Finite Group Databasebeta

← #24915516

Subgroups — #24915516

Order 10624.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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×#14581782C211yesmaximal, characteristic, complemented
2C2×#14581779C211yesmaximal, characteristic
3#19127637C212yesmaximal, complemented
4#19127629C212yesmaximal, complemented
5#19127624C212yesmaximal, complemented
6#19127624C212yesmaximal, complemented
7#19127622C211yesmaximal, characteristic, complemented
8#19127621C211yesmaximal, characteristic, complemented
9#19127615C211yesmaximal, characteristic
10#19127565C211yesmaximal, characteristic, complemented
11#394×C83C211yesmaximal, characteristic
12C22×Dic166C2211yescharacteristic
13#14581782—24no—
14#14581779—24no—
15C2×C4⋊Dic83C2211yescharacteristic
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2211yescharacteristic
18C2×Dic83⋊C4C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C2×Dic83⋊C4C2212yes—
22C2×Dic83⋊C4C2212yes—
23C2×C22⋊Dic83C2212yes—
24C2×C22⋊Dic83C2212yes—
25C2×C22⋊Dic83C2212yes—
26C2×C22⋊Dic83C2212yes—
27C2×C4×Dic83C2211yescharacteristic
28C83⋊#95C2211yescharacteristic
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2212yes—
32C22×D4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34C2×C22⋊C4×C83C2212yes—
35C2×C22⋊C4×C83C2212yes—
36#95×C83C2211yescharacteristic
37C2×Dic166—22no—
38C2×Dic166—42no—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2311yescharacteristic
42C22×Dic83C2311yescharacteristic
43C22×Dic83C2311yescharacteristic
44C22×Dic83C2311yescharacteristic
45C4⋊Dic83—22no—
46C4⋊Dic83—42no—
47Dic83⋊C4—22no—
48Dic83⋊C4—22no—
49Dic83⋊C4—22no—
50Dic83⋊C4—22no—
51Dic83⋊C4—42no—
52Dic83⋊C4—42no—
53C22⋊Dic83—24no—
54C22⋊Dic83—24no—
55C22⋊Dic83—42no—
56C22⋊Dic83—42no—
57C4×Dic83D414yes—
58C24×C83C2312yes—
59C2×D4×C83—22no—
60C2×D4×C83—42no—
61C22×C4×C83C2312yes—
62C22×C4×C83C2311yescharacteristic
63C22×C4×C83C2311yescharacteristic
64C22×C4×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C22⋊C4×C83—42no—
67C22⋊C4×C83—42no—
68C23×C83—22no—
69C23×C83—22no—
70C23×C83—22no—
71C23×C83—22no—
72C23×C83—22no—
73C23×C83—22no—
74C23×C83—22no—
75C23×C83C2411yescharacteristic
76D4×C83—44no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—21no—
85C2×C4×C83—22no—
86C2×C4×C83C2×D412yes—
87C2×Dic83—21no—
88C2×Dic83—22no—
89C2×Dic83—21no—
90C2×Dic83—22no—
91C2×Dic83—21no—
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—22no—
95C2×Dic83—22no—
96C2×Dic83—22no—
97C2×Dic83—22no—
98C2×Dic83—22no—
99C2×Dic83—21no—
100C2×Dic83—22no—
101C2×Dic83C2×D412yes—
102C2×Dic83C2×D412yes—
103Dic166—44no—
104C22×C83—42no—
105C22×C83—42no—
106C22×C83—42no—
107C22×C83—42no—
108C22×C83—42no—
109C22×C83—42no—
110C22×C83—42no—
111C22×C83C22×D411yescharacteristic
112C22×C83H(5,𝔽2)12yes—
113C22×C83H(5,𝔽2)11yescharacteristic
114C22×C832−1+412yes—
115C22×C832−1+411yescharacteristic
116C4×C83—22no—
117C4×C83—41no—
118C4×C83—41no—
119C4×C83—42no—
120Dic83—22no—
121Dic83—22no—
122Dic83—42no—
123Dic83—41no—
124Dic83—41no—
125Dic83—42no—
126C2×C83—82no—
127C2×C83#53511yescharacteristic
128C2×C83#53611yescharacteristic
129C2×C83#53711yescharacteristic
130C2×C83#56212yes—
131C2×C83#56312yes—
132#2820—831nomaximal, Sylow
133C83#282011yescharacteristic, complemented, Sylow
134C2×#125—831no—
135C2×#124—831no—
136#399—832no—
137#397—832no—
138#396—832no—
139#396—832no—
140#395—831no—
141#395—831no—
142#394—831no—
143#394D8311yescharacteristic
144#382—831no—
145C22×Q8—831no—
146C22×D4C2×D8311yescharacteristic
147#125—1664no—
148#124—1664no—
149C2×C4⋊C4—832no—
150C2×C4⋊C4—832no—
151C2×C4⋊C4—832no—
152C2×C4⋊C4—831no—
153C2×C4⋊C4—831no—
154C2×C4⋊C4—831no—
155C2×C4⋊C4—831no—
156C2×C4⋊C4—831no—
157C2×C4⋊C4C2×D8311yescharacteristic
158C2×C22⋊C4—832no—
159C2×C22⋊C4—832no—
160C2×C22⋊C4—832no—
161C2×C22⋊C4—832no—
162C2×C22⋊C4C2×D8312yes—
163C2×C22⋊C4C2×D8312yes—
164C2×C42—831no—
165#95—831no—
166#95—832no—
167#95—832no—
168#95—832no—
169#95C2×D8311yescharacteristic
170C24C22×D8312yes—
171C2×Q8—3322no—
172C2×Q8—1662no—
173C2×D4—22no—
174C2×D4—42no—
175C22×C4—831no—
176C22×C4—831no—
177C22×C4—832no—
178C22×C4—831no—
179C22×C4—832no—
180C22×C4—831no—
181C22×C4C22×D8312yes—
182C22×C4C22×D8311yescharacteristic
183C22×C4C22×D8311yescharacteristic
184C22×C4C22×D8311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—3322no—
187C4⋊C4—3322no—
188C4⋊C4—3322no—
189C4⋊C4—1662no—
190C4⋊C4—1662no—
191C4⋊C4—1662no—
192C4⋊C4—1662no—
193C4⋊C4—1662no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—3322no—
197C22⋊C4—3322no—
198C22⋊C4—1664no—
199C22⋊C4—1664no—
200C42—834no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23C23×D8311yescharacteristic
209Q8—3324no—
210D4—44no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—832no—
221C2×C4—832no—
222C2×C4—1662no—
223C2×C4—1661no—
224C2×C4—1661no—
225C2×C4—1661no—
226C2×C4—1662no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4—1662no—
231C2×C4—1662no—
232C2×C4—1661no—
233C2×C4—1661no—
234C2×C4—1662no—
235C2×C4—1661no—
236C2×C4D4×D8312yes—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D8311yescharacteristic
245C22#1458183711yescharacteristic
246C22#1458183712yes—
247C22#1458183912yes—
248C22#1458183911yescharacteristic
249C4—22no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—3322no—
254C4—3322no—
255C4—3321no—
256C4—3321no—
257C4—1662no—
258C4—1662no—
259C2—82no—
260C2#1912826611yescharacteristic
261C2#1912827111yescharacteristic
262C2#1912828011yescharacteristic
263C2#1912844612yes—
264C2#1912845412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#394×C83C2
prime radicalC22×D4×C83C22
hypercenter#394D83
hypocenterC83#2820
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#14581782C2semidirect
3. #19127637C2semidirect
4. #19127629C2semidirect
5. #19127624C2semidirect
6. #19127624C2semidirect
7. #19127622C2semidirect
8. #19127621C2semidirect
10. #19127565C2semidirect
133. C83#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#145817822
C2×#145817792
#191276372
#191276292
#191276242
#191276222
#191276212
#191276152
#191275652
#394×C832
#282083