Finite Group Databasebeta

← #497669

Subgroups — #497669

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3391C211yesmaximal, characteristic
2C2×#2183C211yesmaximal, characteristic, complemented
3#22727C211yesmaximal, characteristic, complemented
4#13252C211yesmaximal, characteristic, complemented
5#13252C212yesmaximal, complemented
6#13206C212yesmaximal, complemented
7#13202C211yesmaximal, characteristic, complemented
8#12813C212yesmaximal, complemented
9#12811C211yesmaximal, characteristic, complemented
10#12637C212yesmaximal, complemented
11C4×#357C211yesmaximal, characteristic, complemented
12C2×#550—21no—
13#3391—24no—
14#3391C418yescomplemented
15C2×#493C2211yescharacteristic
16C2×#493C2212yes—
17C2×#459—21no—
18C2×#459—22no—
19C2×C8×Q8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2212yes—
22#2355—21no—
23#2183—24no—
24C2×C4⋊C16C2211yescharacteristic
25C2×C4⋊C16C2212yes—
26C2×#357C2211yescharacteristic
27C2×#357C2211yescharacteristic
28C2×#357C2212yes—
29C2×#357C2212yes—
30C2×#357C2212yes—
31C2×#357C2212yes—
32C2×#357C2212yes—
33C2×#346C2212yes—
34C2×C4×C16C2211yescharacteristic
35#1949—22no—
36#1948—21no—
37#1846C2212yes—
38#1845C2211yescharacteristic
39C4×#107C2211yescharacteristic
40C4×#107C2212yes—
41#1380C2212yes—
42#1380C2211yescharacteristic
43#1380C2211yescharacteristic
44C22×#41—41no—
45C22×D8—22no—
46C22×D8—21no—
47C22×D8C2312yes—
48C22×D8C2311yescharacteristic
49#550—28no—
50#550—44no—
51C2×#127—22no—
52C2×#127—21no—
53C2×#121—41no—
54C2×#121—42no—
55C2×C4×Q8D411yescharacteristic
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C22×C16C2311yescharacteristic
59C22×C16C2312yes—
60C22×C16C2311yescharacteristic
61#493—22no—
62#493—24no—
63#493C2×C414yes—
64#493C2×C418yes—
65#459—28no—
66#459—216no—
67C8×Q8C2×C414yes—
68C8×Q8D414yes—
69C4×D8—22no—
70C4×D8—24no—
71C4×D8C2×C414yes—
72C4×D8C2×C418yes—
73C2×#107—21no—
74C2×#107C2312yes—
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#106—22no—
79C2×C4⋊C8D411yescharacteristic
80C2×C4⋊C8D412yes—
81C2×#102—22no—
82C2×#102—21no—
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2312yes—
85#391—41no—
86#387—42no—
87#384—21no—
88#382—22no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—41no—
92C4⋊C16—22no—
93C4⋊C16—24no—
94#357—22no—
95#357—24no—
96#357—22no—
97#357—24no—
98#357—42no—
99#357—42no—
100#357—42no—
101#346—24no—
102C4×C16—22no—
103#340—22no—
104#340—21no—
105C2×#41—48no—
106C2×#41—84no—
107C2×#41—84no—
108C22×Q8D811yescharacteristic
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—42no—
112C22×D4—41no—
113C22×D4—42no—
114C22×D4—41no—
115C23×C4—42no—
116C23×C4—41no—
117C2×D8—24no—
118C2×D8—28no—
119C2×D8—22no—
120C2×D8—21no—
121C2×D8—22no—
122C2×D8—21no—
123C2×D8—22no—
124C2×D8—21no—
125C2×D8—22no—
126C2×D8—21no—
127C2×D8—22no—
128C2×D8—21no—
129C2×D8—82no—
130C2×D8—81no—
131C2×D8—82no—
132C2×D8—81no—
133C2×D8C22×C414yes—
134C2×D8C22×C412yes—
135C22×C8C2×D412yes—
136C22×C8C2×D411yescharacteristic
137C22×C8C2411yescharacteristic
138#127—28no—
139#127—24no—
140#127—44no—
141#127—42no—
142#121—416no—
143#121—48no—
144C4×Q8C22⋊C414yes—
145C4×Q8SD1614yes—
146C4×D4—28no—
147C4×D4—24no—
148C4×D4—44no—
149C4×D4—42no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4D811yescharacteristic
157C2×C4⋊C4D812yes—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—41no—
160C2×C42—41no—
161C2×C42—42no—
162C2×C42C2×D412yes—
163C2×C42C2×D411yescharacteristic
164C2×C16—21no—
165C2×C16—21no—
166C2×C16—21no—
167C2×C16—22no—
168C2×C16—21no—
169C2×C16—22no—
170C2×C16—21no—
171C2×C16—22no—
172C2×C16#4112yescomplemented
173#107—21no—
174#107—21no—
175#107—22no—
176#107—21no—
177#107—22no—
178#107—21no—
179#107—81no—
180#107C22×C412yes—
181#107#4112yescomplemented
182#106—82no—
183C4⋊C8C22⋊C414yes—
184C4⋊C8C22⋊C418yes—
185#102—24no—
186#102—28no—
187C4×C8C22×C412yes—
188C4×C8C22×C414yes—
189C4×C8C2×D412yes—
190C4×C8C2×D414yes—
191#95—41no—
192#95—42no—
193#95—42no—
194#95—42no—
195C24—42no—
196C24—41no—
197#41—832no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8#10214yes—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—48no—
214C2×D4—44no—
215C2×D4—48no—
216C2×D4—44no—
217C2×D4—84no—
218C2×D4—84no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4—162no—
222C2×D4—161no—
223C2×D4—162no—
224C2×D4—161no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—48no—
228C22×C4—44no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—84no—
232C22×C4—84no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4C2×D812yes—
236C22×C4C2×D812yes—
237C22×C4C2×D811yescharacteristic
238C22×C4C2×D811yescharacteristic
239C22×C4C22×D411yescharacteristic
240D8—24no—
241D8—22no—
242D8—24no—
243D8—22no—
244D8—42no—
245D8—41no—
246D8—42no—
247D8—41no—
248D8—88no—
249D8—84no—
250C2×C8—22no—
251C2×C8—21no—
252C2×C8—22no—
253C2×C8—21no—
254C2×C8C2×C22⋊C412yes—
255C2×C8C2×C22⋊C414yes—
256C2×C8C23×C411yescharacteristic
257C2×C8C22×D411yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C8H(5,𝔽2)11yescharacteristic
261C2×C82−1+411yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—82no—
271C4⋊C4—161no—
272C4⋊C4#10218yes—
273C4⋊C4#10214yes—
274C22⋊C4—48no—
275C22⋊C4—44no—
276C42—82no—
277C42—162no—
278C42C2×C22⋊C414yes—
279C42C2×C22⋊C412yes—
280C42C2×SD1614yes—
281C42C2×SD1612yes—
282C16—22no—
283C16—41no—
284C16—42no—
285C23—44no—
286C23—42no—
287C23—82no—
288C23—81no—
289C23—82no—
290C23—81no—
291C23—82no—
292C23—81no—
293C23—82no—
294C23—81no—
295C23—81no—
296C23—82no—
297C23—82no—
298C23—81no—
299C23C22×D811yescharacteristic
300Q8—28no—
301D4—44no—
302D4—42no—
303D4—42no—
304D4—44no—
305D4—816no—
306D4—81no—
307D4—82no—
308D4—82no—
309D4—81no—
310D4—88no—
311D4—168no—
312D4—164no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—816no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—88no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4C2×#10214yes—
334C2×C4C2×#10212yes—
335C2×C4C2×#10214yes—
336C2×C4C2×#10212yes—
337C2×C4#50912yes—
338C2×C4#50912yes—
339C2×C4C22×C22⋊C411yescharacteristic
340C2×C4#54811yescharacteristic
341C2×C4#55011yescharacteristic
342C2×C4C22×SD1611yescharacteristic
343C2×C4#57611yescharacteristic
344C2×C4#57711yescharacteristic
345C8—24no—
346C8—22no—
347C8C4×#4111yescharacteristic
348C8#52011yescharacteristic
349C8#54811yescharacteristic
350C8#55011yescharacteristic
351C22—84no—
352C22—82no—
353C22—84no—
354C22—82no—
355C22—84no—
356C22—82no—
357C22—162no—
358C22—161no—
359C22—162no—
360C22—161no—
361C22—162no—
362C22—161no—
363C22—162no—
364C22—161no—
365C22C22×#10211yescharacteristic
366C22#335811yescharacteristic
367C22#339111yescharacteristic
368C22C2×#50911yescharacteristic
369C22C2×#50911yescharacteristic
370C22#341511yescharacteristic
371C22#341611yescharacteristic
372C4—24no—
373C4—24no—
374C4—22no—
375C4—82no—
376C4—82no—
377C4—161no—
378C4—162no—
379C4#214014yes—
380C4#234111yescharacteristic
381C4#289711yescharacteristic
382C4#336211yescharacteristic
383C4#339211yescharacteristic
384C2—164no—
385C2—162no—
386C2#2254011yescharacteristic
387C2C2×#214011yescharacteristic
388C2#3338111yescharacteristic
389C2#3393311yescharacteristic
390C2#3393511yescharacteristic
391C2#3396411yescharacteristic
392C2#3396611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3391C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2183C2semidirect
3. #22727C2semidirect
4. #13252C2semidirect
5. #13252C2semidirect
6. #13206C2semidirect
7. #13202C2semidirect
8. #12813C2semidirect
9. #12811C2semidirect
10. #12637C2semidirect
11. C4×#357C2semidirect
14. #3391C4semidirect
172. C2×C16#41semidirect
181. #107#41semidirect

Maximal subgroups

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

subgroupindex
C2×#33912
C2×#21832
#227272
#132522
#132062
#132022
#128132
#128112
#126372
C4×#3572