Finite Group Databasebeta

← #28362294

Subgroups — #28362294

Order 13952.

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

normal subgroups only

This group has 234 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×D436C211yesmaximal, characteristic
2#20295422C211yesmaximal, characteristic, complemented
3#20295228C211yesmaximal, characteristic
4#20294773C211yesmaximal, characteristic, complemented
5Q8⋊#11724784C211yesmaximal, characteristic, complemented
6#20294753C211yesmaximal, characteristic
7#475×C109C211yesmaximal, characteristic, complemented
8C2×Q8×D109—21no—
9C2×#13182016—21no—
10#16812699—21no—
11#16812699C2211yescharacteristic
12#16812679—22no—
13#16812679—21no—
14C4×D436—21no—
15C4×D436C2211yescharacteristic
16#16812630—22no—
17C4⋊#11724784C2211yescharacteristic
18#16812623—21no—
19C4×#11724784C2211yescharacteristic
20C4⋊Q8×C109C2211yescharacteristic
21C4×Q8×C109C2211yescharacteristic
22#106×C109—21no—
23C4⋊C8×C109C2211yescharacteristic
24#103×C109—21no—
25C2×Dic218—22no—
26C2×Dic218—21no—
27Q8×D109—42no—
28Q8×D109—42no—
29C2×D436D411yescharacteristic
30C2×C4×D109—22no—
31C2×C4×D109—21no—
32#13182016—42no—
33C2×#11724784—21no—
34C2×#11724784—21no—
35C4⋊Dic109—22no—
36C4⋊Dic109—21no—
37C4⋊Dic109D411yescharacteristic
38C4⋊Dic109D412yes—
39#13182006—22no—
40#13182006—21no—
41C2×Q8×C109—21no—
42C2×Q8×C109D411yescharacteristic
43C2×C8×C109—21no—
44C4⋊C4×C109—21no—
45C4⋊C4×C109—21no—
46C4⋊C4×C109—21no—
47C4⋊C4×C109D411yescharacteristic
48C42×C109—21no—
49C42×C109C2311yescharacteristic
50C22×D109—21no—
51Q8×C109—21no—
52Q8×C109—41no—
53Q8×C109—41no—
54Q8×C109SD1612yescomplemented
55C2×C4×C109—21no—
56C2×C4×C109—21no—
57C2×C4×C109—41no—
58C2×C4×C109C2×D411yescharacteristic
59C2×C4×C109C2×D411yescharacteristic
60C2×C4×C109C2×D411yescharacteristic
61C2×Dic109—22no—
62C2×Dic109—21no—
63D436—21no—
64D436SD1612yes—
65C4×D109—44no—
66C4×D109—41no—
67C4×D109—41no—
68Dic218—44no—
69Dic218—41no—
70Dic218—41no—
71C8×C109—41no—
72#11724784—22no—
73#11724784—41no—
74C22×C109#12011yescharacteristic
75C2×D109—22no—
76C2×D109—41no—
77C2×D109—41no—
78C4×C109—22no—
79C4×C109—21no—
80C4×C109—41no—
81C4×C109—81no—
82C4×C109C2×SD1611yescharacteristic
83C4×C109C2×SD1611yescharacteristic
84C4×C109AGL(1,ℤ/8)11yescharacteristic
85C4×C109#13711yescharacteristic
86Dic109—41no—
87Dic109—42no—
88C2×C109#44811yescharacteristic
89C2×C109#45011yescharacteristic
90C2×C109#45611yescharacteristic
91D109—42no—
92#1651—1091nomaximal, Sylow
93C109#165111yescharacteristic, complemented, Sylow
94D4×Q8—1091no—
95#498—1091no—
96#475D10911yescharacteristic, complemented
97#461—1091no—
98#332—1091no—
99Q8⋊C8—1091no—
100D4⋊C8—1091no—
101C22×Q8—2181no—
102C2×SD16—2181no—
103C4⋊Q8—1091no—
104C4⋊Q8—2181no—
105C4⋊Q8C2×D10911yescharacteristic
106C22⋊Q8—2182no—
107C22⋊Q8—2181no—
108C4×Q8C2×D10911yescharacteristic
109C4×D4—1091no—
110C4×D4—2181no—
111#106—21no—
112#106—2182no—
113C4⋊C8—1091no—
114C4⋊C8C2×D10911yescharacteristic
115#103—21no—
116#103—2181no—
117C4×C8—1091no—
118C2×Q8—21no—
119C2×Q8—4362no—
120C2×Q8—4362no—
121C2×Q8—2182no—
122C2×Q8—2181no—
123C2×Q8C22⋊D10911yescharacteristic
124C2×D4—1091no—
125C22×C4—2182no—
126C22×C4—2181no—
127SD16—4362no—
128C2×C8—21no—
129C2×C8—2181no—
130C2×C8—2181no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1091no—
135C4⋊C4—1092no—
136C4⋊C4—2182no—
137C4⋊C4—2181no—
138C4⋊C4C22⋊D10911yescharacteristic
139C22⋊C4—2182no—
140C22⋊C4—2181no—
141C42—21no—
142C42C22×D10911yescharacteristic
143C23—2181no—
144Q8—21no—
145Q8—41no—
146Q8—41no—
147Q8—4364no—
148Q8—4361no—
149Q8—4361no—
150Q8#1318201612yescomplemented
151D4—1092no—
152D4—2181no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—4364no—
157C2×C4—4361no—
158C2×C4—4361no—
159C2×C4—2182no—
160C2×C4—2181no—
161C2×C4D4×D10911yescharacteristic
162C2×C4D4×D10911yescharacteristic
163C2×C4C2×C22⋊D10911yescharacteristic
164C8—41no—
165C8—4361no—
166C8—2182no—
167C22—4361no—
168C22—4361no—
169C22—2182no—
170C22#1681266711yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—81no—
175C4—4362no—
176C4—4361no—
177C4SD16×D10911yescharacteristic
178C4C2×#1318201611yescharacteristic
179C4#1681272411yescharacteristic
180C4#1681272811yescharacteristic
181C2—4362no—
182C2#2029516311yescharacteristic
183C2#2029517611yescharacteristic
184C2#2029520711yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#475×C109C2
socleC22×C109#120
prime radicalC2×D436D4
hypercenter#475D109
hypocenterC109#1651
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #20295422C2semidirect
4. #20294773C2semidirect
5. Q8⋊#11724784C2semidirect
7. #475×C109C2semidirect
54. Q8×C109SD16semidirect
93. C109#1651semidirect
96. #475D109semidirect
150. Q8#13182016semidirect

Maximal subgroups

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

subgroupindex
Q8×D4362
#202954222
#202952282
#202947732
Q8⋊#117247842
#202947532
#475×C1092
#1651109