Finite Group Databasebeta

← #30670436

Subgroups — #30670436

Order 16768.

544 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16862226C211yesmaximal, characteristic
2C2×C4⋊C4×D131C211yesmaximal, characteristic
3#22553245C214yesmaximal, complemented
4#22553218C211yesmaximal, characteristic, complemented
5#22553217C211yesmaximal, characteristic, complemented
6#22553183C214yesmaximal
7#22553181C211yesmaximal, characteristic, complemented
8C4×(C2×D131)⋊C4C211yesmaximal, characteristic
9#384×C131C211yesmaximal, characteristic, complemented
10C22×Dic262C2211yescharacteristic
11C22×C4×D131C2211yescharacteristic
12C22×C4×D131C2212yes—
13#16862226—24no—
14C2×C4⋊Dic131C2211yescharacteristic
15C2×C4⋊Dic131C2212yes—
16C2×Dic131⋊C4C2212yes—
17C2×Dic131⋊C4C2214yes—
18C4⋊C4×D131—24no—
19C2×(C2×D131)⋊C4C2212yes—
20C2×(C2×D131)⋊C4C2212yes—
21C2×C4×Dic131C2211yescharacteristic
22C131⋊#95C2212yes—
23C131⋊#95C2214yes—
24C131⋊#95C2214yes—
25C2×C4⋊C4×C131C2211yescharacteristic
26C2×C4⋊C4×C131C2211yescharacteristic
27C2×C4⋊C4×C131C2214yes—
28C2×C42×C131C2211yescharacteristic
29C23×D131C2311yescharacteristic
30C2×Dic262—22no—
31C2×Dic262—22no—
32C2×Dic262—22no—
33C22×Dic131C2312yes—
34C22×Dic131C2312yes—
35C22×Dic131C2311yescharacteristic
36C22×Dic131C2312yes—
37C2×C4×D131—24no—
38C2×C4×D131—22no—
39C2×C4×D131—22no—
40C2×C4×D131—44no—
41C2×C4×D131D414yes—
42C4⋊Dic131—24no—
43C4⋊Dic131—22no—
44Dic131⋊C4—24no—
45Dic131⋊C4—44no—
46(C2×D131)⋊C4—24no—
47(C2×D131)⋊C4Q818yescomplemented
48C4×Dic131—22no—
49C22×C4×C131C2312yes—
50C22×C4×C131C2312yes—
51C22×C4×C131C2312yes—
52C22×C4×C131C2311yescharacteristic
53C4⋊C4×C131—22no—
54C4⋊C4×C131—22no—
55C4⋊C4×C131—44no—
56C42×C131—22no—
57C22×D131—21no—
58C22×D131—21no—
59C22×D131—22no—
60C22×D131C2×D412yes—
61C22×D131C2×Q814yes—
62Dic262—44no—
63C2×Dic131—22no—
64C2×Dic131—21no—
65C2×Dic131—21no—
66C2×Dic131—22no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C2×Dic131—22no—
70C2×Dic131—22no—
71C2×Dic131—24no—
72C2×Dic131C2×D412yes—
73C2×Dic131C2×Q814yes—
74C4×D131—28no—
75C4×D131—48no—
76C23×C131C2411yescharacteristic
77C2×C4×C131—22no—
78C2×C4×C131—22no—
79C2×C4×C131—22no—
80C2×C4×C131—22no—
81C2×C4×C131—24no—
82C2×C4×C131—24no—
83C2×C4×C131C2×D412yes—
84C2×C4×C131C2×Q814yes—
85C2×C4×C131#4112yes—
86C2×C4×C131#4112yes—
87C22×C131C22×D411yescharacteristic
88C22×C131C22×Q812yes—
89C22×C131C2×#4111yescharacteristic
90C22×C131C2×#4111yescharacteristic
91C22×C131H(5,𝔽2)12yes—
92C2×D131—22no—
93C2×D131—22no—
94C2×D131—24no—
95C2×D131—24no—
96C2×D131C4⋊Q814yes—
97C4×C131—24no—
98C4×C131—42no—
99C4×C131—42no—
100C4×C131#12314yes—
101Dic131—22no—
102Dic131—24no—
103Dic131—42no—
104Dic131—42no—
105C2×C131C2×#12311yescharacteristic
106C2×C131C2×C4⋊Q811yescharacteristic
107C2×C131#53511yescharacteristic
108C2×C131#55412yes—
109C2×C131#55412yes—
110D131—24no—
111C131#250211yescharacteristic, complemented, Sylow
112#2502—1311nomaximal, Sylow
113C2×C22⋊Q8—1311no—
114C22×C4⋊C4—1311no—
115#398—1314no—
116#391—1311no—
117#391—1311no—
118#385—1314no—
119#384—1311no—
120#384D13111yescharacteristic, complemented
121C4×C22⋊C4—1311no—
122C22×Q8—1311no—
123C23×C4—1311no—
124C23×C4—1312no—
125C22⋊Q8—2624no—
126C2×C4⋊C4—2624no—
127C2×C4⋊C4—1311no—
128C2×C4⋊C4—1312no—
129C2×C4⋊C4—1312no—
130C2×C4⋊C4—1314no—
131C2×C4⋊C4C2×D13111yescharacteristic
132C2×C4⋊C4C2×D13111yescharacteristic
133C2×C4⋊C4C2×D13114yes—
134C2×C22⋊C4—1312no—
135C2×C22⋊C4—1312no—
136C2×C42—1311no—
137C2×C42C2×D13111yescharacteristic
138#95—1312no—
139#95—1314no—
140#95—1314no—
141C24—1311no—
142C2×Q8—2622no—
143C2×Q8—2622no—
144C2×Q8—2622no—
145C22×C4—2624no—
146C22×C4—2622no—
147C22×C4—2622no—
148C22×C4—5244no—
149C22×C4—1312no—
150C22×C4—1312no—
151C22×C4—1312no—
152C22×C4—1311no—
153C22×C4—1314no—
154C22×C4C22×D13112yes—
155C22×C4C22×D13112yes—
156C22×C4C22×D13112yes—
157C22×C4C22×D13111yescharacteristic
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—44no—
161C4⋊C4—2624no—
162C4⋊C4—2624no—
163C4⋊C4—2622no—
164C4⋊C4—5244no—
165C22⋊C4—2624no—
166C22⋊C4—1318no—
167C42—22no—
168C42—2622no—
169C23—2621no—
170C23—2621no—
171C23—2622no—
172C23—1312no—
173C23—1314no—
174C23C23×D13111yescharacteristic
175Q8—5244no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—2621no—
183C2×C4—2622no—
184C2×C4—2621no—
185C2×C4—2622no—
186C2×C4—2628no—
187C2×C4—2622no—
188C2×C4—2622no—
189C2×C4—2622no—
190C2×C4—2622no—
191C2×C4—2624no—
192C2×C4—5248no—
193C2×C4—1312no—
194C2×C4—1314no—
195C2×C4D4×D13112yes—
196C2×C4Q8×D13114yes—
197C2×C4#1452351312yes—
198C2×C4#1452351412yes—
199C22—2622no—
200C22—2624no—
201C22—2624no—
202C22—2622no—
203C22—1314no—
204C22C2×D4×D13111yescharacteristic
205C22C2×Q8×D13112yes—
206C22C2×#1452351311yescharacteristic
207C22C2×#1452351411yescharacteristic
208C22#1686229112yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—2624no—
213C4—2622no—
214C4—5242no—
215C4—5242no—
216C4#1686223114yes—
217C2—2624no—
218C2C2×#1686223111yescharacteristic
219C2C4⋊Q8×D13111yescharacteristic
220C2#2255387511yescharacteristic
221C2#2255401712yes—
222C2#2255402312yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#384×C131C2
prime radicalC23×D131C23
hypercenter#384D131
hypocenterC131#2502
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #22553245C2semidirect
4. #22553218C2semidirect
5. #22553217C2semidirect
7. #22553181C2semidirect
9. #384×C131C2semidirect
47. (C2×D131)⋊C4Q8semidirect
111. C131#2502semidirect
120. #384D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622262
C2×C4⋊C4×D1312
#225532452
#225532182
#225532172
#225531832
#225531812
C4×(C2×D131)⋊C42
#384×C1312
#2502131