Finite Group Databasebeta

← #30665113

Subgroups — #30665113

Order 16768.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#22553965C211yesmaximal, characteristic
2#22553664C211yesmaximal, characteristic, complemented
3#22553603C211yesmaximal, characteristic, complemented
4#22553603C211yesmaximal, characteristic, complemented
5#22553039C211yesmaximal, characteristic, complemented
6#22553039C211yesmaximal, characteristic, complemented
7#332×C131C211yesmaximal, characteristic, complemented
8C2×#14523514—21no—
9#16862246C2211yescharacteristic
10#16862231—22no—
11#16862222—21no—
12#16862221—21no—
13C4×D524C2211yescharacteristic
14C4×D524C2211yescharacteristic
15C4⋊C4×D131—21no—
16#16862178—21no—
17#16862178—21no—
18#16862169—22no—
19#16862168—21no—
20#16862168—21no—
21C4×C131⋊C8C2211yescharacteristic
22C4⋊Q8×C131C2211yescharacteristic
23C4⋊C8×C131C2211yescharacteristic
24C4⋊C8×C131C2211yescharacteristic
25#14523514—42no—
26#14523514—42no—
27C2×Dic262—21no—
28C2×D524—22no—
29C2×D524D411yescharacteristic
30C2×D524D411yescharacteristic
31C2×C4×D131—22no—
32C2×C4×D131—21no—
33C2×C4×D131—21no—
34C2×C131⋊C8—21no—
35C4⋊Dic131—21no—
36C4⋊Dic131—21no—
37C4⋊Dic131D411yescharacteristic
38C4⋊Dic131D411yescharacteristic
39Dic131⋊C4—22no—
40(C2×D131)⋊C4—22no—
41(C2×D131)⋊C4—21no—
42(C2×D131)⋊C4—21no—
43C2×Q8×C131—21no—
44C2×C8×C131—21no—
45C2×C8×C131—21no—
46C4⋊C4×C131—21no—
47C4⋊C4×C131D412yes—
48C42×C131C2311yescharacteristic
49C22×D131—21no—
50C22×D131—21no—
51Dic262—42no—
52D524—21no—
53D524—21no—
54D524—21no—
55D524—44no—
56D524SD1612yes—
57C2×Dic131—21no—
58C2×Dic131—21no—
59C2×Dic131—41no—
60C4×D131—44no—
61C4×D131—41no—
62C4×D131—41no—
63C4×D131—41no—
64C4×D131—41no—
65C131⋊C8—41no—
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—
78C2×D131—41no—
79C2×D131—41no—
80C2×D131—41no—
81C4×C131—21no—
82C4×C131—42no—
83C4×C131C2×SD1611yescharacteristic
84C4×C131#13511yescharacteristic
85C4×C131AGL(1,ℤ/8)11yescharacteristic
86C4×C131#13711yescharacteristic
87Dic131—41no—
88Dic131—41no—
89Dic131—81no—
90C2×C131#45011yescharacteristic
91C2×C131#45211yescharacteristic
92C2×C131#45411yescharacteristic
93D131—42no—
94D131—81no—
95C131#163511yescharacteristic, complemented, Sylow
96#1635—1311nomaximal, Sylow
97#547—1311no—
98#487—1311no—
99#476—1311no—
100#476—1311no—
101#332D13111yescharacteristic, complemented
102D4⋊C8—1311no—
103D4⋊C8—1311no—
104C2×#41—2621no—
105C4⋊Q8—1311no—
106C4⋊Q8C2×D13111yescharacteristic
107#123—2622no—
108C22⋊Q8—2621no—
109#121—2621no—
110C4×D4—1311no—
111C4×D4—1311no—
112C2×C4⋊C4—2621no—
113#106—2621no—
114#106—2621no—
115C4⋊C8C2×D13111yescharacteristic
116C4⋊C8C2×D13111yescharacteristic
117#103—2622no—
118#102—2621no—
119#102—2621no—
120C4×C8—1311no—
121#41—5242no—
122#41—5242no—
123C2×Q8—21no—
124C2×Q8—2621no—
125C2×D4—2622no—
126C2×D4—1311no—
127C2×D4—1311no—
128C22×C4—2622no—
129C22×C4—2621no—
130C22×C4—2621no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2621no—
134C4⋊C4—21no—
135C4⋊C4—2622no—
136C4⋊C4—2621no—
137C4⋊C4—2621no—
138C4⋊C4—1311no—
139C4⋊C4—1311no—
140C4⋊C4D52412yes—
141C22⋊C4—2622no—
142C22⋊C4—2621no—
143C22⋊C4—2621no—
144C42C22×D13111yescharacteristic
145C23—2621no—
146C23—2621no—
147Q8—41no—
148Q8—41no—
149Q8—5242no—
150D4—2621no—
151D4—2621no—
152D4—2621no—
153D4—5244no—
154D4—1312no—
155C2×C4—22no—
156C2×C4—2621no—
157C2×C4—2621no—
158C2×C4—5244no—
159C2×C4—5241no—
160C2×C4—5241no—
161C2×C4—5241no—
162C2×C4—5241no—
163C2×C4—5241no—
164C2×C4D4×D13111yescharacteristic
165C2×C4D4×D13111yescharacteristic
166C2×C4C2×D52411yescharacteristic
167C8—41no—
168C8—41no—
169C8—5241no—
170C22—2622no—
171C22—5241no—
172C22—5241no—
173C22—5241no—
174C22—5241no—
175C22—5241no—
176C22#1686221611yescharacteristic
177C4—21no—
178C4—42no—
179C4—5241no—
180C4—5241no—
181C4—10481no—
182C4SD16×D13111yescharacteristic
183C4#1686226711yescharacteristic
184C4#1686227011yescharacteristic
185C4#1686227811yescharacteristic
186C2—5242no—
187C2—10481no—
188C2#2255346411yescharacteristic
189C2#2255347711yescharacteristic
190C2#2255348711yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#332×C131C2
socleC22×C131#120
prime radical#22553965C2
hypercenter#332D131
hypocenterC131#1635
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. #22553664C2semidirect
3. #22553603C2semidirect
4. #22553603C2semidirect
5. #22553039C2semidirect
6. #22553039C2semidirect
7. #332×C131C2semidirect
95. C131#1635semidirect
101. #332D131semidirect

Maximal subgroups

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

subgroupindex
#225539652
#225536642
#225536032
#225530392
#332×C1312
#1635131