Finite Group Databasebeta

← #483771

Subgroups — #483771

Order 512.

414 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33508C211yesmaximal, characteristic, complemented
2#33505C211yesmaximal, characteristic, complemented
3#17904C211yesmaximal, characteristic
4#12064C211yesmaximal, characteristic, complemented
5#12063C211yesmaximal, characteristic, complemented
6#12060C211yesmaximal, characteristic, complemented
7#12059C211yesmaximal, characteristic, complemented
8#12051C211yesmaximal, characteristic, complemented
9#12050C211yesmaximal, characteristic, complemented
10#12050C211yesmaximal, characteristic, complemented
11C16×SD16C211yesmaximal, characteristic, complemented
12#11997C211yesmaximal, characteristic, complemented
13#11624C211yesmaximal, characteristic, complemented
14#11621C211yesmaximal, characteristic, complemented
15#11412C211yesmaximal, characteristic, complemented
16#3102C411yescharacteristic
17#2966C2211yescharacteristic
18C8×#41C2211yescharacteristic
19#2183C2211yescharacteristic
20Q8×C16C2211yescharacteristic
21Q8×C16C2211yescharacteristic
22Q8×C16C2211yescharacteristic
23#2169—21no—
24#2169—21no—
25#2169C2211yescharacteristic
26#2169C2211yescharacteristic
27#2169C2211yescharacteristic
28#2168—21no—
29D4×C16—21no—
30D4×C16C2211yescharacteristic
31#2164—21no—
32#2162—21no—
33#2162C2211yescharacteristic
34#2162C2211yescharacteristic
35#2151—21no—
36#2150—21no—
37#2150C2211yescharacteristic
38C2×C4⋊C16C2211yescharacteristic
39#1584C411yescharacteristic
40#1583C411yescharacteristic
41#1582C411yescharacteristic
42#1581C411yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×SD16C2211yescharacteristic
45C8×SD16C2211yescharacteristic
46C8×D8C2211yescharacteristic
47#1572C411yescharacteristic
48#1488C411yescharacteristic
49#1482C411yescharacteristic
50#1450C2211yescharacteristic
51#1367C2211yescharacteristic
52#1367C2211yescharacteristic
53#1367C2211yescharacteristic
54#1367C2211yescharacteristic
55Q8⋊C16C2211yescharacteristic
56Q8⋊C16C2211yescharacteristic
57Q8⋊C16C2211yescharacteristic
58Q8⋊C16C2211yescharacteristic
59D4⋊C16C2211yescharacteristic
60D4⋊C16C2211yescharacteristic
61D4⋊C16C2211yescharacteristic
62D4⋊C16C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67#529C2×C411yescharacteristic
68C4×#41C2×C411yescharacteristic
69C2×M32—21no—
70C2×M32—21no—
71C2×M32—21no—
72C2×M32—21no—
73C22×C16—21no—
74C22×C16—21no—
75#485C811yescharacteristic
76#482C811yescharacteristic
77#479C811yescharacteristic
78#478C811yescharacteristic
79#464C811yescharacteristic
80#463C811yescharacteristic
81#452C811yescharacteristic
82#449C811yescharacteristic
83#446C2×C411yescharacteristic
84C8×Q8C2×C411yescharacteristic
85C8×Q8C2311yescharacteristic
86C8×Q8C2311yescharacteristic
87C4×Q16C2×C411yescharacteristic
88C4×SD16C2×C411yescharacteristic
89C4×SD16C2×C411yescharacteristic
90C4×D8C2×C411yescharacteristic
91#436C2×C411yescharacteristic
92#435—21no—
93C8×D4—21no—
94C8×D4—21no—
95C8×D4—21no—
96C8×D4C2×C411yescharacteristic
97C8×D4C2311yescharacteristic
98C8×D4C2311yescharacteristic
99#433—21no—
100#432C2×C411yescharacteristic
101#431—21no—
102#431C2×C411yescharacteristic
103#431C2×C411yescharacteristic
104#431C2×C411yescharacteristic
105#405—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C4⋊C16—21no—
109C4⋊C16—21no—
110C4⋊C16—21no—
111C4⋊C16—21no—
112C4⋊C16—21no—
113C4⋊C16—21no—
114C4⋊C16—21no—
115C4⋊C16D411yescharacteristic
116C4⋊C16D411yescharacteristic
117C4⋊C16C2311yescharacteristic
118C4⋊C16C2311yescharacteristic
119C4⋊C16C2311yescharacteristic
120C4⋊C16C2311yescharacteristic
121C22⋊C16—21no—
122C22⋊C16—21no—
123C22⋊C16—21no—
124C22⋊C16—21no—
125C22⋊C16—21no—
126C22⋊C16D411yescharacteristic
127C22⋊C16D411yescharacteristic
128#346—21no—
129C4×C16—21no—
130C4×C16—21no—
131C4×C16—21no—
132C4×C16C2311yescharacteristic
133C4×C16C2311yescharacteristic
134C4×C16C2311yescharacteristic
135C4×C16C2311yescharacteristic
136#335C2×C411yescharacteristic
137#335C2×C411yescharacteristic
138#334C2×C411yescharacteristic
139#334C2×C411yescharacteristic
140Q8⋊C8C2×C411yescharacteristic
141Q8⋊C8C2×C411yescharacteristic
142Q8⋊C8C2×C411yescharacteristic
143Q8⋊C8C2×C411yescharacteristic
144D4⋊C8C2×C411yescharacteristic
145D4⋊C8C2×C411yescharacteristic
146D4⋊C8C2×C411yescharacteristic
147D4⋊C8C2×C411yescharacteristic
148#322C2×C411yescharacteristic
149#322C2×C411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C2×#41C2×C811yescharacteristic
153C2×Q16C2×C811yescharacteristic
154C2×SD16C2×C811yescharacteristic
155C2×SD16C2×C811yescharacteristic
156C2×D8C2×C811yescharacteristic
157C2×M16—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8#4111yescharacteristic
163#126—21no—
164#125C2×C811yescharacteristic
165#124C2×C811yescharacteristic
166#123—21no—
167C22⋊Q8C2×C811yescharacteristic
168#121C2×C811yescharacteristic
169C4×Q8C2×C811yescharacteristic
170C4×Q8C22×C411yescharacteristic
171C4×Q8C22×C411yescharacteristic
172C4×D4—21no—
173C4×D4—21no—
174C4×D4—21no—
175C4×D4C2×C811yescharacteristic
176C4×D4C22×C411yescharacteristic
177C4×D4C22×C411yescharacteristic
178#117—21no—
179#117—21no—
180#117C2×C811yescharacteristic
181C2×C42—21no—
182C2×C42C22×C411yescharacteristic
183M32—81no—
184M32—81no—
185M32—81no—
186M32—81no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—21no—
193C2×C16—81no—
194C2×C16—81no—
195C2×C16C2×D411yescharacteristic
196C2×C16C2×D411yescharacteristic
197C2×C16C2×D411yescharacteristic
198C2×C16C2×D411yescharacteristic
199#107C2×C811yescharacteristic
200#107C2×C811yescharacteristic
201#106C2×C811yescharacteristic
202#106C2×C811yescharacteristic
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8C2×C811yescharacteristic
209C4⋊C8C2×C811yescharacteristic
210C4⋊C8C22×C411yescharacteristic
211C4⋊C8C22×C411yescharacteristic
212C4⋊C8C22×C411yescharacteristic
213C4⋊C8C22×C411yescharacteristic
214#103C2×C811yescharacteristic
215#103C2×C811yescharacteristic
216#103C2×C811yescharacteristic
217#103C2×C811yescharacteristic
218#102C2×C811yescharacteristic
219#102C2×C811yescharacteristic
220#102C2×C811yescharacteristic
221#102C2×C811yescharacteristic
222C22⋊C8—21no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C22⋊C8—21no—
229C22⋊C8C2×C811yescharacteristic
230C22⋊C8C2×C811yescharacteristic
231#97—21no—
232#97—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8C22×C411yescharacteristic
238C4×C8C22×C411yescharacteristic
239C4×C8C22×C411yescharacteristic
240C4×C8C22×C411yescharacteristic
241C4×C8C22×C411yescharacteristic
242C4×C8C2×D411yescharacteristic
243C4×C8#4111yescharacteristic
244C4×C8C2411yescharacteristic
245#41—22no—
246#41—41no—
247C2×Q8C22×C811yescharacteristic
248C2×Q8C22×C811yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22×C811yescharacteristic
252C2×D4C22×C811yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4C4×D411yescharacteristic
258C22×C4C22×C811yescharacteristic
259C22×C4#13111yescharacteristic
260Q16—22no—
261SD16—22no—
262SD16—41no—
263D8—41no—
264M16—81no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—81no—
279C2×C8C4×D411yescharacteristic
280C2×C8C4×D411yescharacteristic
281C2×C8C22×C811yescharacteristic
282C2×C8C22×C811yescharacteristic
283C2×C8C22×C811yescharacteristic
284C2×C8C22×C811yescharacteristic
285C2×C8C22×D411yescharacteristic
286C2×C8C2×#4111yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4C22×C811yescharacteristic
294C4⋊C4C22×C811yescharacteristic
295C4⋊C4C22×C811yescharacteristic
296C4⋊C4C22×C811yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42C4×D411yescharacteristic
310C42C22×C811yescharacteristic
311C42#13111yescharacteristic
312C42C23×C411yescharacteristic
313C16—21no—
314C16—21no—
315C16—21no—
316C16—21no—
317C16—41no—
318C16—41no—
319C16—41no—
320C16—41no—
321C16—41no—
322C16—41no—
323C23—21no—
324C23—21no—
325C23C8×D411yescharacteristic
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8#50412yes—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—42no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4C8×D411yescharacteristic
354C2×C4C8×D411yescharacteristic
355C2×C4C8×D411yescharacteristic
356C2×C4C2×C4×D411yescharacteristic
357C2×C4C23×C811yescharacteristic
358C2×C4C2×#13111yescharacteristic
359C8—21no—
360C8—21no—
361C8—21no—
362C8—21no—
363C8—21no—
364C8—21no—
365C8—21no—
366C8—41no—
367C8—41no—
368C8—41no—
369C8—41no—
370C8—41no—
371C8—41no—
372C8#57711yescharacteristic
373C8#57711yescharacteristic
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22C2×C8×D411yescharacteristic
384C4—21no—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4#295211yescharacteristic
395C4#295711yescharacteristic
396C4C2×#50411yescharacteristic
397C4#340711yescharacteristic
398C2—41no—
399C2—81no—
400C2—81no—
401C2#1785811yescharacteristic
402C2#3343611yescharacteristic
403C2#3344111yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC4×C8C24
socleC22C2×C8×D4
prime radical#449C8
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ #2162 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33508C2semidirect
2. #33505C2semidirect
4. #12064C2semidirect
5. #12063C2semidirect
6. #12060C2semidirect
7. #12059C2semidirect
8. #12051C2semidirect
9. #12050C2semidirect
10. #12050C2semidirect
11. C16×SD16C2semidirect
12. #11997C2semidirect
13. #11624C2semidirect
14. #11621C2semidirect
15. #11412C2semidirect

Maximal subgroups

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

subgroupindex
#335082
#335052
#179042
#120642
#120632
#120602
#120592
#120512
#120502
C16×SD162
#119972
#116242
#116212
#114122