Finite Group Databasebeta

← #30678634

Subgroups — #30678634

Order 16768.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic262C211yesmaximal, characteristic
2#22554037C211yesmaximal, characteristic
3#22553662C211yesmaximal, characteristic
4#22553661C212yesmaximal
5#22553608C211yesmaximal, characteristic
6#22553605C212yesmaximal
7#22553520C212yesmaximal
8#22553436C211yesmaximal, characteristic
9Q16×Dic131C212yesmaximal
10C4×#14523503C211yesmaximal, characteristic
11#462×C131C211yesmaximal, characteristic
12C2×#14523503C2211yescharacteristic
13#16862246—21no—
14#16862246C2211yescharacteristic
15#16862245—22no—
16#16862245C2212yes—
17#16862244C2212yes—
18#16862237—21no—
19#16862236—22no—
20Q8×Dic131C2212yes—
21Q8×Dic131C2212yes—
22C4×Dic262—21no—
23C4×Dic262C2211yescharacteristic
24C4×Dic262C2211yescharacteristic
25#16862211—21no—
26#16862180C2212yes—
27#16862179C2211yescharacteristic
28Dic131⋊C8C2212yes—
29#16862171C2212yes—
30#16862170C2212yes—
31#16862170C2211yescharacteristic
32#16862170C2212yes—
33#16862169C2211yescharacteristic
34C4×C131⋊C8C2211yescharacteristic
35C8×Dic131C2212yes—
36C2×Q16×C131C2212yes—
37C4⋊Q8×C131C2211yescharacteristic
38C4×Q8×C131C2211yescharacteristic
39C4⋊C8×C131C2211yescharacteristic
40#103×C131C2212yes—
41C2×Dic262—22no—
42C2×Dic262—21no—
43C2×Dic262D411yescharacteristic
44C2×Dic262C2311yescharacteristic
45#14523503—22no—
46C2×C131⋊C8C2311yescharacteristic
47C2×C131⋊C8C2311yescharacteristic
48C4⋊Dic131—22no—
49C4⋊Dic131—22no—
50C4⋊Dic131—21no—
51C4⋊Dic131D411yescharacteristic
52C4⋊Dic131C2311yescharacteristic
53C4⋊Dic131C2312yes—
54Dic131⋊C4—22no—
55Dic131⋊C4—22no—
56Dic131⋊C4—22no—
57Dic131⋊C4—21no—
58Dic131⋊C4D412yes—
59C4×Dic131—22no—
60C4×Dic131—22no—
61C4×Dic131—21no—
62C4×Dic131C2312yes—
63C2×Q8×C131C2312yes—
64C2×Q8×C131C2311yescharacteristic
65Q16×C131—24no—
66C2×C8×C131C2312yes—
67C4⋊C4×C131—21no—
68C4⋊C4×C131—21no—
69C4⋊C4×C131—21no—
70C4⋊C4×C131C2311yescharacteristic
71C42×C131—21no—
72C42×C131C2311yescharacteristic
73Dic262—21no—
74Dic262—21no—
75Dic262—82no—
76Dic262—81no—
77Dic262Q1614yescomplemented
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—22no—
81C2×Dic131C2×D412yes—
82C2×Dic131C2×D412yes—
83C131⋊C8—21no—
84C131⋊C8—21no—
85Q8×C131—21no—
86Q8×C131—22no—
87Q8×C131—22no—
88Q8×C131#4112yes—
89C2×C4×C131—21no—
90C2×C4×C131—21no—
91C2×C4×C131—22no—
92C2×C4×C131C2×D411yescharacteristic
93C2×C4×C131C2×D411yescharacteristic
94C2×C4×C131C2411yescharacteristic
95C8×C131—22no—
96C22×C131C22×D411yescharacteristic
97C4×C131—22no—
98C4×C131—21no—
99C4×C131—42no—
100C4×C131—41no—
101C4×C131C2×Q1612yes—
102C4×C131C2×#4111yescharacteristic
103C4×C1312−1+411yescharacteristic
104Dic131—22no—
105Dic131—41no—
106Dic131—41no—
107Dic131—42no—
108Dic131C2×Q1614yes—
109C2×C131#54811yescharacteristic
110C2×C131C22×Q1611yescharacteristic
111C2×C131#57711yescharacteristic
112C131#336311yescharacteristic, complemented, Sylow
113#3363—1311nomaximal, Sylow
114Q82—1311no—
115#557—1311no—
116#487—1311no—
117#487—1312no—
118#477—1311no—
119#477—1312no—
120#462—1312no—
121#462D13111yescharacteristic
122C8×Q8—1311no—
123C4×Q16—1312no—
124C4×Q16—1311no—
125C2×Q16—1311no—
126C2×Q16C2×D13112yes—
127C4⋊Q8—2621no—
128C4⋊Q8—2622no—
129C4⋊Q8—1312no—
130C4⋊Q8—1312no—
131C4⋊Q8—1311no—
132C4⋊Q8C2×D13111yescharacteristic
133#125—2622no—
134#125—2621no—
135C4×Q8—2621no—
136C4×Q8—2621no—
137C4×Q8—1312no—
138C4×Q8—1312no—
139C4×Q8—1311no—
140C4×Q8—1311no—
141C4×Q8C2×D13111yescharacteristic
142#107—1312no—
143#107—1311no—
144C4⋊C8—1312no—
145C4⋊C8C2×D13111yescharacteristic
146#103—1311no—
147#103—1312no—
148#103—1312no—
149#103—1312no—
150#103—1311no—
151#103C2×D13112yes—
152C4×C8—1312no—
153C4×C8—1311no—
154C2×Q8—2622no—
155C2×Q8—2621no—
156C2×Q8—1311no—
157C2×Q8—1311no—
158C2×Q8C22×D13112yes—
159C2×Q8C22×D13111yescharacteristic
160Q16—24no—
161Q16—2622no—
162C2×C8—1311no—
163C2×C8—1311no—
164C2×C8C22×D13112yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—2622no—
169C4⋊C4—2622no—
170C4⋊C4—2622no—
171C4⋊C4—2622no—
172C4⋊C4—2622no—
173C4⋊C4—2621no—
174C4⋊C4—2621no—
175C4⋊C4—1312no—
176C4⋊C4—1311no—
177C4⋊C4—1312no—
178C4⋊C4—1311no—
179C4⋊C4C22×D13111yescharacteristic
180C42—21no—
181C42—2622no—
182C42—2622no—
183C42—2621no—
184C42—1312no—
185C42C22×D13111yescharacteristic
186Q8—21no—
187Q8—22no—
188Q8—22no—
189Q8—2621no—
190Q8—2621no—
191Q8—10481no—
192Q8—10482no—
193Q8—1314no—
194Q8#1452351212yes—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—2621no—
199C2×C4—2621no—
200C2×C4—2622no—
201C2×C4—1312no—
202C2×C4—1312no—
203C2×C4D4×D13111yescharacteristic
204C2×C4D4×D13111yescharacteristic
205C2×C4C23×D13111yescharacteristic
206C8—22no—
207C8—2621no—
208C8—2621no—
209C22C2×D4×D13111yescharacteristic
210C4—21no—
211C4—22no—
212C4—41no—
213C4—42no—
214C4—2622no—
215C4—5242no—
216C4—5241no—
217C4—5241no—
218C4—1314no—
219C4Q16×D13112yes—
220C4C2×#1452351211yescharacteristic
221C4#1686229311yescharacteristic
222C2#2255396811yescharacteristic
223C2C2×Q16×D13111yescharacteristic
224C2#2255412111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#462×C131C2
socle, prime radicalC22×C131C22×D4
hypercenter#462D131
hypocenterC131#3363
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
77. Dic262Q16semidirect
112. C131#3363semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic2622
#225540372
#225536622
#225536612
#225536082
#225536052
#225535202
#225534362
Q16×Dic1312
C4×#145235032
#462×C1312
#3363131