Finite Group Databasebeta

← #44398900

Subgroups — #44398900

Order 29056.

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

normal subgroups only

This group has 228 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
1#28409697C211yesmaximal, characteristic
2#28409384C211yesmaximal, characteristic, complemented
3#28409328C211yesmaximal, characteristic, complemented
4#28409323C211yesmaximal, characteristic
5#28408762C211yesmaximal, characteristic, complemented
6#28408759C211yesmaximal, characteristic
7#332×C227C211yesmaximal, characteristic, complemented
8C2×Q8×D227—21no—
9#20300288C2211yescharacteristic
10#20300287—21no—
11#20300266—22no—
12#20300264—21no—
13C4×Dic454C2211yescharacteristic
14C4×D908C2211yescharacteristic
15#20300250—21no—
16#20300222—21no—
17#20300220—21no—
18#20300213—21no—
19#20300211—22no—
20#20300210—21no—
21C4×C227⋊C8C2211yescharacteristic
22C4⋊Q8×C227C2211yescharacteristic
23C4⋊C8×C227C2211yescharacteristic
24C4⋊C8×C227C2211yescharacteristic
25C2×Dic454—22no—
26C2×Dic454—21no—
27C2×Dic454D411yescharacteristic
28Q8×D227—42no—
29Q8×D227—42no—
30C2×D908D411yescharacteristic
31C2×C4×D227—22no—
32C2×C4×D227—21no—
33C2×C227⋊C8—21no—
34C4⋊Dic227—21no—
35C4⋊Dic227—21no—
36C4⋊Dic227D411yescharacteristic
37C4⋊Dic227D411yescharacteristic
38Dic227⋊C4—22no—
39Dic227⋊C4—21no—
40(C2×D227)⋊C4—22no—
41(C2×D227)⋊C4—21no—
42C4×Dic227—21no—
43C2×Q8×C227—21no—
44C2×C8×C227—21no—
45C2×C8×C227—21no—
46C4⋊C4×C227—21no—
47C4⋊C4×C227D412yes—
48C42×C227C2311yescharacteristic
49C22×D227—21no—
50Dic454—21no—
51Dic454—44no—
52Dic454—42no—
53Dic454SD1612yes—
54D908—21no—
55D908Q1612yes—
56C2×Dic227—21no—
57C2×Dic227—21no—
58C2×Dic227—21no—
59C2×Dic227—41no—
60C4×D227—44no—
61C4×D227—41no—
62C4×D227—41no—
63C227⋊C8—41no—
64Q8×C227—41no—
65Q8×C227—41no—
66C2×C4×C227—22no—
67C2×C4×C227C2×D411yescharacteristic
68C2×C4×C227C2×D411yescharacteristic
69C2×C4×C227C2×D411yescharacteristic
70C8×C227—41no—
71C8×C227—41no—
72C22×C227#12011yescharacteristic
73C2×D227—22no—
74C2×D227—41no—
75C2×D227—41no—
76C4×C227—21no—
77C4×C227—42no—
78C4×C227C2×SD1611yescharacteristic
79C4×C227C2×Q1611yescharacteristic
80C4×C227AGL(1,ℤ/8)11yescharacteristic
81C4×C227#13711yescharacteristic
82Dic227—22no—
83Dic227—41no—
84Dic227—41no—
85Dic227—81no—
86C2×C227#45011yescharacteristic
87C2×C227#45111yescharacteristic
88C2×C227#45411yescharacteristic
89D227—42no—
90C227#163711yescharacteristic, complemented, Sylow
91#1637—2271nomaximal, Sylow
92D4×Q8—2271no—
93#487—2271no—
94#477—2271no—
95#476—2271no—
96#332D22711yescharacteristic, complemented
97Q8⋊C8—2271no—
98D4⋊C8—2271no—
99C22×Q8—4541no—
100C4⋊Q8—4541no—
101C4⋊Q8—2271no—
102C4⋊Q8C2×D22711yescharacteristic
103C22⋊Q8—4541no—
104C22⋊Q8—4542no—
105C4×Q8—2271no—
106C4×D4—4541no—
107C4×D4—2271no—
108#107—4541no—
109#106—4541no—
110C4⋊C8C2×D22711yescharacteristic
111C4⋊C8C2×D22711yescharacteristic
112#103—4541no—
113#103—4542no—
114#102—4541no—
115C4×C8—2271no—
116C2×Q8—21no—
117C2×Q8—9082no—
118C2×Q8—9082no—
119C2×Q8—4542no—
120C2×Q8—4541no—
121C2×Q8—2271no—
122C2×D4—2271no—
123C22×C4—4542no—
124C22×C4—4541no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—4541no—
128C4⋊C4—21no—
129C4⋊C4—4542no—
130C4⋊C4—4541no—
131C4⋊C4—4541no—
132C4⋊C4—4541no—
133C4⋊C4—2271no—
134C4⋊C4—2271no—
135C4⋊C4D90812yes—
136C22⋊C4—4542no—
137C22⋊C4—4541no—
138C42—4541no—
139C42C22×D22711yescharacteristic
140C23—4541no—
141Q8—41no—
142Q8—41no—
143Q8—9082no—
144Q8—9084no—
145Q8—4541no—
146Q8—2272no—
147D4—4541no—
148D4—2272no—
149C2×C4—22no—
150C2×C4—9084no—
151C2×C4—9081no—
152C2×C4—9081no—
153C2×C4—9081no—
154C2×C4—4541no—
155C2×C4—4541no—
156C2×C4—4541no—
157C2×C4D4×D22711yescharacteristic
158C2×C4D4×D22711yescharacteristic
159C2×C4C2×D90811yescharacteristic
160C8—41no—
161C8—41no—
162C8—9081no—
163C22—9081no—
164C22—9081no—
165C22—4542no—
166C22#2030025811yescharacteristic
167C4—21no—
168C4—42no—
169C4—18161no—
170C4—9081no—
171C4—9081no—
172C4—4542no—
173C4SD16×D22711yescharacteristic
174C4Q16×D22711yescharacteristic
175C4#2030031211yescharacteristic
176C4#2030031711yescharacteristic
177C2—9082no—
178C2#2840918411yescharacteristic
179C2#2840918911yescharacteristic
180C2#2840920711yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC42×C227C23
FrattiniC42C22×D227
Fitting#332×C227C2
socleC22×C227#120
prime radicalC2×D908D4
hypercenter#332D227
hypocenterC227#1637
LCSG ▷ C42×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #28409384C2semidirect
3. #28409328C2semidirect
5. #28408762C2semidirect
7. #332×C227C2semidirect
90. C227#1637semidirect
96. #332D227semidirect

Maximal subgroups

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

subgroupindex
#284096972
#284093842
#284093282
#284093232
#284087622
#284087592
#332×C2272
#1637227