Finite Group Databasebeta

← #30675493

Subgroups — #30675493

Order 16768.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14523503C211yesmaximal, characteristic, complemented
2#22553883C211yesmaximal, characteristic
3#22553790C211yesmaximal, characteristic, complemented
4#22553532C214yesmaximal, complemented
5#22553520C214yesmaximal, complemented
6#22553340C211yesmaximal, characteristic, complemented
7#22553322C211yesmaximal, characteristic, complemented
8#22553308C211yesmaximal, characteristic
9C2×#137×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11C2×#14523503—24no—
12C2×#14523503C2214yes—
13C22×C131⋊C8C2211yescharacteristic
14#16862244C2212yes—
15#16862233C2212yes—
16#16862227—22no—
17#16862225—22no—
18Q8×Dic131—22no—
19Q8×Dic131C2214yes—
20#16862205—22no—
21#16862205C2211yescharacteristic
22#16862205C2211yescharacteristic
23Dic131⋊C8C2214yes—
24#16862171C2214yes—
25#16862170C2212yes—
26#16862167C2212yes—
27C2×#41×C131C2211yescharacteristic
28C22×Q8×C131C2211yescharacteristic
29#137×C131—24no—
30C2×Q16×C131C2212yes—
31C2×SD16×C131C2212yes—
32C2×M16×C131C2211yescharacteristic
33C2×Dic262—21no—
34C2×Dic262—22no—
35C2×Dic262—22no—
36C2×Dic262C2312yes—
37C22×Dic131—21no—
38#14523503—28no—
39C2×C131⋊C8—22no—
40C2×C131⋊C8C2312yes—
41C4⋊Dic131—24no—
42C4⋊Dic131D412yes—
43C4⋊Dic131C2312yes—
44Dic131⋊C4—24no—
45C22⋊Dic131—24no—
46C22⋊Dic131—21no—
47C22⋊Dic131D412yes—
48C4×Dic131—24no—
49C4×Dic131—21no—
50C4×Dic131C2312yes—
51#41×C131—22no—
52#41×C131—22no—
53C2×Q8×C131—21no—
54C2×Q8×C131—22no—
55C2×Q8×C131D414yes—
56C2×Q8×C131C2311yescharacteristic
57C2×Q8×C131C2312yes—
58C2×D4×C131—21no—
59C2×D4×C131C2311yescharacteristic
60C22×C4×C131—21no—
61C22×C4×C131—21no—
62C22×C4×C131C2311yescharacteristic
63Q16×C131—24no—
64SD16×C131—24no—
65M16×C131—22no—
66C2×C8×C131C2312yes—
67Dic262—22no—
68Dic262—22no—
69Dic262—42no—
70C2×Dic131—22no—
71C2×Dic131—22no—
72C2×Dic131—42no—
73C2×Dic131C2×D414yes—
74C131⋊C8—22no—
75C23×C131—21no—
76C23×C131C2×D411yescharacteristic
77Q8×C131—24no—
78Q8×C131—22no—
79Q8×C131—21no—
80Q8×C131—21no—
81Q8×C131C2×D414yes—
82D4×C131—21no—
83D4×C131—21no—
84D4×C131—42no—
85C2×C4×C131—24no—
86C2×C4×C131—22no—
87C2×C4×C131—21no—
88C2×C4×C131—42no—
89C2×C4×C131C2×D411yescharacteristic
90C2×C4×C131C2×D412yes—
91C2×C4×C131#4112yes—
92C2×C4×C131C2411yescharacteristic
93C8×C131—22no—
94C22×C131—22no—
95C22×C131—41no—
96C22×C131—42no—
97C22×C131#12112yes—
98C22×C131C22×D411yescharacteristic
99C4×C131—24no—
100C4×C131—41no—
101C4×C131#12112yes—
102C4×C131C22×D411yescharacteristic
103C4×C131C2×#4111yescharacteristic
104Dic131—24no—
105Dic131—42no—
106Dic131—42no—
107C2×C131—22no—
108C2×C131—81no—
109C2×C131C2×#12111yescharacteristic
110C2×C131#57812yes—
111C131#304411yescharacteristic, complemented, Sylow
112#3044—1311nomaximal, Sylow
113C2×#137D13111yescharacteristic
114C22×Q16—1311no—
115#536—1311no—
116#519—1311no—
117#464—1314no—
118#462—1314no—
119#424—1311no—
120#419—1311no—
121#416—1311no—
122C2×#41C2×D13111yescharacteristic
123C22×Q8—1311no—
124C22×Q8C2×D13111yescharacteristic
125#137—24no—
126C2×Q16—2624no—
127C2×Q16—1314no—
128C2×Q16C2×D13112yes—
129C2×SD16C2×D13112yes—
130C2×M16C2×D13111yescharacteristic
131C22×C8—1311no—
132C4⋊Q8—1312no—
133#124—1312no—
134#123—2622no—
135C22⋊Q8—2622no—
136C4×Q8—2622no—
137C4×Q8—1314no—
138#117—2622no—
139#117—1311no—
140#117—1311no—
141C4⋊C8—1314no—
142#103—1312no—
143#103—1314no—
144#102—1312no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—2621no—
150C2×Q8—2622no—
151C2×Q8—2622no—
152C2×Q8—1312no—
153C2×Q8C22⋊D13114yes—
154C2×Q8C22×D13111yescharacteristic
155C2×Q8C22×D13112yes—
156C2×D4—21no—
157C2×D4C22×D13111yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2621no—
161C22×C4C22×D13111yescharacteristic
162Q16—24no—
163Q16—2628no—
164SD16—24no—
165M16—22no—
166C2×C8—2622no—
167C2×C8—1312no—
168C2×C8C22×D13112yes—
169C4⋊C4—2624no—
170C4⋊C4—2624no—
171C4⋊C4—1312no—
172C4⋊C4—1312no—
173C22⋊C4—2621no—
174C22⋊C4—2624no—
175C22⋊C4—1312no—
176C42—2624no—
177C42—2621no—
178C42—1312no—
179C23—21no—
180C23D4×D13111yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—2622no—
186Q8—2622no—
187Q8—5242no—
188Q8C2×C22⋊D13114yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—2622no—
197C2×C4—2622no—
198C2×C4—5242no—
199C2×C4—1314no—
200C2×C4D4×D13111yescharacteristic
201C2×C4C2×C22⋊D13112yes—
202C2×C4#1452351212yes—
203C2×C4C23×D13111yescharacteristic
204C8—22no—
205C8—2622no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#1686221812yes—
210C22C2×D4×D13111yescharacteristic
211C4—24no—
212C4—41no—
213C4—2624no—
214C4—5242no—
215C4—5242no—
216C4#1686221812yes—
217C4C22×C22⋊D13111yescharacteristic
218C4C2×#1452351211yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1686221811yescharacteristic
222C2#2255412812yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#137×C131C2
socleC22×C131C22×D4
prime radicalC2×#41×C131C22
hypercenterC2×#137D131
hypocenterC131#3044
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#14523503C2semidirect
3. #22553790C2semidirect
4. #22553532C2semidirect
5. #22553520C2semidirect
6. #22553340C2semidirect
7. #22553322C2semidirect
111. C131#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#145235032
#225538832
#225537902
#225535322
#225535202
#225533402
#225533222
#225533082
C2×#137×C1312
#3044131