Finite Group Databasebeta

← #24920271

Subgroups — #24920271

Order 10624.

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

normal subgroups only

This group has 326 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#19128426C211yesmaximal, characteristic
2#19128426C211yesmaximal, characteristic
3#19128068C212yesmaximal
4#19128010C212yesmaximal
5#19127983C211yesmaximal, characteristic
6#19127910C212yesmaximal
7#19127908C211yesmaximal, characteristic
8#19127831C211yesmaximal, characteristic
9#19127805C212yesmaximal
10C4×Dic332C211yesmaximal, characteristic
11#487×C83C211yesmaximal, characteristic
12C2×Dic332C2211yescharacteristic
13C2×#13050580C2212yes—
14#14581790—21no—
15#14581790C2212yes—
16#14581783—21no—
17#14581783C2212yes—
18#14581782—22no—
19#14581782—22no—
20Q8×Dic83—21no—
21Q8×Dic83C2212yes—
22C4×Dic166C2211yescharacteristic
23C4×Dic166C2211yescharacteristic
24C4×Dic166C2211yescharacteristic
25#14581757—21no—
26#14581757C2212yes—
27#14581726C2211yescharacteristic
28#14581723C2212yes—
29C4⋊(C83⋊C8)C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2211yescharacteristic
32#14581717C2211yescharacteristic
33#14581716C2212yes—
34#14581715C2212yes—
35#14581702C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4⋊Q8×C83C2211yescharacteristic
38#103×C83C2212yes—
39#103×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41C2×Dic166D411yescharacteristic
42C2×Dic166C2311yescharacteristic
43C2×Dic166C2311yescharacteristic
44Dic332—22no—
45#13050580—24no—
46C2×C83⋊C8C2312yes—
47C4⋊Dic83—22no—
48C4⋊Dic83—22no—
49C4⋊Dic83D411yescharacteristic
50C4⋊Dic83C2311yescharacteristic
51C4⋊Dic83C2311yescharacteristic
52Dic83⋊C4—21no—
53Dic83⋊C4—21no—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59C4×Dic83—21no—
60C4×Dic83—21no—
61C4×Dic83—22no—
62C4×Dic83—22no—
63C4×Dic83C2312yes—
64C2×Q8×C83—21no—
65C2×Q8×C83C2312yes—
66C2×C8×C83C2311yescharacteristic
67C2×C8×C83C2311yescharacteristic
68C4⋊C4×C83—21no—
69C4⋊C4×C83—21no—
70C4⋊C4×C83—21no—
71C4⋊C4×C83C2312yes—
72C42×C83C2311yescharacteristic
73Q8×C83—22no—
74Q8×C83—22no—
75Q8×C83—42no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2411yescharacteristic
81C2×Dic83—21no—
82C2×Dic83—21no—
83C2×Dic83—21no—
84C2×Dic83—21no—
85C2×Dic83C2×D412yes—
86C2×Dic83C2×D412yes—
87Dic166—22no—
88Dic166—21no—
89Dic166—21no—
90Dic166—21no—
91Dic166#4112yes—
92C8×C83—21no—
93C8×C83—21no—
94C83⋊C8—22no—
95C22×C83C22×D411yescharacteristic
96C4×C83—21no—
97C4×C83—42no—
98C4×C83—42no—
99C4×C83#13712yes—
100C4×C83C2×#4111yescharacteristic
101C4×C832−1+411yescharacteristic
102Dic83—22no—
103Dic83—22no—
104Dic83—22no—
105Dic83—41no—
106Dic83—41no—
107Dic83—41no—
108C2×C83#54811yescharacteristic
109C2×C83C2×#13711yescharacteristic
110C2×C83#57811yescharacteristic
111#3374—831nomaximal, Sylow
112C83#337411yescharacteristic, complemented, Sylow
113#557—831no—
114#557—831no—
115#491—832no—
116#487D8311yescharacteristic
117#479—832no—
118#475—831no—
119#462—831no—
120#462—832no—
121#446—831no—
122#441—832no—
123C4×Q16—831no—
124C2×Q16—832no—
125C2×Q16—831no—
126C4⋊Q8—832no—
127C4⋊Q8—1661no—
128C4⋊Q8C2×D8311yescharacteristic
129C4⋊Q8C2×D8311yescharacteristic
130#125—832no—
131#125—1662no—
132#125—1662no—
133#125—1661no—
134C4×Q8—831no—
135C4×Q8—832no—
136C4×Q8—832no—
137C4×Q8—831no—
138C4×Q8—831no—
139C4×Q8—1661no—
140C4×Q8—1661no—
141#107—831no—
142#106—832no—
143C4⋊C8—832no—
144C4⋊C8—831no—
145#103—832no—
146#103—831no—
147#103—831no—
148#103—832no—
149#103C2×D8312yes—
150#103C2×D8312yes—
151#97—832no—
152C4×C8C2×D8311yescharacteristic
153C2×Q8—21no—
154C2×Q8—831no—
155C2×Q8—831no—
156C2×Q8—831no—
157C2×Q8C22×D8312yes—
158Q16—1662no—
159Q16—1664no—
160C2×C8—832no—
161C2×C8C22×D8311yescharacteristic
162C2×C8C22×D8311yescharacteristic
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—831no—
167C4⋊C4—832no—
168C4⋊C4—831no—
169C4⋊C4—831no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1662no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4—1662no—
176C4⋊C4—1661no—
177C4⋊C4—1661no—
178C4⋊C4C22×D8312yes—
179C42—832no—
180C42—1662no—
181C42—1662no—
182C42—1661no—
183C42—1661no—
184C42C22×D8311yescharacteristic
185Q8—22no—
186Q8—22no—
187Q8—42no—
188Q8—832no—
189Q8—1661no—
190Q8—1661no—
191Q8—1662no—
192Q8—1661no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—832no—
196C2×C4—832no—
197C2×C4—1661no—
198C2×C4—1661no—
199C2×C4—1661no—
200C2×C4—1661no—
201C2×C4D4×D8311yescharacteristic
202C2×C4D4×D8311yescharacteristic
203C2×C4C23×D8311yescharacteristic
204C8—21no—
205C8—21no—
206C8—1662no—
207C22C2×D4×D8311yescharacteristic
208C4—21no—
209C4—42no—
210C4—42no—
211C4—3321no—
212C4—3321no—
213C4—3321no—
214C4—1662no—
215C4—1662no—
216C4—1662no—
217C4#1458182112yes—
218C4#41×D8311yescharacteristic
219C4#1458183911yescharacteristic
220C2#1912835911yescharacteristic
221C2C2×#1458182111yescharacteristic
222C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#487×C83C2
socle, prime radicalC22×C83C22×D4
hypercenter#487D83
hypocenterC83#3374
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
112. C83#3374semidirect

Maximal subgroups

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

subgroupindex
#191284262
#191280682
#191280102
#191279832
#191279102
#191279082
#191278312
#191278052
C4×Dic3322
#487×C832
#337483