Finite Group Databasebeta

← #480973

Subgroups — #480973

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39490C211yesmaximal, characteristic
2#17428C212yesmaximal
3#11574C212yesmaximal, complemented
4#11552C214yesmaximal, complemented
5#11532C211yesmaximal, characteristic, complemented
6#11516C212yesmaximal, complemented
7#11478C211yesmaximal, characteristic, complemented
8#11450C212yesmaximal, complemented
9C2×#556C2211yescharacteristic
10C2×#554C411yescharacteristic
11#2984—22no—
12#2862C411yescharacteristic
13#2689—24no—
14#2621C412yes—
15#2568C411yescharacteristic
16#2567C2211yescharacteristic
17#2503C412yes—
18#2499C2212yes—
19#2453—22no—
20#2447—22no—
21#2317C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23#2293C411yescharacteristic
24#1473—24no—
25#1469—22no—
26#1467C2212yes—
27#1464C2214yes—
28#1461C2212yes—
29#1458C2214yes—
30#1325C2212yes—
31#1325C2214yes—
32#1325C2214yes—
33#1325C2211yescharacteristic
34#1325C2211yescharacteristic
35#1319C2212yes—
36#1319C2212yes—
37#556—42no—
38#556—42no—
39#554—42no—
40#554—42no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#126D414yes—
45C2×#125—21no—
46C2×#124D412yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8C2×C412yes—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8D412yes—
52C2×C4×Q8C2311yescharacteristic
53C2×C4×D4—21no—
54C2×C4×D4C2×C411yescharacteristic
55C2×#117C2×C412yes—
56C2×#117C2312yes—
57C22×C4⋊C4C2×C412yes—
58#446—44no—
59#446—42no—
60C8×Q8—42no—
61#432—24no—
62#432—22no—
63#431—24no—
64#431—22no—
65#400—22no—
66#400—24no—
67#398—24no—
68#397—24no—
69#397—24no—
70#395—22no—
71#391—22no—
72#391—22no—
73#391—22no—
74#391C2×C411yescharacteristic
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2311yescharacteristic
78#390—21no—
79#389—21no—
80#389—22no—
81#389—22no—
82#389—24no—
83#388—22no—
84#387—24no—
85#385—24no—
86#384C2311yescharacteristic
87#384C2312yes—
88#382C2×C412yes—
89#382C2×C412yes—
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—21no—
104C22×Q8C22⋊C412yes—
105C22×Q8C22×C411yescharacteristic
106C22×Q8C22×C411yescharacteristic
107C22×D4C22⋊C411yescharacteristic
108C23×C4—22no—
109C23×C4C22×C412yes—
110C4⋊Q8—22no—
111C4⋊Q8—41no—
112C4⋊Q8—41no—
113C4⋊Q8—41no—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116#126—44no—
117#126—44no—
118#125—42no—
119#125—42no—
120#124—42no—
121#124—42no—
122C22⋊Q8—48no—
123C22⋊Q8—44no—
124C22⋊Q8—44no—
125C4×Q8—22no—
126C4×Q8—24no—
127C4×Q8—22no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—42no—
135C4×D4—42no—
136#117—24no—
137#117—42no—
138#117—44no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4C22⋊C414yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22×C411yescharacteristic
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2411yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C414yes—
162C2×C22⋊C4C2×D414yes—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42C22×C412yes—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D414yes—
171C2×C42C2×D412yes—
172C4⋊C8—44no—
173C4⋊C8—42no—
174C4⋊C8—44no—
175C4⋊C8—42no—
176C22⋊C8—22no—
177C22⋊C8—24no—
178C22⋊C8—22no—
179#97—44no—
180#97—42no—
181C4×C8—44no—
182C4×C8—42no—
183#95—22no—
184#95—22no—
185#95—21no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—22no—
192C24C2×C22⋊C412yes—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—44no—
202C2×Q8—42no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—82no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—42no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—84no—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C414yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C23×C411yescharacteristic
227C22×C4C22×D412yes—
228C2×C8—42no—
229C2×C8—44no—
230C2×C8—42no—
231C4⋊C4—24no—
232C4⋊C4—24no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—41no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—44no—
247C4⋊C4—82no—
248C4⋊C4—84no—
249C4⋊C4#9912yes—
250C4⋊C4#9912yes—
251C22⋊C4—24no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—82no—
257C42—21no—
258C42—24no—
259C42—22no—
260C42—24no—
261C42—22no—
262C42—21no—
263C42—21no—
264C42—22no—
265C42—22no—
266C42—41no—
267C42—42no—
268C42—41no—
269C42—41no—
270C42—81no—
271C42#13112yes—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—42no—
276C23—42no—
277C23C22×C22⋊C411yescharacteristic
278Q8—44no—
279Q8—81no—
280Q8—81no—
281Q8—82no—
282Q8—81no—
283Q8—81no—
284Q8—162no—
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—164no—
320C2×C4C2×#9911yescharacteristic
321C2×C4C2×#9914yes—
322C2×C4C2×#9911yescharacteristic
323C2×C4C2×#13111yescharacteristic
324C2×C4#56811yescharacteristic
325C8—82no—
326C8—84no—
327C8—82no—
328C22—21no—
329C22—21no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—82no—
334C22—82no—
335C22#288011yescharacteristic
336C22C22×#9911yescharacteristic
337C22#288911yescharacteristic
338C4—21no—
339C4—21no—
340C4—24no—
341C4—21no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—41no—
346C4—44no—
347C4—82no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—82no—
352C4#213512yes—
353C2—21no—
354C2—21no—
355C2—82no—
356C2#1741611yescharacteristic
357C2#3336211yescharacteristic
358C2C2×#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. #11574C2semidirect
4. #11552C2semidirect
5. #11532C2semidirect
6. #11516C2semidirect
7. #11478C2semidirect
8. #11450C2semidirect

Maximal subgroups

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

subgroupindex
#394902
#174282
#115742
#115522
#115322
#115162
#114782
#114502