Finite Group Databasebeta

← #315688

Subgroups — #315688

Order 512.

789 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33042C211yesmaximal, characteristic
2#27973C211yesmaximal, characteristic
3#27973C211yesmaximal, characteristic
4#27972C211yesmaximal, characteristic, complemented
5#27948C211yesmaximal, characteristic, complemented
6C2×#1845C211yesmaximal, characteristic, complemented
7#10679C212yesmaximal, complemented
8#10672C212yesmaximal, complemented
9#9830C212yesmaximal, complemented
10#8821C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic, complemented
12C2×#486C2211yescharacteristic
13C22×#107—22no—
14C22×C4×C8C2211yescharacteristic
15#2857C2211yescharacteristic
16#2856C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19#2084—22no—
20#2072—22no—
21#2071—22no—
22#2071—22no—
23#1990C2211yescharacteristic
24#1990C2212yes—
25#1957C2211yescharacteristic
26#1957C2212yes—
27#1957C2212yes—
28#1956C2212yes—
29#1934—22no—
30#1893—22no—
31#1874—22no—
32#1874—22no—
33#1872—22no—
34#1853—22no—
35#1845—22no—
36#1845C2212yes—
37#1845C2212yes—
38#1844C2212yes—
39#1834C2211yescharacteristic
40#1834C2212yes—
41#1834C2212yes—
42#1834C2212yes—
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C2×#322C2212yes—
46#1273C2212yes—
47C23×C8D412yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×#127C2311yescharacteristic
50C2×C22⋊Q8—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C42C2311yescharacteristic
61#486—24no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#107—48no—
65C2×#106—24no—
66C2×#103—24no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—24no—
70C2×#102—24no—
71C2×C22⋊C8D412yes—
72C2×C22⋊C8D412yes—
73C2×C4×C8—24no—
74C2×C4×C8D412yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#391—21no—
80#390—21no—
81#390—21no—
82#390—21no—
83#389—21no—
84#389—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#384C2312yes—
94#384C2312yes—
95#380—21no—
96#380—21no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—22no—
102#340—22no—
103#336D412yes—
104#336D412yes—
105#322—24no—
106C22×Q8—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4—41no—
111C23×C4—41no—
112C23×C4—41no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C22×C8—24no—
117C22×C8—24no—
118C22×C8—24no—
119C22×C8C2×D414yes—
120C22×C8C2×D414yes—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125#127—22no—
126#127—22no—
127C22⋊Q8—44no—
128#121—44no—
129#121—44no—
130#121—44no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—41no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2411yescharacteristic
168#107—48no—
169#107—48no—
170#106—84no—
171#103—84no—
172#102—44no—
173#102—44no—
174#102—84no—
175#102—84no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C4×C8—24no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8#4114yes—
185#95—41no—
186#95—41no—
187C24—41no—
188C24#12711yescharacteristic
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—82no—
223C22×C4#12711yescharacteristic
224C22×C4#12711yescharacteristic
225C22×C4#12711yescharacteristic
226C22×C4#13514yes—
227C22×C4AGL(1,ℤ/8)14yes—
228C22×C4#13714yes—
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232C2×C8—216no—
233C2×C8—22no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—24no—
243C2×C8#12314yes—
244C2×C8#13714yes—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C22⋊C4—82no—
268C22⋊C4—82no—
269C22⋊C4—82no—
270C22⋊C4—82no—
271C42—21no—
272C42—21no—
273C42—22no—
274C42—22no—
275C42C2×#4112yes—
276C422−1+412yes—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—81no—
282C23—81no—
283C23—81no—
284C23—81no—
285C23—81no—
286C23—81no—
287C23—81no—
288C23#49512yes—
289C23#49612yes—
290C23#49712yes—
291C23C2×#12711yescharacteristic
292Q8—84no—
293D4—84no—
294D4—84no—
295D4—82no—
296D4—82no—
297D4—164no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—28no—
305C2×C4—28no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—164no—
324C2×C4—164no—
325C2×C4#49512yes—
326C2×C4#49612yes—
327C2×C4#49712yes—
328C2×C4C2×#12311yescharacteristic
329C2×C4#53611yescharacteristic
330C2×C4#54712yes—
331C2×C4#54712yes—
332C2×C4C2×#13512yes—
333C2×C4C2×AGL(1,ℤ/8)12yes—
334C2×C4C2×#13712yes—
335C2×C4C2×#13712yes—
336C2×C4#57612yes—
337C2×C4#57612yes—
338C8—24no—
339C8—24no—
340C8—44no—
341C8—44no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—161no—
349C22—161no—
350C22—161no—
351C22—161no—
352C22—161no—
353C22—161no—
354C22—161no—
355C22#172014yes—
356C22#250111yescharacteristic
357C22C2×#49511yescharacteristic
358C22C2×#49611yescharacteristic
359C22C2×#49711yescharacteristic
360C22#315211yescharacteristic
361C22#315511yescharacteristic
362C22#315511yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—161no—
367C4—161no—
368C4—161no—
369C4—162no—
370C4—162no—
371C4#309212yes—
372C4#323812yes—
373C4#333914yes—
374C4#334614yes—
375C2—24no—
376C2—321no—
377C2C2×#172011yescharacteristic
378C2#1963811yescharacteristic
379C2#1968311yescharacteristic
380C2#1969811yescharacteristic
381C2#2791411yescharacteristic
382C2#2793211yescharacteristic
383C2#2793811yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #27972C2semidirect
5. #27948C2semidirect
6. C2×#1845C2semidirect
7. #10679C2semidirect
8. #10672C2semidirect
9. #9830C2semidirect
10. #8821C2semidirect
11. #8145C2semidirect

Maximal subgroups

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

subgroupindex
#330422
#279732
#279722
#279482
C2×#18452
#106792
#106722
#98302
#88212
#81452