Finite Group Databasebeta

← #24912939

Subgroups — #24912939

Order 10624.

596 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 139 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×#14581767C211yesmaximal, characteristic
2C2×C4⋊C4×D83C211yesmaximal, characteristic, complemented
3#19127639C211yesmaximal, characteristic, complemented
4#19127619C211yesmaximal, characteristic, complemented
5#19127618C211yesmaximal, characteristic, complemented
6#19127593C212yesmaximal, complemented
7#19127590C212yesmaximal, complemented
8#19127579C212yesmaximal, complemented
9#19127579C212yesmaximal, complemented
10#19127548C211yesmaximal, characteristic, complemented
11#400×C83C211yesmaximal, characteristic, complemented
12C22×D332C2212yes—
13C22×D332C2211yescharacteristic
14C22×C4×D83C2211yescharacteristic
15C22×C4×D83C2211yescharacteristic
16C22×C4×D83C2211yescharacteristic
17#14581767—24no—
18C2×C4⋊Dic83C2211yescharacteristic
19C2×Dic83⋊C4C2211yescharacteristic
20C2×Dic83⋊C4C2211yescharacteristic
21C4⋊C4×D83—24no—
22C2×(C2×D83)⋊C4C2212yes—
23C2×(C2×D83)⋊C4C2212yes—
24C2×(C2×D83)⋊C4C2212yes—
25C2×(C2×D83)⋊C4C2212yes—
26C2×(C2×D83)⋊C4C2212yes—
27C2×(C2×D83)⋊C4C2212yes—
28C2×C4×Dic83C2212yes—
29C83⋊#95C2212yes—
30C83⋊#95C2212yes—
31C83⋊#95C2211yescharacteristic
32C2×C4⋊C4×C83C2211yescharacteristic
33C2×C4⋊C4×C83C2211yescharacteristic
34#95×C83C2212yes—
35#95×C83C2212yes—
36#95×C83C2211yescharacteristic
37C23×D83C2312yes—
38C23×D83C2311yescharacteristic
39C2×D332—24no—
40C2×D332—24no—
41C2×D332—24no—
42C2×D332—22no—
43C2×D332—42no—
44C22×Dic83C2311yescharacteristic
45C22×Dic83C2312yes—
46C22×Dic83C2311yescharacteristic
47C22×Dic83C2311yescharacteristic
48C2×C4×D83—22no—
49C2×C4×D83—22no—
50C2×C4×D83—22no—
51C2×C4×D83—22no—
52C2×C4×D83—42no—
53C2×C4×D83—42no—
54C2×C4×D83D414yes—
55C4⋊Dic83—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58(C2×D83)⋊C4—24no—
59(C2×D83)⋊C4—24no—
60(C2×D83)⋊C4—24no—
61(C2×D83)⋊C4—24no—
62(C2×D83)⋊C4—42no—
63(C2×D83)⋊C4—42no—
64C4×Dic83—42no—
65C22×C4×C83C2311yescharacteristic
66C22×C4×C83C2312yes—
67C22×C4×C83C2312yes—
68C22×C4×C83C2311yescharacteristic
69C22×C4×C83C2311yescharacteristic
70C4⋊C4×C83—22no—
71C4⋊C4×C83—22no—
72C23×C83C2411yescharacteristic
73C22×D83—21no—
74C22×D83—21no—
75C22×D83—22no—
76C22×D83—22no—
77C22×D83—22no—
78C22×D83—22no—
79C22×D83—22no—
80C22×D83—22no—
81C22×D83—22no—
82C22×D83—22no—
83C22×D83C2×D412yes—
84C22×D83#4114yes—
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83—22no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—22no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83C2×D412yes—
97C2×C4×C83#4112yes—
98C2×Dic83—21no—
99C2×Dic83—21no—
100C2×Dic83—22no—
101C2×Dic83—21no—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—22no—
105C2×Dic83—22no—
106C2×Dic83C2×D412yes—
107C2×Dic83#4112yes—
108D332—44no—
109D332—48no—
110C4×D83—28no—
111C4×D83—44no—
112C4×D83—44no—
113C22×C83C22×D411yescharacteristic
114C22×C83C2×#4111yescharacteristic
115C22×C83C2×#4112yes—
116C22×C83C2×#4111yescharacteristic
117C22×C83H(5,𝔽2)12yes—
118C2×D83—22no—
119C2×D83—22no—
120C2×D83—24no—
121C2×D83—24no—
122C2×D83—42no—
123C2×D83—42no—
124C2×D83—42no—
125C2×D83—42no—
126C2×D83—42no—
127C2×D83—42no—
128C2×D83—42no—
129C2×D83#12414yes—
130C4×C83—22no—
131C4×C83—22no—
132C4×C83—42no—
133C4×C83—41no—
134C4×C83—42no—
135Dic83—22no—
136Dic83—22no—
137Dic83—41no—
138Dic83—42no—
139C2×C83#52511yescharacteristic
140C2×C83C2×#12411yescharacteristic
141C2×C83#53411yescharacteristic
142C2×C83#55312yes—
143C2×C83#55312yes—
144D83—24no—
145D83—82no—
146#2591—831nomaximal, Sylow
147C83#259111yescharacteristic, complemented, Sylow
148C2×#121—831no—
149C22×C4⋊C4—831no—
150#400—831no—
151#400D8311yescharacteristic, complemented
152#394—831no—
153#394—831no—
154#388—832no—
155#388—832no—
156#386—832no—
157#386—832no—
158C4×C22⋊C4—831no—
159C22×D4—831no—
160C22×D4—832no—
161C23×C4—831no—
162C23×C4—831no—
163C23×C4—831no—
164#121—1664no—
165C2×C4⋊C4—831no—
166C2×C4⋊C4—831no—
167C2×C4⋊C4—831no—
168C2×C4⋊C4—1664no—
169C2×C4⋊C4C2×D8311yescharacteristic
170C2×C4⋊C4C2×D8311yescharacteristic
171C2×C22⋊C4—832no—
172C2×C22⋊C4—832no—
173C2×C22⋊C4—832no—
174C2×C22⋊C4—832no—
175C2×C22⋊C4—832no—
176C2×C22⋊C4—832no—
177C2×C42—832no—
178#95—831no—
179#95—832no—
180#95—832no—
181#95C2×D8311yescharacteristic
182#95C2×D8312yes—
183#95C2×D8312yes—
184C24—831no—
185C24—832no—
186C2×D4—3322no—
187C2×D4—1664no—
188C2×D4—1664no—
189C2×D4—1664no—
190C2×D4—1662no—
191C22×C4—3322no—
192C22×C4—3322no—
193C22×C4—834no—
194C22×C4—831no—
195C22×C4—831no—
196C22×C4—832no—
197C22×C4—831no—
198C22×C4—1662no—
199C22×C4—1662no—
200C22×C4—1662no—
201C22×C4—1662no—
202C22×C4C22×D8312yes—
203C22×C4C22×D8312yes—
204C22×C4C22×D8311yescharacteristic
205C22×C4C22×D8311yescharacteristic
206C22×C4C22×D8311yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—1662no—
210C4⋊C4—1662no—
211C4⋊C4—1662no—
212C22⋊C4—3322no—
213C22⋊C4—3322no—
214C22⋊C4—1664no—
215C22⋊C4—1664no—
216C22⋊C4—1664no—
217C22⋊C4—1664no—
218C42—3322no—
219C23—832no—
220C23—834no—
221C23—1662no—
222C23—1662no—
223C23—1662no—
224C23—1662no—
225C23—1662no—
226C23—1662no—
227C23—1662no—
228C23—1661no—
229C23—1661no—
230C23—1662no—
231C23C23×D8311yescharacteristic
232D4—3324no—
233D4—3328no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—3324no—
246C2×C4—3324no—
247C2×C4—832no—
248C2×C4—832no—
249C2×C4—1668no—
250C2×C4—1661no—
251C2×C4—1662no—
252C2×C4—1662no—
253C2×C4—1662no—
254C2×C4—1662no—
255C2×C4—1661no—
256C2×C4—1661no—
257C2×C4—1662no—
258C2×C4D4×D8312yes—
259C2×C4#1305059112yes—
260C22—3322no—
261C22—3322no—
262C22—3322no—
263C22—3322no—
264C22—3322no—
265C22—3322no—
266C22—3322no—
267C22—834no—
268C22—1664no—
269C22—1664no—
270C22—1662no—
271C22—1662no—
272C22C2×D4×D8311yescharacteristic
273C22#41×D8311yescharacteristic
274C22#41×D8312yes—
275C22C2×#1305059111yescharacteristic
276C22#1458183812yes—
277C4—22no—
278C4—22no—
279C4—42no—
280C4—42no—
281C4—41no—
282C4—3322no—
283C4—3321no—
284C4—1662no—
285C4—1662no—
286C2—6642no—
287C2—1664no—
288C2#1912821411yescharacteristic
289C2#124×D8311yescharacteristic
290C2#1912826211yescharacteristic
291C2#1912840212yes—
292C2#1912840412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#400×C83C2
prime radicalC2×#14581767C2
hypercenter#400D83
hypocenterC83#2591
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D83C2semidirect
3. #19127639C2semidirect
4. #19127619C2semidirect
5. #19127618C2semidirect
6. #19127593C2semidirect
7. #19127590C2semidirect
8. #19127579C2semidirect
9. #19127579C2semidirect
10. #19127548C2semidirect
11. #400×C83C2semidirect
147. C83#2591semidirect
151. #400D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817672
C2×C4⋊C4×D832
#191276392
#191276192
#191276182
#191275932
#191275902
#191275792
#191275482
#400×C832
#259183