Finite Group Databasebeta

← #356×D83 #24909142

Subgroups — #356×D83 #24909142

Order 10624.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C4⋊Q8×D83C211yesmaximal, characteristic
2#101×D83C212yesmaximal
3#19127500C211yesmaximal, characteristic, complemented
4#19127499C212yesmaximal, complemented
5#356×C83C211yesmaximal, characteristic, complemented
6C2×Q8×D83C411yescharacteristic
7C2×Q8×D83C2211yescharacteristic
8M16×D83—22no—
9#14581792C411yescharacteristic
10#14581791—22no—
11#14581790C411yescharacteristic
12#14581790C2211yescharacteristic
13C4⋊C4×D83—22no—
14C42×D83C411yescharacteristic
15#14581711C2212yes—
16#14581710—22no—
17C4⋊Q8×C83C2211yescharacteristic
18#101×C83C2212yes—
19C2×Dic166—21no—
20C2×Dic166C2×C411yescharacteristic
21C2×Dic166C2×C411yescharacteristic
22C2×Dic166D412yes—
23Q8×D83—41no—
24Q8×D83—41no—
25Q8×D83—41no—
26Q8×D83—41no—
27C2×C4×D83—21no—
28C2×C4×D83—21no—
29C2×C4×D83C2×C411yescharacteristic
30C2×C4×D83D412yes—
31#13050574—22no—
32#13050573—42no—
33C8×D83—42no—
34C4⋊Dic83—22no—
35Dic83⋊C4—22no—
36Dic83⋊C4—22no—
37C4×Dic83—21no—
38C4×Dic83C2×C411yescharacteristic
39C2×Q8×C83C2×C411yescharacteristic
40C2×Q8×C83C2311yescharacteristic
41M16×C83—22no—
42C4⋊C4×C83—22no—
43C42×C83C2×C411yescharacteristic
44C22×D83C22⋊C411yescharacteristic
45Q8×C83—41no—
46Q8×C83—41no—
47C2×C4×C83—21no—
48C2×C4×C83—21no—
49C2×C4×C83C22×C411yescharacteristic
50C2×C4×C83C2×D412yes—
51C2×Dic83—21no—
52C2×Dic83—21no—
53C2×Dic83C22⋊C411yescharacteristic
54C2×Dic83C22⋊C412yes—
55C4×D83—24no—
56C4×D83—22no—
57C4×D83—22no—
58C4×D83—21no—
59C4×D83—21no—
60C4×D83—41no—
61C4×D83—41no—
62Dic166—41no—
63Dic166—42no—
64Dic166—42no—
65Dic166—41no—
66C8×C83—42no—
67C83⋊C8—42no—
68C22×C83C2×C22⋊C411yescharacteristic
69C2×D83—22no—
70C2×D83#9911yescharacteristic
71C2×D83#9911yescharacteristic
72C4×C83—22no—
73C4×C83—22no—
74C4×C83—21no—
75C4×C83—41no—
76Dic83—22no—
77Dic83—21no—
78Dic83—22no—
79Dic83—41no—
80C2×C83—21no—
81C2×C83C2×#9911yescharacteristic
82D83—21no—
83D83#35612yesdirect product factor
84C2×#356—831nomaximal, Sylow
85C83C2×#35611yescharacteristic, complemented, Sylow
86C2×C4⋊Q8—831no—
87C2×#101—832no—
88#356—831no—
89#356—832no—
90#356D8311yescharacteristic, direct product factor
91C22×Q8—831no—
92C22×Q8—831no—
93C2×M16—1662no—
94C4⋊Q8—831no—
95C4⋊Q8—831no—
96C4⋊Q8—831no—
97C4⋊Q8—1662no—
98C4⋊Q8C2×D8311yescharacteristic
99C2×C4⋊C4—1662no—
100C2×C42—831no—
101#101—832no—
102#101—1662no—
103#101C2×D8312yes—
104C2×Q8—3321no—
105C2×Q8—3321no—
106C2×Q8—3321no—
107C2×Q8—3321no—
108C2×Q8—831no—
109C2×Q8—832no—
110C2×Q8—831no—
111C2×Q8—1661no—
112C2×Q8C4×D8311yescharacteristic
113C2×Q8C22×D8311yescharacteristic
114C22×C4—831no—
115C22×C4—832no—
116C22×C4—1661no—
117C22×C4—1661no—
118M16—22no—
119M16—3322no—
120M16—1662no—
121C2×C8—3322no—
122C4⋊C4—22no—
123C4⋊C4—1662no—
124C4⋊C4—1662no—
125C4⋊C4—1662no—
126C42—831no—
127C42—1661no—
128C42C4×D8311yescharacteristic
129C23—831no—
130Q8—41no—
131Q8—41no—
132Q8—3321no—
133Q8—3322no—
134Q8—3321no—
135Q8—3322no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—3321no—
139C2×C4—3321no—
140C2×C4—831no—
141C2×C4—832no—
142C2×C4—1662no—
143C2×C4—1662no—
144C2×C4—1661no—
145C2×C4—1661no—
146C2×C4—1664no—
147C2×C4—1661no—
148C2×C4—1661no—
149C2×C4C2×C4×D8311yescharacteristic
150C2×C4D4×D8312yes—
151C8—42no—
152C8—3322no—
153C22—831no—
154C22—831no—
155C22—1662no—
156C22C22⋊C4×D8311yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—3321no—
162C4—1662no—
163C4—1661no—
164C4—1662no—
165C2—21no—
166C2—832no—
167C2—1661no—
168C2#99×D8311yescharacteristic
subgroupG/H
centerC2#99×D83
commutatorC2×C4×C83C22×C4
FrattiniC2×Q8C22×D83
Fitting#356×C83C2
socleC2×C83C2×#99
prime radicalC22×D83C22⋊C4
cosocleC2×Q8×C83C23
hypercenter#356D83
hypocenterC83C2×#356
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #19127500C2semidirect
4. #19127499C2semidirect
5. #356×C83C2semidirect
83. D83#356direct (N × H)
85. C83C2×#356semidirect
90. #356D83direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D832
#101×D832
#191275002
#191274992
#356×C832
C2×#35683