Finite Group Databasebeta

← #35293232

Subgroups — #35293232

Order 21248.

468 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#24908791C212yesmaximal
2#24908562C211yesmaximal, characteristic
3#24905289C211yesmaximal, characteristic
4#24905286C212yesmaximal
5C83×#2734C211yesmaximal, characteristic
6C22×#13050580—21no—
7C2×#14581792C2211yescharacteristic
8C2×#14581725—22no—
9C2×#14581715—21no—
10#19127463—22no—
11#19127456C2211yescharacteristic
12#19127456C2212yes—
13C2×C4⋊Q8×C83—21no—
14#398×C83—21no—
15#395×C83—22no—
16#391×C83C2211yescharacteristic
17#389×C83C2212yes—
18#387×C83—22no—
19C22×Dic166—21no—
20C2×#13050580—42no—
21C2×#13050580—42no—
22C2×#13050580—42no—
23C22×C83⋊C8—21no—
24C22×C83⋊C8—22no—
25#14581792—42no—
26C2×C4⋊Dic83—22no—
27#14581725—44no—
28#14581715—42no—
29C22×Q8×C83—21no—
30C22×Q8×C83D411yescharacteristic
31C4⋊Q8×C83—42no—
32C4⋊Q8×C83—42no—
33C2×C4⋊C4×C83—22no—
34C2×C4⋊C4×C83—22no—
35C2×C4⋊C4×C83—21no—
36C2×C4⋊C4×C83D412yes—
37C2×C4⋊C4×C83D411yescharacteristic
38C2×C4⋊C4×C83D412yes—
39C2×C42×C83—21no—
40C2×C42×C83C2311yescharacteristic
41#95×C83—21no—
42#95×C83—22no—
43#95×C83—22no—
44#95×C83—22no—
45C2×Dic166—22no—
46C2×Dic166—41no—
47C2×Dic166—41no—
48C2×Dic166—41no—
49C2×Dic166—41no—
50C2×Dic166—41no—
51C22×Dic83—41no—
52#13050580—48no—
53C2×C83⋊C8—24no—
54C2×C83⋊C8—22no—
55C2×C83⋊C8—41no—
56C2×C83⋊C8—42no—
57C2×C83⋊C8—41no—
58C2×C83⋊C8—42no—
59C4⋊Dic83—24no—
60C4⋊Dic83—42no—
61C2×Q8×C83—21no—
62C2×Q8×C83—42no—
63C2×Q8×C83—41no—
64C2×Q8×C83—41no—
65C2×Q8×C83—81no—
66C2×Q8×C83—81no—
67C2×Q8×C83D812yes—
68C22×C4×C83—22no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—22no—
72C22×C4×C83C2×D411yescharacteristic
73C22×C4×C83C2×D412yes—
74C4⋊C4×C83—22no—
75C4⋊C4×C83—41no—
76C4⋊C4×C83—42no—
77C4⋊C4×C83—44no—
78C4⋊C4×C83—42no—
79C4⋊C4×C83—42no—
80C4⋊C4×C83—81no—
81C4⋊C4×C83Q1614yescomplemented
82C42×C83—41no—
83C42×C83—42no—
84C23×C83#12011yescharacteristic
85Q8×C83—42no—
86Q8×C83—81no—
87Q8×C83—84no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—24no—
94C2×C4×C83—42no—
95C2×C4×C83—41no—
96C2×C4×C83—42no—
97C2×C4×C83—41no—
98C2×C4×C83—42no—
99C2×C4×C83—41no—
100C2×C4×C83—42no—
101C2×C4×C83—42no—
102C2×C4×C83—42no—
103C2×C4×C83C2×D811yescharacteristic
104C2×C4×C83C2×Q1612yes—
105C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
106C2×C4×C83#13712yes—
107C2×Dic83—42no—
108C2×Dic83—81no—
109C2×Dic83—81no—
110Dic166—42no—
111Dic166—42no—
112Dic166—81no—
113Dic166—81no—
114C83⋊C8—44no—
115C83⋊C8—42no—
116C22×C83#44711yescharacteristic
117C22×C83#45112yes—
118C22×C83#45311yescharacteristic
119C22×C83#45612yes—
120C22×C83B(4,𝔽2)11yescharacteristic
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—22no—
124C4×C83—22no—
125C4×C83—42no—
126C4×C83—44no—
127C4×C83—82no—
128C4×C83—81no—
129Dic83—82no—
130#7757—831nomaximal, Sylow
131C2×C83#159511yescharacteristic
132C2×C83#160512yes—
133C2×C83#163311yescharacteristic
134C2×C83#164912yes—
135C2×C83#166011yescharacteristic
136#2734D8311yescharacteristic
137#2057—832no—
138#2016—831no—
139#1272—831no—
140#1271—832no—
141C83#775711yescharacteristic, complemented, Sylow
142C22×Q16—1661no—
143C2×C4⋊Q8—21no—
144C2×C4⋊Q8—831no—
145C2×#107—1662no—
146C2×#103—1661no—
147#398—21no—
148#395—22no—
149#391C2×D8311yescharacteristic
150#389C2×D8312yes—
151#387—22no—
152#340—1662no—
153#336—832no—
154#336—831no—
155C22×Q8—21no—
156C22×Q8—1661no—
157C22×Q8C22⋊D8311yescharacteristic
158C2×Q16—3322no—
159C2×Q16—3322no—
160C2×Q16—3322no—
161C22×C8—1661no—
162C22×C8—1662no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—3322no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—1662no—
170C2×C4⋊C4C22⋊D8312yes—
171C2×C4⋊C4C22⋊D8312yes—
172C2×C4⋊C4C22⋊D8311yescharacteristic
173C2×C42—21no—
174C2×C42C22×D8311yescharacteristic
175#107—3324no—
176#103—3322no—
177#95—21no—
178#95—22no—
179#95—22no—
180#95—22no—
181C2×Q8—21no—
182C2×Q8—42no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—81no—
186C2×Q8—81no—
187C2×Q8—3321no—
188C2×Q8—3321no—
189C2×Q8—3321no—
190C2×Q8—3321no—
191C2×Q8—3321no—
192C2×Q8—1662no—
193C2×Q8#1305057512yes—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—3321no—
199C22×C4C2×C22⋊D8311yescharacteristic
200C22×C4C2×C22⋊D8312yes—
201Q16—3328no—
202C2×C8—3322no—
203C2×C8—3321no—
204C2×C8—3321no—
205C2×C8—3322no—
206C2×C8—1662no—
207C2×C8—1664no—
208C4⋊C4—22no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—81no—
215C4⋊C4—3322no—
216C4⋊C4—1664no—
217C4⋊C4#1305058014yescomplemented
218C42—41no—
219C42—42no—
220C23#1458176111yescharacteristic
221Q8—42no—
222Q8—81no—
223Q8—84no—
224Q8—3322no—
225Q8—3322no—
226Q8—6641no—
227Q8—6641no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—3322no—
244C2×C4—6641no—
245C2×C4—6641no—
246C2×C4C2×#1305057511yescharacteristic
247C2×C4C2×#1305058012yes—
248C2×C4#1458181811yescharacteristic
249C2×C4#1458182312yes—
250C8—3322no—
251C8—3324no—
252C22#1912783511yescharacteristic
253C22#1912785712yes—
254C22#1912786911yescharacteristic
255C22#1912788412yes—
256C22#1912788811yescharacteristic
257C4—21no—
258C4—21no—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—44no—
263C4—81no—
264C4—82no—
265C4—6642no—
266C2#2490638111yescharacteristic
267C2#2490643712yes—
268C2#2490660311yescharacteristic
269C2#2490671212yes—
270C2#2490677411yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2734C2
socle, prime radicalC23×C83#120
hypercenter#2734D83
hypocenterC83#7757
LCSG ▷ C2×C42×C83 ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
81. C4⋊C4×C83Q16semidirect
141. C83#7757semidirect
217. C4⋊C4#13050580semidirect

Maximal subgroups

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

subgroupindex
#249087912
#249085622
#249052892
#249052862
C83×#27342
#775783