Finite Group Databasebeta

← #356×D239 #44486227

Subgroups — #356×D239 #44486227

Order 30592.

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×D239C211yesmaximal, characteristic
2#101×D239C212yesmaximal
3#29504571C211yesmaximal, characteristic, complemented
4#29504570C212yesmaximal, complemented
5#356×C239C211yesmaximal, characteristic, complemented
6C2×Q8×D239C411yescharacteristic
7C2×Q8×D239C2211yescharacteristic
8M16×D239—22no—
9#21439960C411yescharacteristic
10#21439959—22no—
11#21439958C411yescharacteristic
12#21439958C2211yescharacteristic
13C4⋊C4×D239—22no—
14C42×D239C411yescharacteristic
15#21439879C2212yes—
16#21439878—22no—
17C4⋊Q8×C239C2211yescharacteristic
18#101×C239C2212yes—
19C2×Dic478—21no—
20C2×Dic478C2×C411yescharacteristic
21C2×Dic478C2×C411yescharacteristic
22C2×Dic478D412yes—
23Q8×D239—41no—
24Q8×D239—41no—
25Q8×D239—41no—
26Q8×D239—41no—
27C2×C4×D239—21no—
28C2×C4×D239—21no—
29C2×C4×D239C2×C411yescharacteristic
30C2×C4×D239D412yes—
31#16838532—22no—
32#16838531—42no—
33C8×D239—42no—
34C4⋊Dic239—22no—
35Dic239⋊C4—22no—
36Dic239⋊C4—22no—
37C4×Dic239—21no—
38C4×Dic239C2×C411yescharacteristic
39C2×Q8×C239C2×C411yescharacteristic
40C2×Q8×C239C2311yescharacteristic
41M16×C239—22no—
42C4⋊C4×C239—22no—
43C42×C239C2×C411yescharacteristic
44C22×D239C22⋊C411yescharacteristic
45Dic478—41no—
46Dic478—42no—
47Dic478—42no—
48Dic478—41no—
49C2×Dic239—21no—
50C2×Dic239—21no—
51C2×Dic239C22⋊C411yescharacteristic
52C2×Dic239C22⋊C412yes—
53C4×D239—24no—
54C4×D239—22no—
55C4×D239—22no—
56C4×D239—21no—
57C4×D239—21no—
58C4×D239—41no—
59C4×D239—41no—
60C239⋊C8—42no—
61Q8×C239—41no—
62Q8×C239—41no—
63C2×C4×C239—21no—
64C2×C4×C239—21no—
65C2×C4×C239C22×C411yescharacteristic
66C2×C4×C239C2×D412yes—
67C8×C239—42no—
68C22×C239C2×C22⋊C411yescharacteristic
69C2×D239—22no—
70C2×D239#9911yescharacteristic
71C2×D239#9911yescharacteristic
72C4×C239—22no—
73C4×C239—22no—
74C4×C239—21no—
75C4×C239—41no—
76Dic239—22no—
77Dic239—22no—
78Dic239—21no—
79Dic239—41no—
80C2×C239—21no—
81C2×C239C2×#9911yescharacteristic
82D239—21no—
83D239#35612yesdirect product factor
84C239C2×#35611yescharacteristic, complemented, Sylow
85C2×#356—2391nomaximal, Sylow
86C2×C4⋊Q8—2391no—
87C2×#101—2392no—
88#356—2391no—
89#356—2392no—
90#356D23911yescharacteristic, direct product factor
91C22×Q8—2391no—
92C22×Q8—2391no—
93C2×M16—4782no—
94C4⋊Q8—4782no—
95C4⋊Q8—2391no—
96C4⋊Q8—2391no—
97C4⋊Q8—2391no—
98C4⋊Q8C2×D23911yescharacteristic
99C2×C4⋊C4—4782no—
100C2×C42—2391no—
101#101—4782no—
102#101—2392no—
103#101C2×D23912yes—
104C2×Q8—9561no—
105C2×Q8—9561no—
106C2×Q8—9561no—
107C2×Q8—9561no—
108C2×Q8—4781no—
109C2×Q8—2391no—
110C2×Q8—2391no—
111C2×Q8—2392no—
112C2×Q8C4×D23911yescharacteristic
113C2×Q8C22×D23911yescharacteristic
114C22×C4—4781no—
115C22×C4—4781no—
116C22×C4—2391no—
117C22×C4—2392no—
118M16—22no—
119M16—9562no—
120M16—4782no—
121C2×C8—9562no—
122C4⋊C4—22no—
123C4⋊C4—4782no—
124C4⋊C4—4782no—
125C4⋊C4—4782no—
126C42—4781no—
127C42—2391no—
128C42C4×D23911yescharacteristic
129C23—2391no—
130Q8—41no—
131Q8—41no—
132Q8—9561no—
133Q8—9562no—
134Q8—9561no—
135Q8—9562no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—9561no—
139C2×C4—9561no—
140C2×C4—4784no—
141C2×C4—4782no—
142C2×C4—4782no—
143C2×C4—4781no—
144C2×C4—4781no—
145C2×C4—4781no—
146C2×C4—4781no—
147C2×C4—2391no—
148C2×C4—2392no—
149C2×C4C2×C4×D23911yescharacteristic
150C2×C4D4×D23912yes—
151C8—42no—
152C8—9562no—
153C22—4782no—
154C22—2391no—
155C22—2391no—
156C22C22⋊C4×D23911yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—9561no—
162C4—4782no—
163C4—4782no—
164C4—4781no—
165C2—21no—
166C2—4781no—
167C2—2392no—
168C2#99×D23911yescharacteristic
subgroupG/H
centerC2#99×D239
commutatorC2×C4×C239C22×C4
FrattiniC2×Q8C22×D239
Fitting#356×C239C2
socleC2×C239C2×#99
prime radicalC22×D239C22⋊C4
cosocleC2×Q8×C239C23
hypercenter#356D239
hypocenterC239C2×#356
LCSG ▷ C2×C4×C239 ▷ C22×C239 ▷ C2×C239 ▷ C239
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #29504571C2semidirect
4. #29504570C2semidirect
5. #356×C239C2semidirect
83. D239#356direct (N × H)
84. C239C2×#356semidirect
90. #356D239direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2392
#101×D2392
#295045712
#295045702
#356×C2392
C2×#356239