Finite Group Databasebeta

← #30675906

Subgroups — #30675906

Order 16768.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C22⋊Q8×D131C211yesmaximal, characteristic
2C2×#16862204C211yesmaximal, characteristic
3#22553655C212yesmaximal, complemented
4#22553651C212yesmaximal, complemented
5#22553647C211yesmaximal, characteristic, complemented
6#22553643C211yesmaximal, characteristic, complemented
7#22553640C211yesmaximal, characteristic, complemented
8#22553353C212yesmaximal, complemented
9#22553319C212yesmaximal, complemented
10#22553274C211yesmaximal, characteristic
11#484×C131C211yesmaximal, characteristic, complemented
12C2×Q8×D131C2211yescharacteristic
13C22×C4×D131C2211yescharacteristic
14C2×#14523496C2212yes—
15#16862226C2211yescharacteristic
16#16862225C2211yescharacteristic
17#16862224—21no—
18#16862223—21no—
19#16862222C2211yescharacteristic
20#16862204—21no—
21#16862204—21no—
22#16862204C2212yes—
23#16862204C2212yes—
24C2×C4⋊Dic131C2211yescharacteristic
25C4⋊C4×D131—21no—
26C4⋊C4×D131C2211yescharacteristic
27C2×(C2×D131)⋊C4—21no—
28C22⋊C4×D131—21no—
29C2×C4×Dic131—21no—
30#16862179C2212yescomplemented
31#16862178C2212yescomplemented
32#16862177C2212yescomplemented
33#16862171C2212yescomplemented
34#16862170C2212yescomplemented
35#16862169C2212yescomplemented
36C22⋊(C131⋊C8)C2211yescharacteristic
37(C2×D131)⋊C8C2212yes—
38C22⋊Q8×C131C2211yescharacteristic
39C2×C4⋊C4×C131C2211yescharacteristic
40#107×C131C2212yescomplemented
41#103×C131C2212yescomplemented
42C22⋊C8×C131C2211yescharacteristic
43C23×D131D411yescharacteristic
44C2×Dic262—21no—
45C2×Dic262C2311yescharacteristic
46Q8×D131—24no—
47C22×Dic131—21no—
48C22×Dic131D411yescharacteristic
49C2×C4×D131—24no—
50C2×C4×D131—21no—
51C2×C4×D131—21no—
52C2×C4×D131D411yescharacteristic
53C2×C4×D131D411yescharacteristic
54C2×C4×D131C2311yescharacteristic
55C2×C4×D131C2311yescharacteristic
56#14523496—24no—
57C2×C131⋊C8C2312yes—
58C4⋊Dic131—21no—
59C4⋊Dic131—21no—
60C4⋊Dic131C2311yescharacteristic
61C4⋊Dic131C2312yes—
62Dic131⋊C4—21no—
63Dic131⋊C4—21no—
64Dic131⋊C4—21no—
65C22⋊Dic131—21no—
66(C2×D131)⋊C4—21no—
67(C2×D131)⋊C4—21no—
68(C2×D131)⋊C4—22no—
69(C2×D131)⋊C4—22no—
70C4×Dic131—22no—
71C4×Dic131—22no—
72C2×Q8×C131C2311yescharacteristic
73C22×C4×C131—21no—
74C22×C4×C131C2311yescharacteristic
75C2×C8×C131C2312yes—
76C4⋊C4×C131—21no—
77C4⋊C4×C131—21no—
78C4⋊C4×C131C2311yescharacteristic
79C4⋊C4×C131C2312yes—
80C22⋊C4×C131—21no—
81C22×D131—22no—
82C22×D131—22no—
83C22×D131—41no—
84C22×D131C2×D411yescharacteristic
85C22×D131C2×D411yescharacteristic
86Dic262—22no—
87Dic262—42no—
88C2×Dic131—22no—
89C2×Dic131—22no—
90C2×Dic131—21no—
91C2×Dic131—21no—
92C2×Dic131—42no—
93C2×Dic131C2×D411yescharacteristic
94C2×Dic131C2×D411yescharacteristic
95C4×D131—22no—
96C4×D131—22no—
97C4×D131—22no—
98C4×D131—42no—
99C4×D131—42no—
100C4×D131#4114yes—
101C131⋊C8—22no—
102C23×C131C2×D411yescharacteristic
103Q8×C131—22no—
104C2×C4×C131—22no—
105C2×C4×C131—21no—
106C2×C4×C131—21no—
107C2×C4×C131—22no—
108C2×C4×C131—42no—
109C2×C4×C131C2×D411yescharacteristic
110C2×C4×C131C2411yescharacteristic
111C8×C131—22no—
112C22×C131—21no—
113C22×C131—21no—
114C22×C131#13712yes—
115C22×C131C22×D411yescharacteristic
116C2×D131—21no—
117C2×D131—21no—
118C2×D131—21no—
119C2×D131—21no—
120C2×D131—21no—
121C2×D131—44no—
122C2×D131#12312yes—
123C4×C131—21no—
124C4×C131—41no—
125C4×C131—42no—
126C4×C131—41no—
127C4×C131C2×#4112yes—
128Dic131—21no—
129Dic131—41no—
130Dic131—42no—
131Dic131—41no—
132Dic131#12312yes—
133C2×C131—22no—
134C2×C131C2×#12311yescharacteristic
135C2×C131C2×#13711yescharacteristic
136C2×C131#57511yescharacteristic
137D131—22no—
138D131—41no—
139C131#309511yescharacteristic, complemented, Sylow
140#3095—1311nomaximal, Sylow
141C2×C22⋊Q8—1311no—
142C2×#117—1311no—
143#485—1312no—
144#485—1312no—
145#484—1311no—
146#484D13111yescharacteristic, complemented
147#483—1311no—
148#483—1311no—
149#428—1312no—
150#419—1312no—
151#407—1311no—
152C22×Q8—1311no—
153C23×C4—1311no—
154C2×M16—1312no—
155C22⋊Q8—2621no—
156C22⋊Q8—2621no—
157C22⋊Q8—1311no—
158C22⋊Q8—1311no—
159C22⋊Q8—1311no—
160C22⋊Q8C2×D13111yescharacteristic
161#117—2621no—
162#117—2621no—
163#117—1312no—
164#117—1312no—
165C2×C4⋊C4—2621no—
166C2×C4⋊C4—1311no—
167C2×C4⋊C4—1311no—
168C2×C4⋊C4C2×D13111yescharacteristic
169C2×C22⋊C4—2621no—
170C2×C22⋊C4—2621no—
171C2×C42—2621no—
172#107—1312no—
173#107C2×D13112yescomplemented
174#106—1312no—
175#106—1312no—
176#103—1312no—
177#103—1312no—
178#103—1312no—
179#103C2×D13112yescomplemented
180C22⋊C8—1311no—
181C22⋊C8—1312no—
182C22⋊C8C2×D13111yescharacteristic
183C24—1311no—
184C2×Q8—2621no—
185C2×Q8—2624no—
186C2×Q8—1311no—
187C2×Q8C22×D13111yescharacteristic
188C22×C4—21no—
189C22×C4—2624no—
190C22×C4—2621no—
191C22×C4—2621no—
192C22×C4—2621no—
193C22×C4—1311no—
194C22×C4—1311no—
195C22×C4—1311no—
196C22×C4—1311no—
197C22×C4—1311no—
198C22×C4C22×D13111yescharacteristic
199M16—2624no—
200C2×C8—1312no—
201C2×C8C22×D13112yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—2621no—
205C4⋊C4—2621no—
206C4⋊C4—2621no—
207C4⋊C4—2621no—
208C4⋊C4—2621no—
209C4⋊C4—1312no—
210C4⋊C4—1311no—
211C4⋊C4C22×D13112yes—
212C4⋊C4C22×D13111yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—2622no—
215C22⋊C4—2622no—
216C22⋊C4—2621no—
217C22⋊C4—2621no—
218C22⋊C4—2621no—
219C42—2622no—
220C42—2622no—
221C23—2622no—
222C23—2622no—
223C23—5241no—
224C23—1311no—
225C23—1311no—
226C23D4×D13111yescharacteristic
227Q8—22no—
228Q8—2622no—
229Q8—5242no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—2622no—
236C2×C4—2622no—
237C2×C4—2622no—
238C2×C4—2622no—
239C2×C4—2622no—
240C2×C4—2621no—
241C2×C4—2621no—
242C2×C4—5242no—
243C2×C4—5242no—
244C2×C4—5242no—
245C2×C4—1311no—
246C2×C4—1311no—
247C2×C4—1314no—
248C2×C4D4×D13111yescharacteristic
249C2×C4C23×D13111yescharacteristic
250C8—22no—
251C8—2622no—
252C22—21no—
253C22—21no—
254C22—2621no—
255C22—2621no—
256C22—2621no—
257C22—2621no—
258C22—2621no—
259C22—5244no—
260C22—1312no—
261C22#1686227812yes—
262C22C2×D4×D13111yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—2621no—
268C4—5241no—
269C4—5242no—
270C4—5241no—
271C4—1312no—
272C4#41×D13112yes—
273C2—22no—
274C2—2622no—
275C2—5241no—
276C2#123×D13111yescharacteristic
277C2C2×#1686227811yescharacteristic
278C2#2255410811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#484×C131C2
socleC22×C131C22×D4
prime radicalC23×D131D4
hypercenter#484D131
hypocenterC131#3095
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #22553655C2semidirect
4. #22553651C2semidirect
5. #22553647C2semidirect
6. #22553643C2semidirect
7. #22553640C2semidirect
8. #22553353C2semidirect
9. #22553319C2semidirect
11. #484×C131C2semidirect
30. #16862179C22semidirect
31. #16862178C22semidirect
32. #16862177C22semidirect
33. #16862171C22semidirect
34. #16862170C22semidirect
35. #16862169C22semidirect
40. #107×C131C22semidirect
41. #103×C131C22semidirect
139. C131#3095semidirect
146. #484D131semidirect
173. #107C2×D131semidirect
179. #103C2×D131semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1312
C2×#168622042
#225536552
#225536512
#225536472
#225536432
#225536402
#225533532
#225533192
#225532742
#484×C1312
#3095131