Finite Group Databasebeta

← #35560992

Subgroups — #35560992

Order 21248.

712 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128135C212yesmaximal, complemented
2#24919149C212yesmaximal, complemented
3#24909724C211yesmaximal, characteristic, complemented
4#24909577C212yesmaximal, complemented
5#24909518C212yesmaximal, complemented
6#24909404C212yesmaximal, complemented
7#24909376C212yesmaximal, complemented
8#24909296C211yesmaximal, characteristic, complemented
9C4×D16×C83C211yesmaximal, characteristic, complemented
10C2×#14581811C2212yes—
11C2×D8×D83C2212yescomplemented
12#19128346—22no—
13#19128135—28no—
14C2×Dic664C2211yescharacteristic, complemented
15C2×#14581737C2212yescomplemented
16C2×#14581736C2212yescomplemented
17C2×D1328C2211yescharacteristic
18C2×#14581733C2212yes—
19C2×#14581731C2212yescomplemented
20#19128085C2212yescomplemented
21#19127943C2212yescomplemented
22#19127938C2212yes—
23#19127916—22no—
24#19127853—22no—
25#19127847—22no—
26C8×D332C2211yescharacteristic
27C4⋊(C83⋊C16)C2211yescharacteristic, complemented
28#19127504C2212yescomplemented
29#19127501C2212yescomplemented
30(C2×D83)⋊C16C2212yescomplemented
31C2×D16×C83C2211yescharacteristic
32C4×D8×C83C2212yescomplemented
33#366×C83C2211yescharacteristic, complemented
34#357×C83C2212yescomplemented
35C4×C16×C83C2211yescharacteristic, complemented
36C2×#13050589—22no—
37C2×D4×D83—22no—
38#14581811—24no—
39#14581811—42no—
40C2×Dic332C2312yes—
41C2×#13050579—22no—
42C2×#13050578—22no—
43C2×D664C2312yes—
44C2×#13050575—22no—
45D8×D83—24no—
46D8×D83—42no—
47C2×C8×D83C2312yes—
48#14581780—22no—
49#14581776—42no—
50#14581772—22no—
51#14581767—22no—
52#14581766—42no—
53#14581762—42no—
54C4×D332D411yescharacteristic
55C2×(C2×D83)⋊C4—42no—
56Dic664—41no—
57#14581737—24no—
58#14581736—24no—
59D1328—41no—
60#14581733—24no—
61#14581731—24no—
62C2×C83⋊C16C2312yes—
63C4⋊(C83⋊C8)D411yescharacteristic
64#14581715—22no—
65#14581712—22no—
66(C2×D83)⋊C8D412yes—
67C2×D8×C83C2312yes—
68C4×D4×C83—22no—
69D16×C83D414yescomplemented
70C2×C16×C83C2311yescharacteristic
71C2×C16×C83C2311yescharacteristic
72#107×C83C2312yes—
73#102×C83—22no—
74C4×C8×C83C2311yescharacteristic
75C23×D83—42no—
76#13050589—44no—
77#13050589—82no—
78C2×Dic166—22no—
79C2×D332—22no—
80C2×D332—42no—
81C2×D332D811yescharacteristic
82C2×C22⋊D83—42no—
83C2×C22⋊D83—42no—
84D4×D83—44no—
85D4×D83—82no—
86C22×Dic83—42no—
87C2×C4×D83C2×D412yes—
88Dic332—22no—
89Dic332—22no—
90#13050579—82no—
91#13050578—44no—
92D664—22no—
93D664—22no—
94#13050575—44no—
95C2×C83⋊C8C2×D412yes—
96C8×D83—22no—
97C8×D83—22no—
98C4⋊Dic83—42no—
99C4⋊Dic83D811yescharacteristic
100(C2×D83)⋊C4—42no—
101(C2×D83)⋊C4—42no—
102(C2×D83)⋊C4—42no—
103(C2×D83)⋊C4—42no—
104(C2×D83)⋊C4D812yes—
105C83⋊C16—22no—
106C2×D4×C83—22no—
107C22×C4×C83—42no—
108D8×C83—22no—
109D8×C83C2×D414yes—
110C2×C8×C83C2×D411yescharacteristic
111C2×C8×C83C2411yescharacteristic
112C4⋊C4×C83—22no—
113C22⋊C4×C83—42no—
114C42×C83C2×D411yescharacteristic
115C16×C83—21no—
116C16×C83C2×D412yes—
117C23×C83—42no—
118C22×D83—42no—
119C22×D83—84no—
120C22×D83—84no—
121C22×D83—162no—
122C22×D83C2×D812yes—
123D4×C83—24no—
124D4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—162no—
127C2×C4×C83C2×D811yescharacteristic
128C2×C4×C83C2×D811yescharacteristic
129C2×C4×C83C22×D411yescharacteristic
130C22⋊D83—84no—
131C22⋊D83—84no—
132C2×Dic83—42no—
133C2×Dic83—84no—
134C2×Dic83C2×D812yes—
135D332—41no—
136D332—42no—
137D332—42no—
138D332—162no—
139C4×D83—22no—
140C4×D83—22no—
141Dic166—42no—
142Dic166—42no—
143C8×C83—21no—
144C8×C83C22×D411yescharacteristic
145C8×C83H(5,𝔽2)11yescharacteristic
146C83⋊C8—22no—
147C22×C83—44no—
148C22×C83—82no—
149C22×C83—82no—
150C22×C83C22×D811yescharacteristic
151C2×D83—22no—
152C2×D83—22no—
153C2×D83—22no—
154C2×D83—82no—
155C2×D83—82no—
156C2×D83—82no—
157C2×D83—168no—
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—82no—
161C4×C83D4211yescharacteristic
162C4×C83#57611yescharacteristic
163Dic83—22no—
164Dic83—82no—
165#33839—831nomaximal, Sylow
166C2×C83—84no—
167C2×C83D4×D811yescharacteristic
168C2×C83#341411yescharacteristic
169C2×C83#341511yescharacteristic
170D83—42no—
171D83—162no—
172C2×#509—832no—
173#3280—832no—
174#2249—831no—
175#2220—832no—
176#2210—832no—
177#2189—832no—
178#2184—832no—
179C4×D16D8311yescharacteristic, complemented
180#2169—831no—
181C83#3383911yescharacteristic, complemented, Sylow
182C2×#135—832no—
183C22×D8—832no—
184#546—1662no—
185#509—1668no—
186C2×Q32—831no—
187C2×SD32—832no—
188C2×SD32—832no—
189C2×D16—832no—
190C2×D16—831no—
191C2×D16C2×D8311yescharacteristic
192C2×M32—832no—
193#495—832no—
194#467—832no—
195#466—832no—
196#463—1662no—
197#450—1662no—
198#449—1662no—
199C4×D8C2×D8312yescomplemented
200C8×D4—831no—
201#366C2×D8311yescharacteristic, complemented
202C4⋊C16—831no—
203#358—832no—
204#357—832no—
205#357C2×D8312yescomplemented
206C22⋊C16—832no—
207C4×C16C2×D8311yescharacteristic, complemented
208C2×#41—1662no—
209C22×D4—1662no—
210#135—3322no—
211#135—1664no—
212C2×Q16—832no—
213C2×SD16—1662no—
214C2×SD16—1662no—
215C2×D8—3322no—
216C2×D8—832no—
217C2×D8—1662no—
218C2×D8—1664no—
219C2×D8C22×D8312yes—
220C22×C8—832no—
221#124—1662no—
222#123—3322no—
223C22⋊Q8—1662no—
224#121—3322no—
225#121—1662no—
226#120—3322no—
227C4×D4—22no—
228C4×D4—831no—
229C2×C22⋊C4—3322no—
230Q32—3321no—
231SD32—1664no—
232SD32—1664no—
233D16—3321no—
234D16—1664no—
235D16D33214yescomplemented
236M32—1664no—
237C2×C16—832no—
238C2×C16C22×D8311yescharacteristic
239C2×C16C22×D8311yescharacteristic
240#107C22×D8312yes—
241C4⋊C8—831no—
242#103—1662no—
243#102—22no—
244#102—1662no—
245C22⋊C8—832no—
246C4×C8C22×D8311yescharacteristic
247C24—3322no—
248#41—3324no—
249#41—6642no—
250C2×Q8—1662no—
251C2×D4—22no—
252C2×D4—3322no—
253C2×D4—3322no—
254C2×D4—3322no—
255C2×D4—3324no—
256C2×D4—831no—
257C2×D4—6642no—
258C2×D4—1662no—
259C22×C4—42no—
260C22×C4—3322no—
261C22×C4—832no—
262Q16—1662no—
263Q16—1662no—
264SD16—3324no—
265SD16—6642no—
266D8—22no—
267D8—3324no—
268D8—1662no—
269D8—1662no—
270D8C2×D33214yes—
271C2×C8—832no—
272C2×C8—1662no—
273C2×C8—1662no—
274C2×C8D4×D8311yescharacteristic
275C2×C8C23×D8311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—3322no—
278C4⋊C4—831no—
279C22⋊C4—42no—
280C22⋊C4—3322no—
281C22⋊C4—3322no—
282C22⋊C4—3322no—
283C22⋊C4—3322no—
284C22⋊C4—832no—
285C42D4×D8311yescharacteristic
286C16—21no—
287C16—1662no—
288C16C2×D33212yes—
289C23—42no—
290C23—13282no—
291C23—3322no—
292C23—832no—
293C23—6644no—
294C23—6644no—
295Q8—3322no—
296Q8—3322no—
297D4—24no—
298D4—42no—
299D4—13282no—
300D4—3321no—
301D4—3322no—
302D4—3322no—
303D4—6644no—
304D4—6644no—
305C2×C4—42no—
306C2×C4—162no—
307C2×C4—3322no—
308C2×C4—832no—
309C2×C4—6644no—
310C2×C4—1662no—
311C2×C4—1662no—
312C2×C4D8×D8311yescharacteristic
313C2×C4D8×D8311yescharacteristic
314C2×C4C2×D4×D8311yescharacteristic
315C8—21no—
316C8—1662no—
317C8C22×D33211yescharacteristic
318C8#1458183811yescharacteristic
319C22—44no—
320C22—82no—
321C22—82no—
322C22—13288no—
323C22—6642no—
324C22—6642no—
325C22—6642no—
326C22—1662no—
327C22—1662no—
328C22—1662no—
329C22C2×D8×D8311yescharacteristic
330C4—21no—
331C4—21no—
332C4—82no—
333C4—6642no—
334C4—1662no—
335C4D4×D33211yescharacteristic
336C4#1912850311yescharacteristic
337C2—84no—
338C2—13282no—
339C2—3322no—
340C2D8×D33211yescharacteristic
341C2#2492059611yescharacteristic
342C2#2492060011yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC4×D16×C83C2
socleC22×C83C22×D8
hypercenterC4×D16D83
hypocenterC83#33839
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×D16

Complements

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

NH ≅ G/Ntype
1. C2×#19128135C2semidirect
2. #24919149C2semidirect
3. #24909724C2semidirect
4. #24909577C2semidirect
5. #24909518C2semidirect
6. #24909404C2semidirect
7. #24909376C2semidirect
8. #24909296C2semidirect
9. C4×D16×C83C2semidirect
11. C2×D8×D83C22semidirect
14. C2×Dic664C22semidirect
15. C2×#14581737C22semidirect
16. C2×#14581736C22semidirect
19. C2×#14581731C22semidirect
20. #19128085C22semidirect
21. #19127943C22semidirect
27. C4⋊(C83⋊C16)C22semidirect
28. #19127504C22semidirect
29. #19127501C22semidirect
30. (C2×D83)⋊C16C22semidirect
32. C4×D8×C83C22semidirect
33. #366×C83C22semidirect
34. #357×C83C22semidirect
35. C4×C16×C83C22semidirect
69. D16×C83D4semidirect
179. C4×D16D83semidirect
181. C83#33839semidirect
199. C4×D8C2×D83semidirect
201. #366C2×D83semidirect
205. #357C2×D83semidirect
207. C4×C16C2×D83semidirect
235. D16D332semidirect

Maximal subgroups

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

subgroupindex
C2×#191281352
#249191492
#249097242
#249095772
#249095182
#249094042
#249093762
#249092962
C4×D16×C832
#3383983