386—22no—
83#386C2×C412yes—
84#386C2×C412yes—
85#385—22no—
86#385C2×C412yes—
87#385C2×C412yes—
88#381C2×C411yescharacteristic
89#381C2×C411yescharacteristic
90#381C2×C412yes—
91#380—22no—
92#380C2312yes—
93#379C2×C412yes—
94C4×C4⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2×C412yes—
97#343—44no—
98#324—24no—
99#324—24no—
100#324—24no—
101#323—24no—
102#323—44no—
103C25C22×C411yescharacteristic
104C2×#41—42no—
105C2×#41—44no—
106C22×Q8C22⋊C411yescharacteristic
107C22×D4—22no—
108C22×D4C22⋊C411yescharacteristic
109C22×D4C22×C412yes—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C412yes—
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#126—48no—
122#126—48no—
123#123—44no—
124#120—44no—
125C4×Q8—44no—
126C4×D4—28no—
127C4×D4—44no—
128C4×D4—44no—
129#117—44no—
130#117—42no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C2×D412yes—
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22×C412yes—
163C2×C22⋊C4C22×C412yes—
164C2×C22⋊C4C22×C412yes—
165C2×C22⋊C4C2×D412yes—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—42no—
169C2×C42C22⋊C412yes—
170C2×C42C22⋊C412yes—
171C2×C42C2×D412yes—
172C4⋊C8—44no—
173C22⋊C8—28no—
174C22⋊C8—28no—
175C22⋊C8—44no—
176C4×C8—44no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—21no—
187#95—22no—
188#95—22no—
189C24—21no—
190C24—22no—
191C24—42no—
192C24—81no—
193C24—81no—
194C24C2×C22⋊C412yes—
195C24C23×C411yescharacteristic
196#41—416no—
197C2×Q8—24no—
198C2×Q8—41no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—44no—
205C2×D4—82no—
206C2×D4—82no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C22×D412yes—
237C22×C4H(5,𝔽2)14yes—
238M16—88no—
239M16—164no—
240C2×C8—48no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
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—44no—
273C42—82no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23—41no—
283C23—42no—
284C23—41no—
285C23—44no—
286C23—41no—
287C23—42no—
288C23—82no—
289C23—82no—
290C23—164no—
291C23C22×C22⋊C411yescharacteristic
292C23#51812yes—
293Q8—44no—
294D4—48no—
295D4—44no—
296D4—164no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—48no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—162no—
332C2×C4#41314yes—
333C2×C4#53414yes—
334C8—88no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—164no—
352C22#230811yescharacteristic
353C22C2×#41312yes—
354C4—24no—
355C4—24no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—41no—
360C4—41no—
361C4—82no—
362C4—82no—
363C4—81no—
364C4—81no—
365C4—81no—
366C2—22no—
367C2—42no—
368C2—82no—
369C2—161no—
370C2#2175411yescharacteristic
371C2#2177711yescharacteristic
372C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radical#2404C4
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. #17370C2semidirect
3. #9516C2semidirect
4. #9510C2semidirect
5. #8758C2semidirect
6. #8556C2semidirect

Maximal subgroups

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

subgroupindex
#354542
#173702
#95162
#95102
#87582
#85562