Finite Group Databasebeta

← #49157948

Subgroups — #49157948

Order 37504.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D1172C211yesmaximal, characteristic
2#34036479C211yesmaximal, characteristic, complemented
3#34036416C211yesmaximal, characteristic
4#34035876C211yesmaximal, characteristic, complemented
5Q8⋊#14556873C211yesmaximal, characteristic, complemented
6#34035856C211yesmaximal, characteristic
7#477×C293C211yesmaximal, characteristic, complemented
8C2×Q8×D293—21no—
9#22627624—21no—
10#22627624C2211yescharacteristic
11#22627604—22no—
12#22627604—21no—
13C4×D1172—21no—
14C4×D1172C2211yescharacteristic
15#22627555—21no—
16C4⋊#14556873C2211yescharacteristic
17#22627548—22no—
18#22627544—21no—
19C4×#14556873C2211yescharacteristic
20C4⋊Q8×C293C2211yescharacteristic
21C4×Q8×C293C2211yescharacteristic
22#107×C293—21no—
23C4⋊C8×C293C2211yescharacteristic
24#103×C293—21no—
25C2×Dic586—22no—
26C2×Dic586—21no—
27Q8×D293—42no—
28Q8×D293—42no—
29C2×D1172D411yescharacteristic
30C2×C4×D293—22no—
31C2×C4×D293—21no—
32C2×#14556873—21no—
33C2×#14556873—21no—
34C4⋊Dic293—22no—
35C4⋊Dic293—21no—
36C4⋊Dic293D411yescharacteristic
37C4⋊Dic293D412yes—
38#17989779—22no—
39#17989779—21no—
40C2×Q8×C293—21no—
41C2×Q8×C293D411yescharacteristic
42C2×C8×C293—21no—
43C4⋊C4×C293—21no—
44C4⋊C4×C293—21no—
45C4⋊C4×C293—21no—
46C4⋊C4×C293D411yescharacteristic
47C42×C293—21no—
48C42×C293C2311yescharacteristic
49C22×D293—21no—
50Dic586—44no—
51Dic586—41no—
52Dic586—41no—
53D1172—21no—
54D1172Q1612yes—
55C2×Dic293—22no—
56C2×Dic293—21no—
57C4×D293—44no—
58C4×D293—41no—
59C4×D293—41no—
60#14556873—41no—
61#14556873—41no—
62Q8×C293—21no—
63Q8×C293—42no—
64Q8×C293SD1612yes—
65C2×C4×C293—21no—
66C2×C4×C293—21no—
67C2×C4×C293—41no—
68C2×C4×C293C2×D411yescharacteristic
69C2×C4×C293C2×D411yescharacteristic
70C2×C4×C293C2×D411yescharacteristic
71C8×C293—41no—
72C22×C293#12011yescharacteristic
73C2×D293—22no—
74C2×D293—41no—
75C2×D293—41no—
76C4×C293—22no—
77C4×C293—21no—
78C4×C293—41no—
79C4×C293—81no—
80C4×C293C2×SD1611yescharacteristic
81C4×C293C2×Q1611yescharacteristic
82C4×C293AGL(1,ℤ/8)11yescharacteristic
83C4×C293#13711yescharacteristic
84Dic293—42no—
85Dic293—41no—
86C2×C293#45011yescharacteristic
87C2×C293#45111yescharacteristic
88C2×C293#45411yescharacteristic
89D293—42no—
90C293#163711yescharacteristic, complemented, Sylow
91#1637—2931nomaximal, Sylow
92D4×Q8—2931no—
93#487—2931no—
94#477D29311yescharacteristic, complemented
95#476—2931no—
96#332—2931no—
97Q8⋊C8—2931no—
98D4⋊C8—2931no—
99C22×Q8—5861no—
100C4⋊Q8—2931no—
101C4⋊Q8—5861no—
102C4⋊Q8C2×D29311yescharacteristic
103C22⋊Q8—5862no—
104C22⋊Q8—5861no—
105C4×Q8C2×D29311yescharacteristic
106C4×D4—2931no—
107C4×D4—5861no—
108#107—21no—
109#106—5861no—
110C4⋊C8—2931no—
111C4⋊C8C2×D29311yescharacteristic
112#103—21no—
113#103—5862no—
114#102—5861no—
115C4×C8—2931no—
116C2×Q8—21no—
117C2×Q8—5862no—
118C2×Q8—5861no—
119C2×Q8—11722no—
120C2×Q8—11722no—
121C2×Q8C22⋊D29311yescharacteristic
122C2×D4—2931no—
123C22×C4—5862no—
124C22×C4—5861no—
125C2×C8—21no—
126C2×C8—5861no—
127C2×C8—5861no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—2932no—
132C4⋊C4—2931no—
133C4⋊C4—5862no—
134C4⋊C4—5861no—
135C4⋊C4C22⋊D29311yescharacteristic
136C22⋊C4—5862no—
137C22⋊C4—5861no—
138C42—21no—
139C42C22×D29311yescharacteristic
140C23—5861no—
141Q8—21no—
142Q8—42no—
143Q8—11724no—
144Q8—11721no—
145Q8—11721no—
146Q8#1798978912yes—
147D4—2932no—
148D4—5861no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—5862no—
153C2×C4—5861no—
154C2×C4—11724no—
155C2×C4—11721no—
156C2×C4—11721no—
157C2×C4D4×D29311yescharacteristic
158C2×C4D4×D29311yescharacteristic
159C2×C4C2×C22⋊D29311yescharacteristic
160C8—41no—
161C8—11721no—
162C8—11721no—
163C22—5862no—
164C22—11721no—
165C22—11721no—
166C22#2262759211yescharacteristic
167C4—22no—
168C4—21no—
169C4—41no—
170C4—81no—
171C4—11722no—
172C4—11721no—
173C4C2×#1798978911yescharacteristic
174C4Q16×D29311yescharacteristic
175C4#2262764911yescharacteristic
176C4#2262765611yescharacteristic
177C2—11722no—
178C2#3403627911yescharacteristic
179C2#3403628311yescharacteristic
180C2#3403630011yescharacteristic
subgroupG/H
centerC22#22627592
commutator, cosocleC42×C293C23
FrattiniC42C22×D293
Fitting#477×C293C2
socleC22×C293#120
prime radicalC2×D1172D4
hypercenter#477D293
hypocenterC293#1637
LCSG ▷ C42×C293 ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #34036479C2semidirect
4. #34035876C2semidirect
5. Q8⋊#14556873C2semidirect
7. #477×C293C2semidirect
90. C293#1637semidirect
94. #477D293semidirect

Maximal subgroups

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

subgroupindex
Q8×D11722
#340364792
#340364162
#340358762
Q8⋊#145568732
#340358562
#477×C2932
#1637293