Finite Group Databasebeta

← #352702

Subgroups — #352702

Order 512.

829 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic, complemented
2#33305C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#33277C211yesmaximal, characteristic
5#30228C211yesmaximal, characteristic, complemented
6#30203C211yesmaximal, characteristic
7#9842C212yesmaximal
8#9839C212yesmaximal, complemented
9#9825C212yesmaximal, complemented
10#9820C212yesmaximal
11C8×#97C211yesmaximal, characteristic, complemented
12C2×#501—22no—
13C2×#497—22no—
14C2×#497—22no—
15C2×#494C2212yes—
16C2×#492C2212yes—
17#2868C2211yescharacteristic
18#2868C2211yescharacteristic
19#2859C2211yescharacteristic
20#2858C2211yescharacteristic
21#2099—22no—
22#2099—22no—
23#2071—22no—
24#2023—22no—
25#1990—22no—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1990C2211yescharacteristic
31#1990C2212yes—
32#1990C2211yescharacteristic
33#1957C2211yescharacteristic
34#1957C2212yes—
35#1957C2211yescharacteristic
36#1934—22no—
37#1934—22no—
38#1847—22no—
39#1845C2212yes—
40#1844C2212yes—
41#1834—22no—
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46C2×C82C2212yes—
47C22×Q16—24no—
48C22×SD16—24no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2312yes—
56C2×#127C2311yescharacteristic
57C2×#125—21no—
58C2×#124—21no—
59C2×#124—21no—
60#501—48no—
61#497—48no—
62#497—48no—
63#494—24no—
64#494—24no—
65#492—24no—
66#492—24no—
67C2×#107—24no—
68C2×#106—24no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—24no—
72C2×#97D412yes—
73C2×#97D412yes—
74C2×C4×C8D412yes—
75C2×C4×C8D412yes—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2312yes—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2312yes—
80#390—21no—
81#390—21no—
82#390—21no—
83#390—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#384C2312yes—
98#382—21no—
99C43C2311yescharacteristic
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—22no—
106#340—22no—
107#336D412yes—
108#336D412yes—
109#322—24no—
110C82—24no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—41no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C2×Q16—48no—
118C2×Q16—48no—
119C2×Q16—48no—
120C2×SD16—48no—
121C2×SD16—48no—
122C2×SD16—48no—
123C22×C8C2×D414yes—
124C22×C8C2×D412yes—
125C22×C8C2×D412yes—
126C22×C8C2×D414yes—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C4⋊Q8—82no—
138#127—22no—
139#127—22no—
140#125—44no—
141#124—44no—
142#124—44no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2411yescharacteristic
176#107—48no—
177#106—48no—
178#102—44no—
179#102—44no—
180#102—84no—
181#97D818yes—
182#97SD1618yes—
183C4×C8—24no—
184C4×C8—22no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—42no—
189C4×C8#4114yes—
190C4×C8#4114yes—
191#95—41no—
192#95—41no—
193C24—41no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—44no—
199C2×Q8—42no—
200C2×Q8—44no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×D4—42no—
205C2×D4—41no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4#12711yescharacteristic
220C22×C4#12711yescharacteristic
221C22×C4#12711yescharacteristic
222C22×C4#12711yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226Q16—816no—
227SD16—816no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—24no—
232C2×C8—22no—
233C2×C8—22no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8#12418yes—
237C2×C8C2×D818yes—
238C2×C8C2×SD1618yes—
239C2×C8#13714yes—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42C2×D814yes—
274C42C2×SD1614yes—
275C42C2×#4112yes—
276C42C2×#4112yes—
277C23—81no—
278C23—81no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23C2×#12711yescharacteristic
285Q8—84no—
286Q8—84no—
287Q8—84no—
288Q8—164no—
289D4—84no—
290D4—82no—
291D4—84no—
292D4—82no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—81no—
311C2×C4—82no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4#49212yes—
317C2×C4#49212yes—
318C2×C4#49312yes—
319C2×C4#49312yes—
320C2×C4C2×#12412yes—
321C2×C4#53211yescharacteristic
322C2×C4#53212yes—
323C2×C4#53211yescharacteristic
324C2×C4C22×D812yes—
325C2×C4C22×SD1612yes—
326C2×C4C2×#13712yes—
327C2×C4#57512yes—
328C2×C4#57712yes—
329C2×C4#57812yes—
330C8—24no—
331C8—42no—
332C8—44no—
333C8#486116yes—
334C22—161no—
335C22—161no—
336C22—161no—
337C22—161no—
338C22—161no—
339C22—161no—
340C22—161no—
341C22#285811yescharacteristic
342C22C2×#49211yescharacteristic
343C22C2×#49311yescharacteristic
344C22#315211yescharacteristic
345C22#315311yescharacteristic
346C22#315411yescharacteristic
347C22#315611yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—22no—
352C4—161no—
353C4—161no—
354C4—162no—
355C4—161no—
356C4—162no—
357C4C2×#48614yes—
358C4#317212yes—
359C4#317814yes—
360C4#318112yes—
361C2—321no—
362C2#1962211yescharacteristic
363C2#1962411yescharacteristic
364C2#1963011yescharacteristic
365C2#1963111yescharacteristic
366C2#3322411yescharacteristic
367C2#3324411yescharacteristic
368C2#3325011yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33316C2semidirect
3. #33289C2semidirect
5. #30228C2semidirect
8. #9839C2semidirect
9. #9825C2semidirect
11. C8×#97C2semidirect

Maximal subgroups

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

subgroupindex
#333162
#333052
#332892
#332772
#302282
#302032
#98422
#98392
#98252
#98202
C8×#972