Finite Group Databasebeta

← #28362216

Subgroups — #28362216

Order 13952.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#20295761C211yesmaximal, characteristic
2#20295374C211yesmaximal, characteristic
3#20295308C211yesmaximal, characteristic, complemented
4#20294769C211yesmaximal, characteristic
5Q8⋊#11724784C211yesmaximal, characteristic, complemented
6#20294756C211yesmaximal, characteristic, complemented
7#462×C109C211yesmaximal, characteristic, complemented
8#16812696C2211yescharacteristic
9#16812688—21no—
10#16812687—21no—
11#16812684—22no—
12C4×Dic218C2211yescharacteristic
13C4×D436—21no—
14#16812661—21no—
15#16812630—21no—
16C4⋊#11724784C2211yescharacteristic
17#16812622—21no—
18#16812619—22no—
19C4×#11724784C2211yescharacteristic
20C2×Q16×C109—21no—
21C4⋊Q8×C109C2211yescharacteristic
22C4×Q8×C109C2211yescharacteristic
23C4⋊C8×C109C2211yescharacteristic
24#103×C109—21no—
25C2×Dic218D411yescharacteristic
26C2×D436—21no—
27C2×D436—21no—
28C2×D436D412yes—
29C2×C4×D109—22no—
30C2×#11724784—21no—
31C2×#11724784—21no—
32C4⋊Dic109D411yescharacteristic
33#13182008—21no—
34#13182006—22no—
35#13182006—22no—
36#13182006—22no—
37C4×Dic109—21no—
38C2×Q8×C109—21no—
39C2×Q8×C109D411yescharacteristic
40Q16×C109—42no—
41C2×C8×C109—21no—
42C4⋊C4×C109—21no—
43C4⋊C4×C109—21no—
44C4⋊C4×C109—21no—
45C4⋊C4×C109D411yescharacteristic
46C42×C109—21no—
47C42×C109C2311yescharacteristic
48C22×D109—22no—
49Q8×C109—21no—
50Q8×C109—21no—
51Q8×C109—41no—
52Q8×C109—41no—
53C2×C4×C109—21no—
54C2×C4×C109—21no—
55C2×C4×C109—41no—
56C2×C4×C109C2×D411yescharacteristic
57C2×C4×C109C2×D411yescharacteristic
58C2×C4×C109C2×D411yescharacteristic
59C2×Dic109—21no—
60C2×Dic109—21no—
61D436—42no—
62D436—41no—
63D436—41no—
64D436—41no—
65D436—41no—
66C4×D109—82no—
67Dic218—21no—
68Dic218—21no—
69C8×C109—41no—
70#11724784—41no—
71#11724784—41no—
72C22×C109#12011yescharacteristic
73C2×D109—42no—
74C2×D109—42no—
75C2×D109—42no—
76C4×C109—21no—
77C4×C109—41no—
78C4×C109—41no—
79C4×C109—81no—
80C4×C109#13511yescharacteristic
81C4×C109#13511yescharacteristic
82C4×C109AGL(1,ℤ/8)11yescharacteristic
83C4×C109#13711yescharacteristic
84Dic109—41no—
85Dic109—41no—
86C2×C109#44911yescharacteristic
87C2×C109#45211yescharacteristic
88C2×C109#45511yescharacteristic
89D109—82no—
90#1640—1091nomaximal, Sylow
91C109#164011yescharacteristic, complemented, Sylow
92#559—1091no—
93#486—1091no—
94#475—1091no—
95#462D10911yescharacteristic, complemented
96#331—1091no—
97Q8⋊C8—1091no—
98Q8⋊C8—1091no—
99C2×Q16—21no—
100C4⋊Q8C2×D10911yescharacteristic
101#127—1091no—
102#124—2181no—
103#124—2181no—
104#123—2182no—
105C4×Q8—1091no—
106C4×Q8C2×D10911yescharacteristic
107C4×D4—2181no—
108C4×D4—2181no—
109#106—2181no—
110C4⋊C8—1091no—
111C4⋊C8C2×D10911yescharacteristic
112#103—21no—
113#103—2181no—
114#102—2182no—
115C4×C8—1091no—
116C2×Q8—21no—
117C2×Q8—1091no—
118C2×Q8C22⋊D10911yescharacteristic
119C2×D4—1092no—
120C2×D4—2181no—
121C2×D4—2181no—
122C22×C4—2182no—
123Q16—42no—
124C2×C8—21no—
125C2×C8—2181no—
126C2×C8—2181no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—1091no—
131C4⋊C4—2181no—
132C4⋊C4C22⋊D10911yescharacteristic
133C22⋊C4—2182no—
134C22⋊C4—2182no—
135C22⋊C4—2182no—
136C42—21no—
137C42—2181no—
138C42C22×D10911yescharacteristic
139C23—2182no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—2181no—
145Q8—2181no—
146D4—4362no—
147D4—4361no—
148D4—4361no—
149D4—4361no—
150D4—4361no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—8722no—
155C2×C4—2181no—
156C2×C4—2181no—
157C2×C4D4×D10911yescharacteristic
158C2×C4D4×D10911yescharacteristic
159C2×C4C2×C22⋊D10911yescharacteristic
160C8—41no—
161C8—4361no—
162C8—4361no—
163C22—4362no—
164C22—4362no—
165C22—4362no—
166C22#1681266711yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—4361no—
172C4—4361no—
173C4#1681271911yescharacteristic
174C4#1681272211yescharacteristic
175C4#1681272511yescharacteristic
176C4#1681273111yescharacteristic
177C2—8722no—
178C2#2029516911yescharacteristic
179C2#2029518711yescharacteristic
180C2#2029520211yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#462×C109C2
socleC22×C109#120
prime radical#16812696C22
hypercenter#462D109
hypocenterC109#1640
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
3. #20295308C2semidirect
5. Q8⋊#11724784C2semidirect
6. #20294756C2semidirect
7. #462×C109C2semidirect
91. C109#1640semidirect
95. #462D109semidirect

Maximal subgroups

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

subgroupindex
#202957612
#202953742
#202953082
#202947692
Q8⋊#117247842
#202947562
#462×C1092
#1640109