Finite Group Databasebeta

← #24920397

Subgroups — #24920397

Order 10624.

326 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 133 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
1Q8×Dic166C211yesmaximal, characteristic
2#19128429C211yesmaximal, characteristic
3#19128110C212yesmaximal
4#19128099C211yesmaximal, characteristic
5#19128009C212yesmaximal
6#19127997C211yesmaximal, characteristic
7#19127987C211yesmaximal, characteristic
8#19127987C212yesmaximal
9#19127809C212yesmaximal
10C4×Dic332C211yesmaximal, characteristic
11#446×C83C211yesmaximal, characteristic
12C2×Dic332C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581792C2212yes—
16#14581791—22no—
17#14581791—21no—
18#14581790—21no—
19#14581784C2212yes—
20#14581782—22no—
21Q8×Dic83—21no—
22C4×Dic166C2212yes—
23C4×Dic166C2212yes—
24C4×Dic166C2211yescharacteristic
25C4×Dic166C2211yescharacteristic
26#14581757—21no—
27#14581726C2211yescharacteristic
28#14581726C2212yes—
29#14581724C2212yes—
30#14581724C2212yes—
31#14581724C2212yes—
32#14581717C2212yes—
33#14581717C2212yes—
34#14581717C2211yescharacteristic
35#14581717C2211yescharacteristic
36C4×Q8×C83C2211yescharacteristic
37C4⋊C8×C83C2211yescharacteristic
38C4⋊C8×C83C2212yes—
39#97×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41C2×Dic166—21no—
42C2×Dic166—22no—
43C2×Dic166—21no—
44C2×Dic166C2311yescharacteristic
45C2×Dic166C2311yescharacteristic
46Dic332Q814yescomplemented
47C4⋊Dic83—21no—
48C4⋊Dic83—22no—
49C4⋊Dic83—21no—
50C4⋊Dic83C2311yescharacteristic
51C4⋊Dic83C2312yes—
52C4⋊Dic83C2312yes—
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—21no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4—22no—
59Dic83⋊C4—21no—
60C4×Dic83—22no—
61C4×Dic83—21no—
62C4×Dic83—22no—
63C4×Dic83—21no—
64C2×Q8×C83D411yescharacteristic
65C2×C8×C83C2311yescharacteristic
66C2×C8×C83C2312yes—
67C2×C8×C83C2311yescharacteristic
68C4⋊C4×C83D412yes—
69C4⋊C4×C83D411yescharacteristic
70C42×C83C2312yes—
71C42×C83C2311yescharacteristic
72Q8×C83—22no—
73C2×C4×C83C2×D412yes—
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2×D412yes—
76C2×C4×C83C2×D411yescharacteristic
77C2×C4×C83C2411yescharacteristic
78C2×Dic83—21no—
79C2×Dic83—21no—
80C2×Dic83—21no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—21no—
84Dic166—21no—
85Dic166—21no—
86Dic166—42no—
87Dic166—82no—
88Dic166—81no—
89Dic166C2×Q812yes—
90Dic166C2×Q812yes—
91C8×C83—21no—
92C8×C83—22no—
93C8×C83C2×Q812yes—
94C22×C83C22×D411yescharacteristic
95C4×C83—22no—
96C4×C83—22no—
97C4×C83—21no—
98C4×C83#13712yes—
99C4×C83C22×Q811yescharacteristic
100C4×C832−1+411yescharacteristic
101Dic83—22no—
102Dic83—22no—
103Dic83—42no—
104Dic83—41no—
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#477—831no—
118#475—832no—
119#475—831no—
120#446D8311yescharacteristic
121#441—832no—
122C4×Q16—831no—
123C2×Q16—831no—
124C4⋊Q8—831no—
125C4⋊Q8—832no—
126C4⋊Q8—831no—
127C4⋊Q8—1661no—
128C4⋊Q8—1661no—
129C4⋊Q8—1662no—
130#125—832no—
131#125—1662no—
132C4×Q8—831no—
133C4×Q8—832no—
134C4×Q8—831no—
135C4×Q8—832no—
136C4×Q8—1661no—
137C4×Q8—1661no—
138C4×Q8C2×D8311yescharacteristic
139#107—831no—
140#107—832no—
141#106—832no—
142#106—832no—
143#106—832no—
144C4⋊C8C2×D8311yescharacteristic
145C4⋊C8C2×D8312yes—
146#103—831no—
147#103—831no—
148#103—832no—
149#103—832no—
150#97C2×D8312yes—
151C4×C8C2×D8311yescharacteristic
152C2×Q8—831no—
153C2×Q8—831no—
154C2×Q8—1661no—
155C2×Q8—1662no—
156C2×Q8—1661no—
157C2×Q8D33211yescharacteristic
158Q16—834no—
159C2×C8C22×D8311yescharacteristic
160C2×C8C22×D8312yes—
161C2×C8C22×D8311yescharacteristic
162C4⋊C4—832no—
163C4⋊C4—831no—
164C4⋊C4—832no—
165C4⋊C4—831no—
166C4⋊C4—1661no—
167C4⋊C4—1662no—
168C4⋊C4—1662no—
169C4⋊C4—1661no—
170C4⋊C4—1662no—
171C4⋊C4—1662no—
172C4⋊C4—1661no—
173C4⋊C4—1662no—
174C4⋊C4—1661no—
175C4⋊C4D33211yescharacteristic
176C4⋊C4D33212yes—
177C42—1662no—
178C42—1661no—
179C42—1662no—
180C42—1661no—
181C42C22×D8311yescharacteristic
182C42C22×D8312yes—
183Q8—22no—
184Q8—3322no—
185Q8—832no—
186Q8—832no—
187Q8—6642no—
188Q8—6641no—
189Q8—1661no—
190Q8—1661no—
191C2×C4—1661no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1661no—
195C2×C4—1661no—
196C2×C4—1661no—
197C2×C4C2×D33212yes—
198C2×C4C2×D33212yes—
199C2×C4C2×D33211yescharacteristic
200C2×C4C2×D33211yescharacteristic
201C2×C4C23×D8311yescharacteristic
202C8—21no—
203C8—22no—
204C8Q8×D8312yes—
205C22C22×D33211yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—3322no—
210C4—3322no—
211C4—3321no—
212C4—3321no—
213C4—1662no—
214C4—1662no—
215C4#1458182512yes—
216C4C2×Q8×D8311yescharacteristic
217C4#1458183911yescharacteristic
218C2Q8×D33211yescharacteristic
219C2C2×#1458182511yescharacteristic
220C2#1912851311yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#446×C83C2
socle, prime radicalC22×C83C22×D4
hypercenter#446D83
hypocenterC83#3387
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
46. Dic332Q8semidirect
111. C83#3387semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic1662
#191284292
#191281102
#191280992
#191280092
#191279972
#191279872
#191278092
C4×Dic3322
#446×C832
#338783