Finite Group Databasebeta

← #30675494

Subgroups — #30675494

Order 16768.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#16862276C211yesmaximal, characteristic
2C22×Dic524C211yesmaximal, characteristic, complemented
3#22553883C211yesmaximal, characteristic
4#22553789C211yesmaximal, characteristic, complemented
5#22553533C214yesmaximal, complemented
6#22553521C214yesmaximal, complemented
7#22553338C211yesmaximal, characteristic, complemented
8#22553320C211yesmaximal, characteristic, complemented
9#416×C131C211yesmaximal, characteristic
10C22×Dic262C2211yescharacteristic
11C22×Dic262C2211yescharacteristic
12#16862276—24no—
13C2×Dic524—24no—
14C2×Dic524C2214yes—
15C2×#14523503C2212yes—
16C2×#14523501C2212yes—
17C2×#14523497C2211yescharacteristic
18#16862244C2212yes—
19#16862233C2212yes—
20#16862227—22no—
21#16862225—22no—
22C4×Dic262—22no—
23#16862211C2214yes—
24#16862205C2211yescharacteristic
25#16862202—22no—
26Dic131⋊C8C2214yes—
27#16862169C2214yes—
28C2×#41×C131C2211yescharacteristic
29C22×C8×C131C2211yescharacteristic
30#117×C131C2211yescharacteristic
31#103×C131C2212yes—
32#102×C131C2212yes—
33C2×Dic262—21no—
34C2×Dic262—21no—
35C2×Dic262—22no—
36C2×Dic262—22no—
37C2×Dic262—22no—
38C2×Dic262D414yes—
39C2×Dic262C2312yes—
40C2×Dic262C2312yes—
41C22×Dic131—21no—
42C22×Dic131—21no—
43Dic524—28no—
44#14523503—24no—
45#14523501—24no—
46#14523497—22no—
47C2×C131⋊C8C2312yes—
48C4⋊Dic131D412yes—
49Dic131⋊C4—24no—
50Dic131⋊C4—24no—
51C22⋊Dic131—24no—
52C22⋊Dic131D412yes—
53C4×Dic131—24no—
54C4×Dic131C2312yes—
55#41×C131—22no—
56#41×C131—22no—
57C2×Q8×C131C2311yescharacteristic
58C2×D4×C131—21no—
59C2×D4×C131C2311yescharacteristic
60C22×C4×C131—21no—
61C22×C4×C131C2311yescharacteristic
62C2×C8×C131—22no—
63C2×C8×C131C2312yes—
64C4⋊C4×C131C2312yes—
65C22⋊C4×C131—21no—
66C42×C131—21no—
67Dic262—24no—
68Dic262—22no—
69Dic262—22no—
70Dic262—22no—
71Dic262—42no—
72Dic262C2×D414yes—
73C2×Dic131—24no—
74C2×Dic131—22no—
75C2×Dic131—22no—
76C2×Dic131—42no—
77C2×Dic131C2×D414yes—
78C131⋊C8—22no—
79C23×C131—21no—
80C23×C131C2×D411yescharacteristic
81Q8×C131—21no—
82Q8×C131—21no—
83D4×C131—21no—
84D4×C131—21no—
85D4×C131—42no—
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—41no—
100C4×C131—42no—
101C4×C131#12112yes—
102C4×C131C22×D411yescharacteristic
103C4×C131C2×#4111yescharacteristic
104Dic131—24no—
105Dic131—24no—
106Dic131—42no—
107C2×C131—22no—
108C2×C131—81no—
109C2×C131C2×#12111yescharacteristic
110C2×C131#57812yes—
111C131#304411yescharacteristic, complemented, Sylow
112#3044—1311nomaximal, Sylow
113C2×#137—1311no—
114C22×Q16—1311no—
115#536—1311no—
116#519—1311no—
117#464—1314no—
118#462—1314no—
119#424—1311no—
120#419—1311no—
121#416D13111yescharacteristic
122C2×#41C2×D13111yescharacteristic
123C22×Q8—1311no—
124C22×Q8—1311no—
125#137—2624no—
126C2×Q16—2624no—
127C2×Q16—1312no—
128C2×Q16—1314no—
129C2×SD16—1312no—
130C2×M16—1311no—
131C22×C8C2×D13111yescharacteristic
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#117C2×D13111yescharacteristic
141C4⋊C8—1314no—
142#103—1314no—
143#103C2×D13112yes—
144#102C2×D13112yes—
145#41—22no—
146#41—22no—
147C2×Q8—2621no—
148C2×Q8—2621no—
149C2×Q8—2622no—
150C2×Q8—2622no—
151C2×Q8—2622no—
152C2×Q8—1314no—
153C2×Q8—1312no—
154C2×Q8—1312no—
155C2×Q8C22×D13111yescharacteristic
156C2×D4—21no—
157C2×D4C22×D13111yescharacteristic
158C22×C4—21no—
159C22×C4—2621no—
160C22×C4—2621no—
161C22×C4C22×D13111yescharacteristic
162Q16—2628no—
163Q16—2624no—
164SD16—2624no—
165M16—2622no—
166C2×C8—22no—
167C2×C8—1312no—
168C2×C8C22×D13112yes—
169C4⋊C4—2624no—
170C4⋊C4—2624no—
171C4⋊C4—1312no—
172C4⋊C4C22×D13112yes—
173C22⋊C4—21no—
174C22⋊C4—2624no—
175C22⋊C4—1312no—
176C42—21no—
177C42—2624no—
178C42—1312no—
179C23—21no—
180C23D4×D13111yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—2624no—
184Q8—2622no—
185Q8—2622no—
186Q8—2622no—
187Q8—5242no—
188Q8—1314no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—2624no—
196C2×C4—2622no—
197C2×C4—2622no—
198C2×C4—5242no—
199C2×C4—1314no—
200C2×C4D4×D13111yescharacteristic
201C2×C4D4×D13112yes—
202C2×C4#1452351312yes—
203C2×C4C23×D13111yescharacteristic
204C8—22no—
205C8—2622no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#1686221912yes—
210C22C2×D4×D13111yescharacteristic
211C4—42no—
212C4—41no—
213C4—2624no—
214C4—2624no—
215C4—5242no—
216C4#1686221912yes—
217C4C2×D4×D13111yescharacteristic
218C4C2×#1452351311yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#1686221911yescharacteristic
222C2#2255412612yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#416×C131C2
socleC22×C131C22×D4
prime radicalC2×#41×C131C22
hypercenter#416D131
hypocenterC131#3044
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic524C2semidirect
4. #22553789C2semidirect
5. #22553533C2semidirect
6. #22553521C2semidirect
7. #22553338C2semidirect
8. #22553320C2semidirect
111. C131#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#168622762
C22×Dic5242
#225538832
#225537892
#225535332
#225535212
#225533382
#225533202
#416×C1312
#3044131