Finite Group Databasebeta

← #30675470

Subgroups — #30675470

Order 16768.

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

normal subgroups only

This group has 494 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
1C2×#16862272C211yesmaximal, characteristic
2#22553886C211yesmaximal, characteristic, complemented
3#22553787C211yesmaximal, characteristic, complemented
4#22553529C214yesmaximal, complemented
5#22553515C214yesmaximal, complemented
6#22553338C211yesmaximal, characteristic, complemented
7#22553321C211yesmaximal, characteristic, complemented
8#22553309C211yesmaximal, characteristic, complemented
9C22×SD16×C131C211yesmaximal, characteristic, complemented
10C2×#14523513C2211yescharacteristic
11#16862272—24no—
12C2×#14523501C2212yes—
13C2×#14523498C2212yes—
14C2×#14523497C2211yescharacteristic
15#16862244C2212yescomplemented
16#16862235C2212yescomplemented
17#16862229—22no—
18#16862222—22no—
19D4×Dic131—22no—
20D4×Dic131C2214yes—
21#16862205C2211yescharacteristic
22#16862205C2211yescharacteristic
23C2×C22⋊Dic131—22no—
24Dic131⋊C8C2214yescomplemented
25#16862171C2212yescomplemented
26#16862170C2214yescomplemented
27#16862168C2212yescomplemented
28C22×Q8×C131C2211yescharacteristic
29C22×D4×C131C2211yescharacteristic
30C2×SD16×C131—24no—
31C2×SD16×C131C2214yes—
32C22×C8×C131C2211yescharacteristic
33#14523513—22no—
34#14523513—22no—
35C2×Dic262C2311yescharacteristic
36C2×D524C2311yescharacteristic
37C2×C22⋊D131—21no—
38C22×Dic131—24no—
39C2×C4×D131—21no—
40#14523501—24no—
41#14523498—24no—
42#14523497—22no—
43C2×C131⋊C8C2312yes—
44C4⋊Dic131D412yes—
45C4⋊Dic131C2312yes—
46Dic131⋊C4—24no—
47C22⋊Dic131—24no—
48C22⋊Dic131—21no—
49C22⋊Dic131D412yes—
50(C2×D131)⋊C4—24no—
51C4×Dic131—21no—
52C4×Dic131C2312yes—
53C24×C131—21no—
54C2×Q8×C131—21no—
55C2×Q8×C131—22no—
56C2×Q8×C131—22no—
57C2×Q8×C131C2312yes—
58C2×D4×C131—21no—
59C2×D4×C131—22no—
60C2×D4×C131D414yes—
61C2×D4×C131C2312yes—
62C22×C4×C131—21no—
63C22×C4×C131C2311yescharacteristic
64SD16×C131—28no—
65C2×C8×C131—22no—
66C2×C8×C131C2312yes—
67C22×D131—21no—
68Dic262—21no—
69Dic262—21no—
70D524—21no—
71D524—21no—
72C22⋊D131—42no—
73C2×Dic131—22no—
74C2×Dic131—21no—
75C2×Dic131—84no—
76C2×Dic131C2×D414yes—
77C4×D131—42no—
78C131⋊C8—22no—
79C23×C131—22no—
80C23×C131—24no—
81C23×C131—82no—
82C23×C131C2×D411yescharacteristic
83Q8×C131—22no—
84Q8×C131—22no—
85Q8×C131—42no—
86D4×C131—24no—
87D4×C131—22no—
88D4×C131C2×D414yes—
89C2×C4×C131—22no—
90C2×C4×C131—42no—
91C2×C4×C131C2×D411yescharacteristic
92C2×C4×C131C2×D412yes—
93C2×C4×C131#4112yes—
94C2×C4×C131C2411yescharacteristic
95C8×C131—22no—
96C22×C131—22no—
97C22×C131—24no—
98C22×C131—44no—
99C22×C131—84no—
100C22×C131#12112yes—
101C22×C131C22×D411yescharacteristic
102C2×D131—41no—
103C2×D131—42no—
104C4×C131—42no—
105C4×C131#12112yes—
106C4×C131C22×D411yescharacteristic
107C4×C131C2×#4111yescharacteristic
108Dic131—24no—
109Dic131—41no—
110Dic131—42no—
111C2×C131—22no—
112C2×C131—44no—
113C2×C131C2×#12111yescharacteristic
114C2×C131#57712yes—
115D131—81no—
116C131#304211yescharacteristic, complemented, Sylow
117#3042—1311nomaximal, Sylow
118C2×AGL(1,ℤ/8)—1311no—
119C22×SD16D13111yescharacteristic, complemented
120#536—1311no—
121#518—1311no—
122#463—1314no—
123#461—1314no—
124#424—1311no—
125#419—1311no—
126#416—1311no—
127C2×#41—1311no—
128C22×Q8C2×D13111yescharacteristic
129C22×D4C2×D13111yescharacteristic
130AGL(1,ℤ/8)—2624no—
131C2×SD16—24no—
132C2×SD16—1312no—
133C2×SD16C2×D13114yes—
134C2×D8—1312no—
135C2×M16—1311no—
136C22×C8C2×D13111yescharacteristic
137C4⋊Q8—1312no—
138#124—1312no—
139#123—2622no—
140C22⋊Q8—2622no—
141C4×D4—2622no—
142C4×D4—1314no—
143#117—1311no—
144#117—1311no—
145C2×C22⋊C4—2622no—
146C4⋊C8—1314no—
147#103—1314no—
148#103—1312no—
149#102—1312no—
150C24—21no—
151#41—2622no—
152#41—2622no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1311no—
157C2×Q8C22×D13112yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—2621no—
161C2×D4—1311no—
162C2×D4C22⋊D13114yes—
163C2×D4C22×D13112yes—
164C22×C4—21no—
165C22×C4—2621no—
166C22×C4—2624no—
167C22×C4C22×D13111yescharacteristic
168SD16—28no—
169SD16—2624no—
170D8—2624no—
171M16—2622no—
172C2×C8—22no—
173C2×C8—1312no—
174C2×C8C22×D13112yes—
175C4⋊C4—2624no—
176C4⋊C4—1312no—
177C4⋊C4—1312no—
178C22⋊C4—2624no—
179C22⋊C4—2621no—
180C22⋊C4—2624no—
181C22⋊C4—1312no—
182C42—2621no—
183C42—1312no—
184C23—22no—
185C23—24no—
186C23—2621no—
187C23—82no—
188C23D4×D13111yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—2621no—
193Q8—2621no—
194D4—24no—
195D4—22no—
196D4—2621no—
197D4—2621no—
198D4—5242no—
199D4C2×C22⋊D13114yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—2622no—
203C2×C4—2621no—
204C2×C4—5242no—
205C2×C4—10484no—
206C2×C4—1314no—
207C2×C4D4×D13111yescharacteristic
208C2×C4C2×C22⋊D13112yes—
209C2×C4#1452351212yes—
210C2×C4C23×D13111yescharacteristic
211C8—22no—
212C8—2622no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—5242no—
218C22—5241no—
219C22#1686221812yes—
220C22C2×D4×D13111yescharacteristic
221C4—42no—
222C4—2624no—
223C4—5242no—
224C4—5241no—
225C4#1686221812yes—
226C4C22×C22⋊D13111yescharacteristic
227C4C2×#1452351211yescharacteristic
228C2—22no—
229C2—44no—
230C2—10481no—
231C2C2×#1686221811yescharacteristic
232C2#2255411812yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC22×SD16×C131C2
socleC22×C131C22×D4
prime radicalC2×#16862272C2
hypercenterC22×SD16D131
hypocenterC131#3042
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #22553886C2semidirect
3. #22553787C2semidirect
4. #22553529C2semidirect
5. #22553515C2semidirect
6. #22553338C2semidirect
7. #22553321C2semidirect
8. #22553309C2semidirect
9. C22×SD16×C131C2semidirect
15. #16862244C22semidirect
16. #16862235C22semidirect
24. Dic131⋊C8C22semidirect
25. #16862171C22semidirect
26. #16862170C22semidirect
27. #16862168C22semidirect
116. C131#3042semidirect
119. C22×SD16D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622722
#225538862
#225537872
#225535292
#225535152
#225533382
#225533212
#225533092
C22×SD16×C1312
#3042131