Finite Group Databasebeta

← #30665261

Subgroups — #30665261

Order 16768.

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

normal subgroups only

This group has 206 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, complemented
2#22553701C211yesmaximal, characteristic
3#22553678C211yesmaximal, characteristic, complemented
4#22553677C211yesmaximal, characteristic, complemented
5#22553053C211yesmaximal, characteristic, complemented
6#22553051C211yesmaximal, characteristic, complemented
7#330×C131C211yesmaximal, characteristic, complemented
8C2×#14523502—21no—
9C2×D1048—21no—
10#16862245—21no—
11#16862243C2211yescharacteristic
12#16862241—21no—
13#16862241—21no—
14#16862238C2211yescharacteristic
15#16862234C2211yescharacteristic
16#16862231—21no—
17#16862226—21no—
18#16862224—21no—
19#16862168—21no—
20#16862168—21no—
21#16862166—21no—
22#16862166—21no—
23#16862157C2211yescharacteristic
24#125×C131C2211yescharacteristic
25#97×C131C2211yescharacteristic
26#97×C131C2211yescharacteristic
27C2×Dic262—21no—
28C2×Dic262D411yescharacteristic
29C2×D524—21no—
30C2×D524—21no—
31C2×D524—21no—
32C2×D524D411yescharacteristic
33C2×C4×D131—21no—
34#14523502—42no—
35D1048—42no—
36C2×C131⋊C8—21no—
37C4⋊Dic131—21no—
38C4⋊Dic131D411yescharacteristic
39C4⋊Dic131D411yescharacteristic
40Dic131⋊C4—21no—
41Dic131⋊C4—21no—
42Dic131⋊C4—21no—
43(C2×D131)⋊C4—21no—
44(C2×D131)⋊C4—21no—
45(C2×D131)⋊C4—21no—
46(C2×D131)⋊C4—21no—
47(C2×D131)⋊C4—21no—
48C4×Dic131—21no—
49C2×C8×C131—21no—
50C2×C8×C131—21no—
51C4⋊C4×C131—21no—
52C4⋊C4×C131—21no—
53C4⋊C4×C131D411yescharacteristic
54C4⋊C4×C131D411yescharacteristic
55C42×C131C2311yescharacteristic
56C22×D131—21no—
57C22×D131—41no—
58Dic262—41no—
59Dic262—81no—
60D524—41no—
61D524—41no—
62D524—41no—
63D524—42no—
64D524—42no—
65C2×Dic131—21no—
66C2×Dic131—21no—
67C2×Dic131—21no—
68C4×D131—81no—
69C131⋊C8—41no—
70C2×C4×C131—21no—
71C2×C4×C131—21no—
72C2×C4×C131C2×D411yescharacteristic
73C2×C4×C131C2×D411yescharacteristic
74C2×C4×C131C2×D411yescharacteristic
75C8×C131—22no—
76C8×C131—41no—
77C22×C131#12011yescharacteristic
78C2×D131—41no—
79C2×D131—41no—
80C2×D131—41no—
81C2×D131—81no—
82C2×D131—81no—
83C2×D131—81no—
84C4×C131—21no—
85C4×C131—21no—
86C4×C131—21no—
87C4×C131—41no—
88C4×C131—41no—
89Dic131—41no—
90Dic131—41no—
91Dic131—41no—
92C2×C131#45311yescharacteristic
93C2×C131#45411yescharacteristic
94C2×C131#45411yescharacteristic
95D131—81no—
96D131—161no—
97C131#165711yescharacteristic, complemented, Sylow
98#1657—1311nomaximal, Sylow
99#563—1311no—
100#496—1311no—
101#490—1311no—
102#490—1311no—
103#330—1311no—
104#330D13111yescharacteristic, complemented
105#329—1311no—
106C2×SD16—2621no—
107C2×D8—2621no—
108C4⋊Q8—2621no—
109#127—1311no—
110#126—2621no—
111#126—2621no—
112#125—1311no—
113#125C2×D13111yescharacteristic
114#124—1311no—
115#123—2621no—
116C22⋊Q8—2621no—
117C22⋊Q8—2621no—
118#102—2621no—
119#102—2621no—
120#102—2621no—
121#102—2621no—
122#97—1311no—
123#97C2×D13111yescharacteristic
124#97C2×D13111yescharacteristic
125C2×Q8—2621no—
126C2×Q8—1311no—
127C2×D4—2621no—
128C2×D4—2621no—
129C2×D4—2621no—
130C2×D4—1311no—
131C22×C4—2621no—
132SD16—5242no—
133D8—5242no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—2621no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—2621no—
140C4⋊C4—2621no—
141C4⋊C4—2621no—
142C4⋊C4—2621no—
143C4⋊C4—1311no—
144C4⋊C4—1311no—
145C4⋊C4D52411yescharacteristic
146C4⋊C4D52411yescharacteristic
147C22⋊C4—2621no—
148C22⋊C4—2621no—
149C22⋊C4—2621no—
150C22⋊C4—2621no—
151C22⋊C4—2621no—
152C42—2621no—
153C42C22×D13111yescharacteristic
154C23—2621no—
155C23—5241no—
156Q8—5241no—
157Q8—10481no—
158D4—5241no—
159D4—5241no—
160D4—5241no—
161D4—5242no—
162D4—5242no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—2621no—
166C2×C4—2621no—
167C2×C4—2621no—
168C2×C4—10481no—
169C2×C4D4×D13111yescharacteristic
170C2×C4D4×D13111yescharacteristic
171C2×C4C2×D52411yescharacteristic
172C8—22no—
173C8—41no—
174C8—5241no—
175C22—5241no—
176C22—5241no—
177C22—5241no—
178C22—10481no—
179C22—10481no—
180C22—10481no—
181C22#1686221611yescharacteristic
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—5241no—
188C4—5241no—
189C4—5241no—
190C2—10481no—
191C2—20961no—
192C2#2255348111yescharacteristic
193C2#2255348611yescharacteristic
194C2#2255348711yescharacteristic
subgroupG/H
centerC22#16862216
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#330×C131C2
socleC22×C131#120
prime radical#22553701C2
hypercenter#330D131
hypocenterC131#1657
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #22554066C2semidirect
3. #22553678C2semidirect
4. #22553677C2semidirect
5. #22553053C2semidirect
6. #22553051C2semidirect
7. #330×C131C2semidirect
97. C131#1657semidirect
104. #330D131semidirect

Maximal subgroups

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

subgroupindex
#225540662
#225537012
#225536782
#225536772
#225530532
#225530512
#330×C1312
#1657131