Finite Group Databasebeta

← #30665277

Subgroups — #30665277

Order 16768.

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

normal subgroups only

This group has 208 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#22554056C211yesmaximal, characteristic
2#22553718C211yesmaximal, characteristic, complemented
3#22553704C211yesmaximal, characteristic, complemented
4#22553053C211yesmaximal, characteristic, complemented
5#22553052C211yesmaximal, characteristic, complemented
6#22553051C211yesmaximal, characteristic, complemented
7#489×C131C211yesmaximal, characteristic, complemented
8C2×#14523503—21no—
9C2×#14523500—21no—
10#16862238C2211yescharacteristic
11#16862234C2211yescharacteristic
12#16862230—21no—
13#16862228—21no—
14#16862223—21no—
15#16862222—21no—
16#16862219—21no—
17#16862218—21no—
18#16862179—21no—
19#16862177—21no—
20#16862157C2211yescharacteristic
21#16862157C2211yescharacteristic
22C4⋊Q8×C131C2211yescharacteristic
23#124×C131C2211yescharacteristic
24#103×C131—21no—
25#102×C131—21no—
26#97×C131C2211yescharacteristic
27C2×Dic262D411yescharacteristic
28C2×D524D411yescharacteristic
29C2×C22⋊D131—21no—
30C2×C22⋊D131—21no—
31C22×Dic131—21no—
32C2×C4×D131—21no—
33#14523503—42no—
34#14523500—42no—
35C2×C131⋊C8—21no—
36C2×C131⋊C8—21no—
37C4⋊Dic131—21no—
38C4⋊Dic131D411yescharacteristic
39C4⋊Dic131D411yescharacteristic
40Dic131⋊C4—21no—
41Dic131⋊C4—21no—
42Dic131⋊C4—21no—
43C22⋊Dic131—21no—
44(C2×D131)⋊C4—21no—
45(C2×D131)⋊C4—21no—
46(C2×D131)⋊C4—21no—
47C2×Q8×C131—21no—
48C2×Q8×C131D411yescharacteristic
49C2×D4×C131D411yescharacteristic
50C2×C8×C131—21no—
51C4⋊C4×C131—21no—
52C4⋊C4×C131—21no—
53C22⋊C4×C131—21no—
54C22⋊C4×C131—21no—
55C42×C131C2311yescharacteristic
56C22×D131—21no—
57Dic262—41no—
58D524—41no—
59C22⋊D131—81no—
60C22⋊D131—81no—
61C2×Dic131—21no—
62C2×Dic131—21no—
63C2×Dic131—21no—
64C2×Dic131—81no—
65C4×D131—81no—
66C131⋊C8—22no—
67C131⋊C8—22no—
68C23×C131—21no—
69Q8×C131—41no—
70Q8×C131—41no—
71Q8×C131—41no—
72D4×C131—41no—
73C2×C4×C131—21no—
74C2×C4×C131—41no—
75C2×C4×C131C2×D411yescharacteristic
76C2×C4×C131C2×D411yescharacteristic
77C2×C4×C131C2×D411yescharacteristic
78C8×C131—41no—
79C22×C131—41no—
80C22×C131—41no—
81C22×C131—41no—
82C22×C131#12011yescharacteristic
83C2×D131—41no—
84C2×D131—41no—
85C2×D131—41no—
86C4×C131—21no—
87C4×C131—21no—
88C4×C131—21no—
89C4×C131—41no—
90C4×C131—81no—
91Dic131—41no—
92Dic131—41no—
93Dic131—41no—
94C2×C131—81no—
95C2×C131#45311yescharacteristic
96C2×C131#45511yescharacteristic
97C2×C131#45611yescharacteristic
98D131—81no—
99C131#166111yescharacteristic, complemented, Sylow
100#1661—1311nomaximal, Sylow
101#562—1311no—
102#501—1311no—
103#497—1311no—
104#489D13111yescharacteristic, complemented
105#330—1311no—
106#329—1311no—
107#329—1311no—
108C2×Q16—2621no—
109C2×SD16—2621no—
110C4⋊Q8C2×D13111yescharacteristic
111#125—1311no—
112#124—1311no—
113#124C2×D13111yescharacteristic
114#123—2621no—
115#123—2621no—
116C22⋊Q8—2621no—
117C22⋊Q8—2621no—
118#121—2621no—
119#121—2621no—
120#107—2621no—
121#106—2621no—
122#103—21no—
123#102—21no—
124#97—1311no—
125#97—1311no—
126#97C2×D13111yescharacteristic
127C2×Q8—21no—
128C2×Q8—1311no—
129C2×Q8C22⋊D13111yescharacteristic
130C2×D4—2621no—
131C2×D4—2621no—
132C2×D4—1311no—
133C2×D4C22⋊D13111yescharacteristic
134C22×C4—2621no—
135C22×C4—2621no—
136Q16—5242no—
137SD16—5242no—
138C2×C8—21no—
139C2×C8—2621no—
140C2×C8—2621no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—2621no—
144C4⋊C4—2621no—
145C4⋊C4—2621no—
146C4⋊C4—2621no—
147C4⋊C4—1311no—
148C4⋊C4—1311no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—2621no—
152C22⋊C4—2621no—
153C22⋊C4—2621no—
154C22⋊C4—2621no—
155C42C22×D13111yescharacteristic
156C23—21no—
157C23—2621no—
158Q8—41no—
159Q8—41no—
160Q8—41no—
161Q8—5241no—
162D4—41no—
163D4—5241no—
164D4—10481no—
165D4—10481no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—2621no—
169C2×C4—2621no—
170C2×C4—2621no—
171C2×C4—10481no—
172C2×C4—10481no—
173C2×C4D4×D13111yescharacteristic
174C2×C4D4×D13111yescharacteristic
175C2×C4C2×C22⋊D13111yescharacteristic
176C8—41no—
177C8—2622no—
178C8—2622no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—5241no—
183C22—5241no—
184C22—5241no—
185C22#1686221411yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—81no—
191C4—5241no—
192C4—5241no—
193C4—5241no—
194C2—81no—
195C2—10481no—
196C2#2255347811yescharacteristic
197C2#2255348911yescharacteristic
198C2#2255349311yescharacteristic
subgroupG/H
centerC22#16862214
commutator, cosocleC42×C131C23
FrattiniC42C22×D131
Fitting#489×C131C2
socleC22×C131#120
prime radical#22554056C2
hypercenter#489D131
hypocenterC131#1661
LCSG ▷ C42×C131 ▷ C22×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #22553718C2semidirect
3. #22553704C2semidirect
4. #22553053C2semidirect
5. #22553052C2semidirect
6. #22553051C2semidirect
7. #489×C131C2semidirect
99. C131#1661semidirect
104. #489D131semidirect

Maximal subgroups

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

subgroupindex
#225540562
#225537182
#225537042
#225530532
#225530522
#225530512
#489×C1312
#1661131