Finite Group Databasebeta

← #24920494

Subgroups — #24920494

Order 10624.

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

normal subgroups only

This group has 350 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#19128430C211yesmaximal, characteristic, complemented
2#19128096C212yesmaximal
3#19128080C211yesmaximal, characteristic, complemented
4#19127920C212yesmaximal
5#19127911C212yesmaximal, complemented
6#19127908C211yesmaximal, characteristic, complemented
7#19127906C211yesmaximal, characteristic
8#19127832C211yesmaximal, characteristic, complemented
9#19127806C212yesmaximal, complemented
10C4×#13050578C211yesmaximal, characteristic
11#556×C83C211yesmaximal, characteristic
12C2×#13050580C2212yes—
13C2×#13050580C2212yes—
14C2×#13050580C2212yes—
15C2×#13050578C2211yescharacteristic
16C2×#13050578C2212yes—
17#14581792C2211yescharacteristic
18#14581792C2212yes—
19#14581783—21no—
20#14581783—22no—
21Q8×Dic83—21no—
22C4×Dic166C2211yescharacteristic
23C4×Dic166C2212yes—
24#14581723C2211yescharacteristic
25C4⋊(C83⋊C8)C2211yescharacteristic
26C4⋊(C83⋊C8)C2212yes—
27#14581716C2212yes—
28#14581715C2212yes—
29#14581715C2211yescharacteristic
30#14581713C2211yescharacteristic
31#14581703C2212yes—
32C4×C83⋊C8C2211yescharacteristic
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×Dic166C2311yescharacteristic
42C2×Dic166C2312yes—
43#13050580—24no—
44#13050580—42no—
45#13050580—42no—
46#13050578—22no—
47#13050578—42no—
48C2×C83⋊C8C2312yes—
49C2×C83⋊C8C2311yescharacteristic
50C2×C83⋊C8C2311yescharacteristic
51C4⋊Dic83—22no—
52C4⋊Dic83—21no—
53C4⋊Dic83—22no—
54C4⋊Dic83—21no—
55C4⋊Dic83C2311yescharacteristic
56Dic83⋊C4—22no—
57Dic83⋊C4—21no—
58C4×Dic83—22no—
59C4×Dic83—21no—
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×C83D411yescharacteristic
68C4⋊C4×C83D412yes—
69C4⋊C4×C83C2311yescharacteristic
70C22⋊C4×C83—22no—
71C22⋊C4×C83—21no—
72C22⋊C4×C83—22no—
73C42×C83—22no—
74C42×C83C2311yescharacteristic
75C42×C83C2312yes—
76C23×C83—21no—
77Q8×C83—22no—
78Q8×C83—22no—
79Q8×C83—22no—
80D4×C83—21no—
81D4×C83—21no—
82C2×C4×C83—21no—
83C2×C4×C83—22no—
84C2×C4×C83—42no—
85C2×C4×C83C2×D412yes—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83C2411yescharacteristic
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93Dic166—21no—
94Dic166—22no—
95Dic166—22no—
96Dic166—21no—
97C83⋊C8—22no—
98C83⋊C8—21no—
99C83⋊C8#4112yes—
100C22×C83—41no—
101C22×C83—41no—
102C22×C83—41no—
103C22×C83C22×D411yescharacteristic
104C4×C83—22no—
105C4×C83—22no—
106C4×C83—21no—
107C4×C83—42no—
108C4×C83—41no—
109C4×C83#13712yes—
110C4×C83C2×#4111yescharacteristic
111C4×C83H(5,𝔽2)11yescharacteristic
112Dic83—41no—
113Dic83—42no—
114Dic83—41no—
115C2×C83—81no—
116C2×C83#55011yescharacteristic
117C2×C83C2×#13711yescharacteristic
118C2×C83#57811yescharacteristic
119#3398—831nomaximal, Sylow
120C83#339811yescharacteristic, complemented, Sylow
121#557—831no—
122#556D8311yescharacteristic
123#497—832no—
124#494—831no—
125#464—832no—
126#462—831no—
127#462—832no—
128#461—831no—
129#446—831no—
130#441—832no—
131C4×SD16—831no—
132C2×Q16—832no—
133C2×Q16—832no—
134C2×Q16—832no—
135C2×SD16—832no—
136C2×SD16—831no—
137C4⋊Q8—831no—
138C4⋊Q8—832no—
139C4⋊Q8C2×D8311yescharacteristic
140#126—22no—
141#125—1661no—
142#125—1662no—
143#124C2×D8312yes—
144C22⋊Q8—21no—
145C4×Q8—831no—
146C4×Q8—832no—
147C4×Q8—1661no—
148C4×Q8C2×D8312yes—
149C4×Q8C2×D8311yescharacteristic
150C4×D4C2×D8311yescharacteristic
151#117—21no—
152#106—831no—
153C4⋊C8—831no—
154C4⋊C8—832no—
155#103—832no—
156#103—831no—
157#103—832no—
158#102—831no—
159#97—832no—
160C4×C8—831no—
161C2×Q8—831no—
162C2×Q8—832no—
163C2×Q8C22⋊D8311yescharacteristic
164C2×Q8C22×D8312yes—
165C2×D4C22×D8311yescharacteristic
166C22×C4—21no—
167Q16—3322no—
168Q16—3322no—
169Q16—1664no—
170SD16—3322no—
171SD16—1662no—
172C2×C8—831no—
173C2×C8—831no—
174C2×C8—832no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—831no—
179C4⋊C4—1662no—
180C4⋊C4—1661no—
181C4⋊C4—1662no—
182C4⋊C4—1661no—
183C4⋊C4—1662no—
184C4⋊C4—1661no—
185C4⋊C4C22⋊D8311yescharacteristic
186C4⋊C4C22⋊D8312yes—
187C4⋊C4C22×D8311yescharacteristic
188C22⋊C4—22no—
189C22⋊C4—21no—
190C22⋊C4—22no—
191C42—22no—
192C42—1662no—
193C42—1661no—
194C42C22×D8311yescharacteristic
195C42C22×D8312yes—
196C23—21no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—1661no—
201Q8—1662no—
202Q8—1662no—
203Q8—1661no—
204D4—21no—
205D4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1661no—
210C2×C4—1661no—
211C2×C4—1662no—
212C2×C4C2×C22⋊D8312yes—
213C2×C4C2×C22⋊D8312yes—
214C2×C4C2×C22⋊D8311yescharacteristic
215C2×C4C2×C22⋊D8311yescharacteristic
216C2×C4C23×D8311yescharacteristic
217C8—832no—
218C8—1662no—
219C8—1661no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22C22×C22⋊D8311yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—42no—
228C4—41no—
229C4—3321no—
230C4—3322no—
231C4—3321no—
232C4#1458182212yes—
233C4#41×D8311yescharacteristic
234C4#1458183811yescharacteristic
235C2—81no—
236C2#1912837011yescharacteristic
237C2C2×#1458182211yescharacteristic
238C2#1912851611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#556×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×C83C23
hypercenter#556D83
hypocenterC83#3398
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. #19128430C2semidirect
3. #19128080C2semidirect
5. #19127911C2semidirect
6. #19127908C2semidirect
8. #19127832C2semidirect
9. #19127806C2semidirect
120. C83#3398semidirect

Maximal subgroups

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

subgroupindex
#191284302
#191280962
#191280802
#191279202
#191279112
#191279082
#191279062
#191278322
#191278062
C4×#130505782
#556×C832
#339883