Finite Group Databasebeta

← #24920447

Subgroups — #24920447

Order 10624.

360 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 123 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#19128424C211yesmaximal, characteristic, complemented
2#19128083C212yesmaximal
3#19128080C211yesmaximal, characteristic, complemented
4#19127923C212yesmaximal, complemented
5#19127920C212yesmaximal, complemented
6#19127906C211yesmaximal, characteristic, complemented
7#19127903C211yesmaximal, characteristic, complemented
8Q8×C83⋊C8C211yesmaximal, characteristic, complemented
9C4×#13050580C212yesmaximal, complemented
10C4×#13050575C211yesmaximal, characteristic
11#556×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050580C2212yes—
14C2×#13050578C2212yes—
15C2×#13050577C2212yes—
16C2×#13050575C2211yescharacteristic
17#14581792C2211yescharacteristic
18#14581787—22no—
19#14581780C2212yes—
20#14581768—21no—
21C4×Dic166C2212yes—
22C4×D332C2211yescharacteristic
23#14581750—21no—
24#14581725C2211yescharacteristic
25C4⋊(C83⋊C8)C2211yescharacteristic
26C4⋊(C83⋊C8)C2212yes—
27#14581716C2212yes—
28#14581715C2212yes—
29#14581713C2211yescharacteristic
30#14581712C2211yescharacteristic
31C4×C83⋊C8C2211yescharacteristic
32C4×C83⋊C8C2212yes—
33C4⋊Q8×C83C2211yescharacteristic
34#126×C83—22no—
35#124×C83C2212yes—
36C22⋊Q8×C83—21no—
37C4×Q8×C83C2212yes—
38C4×Q8×C83C2211yescharacteristic
39C4×D4×C83C2211yescharacteristic
40#117×C83—21no—
41C2×Dic166C2312yes—
42C2×D332C2311yescharacteristic
43C2×C4×D83—21no—
44#13050580—24no—
45#13050580—42no—
46#13050578—42no—
47#13050577—42no—
48#13050575—22no—
49C2×C83⋊C8C2312yes—
50C2×C83⋊C8C2311yescharacteristic
51C2×C83⋊C8C2311yescharacteristic
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83C2311yescharacteristic
55Dic83⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4—21no—
59C4×Dic83—22no—
60C2×Q8×C83D411yescharacteristic
61C2×Q8×C83C2312yes—
62C2×D4×C83C2311yescharacteristic
63C22×C4×C83—21no—
64C4⋊C4×C83—21no—
65C4⋊C4×C83—22no—
66C4⋊C4×C83—21no—
67C4⋊C4×C83D412yes—
68C4⋊C4×C83D411yescharacteristic
69C4⋊C4×C83C2311yescharacteristic
70C22⋊C4×C83—22no—
71C22⋊C4×C83—21no—
72C22⋊C4×C83—22no—
73C42×C83—22no—
74C42×C83C2312yes—
75C42×C83C2311yescharacteristic
76C23×C83—21no—
77C22×D83—21no—
78Q8×C83—22no—
79Q8×C83—22no—
80Q8×C83—22no—
81D4×C83—21no—
82D4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—42no—
86C2×C4×C83C2×D412yes—
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D412yes—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83C2411yescharacteristic
91C2×Dic83—21no—
92C2×Dic83—22no—
93D332—21no—
94D332—21no—
95C4×D83—42no—
96Dic166—22no—
97Dic166—22no—
98C83⋊C8—22no—
99C83⋊C8—21no—
100C83⋊C8#4112yes—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83—41no—
104C22×C83C22×D411yescharacteristic
105C2×D83—41no—
106C2×D83—41no—
107C2×D83—41no—
108C4×C83—22no—
109C4×C83—22no—
110C4×C83—21no—
111C4×C83—42no—
112C4×C83—41no—
113C4×C83#13512yes—
114C4×C83C2×#4111yescharacteristic
115C4×C83H(5,𝔽2)11yescharacteristic
116Dic83—42no—
117Dic83—41no—
118C2×C83—81no—
119C2×C83#55011yescharacteristic
120C2×C83C2×#13511yescharacteristic
121C2×C83#57811yescharacteristic
122D83—81no—
123#3393—831nomaximal, Sylow
124C83#339311yescharacteristic, complemented, Sylow
125#556—831no—
126#556D8311yescharacteristic, complemented
127#495—832no—
128#494—831no—
129#464—832no—
130#464—832no—
131#461—831no—
132#461—831no—
133C8×Q8—831no—
134C4×Q16—832no—
135C4×D8—831no—
136C2×Q16—832no—
137C2×Q16—832no—
138C2×SD16—832no—
139C2×SD16—832no—
140C2×D8—831no—
141C4⋊Q8—831no—
142C4⋊Q8C2×D8311yescharacteristic
143#126—22no—
144#126—1662no—
145#124—832no—
146#124C2×D8312yes—
147C22⋊Q8—21no—
148C22⋊Q8—1661no—
149C4×Q8—832no—
150C4×Q8C2×D8312yes—
151C4×Q8C2×D8311yescharacteristic
152C4×D4—831no—
153C4×D4C2×D8311yescharacteristic
154#117—21no—
155#117—1661no—
156#107—831no—
157C4⋊C8—831no—
158C4⋊C8—832no—
159#103—832no—
160#103—832no—
161#102—831no—
162#102—831no—
163C4×C8—831no—
164C4×C8—832no—
165C2×Q8—832no—
166C2×Q8C22⋊D8311yescharacteristic
167C2×Q8C22×D8312yes—
168C2×D4—831no—
169C2×D4C22×D8311yescharacteristic
170C22×C4—21no—
171C22×C4—1661no—
172Q16—3322no—
173Q16—1664no—
174SD16—3322no—
175SD16—3322no—
176D8—1662no—
177C2×C8—831no—
178C2×C8—831no—
179C2×C8—832no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—831no—
184C4⋊C4—1661no—
185C4⋊C4—1662no—
186C4⋊C4—1661no—
187C4⋊C4C22⋊D8311yescharacteristic
188C4⋊C4C22⋊D8312yes—
189C4⋊C4C22×D8311yescharacteristic
190C22⋊C4—22no—
191C22⋊C4—21no—
192C22⋊C4—22no—
193C22⋊C4—1662no—
194C22⋊C4—1661no—
195C22⋊C4—1662no—
196C42—22no—
197C42—1662no—
198C42C22×D8311yescharacteristic
199C42C22×D8312yes—
200C23—21no—
201C23—1661no—
202Q8—22no—
203Q8—22no—
204Q8—22no—
205Q8—1662no—
206Q8—1662no—
207D4—21no—
208D4—21no—
209D4—1661no—
210D4—1661no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—3322no—
215C2×C4—1661no—
216C2×C4—1662no—
217C2×C4C2×C22⋊D8312yes—
218C2×C4C2×C22⋊D8311yescharacteristic
219C2×C4C2×C22⋊D8312yes—
220C2×C4C2×C22⋊D8311yescharacteristic
221C2×C4C23×D8311yescharacteristic
222C8—832no—
223C8—1662no—
224C8—1661no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—3321no—
229C22—3321no—
230C22—3321no—
231C22C22×C22⋊D8311yescharacteristic
232C4—22no—
233C4—22no—
234C4—21no—
235C4—42no—
236C4—41no—
237C4—3322no—
238C4—3321no—
239C4#1458181212yes—
240C4#41×D8311yescharacteristic
241C4#1458183811yescharacteristic
242C2—81no—
243C2—6641no—
244C2#1912837011yescharacteristic
245C2C2×#1458181211yescharacteristic
246C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#556×C83C2
socleC22×C83C22×D4
prime radicalC2×#13050575C22
hypercenter#556D83
hypocenterC83#3393
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #19128424C2semidirect
3. #19128080C2semidirect
4. #19127923C2semidirect
5. #19127920C2semidirect
6. #19127906C2semidirect
7. #19127903C2semidirect
8. Q8×C83⋊C8C2semidirect
9. C4×#13050580C2semidirect
11. #556×C83C2semidirect
124. C83#3393semidirect
126. #556D83semidirect

Maximal subgroups

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

subgroupindex
#191284242
#191280832
#191280802
#191279232
#191279202
#191279062
#191279032
Q8×C83⋊C82
C4×#130505802
C4×#130505752
#556×C832
#339383