Finite Group Databasebeta

← #35323090

Subgroups — #35323090

Order 21248.

420 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 103 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#24912668C211yesmaximal, characteristic
2C4×#14581711C211yesmaximal, characteristic
3#24905589C214yesmaximal
4C83×#1801C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic
6C2×#14581711C2212yes—
7C4×#13050574—21no—
8#19127621—21no—
9#19127621—21no—
10#19127608C2212yes—
11#19127599—21no—
12#19127599—21no—
13#19127570C412yes—
14C4×C4⋊Dic83C411yescharacteristic
15#19127470—21no—
16C2×C4×Q8×C83C2211yescharacteristic
17#423×C83—21no—
18C2×#101×C83C2212yes—
19#343×C83—21no—
20C22×Dic166—21no—
21C2×#13050574—22no—
22#14581792—41no—
23#14581792—41no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83C2×C412yes—
27C2×Dic83⋊C4—22no—
28C2×Dic83⋊C4—22no—
29C2×C4×Dic83C2×C412yes—
30#14581711—22no—
31#14581711Q814yescomplemented
32#14581703—41no—
33C4×C83⋊C8—41no—
34C83⋊#95—21no—
35C83⋊#95—22no—
36C83⋊#95—22no—
37C22×Q8×C83C2311yescharacteristic
38C2×M16×C83—22no—
39C4×Q8×C83—41no—
40C4×Q8×C83—41no—
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C42×C83C2×C411yescharacteristic
45C2×C42×C83D411yescharacteristic
46C2×C42×C83D411yescharacteristic
47C4⋊C8×C83—41no—
48C4⋊C8×C83—41no—
49#101×C83—24no—
50C2×Dic166—42no—
51C2×Dic166—82no—
52C22×Dic83—22no—
53C22×Dic83—22no—
54#13050574—24no—
55#13050574—42no—
56C2×C83⋊C8—42no—
57C4⋊Dic83—24no—
58C4⋊Dic83—41no—
59C4⋊Dic83—41no—
60C4⋊Dic83—42no—
61Dic83⋊C4—82no—
62Dic83⋊C4—82no—
63C4×Dic83—42no—
64C2×Q8×C83—41no—
65C2×Q8×C83—41no—
66C2×Q8×C83C2×Q812yes—
67C2×Q8×C83#4112yes—
68C22×C4×C83C22⋊C411yescharacteristic
69C22×C4×C83C22⋊C411yescharacteristic
70C22×C4×C83C22⋊C411yescharacteristic
71C22×C4×C83C22⋊C411yescharacteristic
72C22×C4×C83C22×C411yescharacteristic
73C22×C4×C83C2×D411yescharacteristic
74C22×C4×C83C2×D411yescharacteristic
75M16×C83—42no—
76M16×C83—42no—
77C2×C8×C83—42no—
78C4⋊C4×C83—41no—
79C4⋊C4×C83—41no—
80C4⋊C4×C83—41no—
81C42×C83—21no—
82C42×C83—21no—
83C42×C83—21no—
84C42×C83—21no—
85C42×C83—21no—
86C42×C83—21no—
87C23×C83C2×C22⋊C411yescharacteristic
88Q8×C83—42no—
89Q8×C83—42no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—41no—
101C2×C4×C83—41no—
102C2×C4×C83—41no—
103C2×C4×C83#9911yescharacteristic
104C2×C4×C83#9911yescharacteristic
105C2×C4×C83C4×Q812yes—
106C2×C4×C83C22⋊Q812yes—
107C2×C4×C83#12411yescharacteristic
108C2×C4×C83C4⋊Q811yescharacteristic
109C2×Dic83—24no—
110C2×Dic83—42no—
111C2×Dic83—42no—
112C2×Dic83—42no—
113C2×Dic83—42no—
114C2×Dic83—42no—
115Dic166—84no—
116C8×C83—82no—
117C83⋊C8—44no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83#39111yescharacteristic
121C22×C83C2×#9911yescharacteristic
122C22×C83#41011yescharacteristic
123C4×C83—21no—
124C4×C83—22no—
125C4×C83—21no—
126C4×C83—22no—
127C4×C83—21no—
128C4×C83—41no—
129C4×C83—41no—
130C4×C83—41no—
131C4×C83#35612yes—
132Dic83—44no—
133Dic83—82no—
134#13302—831nomaximal, Sylow
135C2×C83—21no—
136C2×C83—21no—
137C2×C83#199611yescharacteristic
138C2×C83C2×#35611yescharacteristic
139C2×C83#213411yescharacteristic
140#2567—831no—
141#1801D8311yescharacteristic
142C4×#101—831no—
143#1394—834no—
144C83#1330211yescharacteristic, complemented, Sylow
145C2×C4⋊Q8—831no—
146C2×C4×Q8C2×D8311yescharacteristic
147#423—21no—
148C2×#101—832no—
149C2×#101C2×D8312yes—
150C4×M16—1661no—
151#395—1661no—
152#395—1661no—
153#391—832no—
154#389—1661no—
155#389—1661no—
156#384—832no—
157C4×C4⋊C4—831no—
158#343—21no—
159#343—1661no—
160C22×Q8—1661no—
161C22×Q8C22×D8311yescharacteristic
162C2×M16—22no—
163C2×M16—1662no—
164C4⋊Q8—3321no—
165C4⋊Q8—3321no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—832no—
169C2×C4⋊C4—1661no—
170C2×C4⋊C4—1662no—
171C2×C4⋊C4—1662no—
172C2×C4⋊C4—1661no—
173C2×C4⋊C4C4×D8311yescharacteristic
174C2×C4⋊C4D33211yescharacteristic
175C2×C4⋊C4C22⋊D8311yescharacteristic
176C2×C42—832no—
177C2×C42C4×D8311yescharacteristic
178C2×C42D33211yescharacteristic
179C2×C42C22⋊D8311yescharacteristic
180C4⋊C8—41no—
181C4⋊C8—41no—
182#101—24no—
183#101—834no—
184#101—1662no—
185#97—3321no—
186C4×C8—3321no—
187#95—1661no—
188#95—1662no—
189#95—1662no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—3322no—
193C2×Q8—6642no—
194C2×Q8Q8×D8312yes—
195C2×Q8#1305058912yes—
196C22×C4—1662no—
197C22×C4—1662no—
198C22×C4(C2×D83)⋊C411yescharacteristic
199C22×C4(C2×D83)⋊C411yescharacteristic
200C22×C4(C2×D83)⋊C411yescharacteristic
201C22×C4(C2×D83)⋊C411yescharacteristic
202C22×C4C2×C4×D8311yescharacteristic
203C22×C4C2×C22⋊D8311yescharacteristic
204C22×C4C2×D33211yescharacteristic
205M16—42no—
206M16—42no—
207M16—3322no—
208M16—1664no—
209C2×C8—42no—
210C2×C8—3322no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—3322no—
215C4⋊C4—3321no—
216C4⋊C4—3321no—
217C4⋊C4—6642no—
218C4⋊C4—6642no—
219C4⋊C4—1664no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—3322no—
227C23C2×(C2×D83)⋊C411yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—6644no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—3322no—
245C2×C4—3322no—
246C2×C4—3322no—
247C2×C4—3322no—
248C2×C4—3322no—
249C2×C4—1664no—
250C2×C4#1458170611yescharacteristic
251C2×C4#1458170611yescharacteristic
252C2×C4#1458175712yes—
253C2×C4#1458177212yes—
254C2×C4#1458177911yescharacteristic
255C2×C4#1458179011yescharacteristic
256C8—82no—
257C8—3324no—
258C22—21no—
259C22—21no—
260C22#1912760611yescharacteristic
261C22C2×#1458170611yescharacteristic
262C22#1912767411yescharacteristic
263C4—21no—
264C4—21no—
265C4—22no—
266C4—21no—
267C4—22no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—3324no—
272C4—6642no—
273C4#1912749912yes—
274C2—21no—
275C2—21no—
276C2#2490844711yescharacteristic
277C2C2×#1912749911yescharacteristic
278C2#2490915411yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC22×Q8C22×D83
FittingC83×#1801C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC22×Q8×C83C23
hypercenter#1801D83
hypocenterC83#13302
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
31. #14581711Q8semidirect
144. C83#13302semidirect

Maximal subgroups

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

subgroupindex
#249126682
C4×#145817112
#249055892
C83×#18012
#1330283