Finite Group Databasebeta

← #24920342

Subgroups — #24920342

Order 10624.

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

normal subgroups only

This group has 338 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#19128426C211yesmaximal, characteristic, complemented
2#19128408C211yesmaximal, characteristic
3#19128102C212yesmaximal, complemented
4#19128097C211yesmaximal, characteristic, complemented
5#19128007C212yesmaximal, complemented
6#19127999C212yesmaximal
7#19127997C211yesmaximal, characteristic, complemented
8#19127825C211yesmaximal, characteristic, complemented
9#19127792C212yesmaximal
10C4×#13050578C211yesmaximal, characteristic
11#474×C83C211yesmaximal, characteristic
12C2×#13050578C2211yescharacteristic
13#14581792C2211yescharacteristic
14#14581790—21no—
15#14581783C2212yes—
16#14581783C2212yes—
17#14581782—22no—
18#14581771—21no—
19#14581769—22no—
20Q8×Dic83—21no—
21C4×Dic166C2211yescharacteristic
22C4×Dic166C2211yescharacteristic
23#14581757C2212yes—
24D4×Dic83C2212yes—
25C2×C4⋊Dic83—21no—
26#14581726C2212yes—
27#14581723C2212yes—
28#14581723C2212yes—
29#14581723C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2212yes—
32#14581715C2211yescharacteristic
33#14581713C2211yescharacteristic
34C4×C83⋊C8C2211yescharacteristic
35C8×Dic83C2212yes—
36C4⋊Q8×C83C2211yescharacteristic
37C4×D4×C83C2211yescharacteristic
38#107×C83C2212yes—
39C4⋊C8×C83C2211yescharacteristic
40#102×C83C2212yes—
41C2×Dic166—21no—
42C2×Dic166D411yescharacteristic
43C2×Dic166C2311yescharacteristic
44C22×Dic83—22no—
45#13050578Q814yescomplemented
46C2×C83⋊C8C2311yescharacteristic
47C2×C83⋊C8C2311yescharacteristic
48C4⋊Dic83—21no—
49C4⋊Dic83—22no—
50C4⋊Dic83—22no—
51C4⋊Dic83D411yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2312yes—
54Dic83⋊C4—21no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4—22no—
58Dic83⋊C4D412yes—
59C22⋊Dic83—22no—
60C4×Dic83—21no—
61C4×Dic83—22no—
62C4×Dic83C2312yes—
63C2×Q8×C83—21no—
64C2×D4×C83C2311yescharacteristic
65C22×C4×C83—21no—
66C2×C8×C83C2312yes—
67C4⋊C4×C83—21no—
68C4⋊C4×C83C2311yescharacteristic
69C4⋊C4×C83C2312yes—
70C22⋊C4×C83—21no—
71C42×C83C2311yescharacteristic
72C23×C83—21no—
73Q8×C83—42no—
74D4×C83—21no—
75D4×C83C2×Q812yes—
76C2×C4×C83—21no—
77C2×C4×C83—22no—
78C2×C4×C83—42no—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2411yescharacteristic
82C2×Dic83—21no—
83C2×Dic83—21no—
84C2×Dic83—22no—
85C2×Dic83—82no—
86C2×Dic83C2×D412yes—
87C2×Dic83C2×D412yes—
88Dic166—22no—
89Dic166—21no—
90Dic166—81no—
91Dic166C2×Q812yes—
92C8×C83—22no—
93C83⋊C8—21no—
94C83⋊C8C2×Q812yes—
95C22×C83—22no—
96C22×C83—41no—
97C22×C83—41no—
98C22×C83C22×D411yescharacteristic
99C4×C83—21no—
100C4×C83—41no—
101C4×C83—42no—
102C4×C83#13512yes—
103C4×C83C22×Q811yescharacteristic
104C4×C832−1+411yescharacteristic
105Dic83—22no—
106Dic83—22no—
107Dic83—22no—
108Dic83—42no—
109Dic83—41no—
110C2×C83—42no—
111C2×C83D4×Q811yescharacteristic
112C2×C83C2×#13511yescharacteristic
113C2×C83#57711yescharacteristic
114#3381—831nomaximal, Sylow
115C83#338111yescharacteristic, complemented, Sylow
116#557—831no—
117#554—831no—
118#499—832no—
119#498—831no—
120#479—832no—
121#478—832no—
122#477—831no—
123#474D8311yescharacteristic
124C8×Q8—831no—
125C4×SD16—831no—
126C4×SD16—832no—
127C2×SD16—831no—
128C4⋊Q8—831no—
129C4⋊Q8—1661no—
130C4⋊Q8C2×D8311yescharacteristic
131#125—832no—
132#125—832no—
133#125—1662no—
134C22⋊Q8—1662no—
135C22⋊Q8—1661no—
136C4×Q8—831no—
137C4×Q8—832no—
138C4×Q8—831no—
139C4×Q8—1661no—
140C4×D4—832no—
141C4×D4C2×D8311yescharacteristic
142C2×C4⋊C4—1661no—
143#107—832no—
144#107C2×D8312yes—
145#106—832no—
146#106—831no—
147#106—832no—
148C4⋊C8—832no—
149C4⋊C8C2×D8311yescharacteristic
150#103—832no—
151#103—831no—
152#102—831no—
153#102C2×D8312yes—
154C4×C8—832no—
155C4×C8—831no—
156C2×Q8—21no—
157C2×Q8—831no—
158C2×Q8—831no—
159C2×Q8—1661no—
160C2×D4C22×D8311yescharacteristic
161C22×C4—21no—
162C22×C4—1662no—
163SD16—834no—
164C2×C8—831no—
165C2×C8—831no—
166C2×C8C22×D8312yes—
167C4⋊C4—21no—
168C4⋊C4—831no—
169C4⋊C4—832no—
170C4⋊C4—831no—
171C4⋊C4—832no—
172C4⋊C4—1662no—
173C4⋊C4—1662no—
174C4⋊C4—1662no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1661no—
178C4⋊C4—1661no—
179C4⋊C4C22×D8312yes—
180C4⋊C4C22×D8311yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—1662no—
183C42—832no—
184C42—1662no—
185C42—1661no—
186C42C22×D8311yescharacteristic
187C23—21no—
188Q8—42no—
189Q8—832no—
190Q8—6641no—
191Q8—1662no—
192Q8—1661no—
193D4—21no—
194D4Q8×D8312yes—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—42no—
198C2×C4—832no—
199C2×C4—832no—
200C2×C4—6642no—
201C2×C4—1661no—
202C2×C4—1662no—
203C2×C4—1661no—
204C2×C4D4×D8311yescharacteristic
205C2×C4D4×D8311yescharacteristic
206C2×C4C23×D8311yescharacteristic
207C8—22no—
208C8—832no—
209C8—1661no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22C2×D4×D8311yescharacteristic
214C4—21no—
215C4—41no—
216C4—42no—
217C4—3322no—
218C4—3321no—
219C4—1662no—
220C4—1662no—
221C4—1662no—
222C4#1458181112yes—
223C4C2×Q8×D8311yescharacteristic
224C4#1458183911yescharacteristic
225C2—42no—
226C2#1912836411yescharacteristic
227C2C2×#1458181111yescharacteristic
228C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#474×C83C2
socleC22×C83C22×D4
prime radicalC2×D4×C83C23
hypercenter#474D83
hypocenterC83#3381
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #19128426C2semidirect
3. #19128102C2semidirect
4. #19128097C2semidirect
5. #19128007C2semidirect
7. #19127997C2semidirect
8. #19127825C2semidirect
45. #13050578Q8semidirect
115. C83#3381semidirect

Maximal subgroups

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

subgroupindex
#191284262
#191284082
#191281022
#191280972
#191280072
#191279992
#191279972
#191278252
#191277922
C4×#130505782
#474×C832
#338183