Finite Group Databasebeta

← #356×D227 #44401413

Subgroups — #356×D227 #44401413

Order 29056.

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×D227C211yesmaximal, characteristic
2#101×D227C212yesmaximal
3#28408831C211yesmaximal, characteristic, complemented
4#28408830C212yesmaximal, complemented
5#356×C227C211yesmaximal, characteristic, complemented
6C2×Q8×D227C411yescharacteristic
7C2×Q8×D227C2211yescharacteristic
8M16×D227—22no—
9#20300288C411yescharacteristic
10#20300287—22no—
11#20300286C411yescharacteristic
12#20300286C2211yescharacteristic
13C4⋊C4×D227—22no—
14C42×D227C411yescharacteristic
15#20300207C2212yes—
16#20300206—22no—
17C4⋊Q8×C227C2211yescharacteristic
18#101×C227C2212yes—
19C2×Dic454—21no—
20C2×Dic454C2×C411yescharacteristic
21C2×Dic454C2×C411yescharacteristic
22C2×Dic454D412yes—
23Q8×D227—41no—
24Q8×D227—41no—
25Q8×D227—41no—
26Q8×D227—41no—
27C2×C4×D227—21no—
28C2×C4×D227—21no—
29C2×C4×D227C2×C411yescharacteristic
30C2×C4×D227D412yes—
31#16813868—22no—
32#16813867—42no—
33C8×D227—42no—
34C4⋊Dic227—22no—
35Dic227⋊C4—22no—
36Dic227⋊C4—22no—
37C4×Dic227—21no—
38C4×Dic227C2×C411yescharacteristic
39C2×Q8×C227C2×C411yescharacteristic
40C2×Q8×C227C2311yescharacteristic
41M16×C227—22no—
42C4⋊C4×C227—22no—
43C42×C227C2×C411yescharacteristic
44C22×D227C22⋊C411yescharacteristic
45Dic454—41no—
46Dic454—42no—
47Dic454—42no—
48Dic454—41no—
49C2×Dic227—21no—
50C2×Dic227—21no—
51C2×Dic227C22⋊C411yescharacteristic
52C2×Dic227C22⋊C412yes—
53C4×D227—24no—
54C4×D227—22no—
55C4×D227—22no—
56C4×D227—21no—
57C4×D227—21no—
58C4×D227—41no—
59C4×D227—41no—
60C227⋊C8—42no—
61Q8×C227—41no—
62Q8×C227—41no—
63C2×C4×C227—21no—
64C2×C4×C227—21no—
65C2×C4×C227C22×C411yescharacteristic
66C2×C4×C227C2×D412yes—
67C8×C227—42no—
68C22×C227C2×C22⋊C411yescharacteristic
69C2×D227—22no—
70C2×D227#9911yescharacteristic
71C2×D227#9911yescharacteristic
72C4×C227—22no—
73C4×C227—22no—
74C4×C227—21no—
75C4×C227—41no—
76Dic227—22no—
77Dic227—22no—
78Dic227—21no—
79Dic227—41no—
80C2×C227—21no—
81C2×C227C2×#9911yescharacteristic
82D227—21no—
83D227#35612yesdirect product factor
84C227C2×#35611yescharacteristic, complemented, Sylow
85C2×#356—2271nomaximal, Sylow
86C2×C4⋊Q8—2271no—
87C2×#101—2272no—
88#356—2271no—
89#356—2272no—
90#356D22711yescharacteristic, direct product factor
91C22×Q8—2271no—
92C22×Q8—2271no—
93C2×M16—4542no—
94C4⋊Q8—4542no—
95C4⋊Q8—2271no—
96C4⋊Q8—2271no—
97C4⋊Q8—2271no—
98C4⋊Q8C2×D22711yescharacteristic
99C2×C4⋊C4—4542no—
100C2×C42—2271no—
101#101—4542no—
102#101—2272no—
103#101C2×D22712yes—
104C2×Q8—9081no—
105C2×Q8—9081no—
106C2×Q8—9081no—
107C2×Q8—9081no—
108C2×Q8—4541no—
109C2×Q8—2271no—
110C2×Q8—2271no—
111C2×Q8—2272no—
112C2×Q8C4×D22711yescharacteristic
113C2×Q8C22×D22711yescharacteristic
114C22×C4—4541no—
115C22×C4—4541no—
116C22×C4—2271no—
117C22×C4—2272no—
118M16—22no—
119M16—9082no—
120M16—4542no—
121C2×C8—9082no—
122C4⋊C4—22no—
123C4⋊C4—4542no—
124C4⋊C4—4542no—
125C4⋊C4—4542no—
126C42—4541no—
127C42—2271no—
128C42C4×D22711yescharacteristic
129C23—2271no—
130Q8—41no—
131Q8—41no—
132Q8—9081no—
133Q8—9082no—
134Q8—9081no—
135Q8—9082no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—9081no—
139C2×C4—9081no—
140C2×C4—4544no—
141C2×C4—4542no—
142C2×C4—4542no—
143C2×C4—4541no—
144C2×C4—4541no—
145C2×C4—4541no—
146C2×C4—4541no—
147C2×C4—2271no—
148C2×C4—2272no—
149C2×C4C2×C4×D22711yescharacteristic
150C2×C4D4×D22712yes—
151C8—42no—
152C8—9082no—
153C22—4542no—
154C22—2271no—
155C22—2271no—
156C22C22⋊C4×D22711yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—9081no—
162C4—4542no—
163C4—4542no—
164C4—4541no—
165C2—21no—
166C2—4541no—
167C2—2272no—
168C2#99×D22711yescharacteristic
subgroupG/H
centerC2#99×D227
commutatorC2×C4×C227C22×C4
FrattiniC2×Q8C22×D227
Fitting#356×C227C2
socleC2×C227C2×#99
prime radicalC22×D227C22⋊C4
cosocleC2×Q8×C227C23
hypercenter#356D227
hypocenterC227C2×#356
LCSG ▷ C2×C4×C227 ▷ C22×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #28408831C2semidirect
4. #28408830C2semidirect
5. #356×C227C2semidirect
83. D227#356direct (N × H)
84. C227C2×#356semidirect
90. #356D227direct (N × H)

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2272
#101×D2272
#284088312
#284088302
#356×C2272
C2×#356227