Finite Group Databasebeta

← #491334

Subgroups — #491334

Order 512.

388 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33797C211yesmaximal, characteristic
2#33498C211yesmaximal, characteristic
3#19262C211yesmaximal, characteristic
4#19118C211yesmaximal, characteristic
5#12482C211yesmaximal, characteristic, complemented
6#12481C211yesmaximal, characteristic, complemented
7#12478C211yesmaximal, characteristic, complemented
8#12477C211yesmaximal, characteristic, complemented
9#12462C211yesmaximal, characteristic, complemented
10#12461C211yesmaximal, characteristic, complemented
11#12459C211yesmaximal, characteristic, complemented
12#12458C211yesmaximal, characteristic, complemented
13#12036C211yesmaximal, characteristic
14#11976C211yesmaximal, characteristic
15#11953C211yesmaximal, characteristic
16#3170C2211yescharacteristic
17#3160C2211yescharacteristic
18#3104—21no—
19#3078—21no—
20C2×C8×Q8C2211yescharacteristic
21#2254C2211yescharacteristic
22#2252C2211yescharacteristic
23#2241C2211yescharacteristic
24#2241C2211yescharacteristic
25#2237—21no—
26#2237—21no—
27#2229—21no—
28#2229—21no—
29#2183—21no—
30#2183—21no—
31#2183C2211yescharacteristic
32#2183C2211yescharacteristic
33Q8×C16C2211yescharacteristic
34Q8×C16C2211yescharacteristic
35#2163—21no—
36#2163C2211yescharacteristic
37#2162—21no—
38#2162C2211yescharacteristic
39#1684—21no—
40#1684—21no—
41#1676C2211yescharacteristic
42#1676C2211yescharacteristic
43#1676C2211yescharacteristic
44#1676C2211yescharacteristic
45#1674C2211yescharacteristic
46#1674C2211yescharacteristic
47#1672—21no—
48#1672—21no—
49#1670C2211yescharacteristic
50#1670C2211yescharacteristic
51#1567C2211yescharacteristic
52#1564C2211yescharacteristic
53#1550—21no—
54#1542—21no—
55#1372C2211yescharacteristic
56#1372C2211yescharacteristic
57#1371C2211yescharacteristic
58#1371C2211yescharacteristic
59#1365C2211yescharacteristic
60#1365C2211yescharacteristic
61#1365C2211yescharacteristic
62#1365C2211yescharacteristic
63#1364C2211yescharacteristic
64#1364C2211yescharacteristic
65#1362C2211yescharacteristic
66#1362C2211yescharacteristic
67#533—21no—
68#533—21no—
69C2×C4×Q8D411yescharacteristic
70#500C2311yescharacteristic
71#500C2311yescharacteristic
72#500C2311yescharacteristic
73#499—21no—
74#499—21no—
75#498—21no—
76#494C2311yescharacteristic
77#487—21no—
78#484—41no—
79#479—41no—
80#479—41no—
81#477—21no—
82#477—21no—
83#477—21no—
84#477—21no—
85#475—21no—
86#475—21no—
87#467—21no—
88#467—21no—
89#462—21no—
90#462—21no—
91#451—41no—
92C8×Q8—21no—
93C8×Q8—21no—
94C8×Q8D411yescharacteristic
95C8×Q8D411yescharacteristic
96C8×Q8C2311yescharacteristic
97C8×Q8C2311yescharacteristic
98C4×Q16—21no—
99C4×Q16—21no—
100C4×Q16—41no—
101C4×Q16—41no—
102#431—21no—
103#428—41no—
104#427—21no—
105#427—21no—
106#423—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8D411yescharacteristic
109#419—41no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112#367—21no—
113#367—21no—
114#366—21no—
115#366—21no—
116C4⋊C16—21no—
117C4⋊C16—21no—
118C4⋊C16—21no—
119C4⋊C16—21no—
120C4⋊C16D411yescharacteristic
121C4⋊C16D411yescharacteristic
122C4⋊C16C2311yescharacteristic
123C4⋊C16C2311yescharacteristic
124#358—21no—
125#358—21no—
126#358—21no—
127#358—21no—
128C22⋊C16—21no—
129C22⋊C16D411yescharacteristic
130C22⋊C16D411yescharacteristic
131#346—21no—
132#346—21no—
133#346—21no—
134C4×C16—21no—
135C4×C16—21no—
136C4×C16C2311yescharacteristic
137C4×C16C2311yescharacteristic
138#335—21no—
139#335C2311yescharacteristic
140#335C2311yescharacteristic
141#335C2311yescharacteristic
142#335C2311yescharacteristic
143#334—21no—
144#333—21no—
145#333—21no—
146#333—21no—
147#333—21no—
148#332—21no—
149#332—21no—
150#332—21no—
151#332—21no—
152C22×Q8D811yescharacteristic
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16—41no—
156C2×Q16—41no—
157C2×M16—41no—
158C22×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8#4111yescharacteristic
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165#125—41no—
166#125—41no—
167C22⋊Q8—41no—
168C22⋊Q8—41no—
169C22⋊Q8—41no—
170C22⋊Q8—41no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8D811yescharacteristic
178C4×Q8D811yescharacteristic
179C4×Q8C2×D411yescharacteristic
180C4×Q8C2×D411yescharacteristic
181#117—41no—
182#117—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4D811yescharacteristic
185C2×C42—21no—
186C2×C42C2×D411yescharacteristic
187C2×C16—21no—
188C2×C16—21no—
189C2×C16C2×D411yescharacteristic
190C2×C16C2×D411yescharacteristic
191C2×C16C2×D411yescharacteristic
192C2×C16C2×D411yescharacteristic
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—41no—
200#107—41no—
201#107—41no—
202#107—41no—
203#106—41no—
204#106—41no—
205#106—41no—
206#106—41no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—41no—
217C4⋊C8C2×D411yescharacteristic
218C4⋊C8C2×D411yescharacteristic
219#103—41no—
220#103—41no—
221#103—41no—
222#103—41no—
223#103—41no—
224#103—41no—
225#103—41no—
226#103—41no—
227#103—41no—
228#103—41no—
229C22⋊C8—41no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—41no—
235C4×C8C2×D411yescharacteristic
236C4×C8C2×D411yescharacteristic
237C4×C8#4111yescharacteristic
238C4×C8C2411yescharacteristic
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C2×D811yescharacteristic
247C2×Q8C2×D811yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4#12111yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4#13511yescharacteristic
253Q16—41no—
254Q16—41no—
255Q16—81no—
256Q16—82no—
257Q16—82no—
258M16—82no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—81no—
269C2×C8#12111yescharacteristic
270C2×C8#12111yescharacteristic
271C2×C8C22×D411yescharacteristic
272C2×C8C2×#4111yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—81no—
287C4⋊C4—81no—
288C4⋊C4—81no—
289C4⋊C4—81no—
290C4⋊C4C2×D811yescharacteristic
291C4⋊C4C2×D811yescharacteristic
292C22⋊C4—81no—
293C22⋊C4—81no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—81no—
299C42—81no—
300C42#12111yescharacteristic
301C42C2×D811yescharacteristic
302C42#13511yescharacteristic
303C42C22×D411yescharacteristic
304C16—21no—
305C16—21no—
306C16—21no—
307C16—21no—
308C16—41no—
309C16—41no—
310C23#46611yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—42no—
315Q8—81no—
316Q8—81no—
317Q8—81no—
318Q8—81no—
319Q8—161no—
320Q8—161no—
321Q8#51012yes—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4#46611yescharacteristic
338C2×C4#46611yescharacteristic
339C2×C4#46611yescharacteristic
340C2×C4C2×#12111yescharacteristic
341C2×C4C22×D811yescharacteristic
342C2×C4C2×#13511yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—41no—
347C8—41no—
348C8—81no—
349C8—81no—
350C8—81no—
351C8—81no—
352C8#57811yescharacteristic
353C8#57811yescharacteristic
354C22—21no—
355C22—21no—
356C22—21no—
357C22C2×#46611yescharacteristic
358C4—21no—
359C4—22no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—161no—
367C4—161no—
368C4—161no—
369C4—161no—
370C4#304411yescharacteristic
371C4#306911yescharacteristic
372C4C2×#51011yescharacteristic
373C4#341411yescharacteristic
374C2—41no—
375C2#1940411yescharacteristic
376C2#3360611yescharacteristic
377C2#3361711yescharacteristic
subgroupG/H
center, socleC22C2×#466
commutator, Frattini, cosocleC4×C8C24
prime radicalC23#466
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #12482C2semidirect
6. #12481C2semidirect
7. #12478C2semidirect
8. #12477C2semidirect
9. #12462C2semidirect
10. #12461C2semidirect
11. #12459C2semidirect
12. #12458C2semidirect

Maximal subgroups

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

subgroupindex
#337972
#334982
#192622
#191182
#124822
#124812
#124782
#124772
#124622
#124612
#124592
#124582
#120362
#119762
#119532