Finite Group Databasebeta

← #30670945

Subgroups — #30670945

Order 16768.

540 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 171 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×#16862245C212yesmaximal
2C2×#16862245C211yesmaximal, characteristic
3C2×#16862211C211yesmaximal, characteristic
4#22553244C211yesmaximal, characteristic
5#22553243C211yesmaximal, characteristic
6#22553242C211yesmaximal, characteristic
7#22553216C212yesmaximal
8#22553209C212yesmaximal
9#22553207C212yesmaximal
10#22553163C211yesmaximal, characteristic
11#398×C131C211yesmaximal, characteristic
12C22×Dic262C2212yes—
13C22×Dic262C2211yescharacteristic
14#16862245—28no—
15#16862245—24no—
16#16862211—24no—
17C2×C4⋊Dic131C2212yes—
18C2×C4⋊Dic131C2211yescharacteristic
19C2×Dic131⋊C4C2212yes—
20C2×Dic131⋊C4C2212yes—
21C2×Dic131⋊C4C2212yes—
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2211yescharacteristic
24C2×Dic131⋊C4C2212yes—
25C2×C4×Dic131C2212yes—
26C2×C4×Dic131C2211yescharacteristic
27C2×C4×Dic131C2211yescharacteristic
28C2×C4×Dic131C2211yescharacteristic
29C131⋊#95C2211yescharacteristic
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C131⋊#95C2212yes—
33C2×C4⋊C4×C131C2212yes—
34C2×C4⋊C4×C131C2211yescharacteristic
35C2×C4⋊C4×C131C2211yescharacteristic
36#95×C131C2211yescharacteristic
37#95×C131C2212yes—
38C2×Dic262—22no—
39C2×Dic262—24no—
40C2×Dic262—24no—
41C2×Dic262—22no—
42C2×Dic262—24no—
43C2×Dic262D414yes—
44C22×Dic131C2312yes—
45C22×Dic131C2311yescharacteristic
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2312yes—
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2311yescharacteristic
50C4⋊Dic131—24no—
51C4⋊Dic131—22no—
52Dic131⋊C4—22no—
53Dic131⋊C4—24no—
54Dic131⋊C4—24no—
55Dic131⋊C4—22no—
56Dic131⋊C4—24no—
57Dic131⋊C4Q818yescomplemented
58C4×Dic131—22no—
59C4×Dic131—22no—
60C4×Dic131—24no—
61C4×Dic131D414yes—
62C22×C4×C131C2312yes—
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2311yescharacteristic
65C22×C4×C131C2311yescharacteristic
66C22×C4×C131C2312yes—
67C4⋊C4×C131—22no—
68C4⋊C4×C131—24no—
69C4⋊C4×C131—22no—
70Dic262—28no—
71Dic262—48no—
72C2×Dic131—22no—
73C2×Dic131—21no—
74C2×Dic131—21no—
75C2×Dic131—22no—
76C2×Dic131—21no—
77C2×Dic131—22no—
78C2×Dic131—22no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131C2×D412yes—
82C2×Dic131C2×D412yes—
83C2×Dic131C2×D412yes—
84C2×Dic131C2×D412yes—
85C2×Dic131C2×Q814yes—
86C2×Dic131C2×Q814yes—
87C2×Dic131#4112yes—
88C23×C131C2411yescharacteristic
89C2×C4×C131—22no—
90C2×C4×C131—21no—
91C2×C4×C131—21no—
92C2×C4×C131—22no—
93C2×C4×C131—22no—
94C2×C4×C131—21no—
95C2×C4×C131—22no—
96C2×C4×C131—22no—
97C2×C4×C131—22no—
98C2×C4×C131—22no—
99C2×C4×C131C2×D412yes—
100C2×C4×C131C2×D412yes—
101C2×C4×C131C2×Q814yes—
102C22×C131C22×D411yescharacteristic
103C22×C131C22×D411yescharacteristic
104C22×C131C22×Q812yes—
105C22×C131C2×#4111yescharacteristic
106C22×C1312−1+412yes—
107C4×C131—22no—
108C4×C131—22no—
109C4×C131—24no—
110C4×C131—42no—
111C4×C131—41no—
112Dic131—22no—
113Dic131—24no—
114Dic131—42no—
115Dic131—41no—
116Dic131#12114yes—
117Dic131C4⋊Q814yescomplemented
118C2×C131C2×#12111yescharacteristic
119C2×C131C2×C4⋊Q811yescharacteristic
120C2×C131#53611yescharacteristic
121C2×C131D4×Q812yes—
122C2×C131#55712yes—
123C131#254611yescharacteristic, complemented, Sylow
124#2546—1311nomaximal, Sylow
125C2×C4⋊Q8—1312no—
126C2×C4⋊Q8—1311no—
127C2×C4×Q8—1311no—
128#398—1311no—
129#398—1311no—
130#398—1311no—
131#398D13111yescharacteristic
132#391—1312no—
133#389—1312no—
134#389—1312no—
135C4×C4⋊C4—1311no—
136C22×Q8—1312no—
137C22×Q8—1311no—
138C4⋊Q8—2628no—
139C4⋊Q8—2624no—
140C4×Q8—2624no—
141C2×C4⋊C4—1311no—
142C2×C4⋊C4—1311no—
143C2×C4⋊C4—1312no—
144C2×C4⋊C4—1312no—
145C2×C4⋊C4—1312no—
146C2×C4⋊C4—1312no—
147C2×C4⋊C4—1311no—
148C2×C4⋊C4—1312no—
149C2×C4⋊C4C2×D13111yescharacteristic
150C2×C4⋊C4C2×D13112yes—
151C2×C4⋊C4C2×D13111yescharacteristic
152C2×C42—1311no—
153C2×C42—1311no—
154C2×C42—1312no—
155C2×C42—1311no—
156#95—1311no—
157#95—1312no—
158#95—1312no—
159#95—1312no—
160#95C2×D13111yescharacteristic
161#95C2×D13112yes—
162C2×Q8—2622no—
163C2×Q8—2622no—
164C2×Q8—2624no—
165C2×Q8—2624no—
166C2×Q8—2624no—
167C2×Q8—1314no—
168C22×C4—1311no—
169C22×C4—1312no—
170C22×C4—1312no—
171C22×C4—1311no—
172C22×C4—1311no—
173C22×C4—1311no—
174C22×C4C22×D13111yescharacteristic
175C22×C4C22×D13112yes—
176C22×C4C22×D13111yescharacteristic
177C22×C4C22×D13111yescharacteristic
178C22×C4C22×D13112yes—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—2622no—
183C4⋊C4—2624no—
184C4⋊C4—2624no—
185C4⋊C4—2624no—
186C4⋊C4—2622no—
187C4⋊C4—2624no—
188C4⋊C4—2622no—
189C4⋊C4—1318no—
190C42—2622no—
191C42—2622no—
192C42—2624no—
193C42—1314no—
194C23C23×D13111yescharacteristic
195Q8—2628no—
196Q8—5248no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—2622no—
208C2×C4—2621no—
209C2×C4—2621no—
210C2×C4—2622no—
211C2×C4—2621no—
212C2×C4—2622no—
213C2×C4—2622no—
214C2×C4—2622no—
215C2×C4—2622no—
216C2×C4—1312no—
217C2×C4—1314no—
218C2×C4—1312no—
219C2×C4—1312no—
220C2×C4—1312no—
221C2×C4—1314no—
222C2×C4—1312no—
223C2×C4D4×D13112yes—
224C2×C4D4×D13112yes—
225C2×C4Q8×D13114yes—
226C22C2×D4×D13111yescharacteristic
227C22C2×D4×D13111yescharacteristic
228C22C2×Q8×D13112yes—
229C22#41×D13111yescharacteristic
230C22#1686229312yes—
231C4—22no—
232C4—24no—
233C4—22no—
234C4—41no—
235C4—42no—
236C4—2624no—
237C4—2622no—
238C4—5241no—
239C4—5242no—
240C4—1314no—
241C4—1314no—
242C2#121×D13111yescharacteristic
243C2C4⋊Q8×D13111yescharacteristic
244C2#2255388411yescharacteristic
245C2#2255397512yes—
246C2#2255403912yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, prime radical, cosocleC23×C131C24
Fitting#398×C131C2
hypercenter#398D131
hypocenterC131#2546
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
57. Dic131⋊C4Q8semidirect
117. Dic131C4⋊Q8semidirect
123. C131#2546semidirect

Maximal subgroups

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

subgroupindex
C2×#168622452
C2×#168622112
#225532442
#225532432
#225532422
#225532162
#225532092
#225532072
#225531632
#398×C1312
#2546131