Finite Group Databasebeta

← #496312

Subgroups — #496312

Order 512.

1146 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1146 conjugacy classes of subgroups — too many to draw. Only its 198 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#35327C211yesmaximal, characteristic
2#20723C212yesmaximal, complemented
3#13225C218yesmaximal, complemented
4#12698C212yesmaximal, complemented
5#12697C212yesmaximal
6C2×#536C2211yescharacteristic
7C2×#535C2211yescharacteristic
8C22×#124C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10#2850—22no—
11#2620—22no—
12#2545C414yescomplemented
13#2544C2214yes—
14#2534—22no—
15#2534—22no—
16#2310C411yescharacteristic, complemented
17#2309C411yescharacteristic, complemented
18#2285C411yescharacteristic, complemented
19#2283—22no—
20C2×#354C2218yescomplemented
21C2×#353C2218yes—
22#1820—22no—
23#1799—22no—
24#1799C2214yescomplemented
25#1782C2214yes—
26C4×#99—22no—
27C4×#99C2214yescomplemented
28C22×#41C2311yescharacteristic
29C23×Q8C2×C411yescharacteristic
30C23×D4C2×C411yescharacteristic
31#536—44no—
32#535—44no—
33C2×C4⋊Q8C2×C412yes—
34C2×#127C2312yes—
35C2×#124—21no—
36C2×#124—21no—
37C2×#124—41no—
38C2×#124—42no—
39C2×#124—41no—
40C2×#124C2×C414yes—
41C2×#124C2×C412yes—
42C2×#124C2312yes—
43C2×#123—22no—
44C2×C22⋊Q8—22no—
45C2×#121—22no—
46C2×#120—22no—
47C4×#41—41no—
48C4×#41—42no—
49C4×#41—41no—
50C2×C4×Q8C2×C412yes—
51C2×C4×D4D412yes—
52C2×C4×D4D412yes—
53C2×C4×D4C2312yes—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117C2×C411yescharacteristic
57C2×#117D412yes—
58C22×C22⋊C4—22no—
59C22×C42C2×C411yescharacteristic
60C22×C42D412yes—
61C2×#99—24no—
62C2×#99C2318yes—
63#396—24no—
64#396—24no—
65#394—24no—
66#394—24no—
67#391—22no—
68#391C2×C414yes—
69#390—22no—
70#390C2×C414yes—
71#388—24no—
72#388—24no—
73#386—24no—
74#386—24no—
75#384—21no—
76#383—42no—
77#382—21no—
78#382—42no—
79#381—21no—
80#381—21no—
81#381—24no—
82#380—22no—
83#380—24no—
84#379—21no—
85#379—21no—
86C4×C4⋊C4—42no—
87C4×C4⋊C4C2×C414yes—
88C4×C22⋊C4—24no—
89#376—42no—
90#354—48no—
91#353—48no—
92#342—24no—
93#342—24no—
94C25—21no—
95C2×#41—24no—
96C2×#41—41no—
97C2×#41—41no—
98C2×#41—42no—
99C22×Q8—21no—
100C22×Q8—42no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—42no—
104C22×Q8C22×C412yes—
105C22×Q8C22×C411yescharacteristic
106C22×D4—24no—
107C22×D4—21no—
108C22×D4—42no—
109C22×D4—42no—
110C22×D4—41no—
111C22×D4—41no—
112C22×D4C22×C412yes—
113C22×D4C2×D412yes—
114C22×D4C2411yescharacteristic
115C23×C4—21no—
116C23×C4C22⋊C412yes—
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C4⋊Q8—44no—
122#127—28no—
123#124—28no—
124#124—44no—
125#124—42no—
126#124—42no—
127#124—44no—
128#124—44no—
129#124—41no—
130#124—42no—
131#124—41no—
132#123—48no—
133C22⋊Q8—48no—
134#121—48no—
135#120—48no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×D4—42no—
139C4×D4—42no—
140C4×D4—42no—
141C4×D4—42no—
142C4×D4—42no—
143C4×D4—42no—
144#117—41no—
145#117—41no—
146#117—42no—
147#117—42no—
148#117—42no—
149#117—42no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4C22⋊C411yescharacteristic
170C2×C22⋊C4C22⋊C411yescharacteristic
171C2×C22⋊C4C2×D412yes—
172C2×C22⋊C4C2×D412yes—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—44no—
181C2×C42C22⋊C412yes—
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C412yes—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22×C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C412yes—
189#99—216no—
190#99—48no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198C24—22no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24—84no—
203C24C2×C22⋊C411yescharacteristic
204C24C22×D412yes—
205#41—48no—
206#41—48no—
207C2×Q8—22no—
208C2×Q8—24no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—84no—
218C2×Q8—84no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—48no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—84no—
233C2×D4—84no—
234C2×D4—84no—
235C2×D4—84no—
236C2×D4H(5,𝔽2)14yes—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C23×C411yescharacteristic
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—48no—
274C4⋊C4—84no—
275C4⋊C4—88no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—48no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—48no—
288C22⋊C4—48no—
289C22⋊C4—84no—
290C22⋊C4—84no—
291C42—22no—
292C42—22no—
293C42—21no—
294C42—21no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—24no—
302C42—24no—
303C42—88no—
304C42#9911yescharacteristic
305C42#9912yes—
306C42#9911yescharacteristic
307C23—21no—
308C23—22no—
309C23—21no—
310C23—22no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—44no—
315C23—44no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—84no—
320C23—88no—
321C23—164no—
322C23—164no—
323C23#41012yes—
324C23C22×C22⋊C411yescharacteristic
325C23#53514yes—
326Q8—44no—
327Q8—84no—
328Q8—82no—
329Q8—82no—
330D4—44no—
331D4—48no—
332D4—84no—
333D4—82no—
334D4—82no—
335D4—168no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—88no—
369C2×C4—88no—
370C2×C4—88no—
371C2×C4C2×#9912yes—
372C2×C4C2×#9912yes—
373C2×C4C2×#9911yescharacteristic
374C2×C4C2×#9911yescharacteristic
375C2×C4#51812yes—
376C22—22no—
377C22—22no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—82no—
386C22—84no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22—84no—
391C22—168no—
392C22#230911yescharacteristic
393C22C22×#9911yescharacteristic
394C22C2×#41011yescharacteristic
395C4—21no—
396C4—21no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—168no—
408C2—21no—
409C2—21no—
410C2—44no—
411C2—42no—
412C2—162no—
413C2#2179311yescharacteristic
414C2#3335812yes—
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20723C2semidirect
3. #13225C2semidirect
4. #12698C2semidirect
8. C22×#124C4semidirect
12. #2545C4semidirect
16. #2310C4semidirect
17. #2309C4semidirect
18. #2285C4semidirect
20. C2×#354C22semidirect
24. #1799C22semidirect
27. C4×#99C22semidirect

Maximal subgroups

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

subgroupindex
#353272
#207232
#132252
#126982
#126972