Finite Group Databasebeta

← #30675594

Subgroups — #30675594

Order 16768.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16862276C211yesmaximal, characteristic, complemented
2C2×M16×D131C211yesmaximal, characteristic, complemented
3#22553819C211yesmaximal, characteristic
4C2×#16862226C211yesmaximal, characteristic, complemented
5#22553569C212yesmaximal, complemented
6#22553557C212yesmaximal, complemented
7#22553552C212yesmaximal, complemented
8#22553540C212yesmaximal, complemented
9#22553352C211yesmaximal, characteristic, complemented
10#22553322C211yesmaximal, characteristic, complemented
11#417×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2211yescharacteristic
13C22×C4×D131C2211yescharacteristic
14#16862276—24no—
15C2×#14523503C2212yescomplemented
16C2×#14523501C2212yescomplemented
17C2×#14523497C2211yescharacteristic
18C2×#14523496C2212yes—
19M16×D131—24no—
20C2×C8×D131C2212yes—
21#16862227—21no—
22#16862226—21no—
23#16862226—21no—
24#16862226C2212yes—
25#16862226C2212yes—
26#16862222C2211yescharacteristic
27#16862222C2211yescharacteristic
28#16862217C2211yescharacteristic
29#16862217C2211yescharacteristic
30#16862214—21no—
31C4×C22⋊D131—21no—
32C4×C22⋊D131—21no—
33#16862205C2211yescharacteristic
34C2×C4⋊Dic131—21no—
35C2×(C2×D131)⋊C4—21no—
36#16862179C2212yescomplemented
37#16862177C2212yescomplemented
38#16862171C2212yescomplemented
39#16862171C2212yescomplemented
40C2×#41×C131C2211yescharacteristic
41C2×M16×C131C2211yescharacteristic
42C2×C4⋊C4×C131C2211yescharacteristic
43#103×C131C2212yescomplemented
44#102×C131C2212yescomplemented
45C23×D131D411yescharacteristic
46C2×Dic262—21no—
47C2×Dic262—22no—
48C2×Dic262—22no—
49C2×Dic262C2312yes—
50C2×C22⋊D131—21no—
51C2×C22⋊D131—21no—
52C22×Dic131—21no—
53C22×Dic131D411yescharacteristic
54C2×C4×D131—22no—
55C2×C4×D131—22no—
56C2×C4×D131D411yescharacteristic
57C2×C4×D131D411yescharacteristic
58C2×C4×D131C2311yescharacteristic
59C2×C4×D131C2311yescharacteristic
60#14523503—24no—
61#14523501—24no—
62#14523497D414yescomplemented
63#14523496—24no—
64C2×C131⋊C8C2312yes—
65C8×D131—24no—
66C4⋊Dic131—22no—
67C4⋊Dic131—22no—
68C4⋊Dic131C2311yescharacteristic
69C4⋊Dic131C2311yescharacteristic
70Dic131⋊C4—21no—
71Dic131⋊C4—21no—
72C22⋊Dic131—21no—
73C22⋊Dic131—21no—
74C22⋊Dic131—21no—
75(C2×D131)⋊C4—21no—
76(C2×D131)⋊C4—21no—
77(C2×D131)⋊C4—22no—
78(C2×D131)⋊C4—22no—
79C4×Dic131—21no—
80#41×C131—22no—
81#41×C131—22no—
82C2×Q8×C131C2311yescharacteristic
83C2×D4×C131—21no—
84C2×D4×C131C2311yescharacteristic
85C22×C4×C131—21no—
86C22×C4×C131—21no—
87C22×C4×C131C2311yescharacteristic
88M16×C131—22no—
89C2×C8×C131C2312yes—
90C4⋊C4×C131—21no—
91C4⋊C4×C131C2312yes—
92C22×D131—22no—
93C22×D131—22no—
94C22×D131—41no—
95C22×D131C2×D411yescharacteristic
96C22×D131C2×D411yescharacteristic
97Dic262—22no—
98Dic262—22no—
99Dic262—42no—
100C22⋊D131—42no—
101C22⋊D131—42no—
102C2×Dic131—22no—
103C2×Dic131—21no—
104C2×Dic131—21no—
105C2×Dic131—22no—
106C2×Dic131—42no—
107C2×Dic131C2×D411yescharacteristic
108C2×Dic131C2×D411yescharacteristic
109C4×D131—22no—
110C4×D131—22no—
111C4×D131—21no—
112C4×D131—21no—
113C4×D131—21no—
114C4×D131—21no—
115C131⋊C8C2×D414yes—
116C23×C131—21no—
117C23×C131C2×D411yescharacteristic
118Q8×C131—21no—
119Q8×C131—21no—
120D4×C131—21no—
121D4×C131—21no—
122D4×C131—42no—
123C2×C4×C131—22no—
124C2×C4×C131—21no—
125C2×C4×C131—42no—
126C2×C4×C131—42no—
127C2×C4×C131C2×D411yescharacteristic
128C2×C4×C131C2×D412yes—
129C2×C4×C131#4112yes—
130C2×C4×C131C2411yescharacteristic
131C8×C131—22no—
132C22×C131—21no—
133C22×C131—21no—
134C22×C131—41no—
135C22×C131—41no—
136C22×C131—41no—
137C22×C131#12112yes—
138C22×C131C22×D411yescharacteristic
139C2×D131—21no—
140C2×D131—21no—
141C2×D131—21no—
142C2×D131—21no—
143C2×D131—21no—
144C2×D131—44no—
145C2×D131#13712yescomplemented
146C4×C131—41no—
147C4×C131—42no—
148C4×C131#12112yes—
149C4×C131C22×D411yescharacteristic
150C4×C131C2×#4111yescharacteristic
151Dic131—21no—
152Dic131—21no—
153Dic131—42no—
154Dic131—41no—
155Dic131—41no—
156C2×C131—22no—
157C2×C131—81no—
158C2×C131C2×#12111yescharacteristic
159C2×C131C2×#13711yescharacteristic
160C2×C131#57511yescharacteristic
161D131—22no—
162D131—41no—
163C131#305611yescharacteristic, complemented, Sylow
164#3056—1311nomaximal, Sylow
165C2×#137—1311no—
166C22×M16—1311no—
167#525—1311no—
168C2×C22⋊Q8—1311no—
169#470—1312no—
170#468—1312no—
171#467—1312no—
172#465—1312no—
173#428—1311no—
174#419—1311no—
175#417D13111yescharacteristic, complemented
176C2×#41C2×D13111yescharacteristic
177C22×Q8—1311no—
178C23×C4—1311no—
179#137—2624no—
180C2×Q16—1312no—
181C2×SD16—1312no—
182C2×M16—2624no—
183C2×M16—1311no—
184C2×M16—1312no—
185C2×M16C2×D13111yescharacteristic
186C22×C8—1312no—
187#123—2621no—
188C22⋊Q8—2621no—
189C22⋊Q8—2621no—
190C22⋊Q8—1311no—
191C22⋊Q8—1311no—
192C22⋊Q8—1312no—
193C22⋊Q8—1312no—
194#121—1311no—
195#121—1311no—
196#120—2621no—
197C4×D4—2621no—
198C4×D4—2621no—
199#117—1311no—
200C2×C4⋊C4—2621no—
201C2×C4⋊C4C2×D13111yescharacteristic
202C2×C22⋊C4—2621no—
203#107—1312no—
204#106—1312no—
205#103—1312no—
206#103—1312no—
207#103C2×D13112yescomplemented
208#102C2×D13112yescomplemented
209C24—1311no—
210#41—22no—
211#41—22no—
212C2×Q8—2621no—
213C2×Q8—2622no—
214C2×Q8—2622no—
215C2×Q8—1312no—
216C2×Q8C22×D13111yescharacteristic
217C2×D4—21no—
218C2×D4—2621no—
219C2×D4—2621no—
220C2×D4C22×D13111yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—2622no—
224C22×C4—2622no—
225C22×C4—2621no—
226C22×C4—1311no—
227C22×C4—1311no—
228C22×C4—1311no—
229C22×C4—1311no—
230C22×C4—1311no—
231C22×C4C22×D13111yescharacteristic
232Q16—2624no—
233SD16—2624no—
234M16—22no—
235M16—2624no—
236M16—1314no—
237C2×C8—2624no—
238C2×C8—1312no—
239C2×C8C22×D13112yes—
240C4⋊C4—21no—
241C4⋊C4—2621no—
242C4⋊C4—2621no—
243C4⋊C4—2622no—
244C4⋊C4—2622no—
245C4⋊C4—1311no—
246C4⋊C4—1311no—
247C4⋊C4C22×D13112yes—
248C22⋊C4—2622no—
249C22⋊C4—2622no—
250C22⋊C4—2621no—
251C22⋊C4—2621no—
252C22⋊C4—2621no—
253C22⋊C4—2621no—
254C22⋊C4—2621no—
255C42—2621no—
256C23—21no—
257C23—2622no—
258C23—2622no—
259C23—5241no—
260C23—1311no—
261C23—1311no—
262C23D4×D13111yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—2622no—
266Q8—2622no—
267Q8—5242no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—5242no—
272D4—5242no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—2622no—
278C2×C4—2622no—
279C2×C4—2622no—
280C2×C4—2621no—
281C2×C4—2621no—
282C2×C4—2621no—
283C2×C4—2621no—
284C2×C4—2621no—
285C2×C4—2621no—
286C2×C4—2622no—
287C2×C4—5242no—
288C2×C4—1311no—
289C2×C4—1311no—
290C2×C4D4×D13111yescharacteristic
291C2×C4D4×D13112yes—
292C2×C4#1452351312yes—
293C2×C4C23×D13111yescharacteristic
294C8—22no—
295C8—1314no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—2621no—
302C22—2621no—
303C22—2621no—
304C22—2621no—
305C22—2621no—
306C22—5244no—
307C22—1312no—
308C22#1686221912yes—
309C22C2×D4×D13111yescharacteristic
310C4—41no—
311C4—42no—
312C4—2621no—
313C4—2621no—
314C4—5242no—
315C4—5241no—
316C4—5241no—
317C4#1686221912yes—
318C4C2×D4×D13111yescharacteristic
319C4C2×#1452351311yescharacteristic
320C2—22no—
321C2—2622no—
322C2—81no—
323C2—5241no—
324C2C2×#1686221911yescharacteristic
325C2#137×D13111yescharacteristic
326C2#2255410811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#417×C131C2
socleC22×C131C22×D4
prime radical#22553819C2
hypercenter#417D131
hypocenterC131#3056
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16862276C2semidirect
2. C2×M16×D131C2semidirect
4. C2×#16862226C2semidirect
5. #22553569C2semidirect
6. #22553557C2semidirect
7. #22553552C2semidirect
8. #22553540C2semidirect
9. #22553352C2semidirect
10. #22553322C2semidirect
11. #417×C131C2semidirect
15. C2×#14523503C22semidirect
16. C2×#14523501C22semidirect
36. #16862179C22semidirect
37. #16862177C22semidirect
38. #16862171C22semidirect
39. #16862171C22semidirect
43. #103×C131C22semidirect
44. #102×C131C22semidirect
62. #14523497D4semidirect
145. C2×D131#137semidirect
163. C131#3056semidirect
175. #417D131semidirect
207. #103C2×D131semidirect
208. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622762
C2×M16×D1312
#225538192
C2×#168622262
#225535692
#225535572
#225535522
#225535402
#225533522
#225533222
#417×C1312
#3056131