Finite Group Databasebeta

← #30665211

Subgroups — #30665211

Order 16768.

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

normal subgroups only

This group has 234 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#22553977C211yesmaximal, characteristic
2#22553715C211yesmaximal, characteristic, complemented
3#22553593C211yesmaximal, characteristic
4#22553523C211yesmaximal, characteristic, complemented
5#22553042C211yesmaximal, characteristic, complemented
6#22553039C211yesmaximal, characteristic
7#332×C131C211yesmaximal, characteristic, complemented
8C2×Q8×D131—21no—
9C2×Dic524—21no—
10#16862246C2211yescharacteristic
11#16862245—21no—
12#16862224—22no—
13#16862222—21no—
14C4×Dic262C2211yescharacteristic
15C4×D524C2211yescharacteristic
16#16862208—21no—
17#16862180—21no—
18#16862179—22no—
19#16862171—21no—
20#16862168—21no—
21C4×C131⋊C8C2211yescharacteristic
22C4⋊Q8×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24C4⋊C8×C131C2211yescharacteristic
25C2×Dic262—22no—
26C2×Dic262—21no—
27C2×Dic262D411yescharacteristic
28Q8×D131—42no—
29Q8×D131—42no—
30C2×D524D411yescharacteristic
31C2×C4×D131—22no—
32C2×C4×D131—21no—
33Dic524—42no—
34C2×C131⋊C8—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131—21no—
37C4⋊Dic131D411yescharacteristic
38C4⋊Dic131D411yescharacteristic
39Dic131⋊C4—22no—
40Dic131⋊C4—21no—
41(C2×D131)⋊C4—22no—
42(C2×D131)⋊C4—21no—
43C4×Dic131—21no—
44C2×Q8×C131—21no—
45C2×C8×C131—21no—
46C2×C8×C131—21no—
47C4⋊C4×C131—21no—
48C4⋊C4×C131D412yes—
49C42×C131C2311yescharacteristic
50C22×D131—21no—
51Dic262—21no—
52Dic262—44no—
53Dic262—41no—
54Dic262—41no—
55Dic262D812yes—
56D524—21no—
57D524Q1612yescomplemented
58C2×Dic131—21no—
59C2×Dic131—21no—
60C2×Dic131—21no—
61C2×Dic131—41no—
62C4×D131—44no—
63C4×D131—41no—
64C4×D131—41no—
65C131⋊C8—22no—
66Q8×C131—41no—
67Q8×C131—41no—
68C2×C4×C131—22no—
69C2×C4×C131C2×D411yescharacteristic
70C2×C4×C131C2×D411yescharacteristic
71C2×C4×C131C2×D411yescharacteristic
72C8×C131—41no—
73C8×C131—41no—
74C22×C131#12011yescharacteristic
75C2×D131—22no—
76C2×D131—41no—
77C2×D131—41no—
78C4×C131—21no—
79C4×C131—42no—
80C4×C131C2×D811yescharacteristic
81C4×C131C2×Q1611yescharacteristic
82C4×C131AGL(1,ℤ/8)11yescharacteristic
83C4×C131#13711yescharacteristic
84Dic131—22no—
85Dic131—41no—
86Dic131—41no—
87Dic131—81no—
88C2×C131#44711yescharacteristic
89C2×C131#45111yescharacteristic
90C2×C131#45611yescharacteristic
91D131—42no—
92C131#164911yescharacteristic, complemented, Sylow
93#1649—1311nomaximal, Sylow
94D4×Q8—1311no—
95#500—1311no—
96#474—1311no—
97#462—1311no—
98#332D13111yescharacteristic, complemented
99Q8⋊C8—1311no—
100D4⋊C8—1311no—
101C22×Q8—2621no—
102C2×Q16—2621no—
103C4⋊Q8—2621no—
104C4⋊Q8—1311no—
105C4⋊Q8C2×D13111yescharacteristic
106C22⋊Q8—2621no—
107C22⋊Q8—2622no—
108C4×Q8—1311no—
109C4×D4—2621no—
110C4×D4—1311no—
111#107—2621no—
112#107—2622no—
113C4⋊C8C2×D13111yescharacteristic
114C4⋊C8C2×D13111yescharacteristic
115#103—2621no—
116#102—2621no—
117C4×C8—1311no—
118C2×Q8—21no—
119C2×Q8—2622no—
120C2×Q8—2621no—
121C2×Q8—5242no—
122C2×Q8—5242no—
123C2×Q8—1311no—
124C2×D4—1311no—
125C22×C4—2622no—
126C22×C4—2621no—
127Q16—5242no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—2621no—
131C4⋊C4—21no—
132C4⋊C4—2622no—
133C4⋊C4—2621no—
134C4⋊C4—2621no—
135C4⋊C4—2621no—
136C4⋊C4—1311no—
137C4⋊C4—1311no—
138C4⋊C4D52412yes—
139C22⋊C4—2622no—
140C22⋊C4—2621no—
141C42—2621no—
142C42C22×D13111yescharacteristic
143C23—2621no—
144Q8—41no—
145Q8—41no—
146Q8—2621no—
147Q8—5244no—
148Q8—5241no—
149Q8—5241no—
150Q8—1312no—
151D4—2621no—
152D4—1312no—
153C2×C4—22no—
154C2×C4—2621no—
155C2×C4—2621no—
156C2×C4—2621no—
157C2×C4—5244no—
158C2×C4—5241no—
159C2×C4—5241no—
160C2×C4—5241no—
161C2×C4D4×D13111yescharacteristic
162C2×C4D4×D13111yescharacteristic
163C2×C4C2×D52411yescharacteristic
164C8—41no—
165C8—41no—
166C8—2622no—
167C22—2622no—
168C22—5241no—
169C22—5241no—
170C22#1686221611yescharacteristic
171C4—21no—
172C4—42no—
173C4—2622no—
174C4—5241no—
175C4—5241no—
176C4—10481no—
177C4D8×D13111yescharacteristic
178C4Q16×D13111yescharacteristic
179C4#1686227311yescharacteristic
180C4#1686227511yescharacteristic
181C2—5242no—
182C2#2255344711yescharacteristic
183C2#2255346911yescharacteristic
184C2#2255349711yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#332×C131C2
socleC22×C131#120
prime radicalC2×D524D4
hypercenter#332D131
hypocenterC131#1649
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22553715C2semidirect
4. #22553523C2semidirect
5. #22553042C2semidirect
7. #332×C131C2semidirect
57. D524Q16semidirect
92. C131#1649semidirect
98. #332D131semidirect

Maximal subgroups

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

subgroupindex
#225539772
#225537152
#225535932
#225535232
#225530422
#225530392
#332×C1312
#1649131