Finite Group Databasebeta

← #24919390

Subgroups — #24919390

Order 10624.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128417C211yesmaximal, characteristic, complemented
2#19128354C211yesmaximal, characteristic, complemented
3#19128041C212yesmaximal, complemented
4#19127988C211yesmaximal, characteristic, complemented
5#19127933C212yesmaximal, complemented
6#19127903C211yesmaximal, characteristic, complemented
7#19127849C212yesmaximal, complemented
8C4×D664C211yesmaximal, characteristic, complemented
9C8×D332C211yesmaximal, characteristic, complemented
10#19127696C212yesmaximal, complemented
11#487×C83C211yesmaximal, characteristic, complemented
12C2×#13050591C2212yes—
13C2×#13050577C2212yes—
14C2×D664C2211yescharacteristic
15C2×C8×D83C2212yes—
16#14581781—22no—
17#14581777—22no—
18#14581768C2212yes—
19#14581767—21no—
20#14581767—21no—
21#14581767C2212yes—
22C4×D332C2211yescharacteristic
23C4×D332C2211yescharacteristic
24C4×D332C2211yescharacteristic
25#14581750—21no—
26#14581750C2212yes—
27C4⋊C4×D83—21no—
28#14581726C2211yescharacteristic
29#14581723C2212yes—
30C4⋊(C83⋊C8)C2211yescharacteristic
31#14581716C2212yes—
32#14581714C2211yescharacteristic
33#14581714C2211yescharacteristic
34#14581712C2212yes—
35(C2×D83)⋊C8C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4⋊Q8×C83C2211yescharacteristic
38#103×C83C2212yes—
39#103×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41#13050591—24no—
42#13050591—42no—
43C2×D332—22no—
44C2×D332—22no—
45C2×D332D411yescharacteristic
46C2×D332C2311yescharacteristic
47C2×D332C2311yescharacteristic
48C2×C4×D83—21no—
49C2×C4×D83—21no—
50C2×C4×D83—22no—
51C2×C4×D83C2312yes—
52#13050577—42no—
53D664—22no—
54C2×C83⋊C8C2312yes—
55C8×D83—42no—
56C4⋊Dic83D411yescharacteristic
57C4⋊Dic83C2311yescharacteristic
58C4⋊Dic83C2311yescharacteristic
59Dic83⋊C4—22no—
60(C2×D83)⋊C4—22no—
61(C2×D83)⋊C4—22no—
62(C2×D83)⋊C4—21no—
63(C2×D83)⋊C4—21no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4D412yes—
66C4×Dic83—22no—
67C2×Q8×C83—21no—
68C2×Q8×C83C2312yes—
69C2×C8×C83C2311yescharacteristic
70C2×C8×C83C2311yescharacteristic
71C4⋊C4×C83—21no—
72C4⋊C4×C83—21no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83C2312yes—
75C42×C83C2311yescharacteristic
76C22×D83—21no—
77C22×D83—21no—
78C22×D83C2×D412yes—
79Q8×C83—22no—
80Q8×C83—22no—
81Q8×C83—81no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2411yescharacteristic
87C2×Dic83—21no—
88C2×Dic83—21no—
89C2×Dic83C2×D412yes—
90D332—22no—
91D332—21no—
92D332—21no—
93D332—21no—
94D332—44no—
95D332—82no—
96D332#4112yes—
97C4×D83—22no—
98C4×D83—22no—
99C4×D83—44no—
100C4×D83—42no—
101C4×D83—42no—
102C8×C83—21no—
103C8×C83—21no—
104C83⋊C8—22no—
105C22×C83C22×D411yescharacteristic
106C2×D83—22no—
107C2×D83—22no—
108C2×D83—22no—
109C2×D83—22no—
110C2×D83—41no—
111C2×D83—41no—
112C2×D83—41no—
113C2×D83—41no—
114C2×D83—41no—
115C4×C83—21no—
116C4×C83—42no—
117C4×C83—42no—
118C4×C83#13512yes—
119C4×C83C2×#4111yescharacteristic
120C4×C83H(5,𝔽2)11yescharacteristic
121Dic83—22no—
122Dic83—41no—
123Dic83—41no—
124C2×C83#54611yescharacteristic
125C2×C83C2×#13511yescharacteristic
126C2×C83#57711yescharacteristic
127D83—42no—
128D83—42no—
129D83—81no—
130#3303—831nomaximal, Sylow
131C83#330311yescharacteristic, complemented, Sylow
132#555—831no—
133#547—831no—
134#487D8311yescharacteristic, complemented
135#485—832no—
136#476—831no—
137#465—832no—
138#461—831no—
139#449—832no—
140C4×D8—831no—
141C8×D4—831no—
142#416—832no—
143C2×#41—832no—
144C2×SD16—832no—
145C2×D8—831no—
146C22×C8—832no—
147C4⋊Q8C2×D8311yescharacteristic
148C4⋊Q8C2×D8311yescharacteristic
149#124—1662no—
150#123—1662no—
151C22⋊Q8—832no—
152#121—832no—
153#121—1661no—
154#121—1661no—
155C4×D4—831no—
156C4×D4—831no—
157C4×D4—831no—
158#117—832no—
159#117—1661no—
160C2×C4⋊C4—1661no—
161#107—831no—
162#106—832no—
163C4⋊C8—831no—
164#103—832no—
165#103C2×D8312yes—
166#103C2×D8312yes—
167#102—831no—
168#102—831no—
169#102—832no—
170C22⋊C8—832no—
171C4×C8C2×D8311yescharacteristic
172#41—3322no—
173#41—1664no—
174C2×Q8—21no—
175C2×Q8C22×D8312yes—
176C2×D4—831no—
177C2×D4—831no—
178C2×D4—831no—
179C2×D4—1662no—
180C2×D4—1662no—
181C22×C4—832no—
182C22×C4—1662no—
183C22×C4—1661no—
184C22×C4—1661no—
185SD16—3322no—
186D8—1662no—
187C2×C8—3322no—
188C2×C8—832no—
189C2×C8C22×D8311yescharacteristic
190C2×C8C22×D8311yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—831no—
195C4⋊C4—831no—
196C4⋊C4—831no—
197C4⋊C4—1662no—
198C4⋊C4C22×D8312yes—
199C22⋊C4—832no—
200C22⋊C4—1662no—
201C22⋊C4—1662no—
202C22⋊C4—1662no—
203C22⋊C4—1661no—
204C22⋊C4—1661no—
205C42—1662no—
206C42C22×D8311yescharacteristic
207C23—832no—
208C23—1661no—
209C23—1661no—
210Q8—22no—
211Q8—22no—
212Q8—81no—
213D4—3324no—
214D4—832no—
215D4—6642no—
216D4—1662no—
217D4—1661no—
218D4—1661no—
219D4—1661no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—3324no—
223C2×C4—3322no—
224C2×C4—3322no—
225C2×C4—832no—
226C2×C4—1661no—
227C2×C4—1661no—
228C2×C4—1662no—
229C2×C4—1662no—
230C2×C4D4×D8311yescharacteristic
231C2×C4D4×D8311yescharacteristic
232C2×C4C23×D8311yescharacteristic
233C8—21no—
234C8—21no—
235C8—1662no—
236C22—3321no—
237C22—3321no—
238C22—3321no—
239C22—3321no—
240C22—3321no—
241C22—1662no—
242C22—1662no—
243C22—1662no—
244C22—1662no—
245C22C2×D4×D8311yescharacteristic
246C4—21no—
247C4—42no—
248C4—42no—
249C4—3321no—
250C4—3321no—
251C4—1662no—
252C4#1458181512yes—
253C4#41×D8311yescharacteristic
254C4#1458183811yescharacteristic
255C2—3322no—
256C2—3322no—
257C2—6641no—
258C2#1912834411yescharacteristic
259C2C2×#1458181511yescharacteristic
260C2#1912850711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#487×C83C2
socleC22×C83C22×D4
hypercenter#487D83
hypocenterC83#3303
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #19128417C2semidirect
2. #19128354C2semidirect
3. #19128041C2semidirect
4. #19127988C2semidirect
5. #19127933C2semidirect
6. #19127903C2semidirect
7. #19127849C2semidirect
8. C4×D664C2semidirect
9. C8×D332C2semidirect
10. #19127696C2semidirect
11. #487×C83C2semidirect
131. C83#3303semidirect
134. #487D83semidirect

Maximal subgroups

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

subgroupindex
#191284172
#191283542
#191280412
#191279882
#191279332
#191279032
#191278492
C4×D6642
C8×D3322
#191276962
#487×C832
#330383