Finite Group Databasebeta

← #480970

Subgroups — #480970

Order 512.

842 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39488C211yesmaximal, characteristic
2#17428C212yesmaximal, complemented
3#11573C212yesmaximal, complemented
4#11550C214yesmaximal, complemented
5#11530C211yesmaximal, characteristic, complemented
6#11515C212yesmaximal, complemented
7#11476C211yesmaximal, characteristic, complemented
8#11449C212yesmaximal, complemented
9C2×#556C2211yescharacteristic
10C2×#547C411yescharacteristic
11#2984—22no—
12#2868C411yescharacteristic
13#2718—24no—
14#2620C412yes—
15#2568C2211yescharacteristic
16#2542C411yescharacteristic
17#2502C412yes—
18#2500C2212yes—
19#2432—22no—
20#2429—22no—
21#2317C2212yes—
22C4×#124C2211yescharacteristic
23#2293C411yescharacteristic
24#1473—24no—
25#1469—22no—
26#1467C2212yes—
27#1464C2214yes—
28#1461C2212yes—
29#1458C2214yes—
30#1324C2212yes—
31#1324C2214yes—
32#1324C2214yes—
33#1324C2211yescharacteristic
34#1324C2211yescharacteristic
35#1318C2212yes—
36#1318C2212yes—
37C22×#41—21no—
38#556—42no—
39#556—42no—
40#547—42no—
41#547—42no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#126D414yes—
46C2×#124C2×C411yescharacteristic
47C2×#124D412yes—
48C2×#123—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8C2×C412yes—
52C2×#121C2×C412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8C2311yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×#117C2×C412yes—
58C2×#117C2312yes—
59C22×C4⋊C4C2×C412yes—
60#446—44no—
61#446—42no—
62C8×Q8—42no—
63#432—24no—
64#432—22no—
65#431—24no—
66#431—22no—
67#400—22no—
68#399—22no—
69#398—24no—
70#396—24no—
71#394—24no—
72#394—24no—
73#391—22no—
74#391C2×C412yes—
75#391C2×C411yescharacteristic
76#390—21no—
77#390C2×C412yes—
78#390C2311yescharacteristic
79#389—21no—
80#389—24no—
81#388—22no—
82#388—22no—
83#388—24no—
84#385—24no—
85#385—24no—
86#384—22no—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2×C412yes—
90#382C2312yes—
91#382C2311yescharacteristic
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94C4×C22⋊C4C2×C412yes—
95C43C2×C411yescharacteristic
96#324—22no—
97#324—22no—
98#324C2312yes—
99#324C2314yes—
100#324C2312yes—
101#323—24no—
102#323—24no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—84no—
106C22×Q8—21no—
107C22×Q8C22⋊C412yes—
108C22×Q8C22×C411yescharacteristic
109C22×D4—22no—
110C22×D4C22⋊C411yescharacteristic
111C22×D4C22×C411yescharacteristic
112C23×C4—22no—
113C23×C4—21no—
114C23×C4C22×C412yes—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—84no—
120#126—44no—
121#126—44no—
122#124—22no—
123#124—41no—
124#124—42no—
125#124—42no—
126#123—48no—
127C22⋊Q8—44no—
128C22⋊Q8—42no—
129C22⋊Q8—42no—
130C22⋊Q8—84no—
131#121—44no—
132C4×Q8—22no—
133C4×Q8—24no—
134C4×Q8—22no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×D4—22no—
138C4×D4—22no—
139C4×D4—41no—
140C4×D4—41no—
141#117—24no—
142#117—42no—
143#117—44no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C22⋊C414yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4C22⋊C412yes—
168C2×C22⋊C4C22⋊C414yes—
169C2×C22⋊C4C2×D414yes—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42C22×C412yes—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D414yes—
177C2×C42C2×D412yes—
178C4⋊C8—44no—
179C4⋊C8—42no—
180C4⋊C8—44no—
181C4⋊C8—42no—
182C22⋊C8—22no—
183C22⋊C8—24no—
184C22⋊C8—22no—
185#97—44no—
186#97—42no—
187C4×C8—44no—
188C4×C8—42no—
189#95—22no—
190#95—22no—
191#95—21no—
192#95—24no—
193#95—22no—
194#95—24no—
195C24—21no—
196C24C2×C22⋊C412yes—
197#41—88no—
198#41—88no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—42no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—44no—
217C2×D4—84no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—84no—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C414yes—
237C22×C4C23×C411yescharacteristic
238C22×C4C22×D412yes—
239C2×C8—42no—
240C2×C8—44no—
241C2×C8—42no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—44no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4#9912yes—
258C4⋊C4#9912yes—
259C22⋊C4—24no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—44no—
268C22⋊C4—84no—
269C42—21no—
270C42—24no—
271C42—22no—
272C42—24no—
273C42—22no—
274C42—21no—
275C42—21no—
276C42—22no—
277C42—22no—
278C42—41no—
279C42—42no—
280C42—42no—
281C42#13112yes—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23C22×C22⋊C411yescharacteristic
293Q8—44no—
294Q8—81no—
295Q8—81no—
296Q8—82no—
297Q8—82no—
298Q8—82no—
299D4—42no—
300D4—42no—
301D4—81no—
302D4—81no—
303D4—88no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—82no—
334C2×C4—88no—
335C2×C4—82no—
336C2×C4C2×#9911yescharacteristic
337C2×C4C2×#9914yes—
338C2×C4C2×#9911yescharacteristic
339C2×C4C2×#13111yescharacteristic
340C2×C4#56811yescharacteristic
341C8—82no—
342C8—84no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#288011yescharacteristic
357C22C22×#9911yescharacteristic
358C22#288911yescharacteristic
359C4—21no—
360C4—21no—
361C4—24no—
362C4—21no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—41no—
368C4—44no—
369C4—82no—
370C4—81no—
371C4—82no—
372C4#213412yes—
373C2—21no—
374C2—21no—
375C2—82no—
376C2—161no—
377C2#1741611yescharacteristic
378C2#3336011yescharacteristic
379C2C2×#213411yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#556 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17428C2semidirect
3. #11573C2semidirect
4. #11550C2semidirect
5. #11530C2semidirect
6. #11515C2semidirect
7. #11476C2semidirect
8. #11449C2semidirect

Maximal subgroups

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

subgroupindex
#394882
#174282
#115732
#115502
#115302
#115152
#114762
#114492