Finite Group Databasebeta

← #24912440

Subgroups — #24912440

Order 10624.

576 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581781C211yesmaximal, characteristic
2C2×#14581781C211yesmaximal, characteristic
3C2×Q8×Dic83C211yesmaximal, characteristic, complemented
4#19127623C214yesmaximal, complemented
5#19127608C211yesmaximal, characteristic, complemented
6#19127603C211yesmaximal, characteristic
7#19127591C214yesmaximal, complemented
8C4×Dic83⋊C4C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11#14581781—24no—
12#14581781—24no—
13Q8×Dic83—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×(C2×D83)⋊C4C2212yes—
18C2×(C2×D83)⋊C4C2214yes—
19C2×(C2×D83)⋊C4C2214yes—
20C2×(C2×D83)⋊C4C2212yes—
21C2×C4×Dic83C2211yescharacteristic
22C2×C4×Dic83C2212yes—
23C2×C4×Dic83C2211yescharacteristic
24C83⋊#95C2212yes—
25C83⋊#95C2214yes—
26C22×Q8×C83C2211yescharacteristic
27C22×Q8×C83C2211yescharacteristic
28C4⋊Q8×C83—24no—
29C2×C4⋊C4×C83C2214yes—
30C2×C42×C83C2211yescharacteristic
31C23×D83C2312yes—
32C2×D332—22no—
33C2×D332—22no—
34C2×D332—22no—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2311yescharacteristic
38C22×Dic83C2311yescharacteristic
39C4⋊Dic83—22no—
40C4⋊Dic83—24no—
41Dic83⋊C4—24no—
42(C2×D83)⋊C4—28no—
43(C2×D83)⋊C4—28no—
44(C2×D83)⋊C4—24no—
45(C2×D83)⋊C4—24no—
46C4×Dic83—22no—
47C4×Dic83—24no—
48C4×Dic83D414yes—
49C2×Q8×C83—22no—
50C2×Q8×C83—22no—
51C2×Q8×C83—22no—
52C2×Q8×C83—22no—
53C2×Q8×C83—22no—
54C2×Q8×C83D414yes—
55C22×C4×C83C2312yes—
56C22×C4×C83C2312yes—
57C22×C4×C83C2312yes—
58C22×C4×C83C2311yescharacteristic
59C4⋊C4×C83—28no—
60C42×C83—22no—
61C23×C83C2411yescharacteristic
62C22×D83—22no—
63C22×D83—22no—
64C22×D83—24no—
65C22×D83—22no—
66C22×D83—24no—
67Q8×C83—28no—
68Q8×C83—44no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83—22no—
75C2×C4×C83—22no—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2×D414yes—
78C2×C4×C83C2×D412yes—
79C2×C4×C83#4112yes—
80C2×C4×C83#4114yes—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—24no—
86C2×Dic83—22no—
87C2×Dic83—22no—
88C2×Dic83C2×D412yes—
89C2×Dic83C2×D412yes—
90D332—44no—
91C22×C83C22×D411yescharacteristic
92C22×C83C22×D411yescharacteristic
93C22×C83C2×#4111yescharacteristic
94C22×C83C2×#4112yes—
95C22×C832−1+412yes—
96C2×D83—42no—
97C2×D83—42no—
98C2×D83—42no—
99C2×D83—44no—
100C2×D83—44no—
101C4×C83—24no—
102C4×C83—24no—
103C4×C83—42no—
104C4×C83#12114yes—
105Dic83—22no—
106Dic83—22no—
107Dic83—42no—
108Dic83—42no—
109C2×C83C2×#12111yescharacteristic
110C2×C83#53211yescharacteristic
111C2×C83#53611yescharacteristic
112C2×C83#54812yes—
113C2×C83#55612yes—
114D83—82no—
115#2545—831nomaximal, Sylow
116C83#254511yescharacteristic, complemented, Sylow
117C2×C4⋊Q8D8311yescharacteristic, complemented
118C2×#124—831no—
119C2×#124—831no—
120C2×C4×Q8—831no—
121#396—834no—
122#391—831no—
123#390—831no—
124#388—834no—
125C4×C4⋊C4—831no—
126C22×Q8C2×D8311yescharacteristic
127C22×Q8C2×D8311yescharacteristic
128C22×D4—831no—
129C4⋊Q8—24no—
130#124—1664no—
131#124—1664no—
132C4×Q8—1664no—
133C2×C4⋊C4—831no—
134C2×C4⋊C4—832no—
135C2×C4⋊C4—832no—
136C2×C4⋊C4C2×D8314yes—
137C2×C22⋊C4—832no—
138C2×C22⋊C4—834no—
139C2×C22⋊C4—832no—
140C2×C22⋊C4—834no—
141C2×C42—831no—
142C2×C42—832no—
143C2×C42—831no—
144C2×C42C2×D8311yescharacteristic
145#95—832no—
146#95—834no—
147C24—832no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8C22⋊D8314yes—
154C2×D4—1662no—
155C2×D4—1662no—
156C2×D4—1662no—
157C22×C4—831no—
158C22×C4—831no—
159C22×C4—832no—
160C22×C4—832no—
161C22×C4C22×D8312yes—
162C22×C4C22×D8312yes—
163C22×C4C22×D8312yes—
164C22×C4C22×D8311yescharacteristic
165C4⋊C4—28no—
166C4⋊C4—1664no—
167C4⋊C4—1662no—
168C4⋊C4—1664no—
169C22⋊C4—1664no—
170C22⋊C4—1664no—
171C22⋊C4—1668no—
172C22⋊C4—1668no—
173C42—22no—
174C42—834no—
175C42—1664no—
176C42—1662no—
177C23—1662no—
178C23—1662no—
179C23—1662no—
180C23—1664no—
181C23—1664no—
182C23C23×D8311yescharacteristic
183Q8—28no—
184Q8—44no—
185D4—3324no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—832no—
194C2×C4—832no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1664no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4—1662no—
202C2×C4D4×D8312yes—
203C2×C4C2×C22⋊D8314yes—
204C2×C4C2×C22⋊D8312yes—
205C2×C4#1305058912yes—
206C2×C4#1305059114yes—
207C22—3324no—
208C22—3324no—
209C22—3322no—
210C22—3322no—
211C22—3322no—
212C22C2×D4×D8311yescharacteristic
213C22C22×C22⋊D8311yescharacteristic
214C22C2×#1305058911yescharacteristic
215C22C2×#1305059112yes—
216C22#1458184012yes—
217C4—24no—
218C4—24no—
219C4—42no—
220C4—3322no—
221C4—3322no—
222C4—1662no—
223C4—1662no—
224C4#1458176414yes—
225C2—6642no—
226C2C2×#1458176411yescharacteristic
227C2#1912825211yescharacteristic
228C2#1912827611yescharacteristic
229C2#1912836212yes—
230C2#1912842512yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4⋊Q8×C83C2
prime radicalC22×D332C22
hypercenterC2×C4⋊Q8D83
hypocenterC83#2545
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×Q8×Dic83C2semidirect
4. #19127623C2semidirect
5. #19127608C2semidirect
7. #19127591C2semidirect
8. C4×Dic83⋊C4C2semidirect
9. C2×C4⋊Q8×C83C2semidirect
116. C83#2545semidirect
117. C2×C4⋊Q8D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817812
C2×Q8×Dic832
#191276232
#191276082
#191276032
#191275912
C4×Dic83⋊C42
C2×C4⋊Q8×C832
#254583