Finite Group Databasebeta

← #24920399

Subgroups — #24920399

Order 10624.

326 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128431C211yesmaximal, characteristic
2#19128430C211yesmaximal, characteristic
3#19128109C212yesmaximal
4#19128097C211yesmaximal, characteristic
5#19128007C212yesmaximal
6#19127987C211yesmaximal, characteristic
7#19127984C212yesmaximal
8#19127828C211yesmaximal, characteristic
9#19127805C212yesmaximal
10C4×#13050580C211yesmaximal, characteristic
11#477×C83C211yesmaximal, characteristic
12C2×#13050580C2211yescharacteristic
13#14581792—21no—
14#14581792C2211yescharacteristic
15#14581791—21no—
16#14581791C2212yes—
17#14581790—22no—
18#14581783—22no—
19#14581783C2212yes—
20Q8×Dic83C2212yes—
21C4×Dic166—21no—
22C4×Dic166C2211yescharacteristic
23C4×Dic166C2211yescharacteristic
24#14581757—21no—
25#14581757C2212yes—
26#14581725C2211yescharacteristic
27#14581724C2212yes—
28#14581723C2212yes—
29#14581723C2212yes—
30Dic83⋊C8C2212yes—
31#14581717C2212yes—
32#14581716C2211yescharacteristic
33#14581715C2211yescharacteristic
34#14581702C2212yes—
35C4×C83⋊C8C2211yescharacteristic
36C4⋊Q8×C83C2211yescharacteristic
37C4×Q8×C83C2211yescharacteristic
38#107×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41C2×Dic166—21no—
42C2×Dic166—22no—
43C2×Dic166D411yescharacteristic
44C2×Dic166C2311yescharacteristic
45#13050580Q814yescomplemented
46C2×C83⋊C8C2311yescharacteristic
47C2×C83⋊C8C2311yescharacteristic
48C4⋊Dic83—21no—
49C4⋊Dic83—22no—
50C4⋊Dic83—22no—
51C4⋊Dic83D411yescharacteristic
52C4⋊Dic83C2312yes—
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—21no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59C4×Dic83—21no—
60C4×Dic83—22no—
61C4×Dic83—22no—
62C4×Dic83C2312yes—
63C2×Q8×C83—21no—
64C2×Q8×C83C2311yescharacteristic
65C2×C8×C83C2312yes—
66C4⋊C4×C83—21no—
67C4⋊C4×C83—21no—
68C4⋊C4×C83C2311yescharacteristic
69C4⋊C4×C83C2312yes—
70C42×C83—21no—
71C42×C83C2311yescharacteristic
72Q8×C83—21no—
73Q8×C83—81no—
74Q8×C83C2×Q812yes—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2411yescharacteristic
81C2×Dic83—21no—
82C2×Dic83—21no—
83C2×Dic83—22no—
84C2×Dic83C2×D412yes—
85C2×Dic83C2×D412yes—
86Dic166—22no—
87Dic166—21no—
88Dic166—42no—
89Dic166—82no—
90Dic166C2×Q812yes—
91C8×C83—22no—
92C83⋊C8—21no—
93C83⋊C8C2×Q812yes—
94C22×C83C22×D411yescharacteristic
95C4×C83—22no—
96C4×C83—21no—
97C4×C83—42no—
98C4×C83—41no—
99C4×C83#13712yes—
100C4×C83C22×Q811yescharacteristic
101C4×C832−1+411yescharacteristic
102Dic83—22no—
103Dic83—22no—
104Dic83—22no—
105Dic83—42no—
106Dic83—41no—
107C2×C83D4×Q811yescharacteristic
108C2×C83C2×#13711yescharacteristic
109C2×C83#57711yescharacteristic
110#3387—831nomaximal, Sylow
111C83#338711yescharacteristic, complemented, Sylow
112Q82—831no—
113#557—831no—
114#501—832no—
115#498—831no—
116#479—832no—
117#477D8311yescharacteristic
118#475—831no—
119#475—832no—
120#446—831no—
121#441—832no—
122C4×Q16—831no—
123C2×Q16—831no—
124C4⋊Q8—832no—
125C4⋊Q8—831no—
126C4⋊Q8—1661no—
127C4⋊Q8—1661no—
128C4⋊Q8—1662no—
129C4⋊Q8C2×D8311yescharacteristic
130#125—832no—
131#125—1662no—
132C4×Q8—831no—
133C4×Q8—832no—
134C4×Q8—832no—
135C4×Q8—831no—
136C4×Q8—1661no—
137C4×Q8—1661no—
138C4×Q8C2×D8311yescharacteristic
139#107—831no—
140#107C2×D8312yes—
141#106—832no—
142#106—832no—
143#106—832no—
144C4⋊C8—832no—
145C4⋊C8C2×D8311yescharacteristic
146#103—831no—
147#103—832no—
148#103—831no—
149#103C2×D8312yes—
150#97—832no—
151C4×C8—831no—
152C2×Q8—21no—
153C2×Q8—831no—
154C2×Q8—831no—
155C2×Q8—1662no—
156C2×Q8—1661no—
157C2×Q8C22×D8311yescharacteristic
158Q16—834no—
159C2×C8—831no—
160C2×C8—831no—
161C2×C8C22×D8312yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—831no—
165C4⋊C4—832no—
166C4⋊C4—832no—
167C4⋊C4—831no—
168C4⋊C4—1662no—
169C4⋊C4—1662no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1662no—
173C4⋊C4—1661no—
174C4⋊C4—1661no—
175C4⋊C4C22×D8311yescharacteristic
176C4⋊C4C22×D8312yes—
177C42—21no—
178C42—832no—
179C42—1662no—
180C42—1662no—
181C42—1661no—
182C42C22×D8311yescharacteristic
183Q8—21no—
184Q8—81no—
185Q8—3322no—
186Q8—832no—
187Q8—6642no—
188Q8—1662no—
189Q8—1661no—
190Q8Q8×D8312yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—832no—
195C2×C4—832no—
196C2×C4—1662no—
197C2×C4—1661no—
198C2×C4—1661no—
199C2×C4D4×D8311yescharacteristic
200C2×C4D4×D8311yescharacteristic
201C2×C4C23×D8311yescharacteristic
202C8—22no—
203C8—832no—
204C8—1661no—
205C22C2×D4×D8311yescharacteristic
206C4—22no—
207C4—21no—
208C4—42no—
209C4—41no—
210C4—3322no—
211C4—3321no—
212C4—1662no—
213C4—1662no—
214C4—1662no—
215C4#1458182412yes—
216C4C2×Q8×D8311yescharacteristic
217C4#1458183911yescharacteristic
218C2#1912836411yescharacteristic
219C2C2×#1458182411yescharacteristic
220C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#477×C83C2
socle, prime radicalC22×C83C22×D4
hypercenter#477D83
hypocenterC83#3387
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
45. #13050580Q8semidirect
111. C83#3387semidirect

Maximal subgroups

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

subgroupindex
#191284312
#191284302
#191281092
#191280972
#191280072
#191279872
#191279842
#191278282
#191278052
C4×#130505802
#477×C832
#338783