Finite Group Databasebeta

← #480895

Subgroups — #480895

Order 512.

807 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38954C211yesmaximal, characteristic
2#17428C212yesmaximal
3#11568C212yesmaximal, complemented
4#11552C214yesmaximal, complemented
5#11532C211yesmaximal, characteristic, complemented
6#11510C212yesmaximal, complemented
7#11478C211yesmaximal, characteristic, complemented
8#11434C212yesmaximal, complemented
9C2×#556C2211yescharacteristic
10C2×D4×Q8C411yescharacteristic
11#2984—22no—
12#2862C411yescharacteristic
13#2698—24no—
14#2621C412yes—
15#2568C411yescharacteristic
16#2546C2211yescharacteristic
17#2499C2212yes—
18#2451—22no—
19#2447—22no—
20#2434C412yes—
21#2310C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23#2293C411yescharacteristic
24#1473—24no—
25#1469—22no—
26#1467C2212yes—
27#1464C2214yes—
28#1461C2212yes—
29#1458C2214yes—
30#1325C2212yes—
31#1325C2212yes—
32#1319C2212yes—
33#1319C2214yes—
34#1319C2214yes—
35#1319C2211yescharacteristic
36#1319C2211yescharacteristic
37C23×Q8C2×C412yes—
38#556—42no—
39#556—42no—
40D4×Q8—42no—
41D4×Q8—42no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#126D414yes—
48C2×#124D412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2×C412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8C2311yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2×C411yescharacteristic
57C2×#117C2×C412yes—
58C2×#117C2312yes—
59#446—44no—
60#446—42no—
61C8×Q8—42no—
62#432—24no—
63#432—22no—
64#431—24no—
65#431—22no—
66#400—22no—
67#398—24no—
68#398—22no—
69#397—24no—
70#397—24no—
71#395—24no—
72#391—22no—
73#391—22no—
74#391—24no—
75#391C2×C411yescharacteristic
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2×C412yes—
79#391C2311yescharacteristic
80#391C2312yes—
81#390—21no—
82#389—21no—
83#389—22no—
84#389—22no—
85#388—22no—
86#387—24no—
87#385—24no—
88#384C2311yescharacteristic
89#382—22no—
90#382C2×C412yes—
91#381—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C22⋊C4C2×C412yes—
95C43C2×C411yescharacteristic
96#324—22no—
97#324—22no—
98#324C2312yes—
99#324C2314yes—
100#324C2312yes—
101#323—24no—
102#323—24no—
103C22×Q8—22no—
104C22×Q8—24no—
105C22×Q8—42no—
106C22×Q8—44no—
107C22×Q8—82no—
108C22×Q8C22⋊C412yes—
109C22×Q8C22×C412yes—
110C22×Q8C22×C411yescharacteristic
111C22×Q8C22×C411yescharacteristic
112C22×D4C22⋊C411yescharacteristic
113C23×C4—22no—
114C23×C4C22×C412yes—
115C4⋊Q8—22no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—42no—
122C4⋊Q8—42no—
123C4⋊Q8—82no—
124#126—44no—
125#126—44no—
126#124—42no—
127#124—42no—
128C22⋊Q8—48no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C4×Q8—22no—
132C4×Q8—24no—
133C4×Q8—22no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4—42no—
141C4×D4—42no—
142#117—44no—
143#117—82no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4C22⋊C414yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C411yescharacteristic
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C22⋊C414yes—
165C2×C22⋊C4C2×D414yes—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C412yes—
173C2×C42C2×D414yes—
174C2×C42C2×D412yes—
175C4⋊C8—44no—
176C4⋊C8—42no—
177C4⋊C8—44no—
178C4⋊C8—42no—
179C22⋊C8—22no—
180C22⋊C8—24no—
181C22⋊C8—22no—
182#97—44no—
183#97—42no—
184C4×C8—44no—
185C4×C8—42no—
186#95—22no—
187#95—22no—
188#95—21no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—24no—
195C24C2×C22⋊C412yes—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—22no—
200C2×Q8—22no—
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×Q8—42no—
210C2×Q8—42no—
211C2×Q8—44no—
212C2×Q8—82no—
213C2×Q8—88no—
214C2×Q8—88no—
215C2×Q8#13112yes—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—42no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—82no—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C414yes—
236C22×C4C2×C22⋊C411yescharacteristic
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—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—44no—
257C4⋊C4—82no—
258C4⋊C4—84no—
259C4⋊C4#9912yes—
260C4⋊C4#9912yes—
261C22⋊C4—24no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—82no—
267C42—24no—
268C42—22no—
269C42—24no—
270C42—22no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—41no—
276C42—42no—
277C42—42no—
278C42—42no—
279C42—41no—
280C42—41no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—42no—
285C23—42no—
286C23C22×C22⋊C411yescharacteristic
287Q8—42no—
288Q8—42no—
289Q8—44no—
290Q8—81no—
291Q8—81no—
292Q8—82no—
293Q8—88no—
294Q8—82no—
295Q8—82no—
296D4—42no—
297D4—42no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—88no—
332C2×C4C2×#9911yescharacteristic
333C2×C4C2×#9914yes—
334C2×C4C2×#9911yescharacteristic
335C2×C4C2×#13111yescharacteristic
336C2×C4#56811yescharacteristic
337C8—82no—
338C8—84no—
339C8—82no—
340C22—21no—
341C22—21no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—82no—
346C22—82no—
347C22#288011yescharacteristic
348C22C22×#9911yescharacteristic
349C22#288811yescharacteristic
350C4—21no—
351C4—21no—
352C4—24no—
353C4—21no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—41no—
359C4—44no—
360C4—81no—
361C4—82no—
362C4—82no—
363C4—81no—
364C4#213512yes—
365C2—21no—
366C2—21no—
367C2—82no—
368C2#1741111yescharacteristic
369C2#3336011yescharacteristic
370C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×D4C22⋊C4
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
3. #11568C2semidirect
4. #11552C2semidirect
5. #11532C2semidirect
6. #11510C2semidirect
7. #11478C2semidirect
8. #11434C2semidirect

Maximal subgroups

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

subgroupindex
#389542
#174282
#115682
#115522
#115322
#115102
#114782
#114342