Finite Group Databasebeta

← #30665271

Subgroups — #30665271

Order 16768.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#22554066C211yesmaximal, characteristic
2#22553718C211yesmaximal, characteristic, complemented
3#22553704C211yesmaximal, characteristic
4#22553054C211yesmaximal, characteristic, complemented
5#22553053C211yesmaximal, characteristic, complemented
6#22553051C211yesmaximal, characteristic
7#501×C131C211yesmaximal, characteristic, complemented
8C2×#14523503—21no—
9C2×#14523500—21no—
10#16862245—21no—
11#16862241—21no—
12#16862241—21no—
13#16862238C2211yescharacteristic
14#16862234C2211yescharacteristic
15#16862231—21no—
16#16862226—21no—
17#16862224—21no—
18#16862179—21no—
19#16862177—21no—
20#16862157C2211yescharacteristic
21#16862157C2211yescharacteristic
22C4⋊Q8×C131C2211yescharacteristic
23#125×C131C2211yescharacteristic
24#107×C131—21no—
25#106×C131—21no—
26#97×C131C2211yescharacteristic
27C2×Dic262—21no—
28C2×Dic262D411yescharacteristic
29C2×D524D411yescharacteristic
30C2×C4×D131—21no—
31#14523503—42no—
32#14523500—42no—
33C2×C131⋊C8—21no—
34C2×C131⋊C8—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131D411yescharacteristic
37C4⋊Dic131D411yescharacteristic
38Dic131⋊C4—21no—
39Dic131⋊C4—21no—
40Dic131⋊C4—21no—
41(C2×D131)⋊C4—21no—
42(C2×D131)⋊C4—21no—
43(C2×D131)⋊C4—21no—
44(C2×D131)⋊C4—21no—
45(C2×D131)⋊C4—21no—
46C4×Dic131—21no—
47C2×Q8×C131—21no—
48C2×C8×C131—21no—
49C4⋊C4×C131—21no—
50C4⋊C4×C131—21no—
51C4⋊C4×C131—21no—
52C4⋊C4×C131—21no—
53C4⋊C4×C131D411yescharacteristic
54C4⋊C4×C131D411yescharacteristic
55C42×C131C2311yescharacteristic
56C22×D131—21no—
57Dic262—41no—
58Dic262—81no—
59D524—41no—
60C2×Dic131—21no—
61C2×Dic131—21no—
62C2×Dic131—21no—
63C4×D131—81no—
64C131⋊C8—22no—
65C131⋊C8—22no—
66Q8×C131—41no—
67Q8×C131—41no—
68C2×C4×C131—21no—
69C2×C4×C131—21no—
70C2×C4×C131—41no—
71C2×C4×C131C2×D411yescharacteristic
72C2×C4×C131C2×D411yescharacteristic
73C2×C4×C131C2×D411yescharacteristic
74C8×C131—22no—
75C22×C131#12011yescharacteristic
76C2×D131—41no—
77C2×D131—41no—
78C2×D131—41no—
79C4×C131—21no—
80C4×C131—21no—
81C4×C131—21no—
82C4×C131—41no—
83C4×C131—41no—
84C4×C131—81no—
85Dic131—41no—
86Dic131—41no—
87Dic131—41no—
88C2×C131#45311yescharacteristic
89C2×C131#45611yescharacteristic
90C2×C131#45611yescharacteristic
91D131—81no—
92C131#166011yescharacteristic, complemented, Sylow
93#1660—1311nomaximal, Sylow
94#563—1311no—
95#501—1311no—
96#501D13111yescharacteristic, complemented
97#497—1311no—
98#330—1311no—
99#330—1311no—
100#329—1311no—
101C2×Q16—2621no—
102C2×SD16—2621no—
103C4⋊Q8—2621no—
104C4⋊Q8C2×D13111yescharacteristic
105#126—2621no—
106#126—2621no—
107#125—1311no—
108#125C2×D13111yescharacteristic
109#124—1311no—
110#123—2621no—
111C22⋊Q8—2621no—
112C22⋊Q8—2621no—
113#107—21no—
114#107—2621no—
115#106—21no—
116#106—2621no—
117#97—1311no—
118#97—1311no—
119#97C2×D13111yescharacteristic
120C2×Q8—21no—
121C2×Q8—2621no—
122C2×Q8—1311no—
123C2×D4—1311no—
124C22×C4—2621no—
125Q16—5242no—
126SD16—5242no—
127C2×C8—21no—
128C2×C8—2621no—
129C2×C8—2621no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—2621no—
135C4⋊C4—2621no—
136C4⋊C4—2621no—
137C4⋊C4—2621no—
138C4⋊C4—1311no—
139C4⋊C4—1311no—
140C4⋊C4C22⋊D13111yescharacteristic
141C4⋊C4C22⋊D13111yescharacteristic
142C22⋊C4—2621no—
143C22⋊C4—2621no—
144C22⋊C4—2621no—
145C22⋊C4—2621no—
146C22⋊C4—2621no—
147C42—2621no—
148C42C22×D13111yescharacteristic
149C23—2621no—
150Q8—41no—
151Q8—41no—
152Q8—5241no—
153Q8—10481no—
154D4—5241no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—2621no—
159C2×C4—2621no—
160C2×C4—2621no—
161C2×C4—10481no—
162C2×C4D4×D13111yescharacteristic
163C2×C4D4×D13111yescharacteristic
164C2×C4C2×C22⋊D13111yescharacteristic
165C8—22no—
166C8—2622no—
167C8—2622no—
168C22—5241no—
169C22—5241no—
170C22—5241no—
171C22#1686221411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—81no—
178C4—5241no—
179C4—5241no—
180C4—5241no—
181C2—10481no—
182C2#2255347811yescharacteristic
183C2#2255349311yescharacteristic
184C2#2255349411yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#501×C131C2
socleC22×C131#120
prime radicalC2×D524D4
hypercenter#501D131
hypocenterC131#1660
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
2. #22553718C2semidirect
4. #22553054C2semidirect
5. #22553053C2semidirect
7. #501×C131C2semidirect
92. C131#1660semidirect
96. #501D131semidirect

Maximal subgroups

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

subgroupindex
#225540662
#225537182
#225537042
#225530542
#225530532
#225530512
#501×C1312
#1660131