Finite Group Databasebeta

← #481013

Subgroups — #481013

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#17420C212yesmaximal, complemented
3#11565C214yesmaximal, complemented
4#11564C212yesmaximal, complemented
5#11532C212yesmaximal, complemented
6#11529C211yesmaximal, characteristic
7#11478C212yesmaximal, complemented
8#11475C211yesmaximal, characteristic
9C2×#556C411yescharacteristic
10C2×#556C2211yescharacteristic
11#2924—22no—
12#2868C411yescharacteristic
13#2858C412yes—
14#2547C411yescharacteristic
15#2547C2211yescharacteristic
16#2545C412yes—
17#2545C2212yes—
18#2538—22no—
19#2512—24no—
20#2354—22no—
21C4×C4⋊Q8C2212yes—
22C4×#127C2211yescharacteristic
23C42×Q8C411yescharacteristic
24#1473—24no—
25#1472—22no—
26#1467C2214yes—
27#1465C2212yes—
28#1461C2214yes—
29#1459C2212yes—
30#1325C2212yes—
31#1325C2211yescharacteristic
32#1325C2211yescharacteristic
33#1319C2214yes—
34#1319C2214yes—
35#1318C2212yes—
36#1318C2212yes—
37#556—41no—
38#556—41no—
39#556—42no—
40#556—42no—
41#556—41no—
42#556—41no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#127C2×C411yescharacteristic
48C2×#126—22no—
49C2×#126D414yes—
50C2×#124C2×C412yes—
51C2×#124C2×C412yes—
52C2×#124C2×C412yes—
53C2×#124C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8D412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2312yes—
60C2×C4×D4—22no—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117D412yes—
65#433—44no—
66#432—24no—
67#431—24no—
68#424—42no—
69#407—22no—
70C2×C22⋊C8—22no—
71#405—42no—
72#397—22no—
73#397—22no—
74#396—24no—
75#396—24no—
76#391C2×C412yes—
77#391C2312yes—
78#390—24no—
79#390—24no—
80#390—22no—
81#390C2×C411yescharacteristic
82#390C2311yescharacteristic
83#389—24no—
84#388—24no—
85#388—24no—
86#384—22no—
87#384—22no—
88#384C2311yescharacteristic
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C411yescharacteristic
96C43C2×C411yescharacteristic
97C43C2×C412yes—
98#324—24no—
99#324—24no—
100#324C2312yes—
101#324C2312yes—
102#324C2314yes—
103#323—22no—
104#323—22no—
105C22×Q8C22⋊C411yescharacteristic
106C22×Q8C22×C412yes—
107C22×Q8C22×C412yes—
108C22×D4—22no—
109C22×D4C22×C411yescharacteristic
110C22×D4C22×C411yescharacteristic
111C23×C4—21no—
112C23×C4C2×D412yes—
113C2×M16—42no—
114C22×C8—42no—
115C4⋊Q8—24no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—42no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—82no—
122#127—22no—
123#127—41no—
124#126—44no—
125#126—44no—
126#126—44no—
127#126—44no—
128#124—42no—
129#124—42no—
130#124—42no—
131#124—42no—
132#124—82no—
133#124—82no—
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138C4×Q8—22no—
139C4×Q8—21no—
140C4×Q8—21no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—44no—
144C4×Q8—42no—
145C4×Q8—42no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—48no—
149C4×D4—41no—
150C4×D4—41no—
151#117—22no—
152#117—22no—
153#117—24no—
154#117—42no—
155#117—41no—
156#117—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2411yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4C22⋊C414yes—
180C2×C22⋊C4C22⋊C414yes—
181C2×C22⋊C4C2×D412yes—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D414yes—
193C4⋊C8—44no—
194C4⋊C8—44no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—24no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200#97—44no—
201C4×C8—44no—
202#95—22no—
203#95—21no—
204#95—24no—
205#95—24no—
206C24—21no—
207C24C2×C22⋊C412yes—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—21no—
211C2×Q8—21no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—44no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—44no—
229C2×D4—44no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4C2×C22⋊C414yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248M16—84no—
249C2×C8—42no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—84no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—41no—
260C4⋊C4—48no—
261C4⋊C4—41no—
262C4⋊C4—44no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—84no—
270C4⋊C4#9912yes—
271C4⋊C4#9911yescharacteristic
272C4⋊C4#9911yescharacteristic
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—84no—
284C22⋊C4—84no—
285C42—24no—
286C42—21no—
287C42—24no—
288C42—24no—
289C42—21no—
290C42—21no—
291C42—22no—
292C42—22no—
293C42—42no—
294C42—41no—
295C42—44no—
296C42—42no—
297C42—44no—
298C42—42no—
299C42—42no—
300C42#13112yes—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23C22×C22⋊C411yescharacteristic
312Q8—42no—
313Q8—42no—
314Q8—82no—
315Q8—82no—
316Q8—82no—
317Q8—82no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—88no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4C2×#9912yes—
355C2×C4C2×#9912yes—
356C2×C4C2×#9911yescharacteristic
357C2×C4C2×#9911yescharacteristic
358C2×C4C2×#13111yescharacteristic
359C2×C4#56811yescharacteristic
360C8—82no—
361C8—82no—
362C8—84no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22#288011yescharacteristic
376C22C22×#9911yescharacteristic
377C22#288811yescharacteristic
378C4—24no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—22no—
383C4—22no—
384C4—42no—
385C4—44no—
386C4—42no—
387C4—42no—
388C4—41no—
389C4—82no—
390C4—82no—
391C4—81no—
392C4#212612yes—
393C2—21no—
394C2—21no—
395C2—82no—
396C2—161no—
397C2#1741911yescharacteristic
398C2C2×#212611yescharacteristic
399C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#127C2×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
2. #17420C2semidirect
3. #11565C2semidirect
4. #11564C2semidirect
5. #11532C2semidirect
7. #11478C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#174202
#115652
#115642
#115322
#115292
#114782
#114752