Finite Group Databasebeta

← #30675216

Subgroups — #30675216

Order 16768.

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

normal subgroups only

This group has 754 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
1C22×#14523512C211yesmaximal, characteristic
2C2×#16862275C212yesmaximal, complemented
3C22⋊Q8×D131C211yesmaximal, characteristic, complemented
4#22553475C212yesmaximal, complemented
5#22553471C212yesmaximal, complemented
6#22553470C211yesmaximal, characteristic, complemented
7#22553467C211yesmaximal, characteristic, complemented
8#22553463C211yesmaximal, characteristic, complemented
9#22553310C212yesmaximal, complemented
10#22553274C211yesmaximal, characteristic, complemented
11#450×C131C211yesmaximal, characteristic, complemented
12C2×#14523512—28no—
13C2×#14523512—21no—
14C2×#14523512—21no—
15C2×#14523512C2212yes—
16C2×#14523512C2212yes—
17C22×Dic262C2211yescharacteristic
18C2×Q8×D131C2211yescharacteristic, complemented
19C22×C22⋊D131—21no—
20C23×Dic131—21no—
21C22×C4×D131C2211yescharacteristic
22#16862275—28no—
23C2×Dic524C2212yescomplemented
24C2×#14523503C2212yescomplemented
25C2×#14523501C2212yescomplemented
26C2×#14523496C2212yescomplemented
27#16862226C2211yescharacteristic, complemented
28#16862225C2211yescharacteristic
29#16862224—21no—
30#16862223—21no—
31#16862222C2211yescharacteristic, complemented
32C4⋊C4×D131—21no—
33C4⋊C4×D131C2211yescharacteristic, complemented
34C22⋊C4×D131—21no—
35#16862171C2212yescomplemented
36#16862169C2212yescomplemented
37#16862167C2212yescomplemented
38C22⋊(C131⋊C8)C2211yescharacteristic
39(C2×D131)⋊C8C2212yescomplemented
40C22×D4×C131C2211yescharacteristic
41C2×SD16×C131C2212yescomplemented
42C22⋊Q8×C131C2211yescharacteristic
43#102×C131C2212yescomplemented
44C22⋊C8×C131C2211yescharacteristic
45C23×D131D411yescharacteristic
46#14523512—28no—
47#14523512—28no—
48#14523512—24no—
49#14523512—24no—
50#14523512—24no—
51#14523512D418yes—
52C2×Dic262—24no—
53C2×Dic262—21no—
54C2×Dic262—21no—
55C2×Dic262C2311yescharacteristic
56C2×Dic262C2312yes—
57Q8×D131—24no—
58C2×C22⋊D131—28no—
59C2×C22⋊D131—22no—
60C2×C22⋊D131—22no—
61C22×Dic131—28no—
62C22×Dic131—22no—
63C22×Dic131—22no—
64C22×Dic131—21no—
65C22×Dic131D411yescharacteristic
66C2×C4×D131—24no—
67C2×C4×D131—21no—
68C2×C4×D131—21no—
69C2×C4×D131D411yescharacteristic
70C2×C4×D131D411yescharacteristic
71C2×C4×D131C2311yescharacteristic
72C2×C4×D131C2311yescharacteristic
73Dic524—24no—
74#14523503—24no—
75#14523501—24no—
76#14523496—24no—
77C2×C131⋊C8C2312yes—
78C4⋊Dic131—21no—
79C4⋊Dic131C2311yescharacteristic
80Dic131⋊C4—21no—
81Dic131⋊C4—21no—
82Dic131⋊C4—21no—
83C22⋊Dic131—21no—
84(C2×D131)⋊C4—21no—
85(C2×D131)⋊C4—21no—
86C24×C131—21no—
87C2×Q8×C131C2311yescharacteristic
88C2×D4×C131—24no—
89C2×D4×C131—21no—
90C2×D4×C131C2312yes—
91C22×C4×C131C2311yescharacteristic
92SD16×C131—24no—
93C2×C8×C131C2312yes—
94C4⋊C4×C131—21no—
95C4⋊C4×C131C2311yescharacteristic
96C22⋊C4×C131—21no—
97C22×D131—22no—
98C22×D131—22no—
99C22×D131—41no—
100C22×D131C2×D411yescharacteristic
101C22×D131C2×D411yescharacteristic
102Dic262—24no—
103Dic262—22no—
104Dic262—22no—
105Dic262—42no—
106Dic262C2×D414yes—
107C22⋊D131—28no—
108C22⋊D131—28no—
109C2×Dic131—24no—
110C2×Dic131—28no—
111C2×Dic131—22no—
112C2×Dic131—22no—
113C2×Dic131—28no—
114C2×Dic131—21no—
115C2×Dic131—21no—
116C2×Dic131C2×D411yescharacteristic
117C2×Dic131C2×D411yescharacteristic
118C4×D131—22no—
119C4×D131—22no—
120C4×D131—22no—
121C4×D131—42no—
122C4×D131—42no—
123C4×D131C2×D414yes—
124C131⋊C8—22no—
125C23×C131—24no—
126C23×C131—22no—
127C23×C131—42no—
128C23×C131—42no—
129C23×C131C2×D411yescharacteristic
130Q8×C131—22no—
131D4×C131—24no—
132D4×C131—22no—
133D4×C131C2×D414yes—
134C2×C4×C131—22no—
135C2×C4×C131—21no—
136C2×C4×C131—21no—
137C2×C4×C131C2×D411yescharacteristic
138C2×C4×C131C2411yescharacteristic
139C8×C131—22no—
140C22×C131—21no—
141C22×C131—21no—
142C22×C131—24no—
143C22×C131—48no—
144C22×C131—42no—
145C22×C131—42no—
146C22×C131#13712yescomplemented
147C22×C131C22×D411yescharacteristic
148C2×D131—21no—
149C2×D131—21no—
150C2×D131—21no—
151C2×D131—21no—
152C2×D131—21no—
153C2×D131—44no—
154C2×D131#12012yes—
155C4×C131—21no—
156C4×C131—41no—
157C4×C131—41no—
158C4×C131C22×D412yes—
159Dic131—21no—
160Dic131—24no—
161Dic131—41no—
162Dic131—41no—
163Dic131#12012yes—
164C2×C131—22no—
165C2×C131—44no—
166C2×C131C2×#12011yescharacteristic
167C2×C131C2×#13711yescharacteristic
168C2×C131#57511yescharacteristic
169D131—22no—
170D131—41no—
171C131#300711yescharacteristic, complemented, Sylow
172#3007—1311nomaximal, Sylow
173C22×#41—1311no—
174C2×#137—1312no—
175C2×C22⋊Q8—1311no—
176#452—1312no—
177#452—1312no—
178#451—1311no—
179#451—1311no—
180#450—1311no—
181#450D13111yescharacteristic, complemented
182#417—1312no—
183#407—1311no—
184C2×#41—2628no—
185C2×#41—2621no—
186C2×#41—2621no—
187C2×#41—1312no—
188C2×#41—1312no—
189C22×Q8—1311no—
190C22×Q8—1311no—
191C22×D4—2621no—
192C22×D4C2×D13111yescharacteristic
193C23×C4—2621no—
194C23×C4—1311no—
195#137—2628no—
196C2×Q16—1312no—
197C2×Q16—1312no—
198C2×SD16—1312no—
199C2×SD16C2×D13112yescomplemented
200C2×M16—1312no—
201C22⋊Q8—2621no—
202C22⋊Q8—2621no—
203C22⋊Q8—1311no—
204C22⋊Q8—1311no—
205C22⋊Q8—1311no—
206C22⋊Q8C2×D13111yescharacteristic
207C2×C4⋊C4—2621no—
208C2×C4⋊C4—1311no—
209C2×C22⋊C4—2621no—
210#103—1312no—
211#103—1312no—
212#102—1312no—
213#102C2×D13112yescomplemented
214C22⋊C8—1311no—
215C22⋊C8—1312no—
216C22⋊C8C2×D13111yescharacteristic
217C24—21no—
218C24—1311no—
219#41—2628no—
220#41—2628no—
221#41—2624no—
222#41—2624no—
223#41—2624no—
224#41—1318no—
225C2×Q8—2624no—
226C2×Q8—2621no—
227C2×Q8—2621no—
228C2×Q8—2624no—
229C2×Q8—1311no—
230C2×Q8—1312no—
231C2×Q8C22×D13111yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—2628no—
235C2×D4—2622no—
236C2×D4—2622no—
237C2×D4C22×D13112yes—
238C22×C4—2624no—
239C22×C4—2621no—
240C22×C4—2621no—
241C22×C4—2628no—
242C22×C4—2621no—
243C22×C4—2622no—
244C22×C4—2622no—
245C22×C4—1311no—
246C22×C4—1311no—
247C22×C4—1311no—
248C22×C4—1311no—
249C22×C4—1311no—
250C22×C4C22×D13111yescharacteristic
251Q16—2624no—
252Q16—2624no—
253SD16—24no—
254SD16—2624no—
255M16—2624no—
256C2×C8—1312no—
257C2×C8C22×D13112yes—
258C4⋊C4—21no—
259C4⋊C4—2621no—
260C4⋊C4—2621no—
261C4⋊C4—2621no—
262C4⋊C4—2621no—
263C4⋊C4—1311no—
264C4⋊C4C22×D13111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2621no—
267C22⋊C4—2621no—
268C22⋊C4—2621no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—2622no—
274C23—2622no—
275C23—5241no—
276C23—1311no—
277C23—1311no—
278C23D4×D13111yescharacteristic
279Q8—22no—
280Q8—2624no—
281Q8—2622no—
282Q8—2622no—
283Q8—5242no—
284Q8—1314no—
285D4—24no—
286D4—22no—
287D4—2628no—
288D4—2628no—
289D4D4×D13114yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—2622no—
294C2×C4—2622no—
295C2×C4—2624no—
296C2×C4—2628no—
297C2×C4—2622no—
298C2×C4—2622no—
299C2×C4—2628no—
300C2×C4—2622no—
301C2×C4—2621no—
302C2×C4—2621no—
303C2×C4—5242no—
304C2×C4—5242no—
305C2×C4—1311no—
306C2×C4—1311no—
307C2×C4—1314no—
308C2×C4D4×D13111yescharacteristic
309C2×C4C23×D13111yescharacteristic
310C8—22no—
311C8—2622no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—2621no—
319C22—2621no—
320C22—2621no—
321C22—2621no—
322C22—2621no—
323C22—5244no—
324C22—1312no—
325C22#1686227512yescomplemented
326C22C2×D4×D13111yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—2621no—
331C4—2624no—
332C4—5241no—
333C4—5241no—
334C4—1312no—
335C4C2×D4×D13112yes—
336C2—22no—
337C2—44no—
338C2—2622no—
339C2—5241no—
340C2#120×D13111yescharacteristic
341C2C2×#1686227511yescharacteristic
342C2#2255410811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#450×C131C2
socleC22×C131C22×D4
prime radicalC22×#14523512C2
hypercenter#450D131
hypocenterC131#3007
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#16862275C2semidirect
3. C22⋊Q8×D131C2semidirect
4. #22553475C2semidirect
5. #22553471C2semidirect
6. #22553470C2semidirect
7. #22553467C2semidirect
8. #22553463C2semidirect
9. #22553310C2semidirect
10. #22553274C2semidirect
11. #450×C131C2semidirect
18. C2×Q8×D131C22semidirect
23. C2×Dic524C22semidirect
24. C2×#14523503C22semidirect
25. C2×#14523501C22semidirect
26. C2×#14523496C22semidirect
27. #16862226C22semidirect
31. #16862222C22semidirect
33. C4⋊C4×D131C22semidirect
35. #16862171C22semidirect
36. #16862169C22semidirect
37. #16862167C22semidirect
39. (C2×D131)⋊C8C22semidirect
41. C2×SD16×C131C22semidirect
43. #102×C131C22semidirect
146. C22×C131#137semidirect
171. C131#3007semidirect
181. #450D131semidirect
199. C2×SD16C2×D131semidirect
213. #102C2×D131semidirect
325. C22#16862275semidirect

Maximal subgroups

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

subgroupindex
C22×#145235122
C2×#168622752
C22⋊Q8×D1312
#225534752
#225534712
#225534702
#225534672
#225534632
#225533102
#225532742
#450×C1312
#3007131