Finite Group Databasebeta

← #483697

Subgroups — #483697

Order 512.

403 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33503C211yesmaximal, characteristic
2#33499C211yesmaximal, characteristic, complemented
3#17913C211yesmaximal, characteristic
4#12061C211yesmaximal, characteristic, complemented
5#12060C211yesmaximal, characteristic, complemented
6#12058C211yesmaximal, characteristic, complemented
7#12056C211yesmaximal, characteristic, complemented
8#12054C211yesmaximal, characteristic, complemented
9#12053C211yesmaximal, characteristic, complemented
10#12051C211yesmaximal, characteristic, complemented
11#12050C211yesmaximal, characteristic, complemented
12#12016C211yesmaximal, characteristic, complemented
13#11615C211yesmaximal, characteristic
14#11606C211yesmaximal, characteristic, complemented
15#11409C211yesmaximal, characteristic, complemented
16#3107C411yescharacteristic
17C8×#41C2211yescharacteristic
18#2960C2211yescharacteristic
19#2183—21no—
20#2183—21no—
21#2183C2211yescharacteristic
22#2183C2211yescharacteristic
23#2183C2211yescharacteristic
24#2183C2211yescharacteristic
25Q8×C16C2211yescharacteristic
26Q8×C16C2211yescharacteristic
27#2169C2211yescharacteristic
28#2169C2211yescharacteristic
29#2168—21no—
30#2168—21no—
31#2164—21no—
32#2164—21no—
33#2163—21no—
34#2163C2211yescharacteristic
35#2162C2211yescharacteristic
36#2150—21no—
37C2×C4⋊C16C2211yescharacteristic
38C4×M32C2211yescharacteristic
39#1587C411yescharacteristic
40#1587C411yescharacteristic
41#1585C411yescharacteristic
42#1585C411yescharacteristic
43C8×Q16C2211yescharacteristic
44C8×Q16C2211yescharacteristic
45C8×SD16C2211yescharacteristic
46C8×SD16C2211yescharacteristic
47#1574C411yescharacteristic
48#1489C411yescharacteristic
49#1485C411yescharacteristic
50C8×M16C2211yescharacteristic
51#1370C2211yescharacteristic
52#1370C2211yescharacteristic
53#1367C2211yescharacteristic
54#1367C2211yescharacteristic
55Q8⋊C16C2211yescharacteristic
56Q8⋊C16C2211yescharacteristic
57Q8⋊C16C2211yescharacteristic
58Q8⋊C16C2211yescharacteristic
59Q8⋊C16C2211yescharacteristic
60Q8⋊C16C2211yescharacteristic
61D4⋊C16C2211yescharacteristic
62D4⋊C16C2211yescharacteristic
63#1312C2211yescharacteristic
64#1312C2211yescharacteristic
65#1311C2211yescharacteristic
66C8×C16C2211yescharacteristic
67#533C2×C411yescharacteristic
68C4×#41C2×C411yescharacteristic
69C2×M32—21no—
70C2×M32—21no—
71C2×M32—21no—
72C22×C16—21no—
73#485C811yescharacteristic
74#485C811yescharacteristic
75#475C811yescharacteristic
76#474C811yescharacteristic
77#462C811yescharacteristic
78#461C811yescharacteristic
79#452C811yescharacteristic
80#452C811yescharacteristic
81#446—21no—
82#446C2×C411yescharacteristic
83#446C2×C411yescharacteristic
84C8×Q8—21no—
85C8×Q8C2×C411yescharacteristic
86C8×Q8C2311yescharacteristic
87C8×Q8C2311yescharacteristic
88C8×Q8C2311yescharacteristic
89C4×Q16C2×C411yescharacteristic
90C4×Q16C2×C411yescharacteristic
91C4×SD16C2×C411yescharacteristic
92C4×SD16C2×C411yescharacteristic
93C8×D4—21no—
94C8×D4—21no—
95C8×D4C2×C411yescharacteristic
96C8×D4C2311yescharacteristic
97#431—21no—
98#431C2×C411yescharacteristic
99#431C2×C411yescharacteristic
100#424—21no—
101#423C2×C411yescharacteristic
102#405—21no—
103C4×M16C2×C411yescharacteristic
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C4⋊C16—21no—
107C4⋊C16—21no—
108C4⋊C16—21no—
109C4⋊C16—21no—
110C4⋊C16—21no—
111C4⋊C16—21no—
112C4⋊C16—21no—
113C4⋊C16D411yescharacteristic
114C4⋊C16D411yescharacteristic
115C4⋊C16C2311yescharacteristic
116C4⋊C16C2311yescharacteristic
117C4⋊C16C2311yescharacteristic
118C4⋊C16C2311yescharacteristic
119C22⋊C16—21no—
120C22⋊C16—21no—
121C22⋊C16—21no—
122C22⋊C16D411yescharacteristic
123C22⋊C16D411yescharacteristic
124#346—21no—
125#346—21no—
126#346—21no—
127#346—21no—
128#346—21no—
129#346—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
144Q8⋊C8C2×C411yescharacteristic
145Q8⋊C8C2×C411yescharacteristic
146D4⋊C8C2×C411yescharacteristic
147D4⋊C8C2×C411yescharacteristic
148#322C2×C411yescharacteristic
149#322C2×C411yescharacteristic
150C82C2311yescharacteristic
151C82C2311yescharacteristic
152C2×#41C2×C811yescharacteristic
153C2×Q16C2×C811yescharacteristic
154C2×Q16C2×C811yescharacteristic
155C2×SD16C2×C811yescharacteristic
156C2×SD16C2×C811yescharacteristic
157C2×M16—21no—
158C2×M16—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8#4111yescharacteristic
163C4⋊Q8C2×C811yescharacteristic
164C4⋊Q8C2×C811yescharacteristic
165#125—21no—
166C22⋊Q8—21no—
167C22⋊Q8C2×C811yescharacteristic
168C22⋊Q8C2×C811yescharacteristic
169C4×Q8—21no—
170C4×Q8C2×C811yescharacteristic
171C4×Q8C22×C411yescharacteristic
172C4×Q8C22×C411yescharacteristic
173C4×Q8C22×C411yescharacteristic
174C4×D4—21no—
175C4×D4—21no—
176C4×D4C2×C811yescharacteristic
177C4×D4C22×C411yescharacteristic
178#117—21no—
179#117—21no—
180#117C2×C811yescharacteristic
181C2×C42—21no—
182C2×C42C22×C411yescharacteristic
183M32—42no—
184M32—81no—
185M32—81no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16—21no—
189C2×C16—21no—
190C2×C16—21no—
191C2×C16—21no—
192C2×C16—81no—
193C2×C16C2×D411yescharacteristic
194C2×C16C2×D411yescharacteristic
195C2×C16C2×D411yescharacteristic
196C2×C16C2×D411yescharacteristic
197#107C2×C811yescharacteristic
198#107C2×C811yescharacteristic
199#106C2×C811yescharacteristic
200#106C2×C811yescharacteristic
201C4⋊C8—21no—
202C4⋊C8—21no—
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#103C2×C811yescharacteristic
219#103C2×C811yescharacteristic
220#102C2×C811yescharacteristic
221#102C2×C811yescharacteristic
222C22⋊C8—21no—
223C22⋊C8—21no—
224C22⋊C8—21no—
225C22⋊C8C2×C811yescharacteristic
226C22⋊C8C2×C811yescharacteristic
227#97—21no—
228#97—21no—
229#97—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8C22×C411yescharacteristic
236C4×C8C22×C411yescharacteristic
237C4×C8C22×C411yescharacteristic
238C4×C8C22×C411yescharacteristic
239C4×C8C22×C411yescharacteristic
240C4×C8C2×D411yescharacteristic
241C4×C8#4111yescharacteristic
242C4×C8C2411yescharacteristic
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8C22×C811yescharacteristic
247C2×Q8C22×C811yescharacteristic
248C2×Q8C22×C811yescharacteristic
249C2×D4—21no—
250C2×D4C22×C811yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C4×D411yescharacteristic
255C22×C4C22×C811yescharacteristic
256C22×C4#13111yescharacteristic
257Q16—41no—
258Q16—41no—
259SD16—41no—
260SD16—41no—
261M16—42no—
262M16—81no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—81no—
275C2×C8C4×D411yescharacteristic
276C2×C8C4×D411yescharacteristic
277C2×C8C22×C811yescharacteristic
278C2×C8C22×C811yescharacteristic
279C2×C8C22×C811yescharacteristic
280C2×C8C22×C811yescharacteristic
281C2×C8C22×D411yescharacteristic
282C2×C8C2×#4111yescharacteristic
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4C22×C811yescharacteristic
293C4⋊C4C22×C811yescharacteristic
294C4⋊C4C22×C811yescharacteristic
295C4⋊C4C22×C811yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42C4×D411yescharacteristic
307C42C22×C811yescharacteristic
308C42#13111yescharacteristic
309C42C23×C411yescharacteristic
310C16—22no—
311C16—21no—
312C16—21no—
313C16—21no—
314C16—21no—
315C16—41no—
316C16—41no—
317C16—41no—
318C16—41no—
319C16—41no—
320C23—21no—
321C23C8×D411yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—21no—
326Q8—21no—
327Q8—21no—
328Q8—81no—
329D4—21no—
330D4—21no—
331D4—42no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4C8×D411yescharacteristic
349C2×C4C8×D411yescharacteristic
350C2×C4C8×D411yescharacteristic
351C2×C4C2×C4×D411yescharacteristic
352C2×C4C23×C811yescharacteristic
353C2×C4C2×#13111yescharacteristic
354C8—22no—
355C8—21no—
356C8—21no—
357C8—21no—
358C8—21no—
359C8—21no—
360C8—21no—
361C8—21no—
362C8—41no—
363C8—41no—
364C8—41no—
365C8—41no—
366C8—41no—
367C8#57711yescharacteristic
368C8#57811yescharacteristic
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22C2×C8×D411yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4#294711yescharacteristic
388C4#295611yescharacteristic
389C4#340711yescharacteristic
390C4#340711yescharacteristic
391C2—41no—
392C2—81no—
393C2#1785311yescharacteristic
394C2#3343311yescharacteristic
395C2#3344311yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC4×C8C24
socleC22C2×C8×D4
prime radicalC2×#41C2×C8
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
2. #33499C2semidirect
4. #12061C2semidirect
5. #12060C2semidirect
6. #12058C2semidirect
7. #12056C2semidirect
8. #12054C2semidirect
9. #12053C2semidirect
10. #12051C2semidirect
11. #12050C2semidirect
12. #12016C2semidirect
14. #11606C2semidirect
15. #11409C2semidirect

Maximal subgroups

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

subgroupindex
#335032
#334992
#179132
#120612
#120602
#120582
#120562
#120542
#120532
#120512
#120502
#120162
#116152
#116062
#114092