Finite Group Databasebeta

← #29711974

Subgroups — #29711974

Order 15616.

1318 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 191 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×#16840374C211yesmaximal, characteristic, complemented
2C2×#16839915C212yesmaximal
3#21448399C211yesmaximal, characteristic
4#21446966C212yesmaximal, complemented
5#21446904C212yesmaximal, complemented
6#21446173C214yesmaximal, complemented
7#21446129C212yesmaximal, complemented
8C61×#1818C211yesmaximal, characteristic, complemented
9C23×D244C2211yescharacteristic
10C22×#11758190C2214yes—
11#16840374—28no—
12C2×#14519777C2211yescharacteristic, complemented
13C2×#14519770—22no—
14C2×#14519769C2211yescharacteristic, complemented
15C2×#14519767—22no—
16C2×#14519756C2212yes—
17C2×C4×Dic122C2212yescomplemented
18C2×#14519736C2211yescharacteristic
19#16839915—28no—
20C2×#14519702C2214yescomplemented
21C2×#14519699C2214yes—
22#16839664C2212yes—
23#16839664C2212yes—
24#16839663—22no—
25#16839617—22no—
26#16839525C2214yescomplemented
27C2×#117×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2212yes—
29C61×#336C2212yescomplemented
30C61×#336C2212yescomplemented
31C24×D61—21no—
32C22×Dic122C2312yes—
33C22×D244—21no—
34C22×D244—24no—
35C22×D244—24no—
36C22×D244—42no—
37C22×D244C2312yes—
38C2×#11758190—28no—
39C2×#11758190—28no—
40C2×#11758190—44no—
41#14519777—24no—
42#14519770—216no—
43#14519769—24no—
44#14519767—216no—
45#14519756—24no—
46#14519756D418yes—
47C4×Dic122—24no—
48C4×Dic122—24no—
49#14519736—42no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61C2312yes—
52C2×C4⋊Dic61C2312yes—
53C2×#11758196—22no—
54C2×#11758196—22no—
55C2×#11758196—22no—
56C2×C22⋊Dic61—22no—
57C2×C22⋊Dic61—21no—
58C2×#11758198—24no—
59C2×#11758198—24no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61—21no—
62#14519702—44no—
63#14519699—28no—
64C23×C4×C61C2311yescharacteristic
65C22×C8×C61C2314yes—
66#117×C61—24no—
67C2×C4⋊C4×C61D412yes—
68C2×C22⋊C4×C61D412yes—
69C2×C42×C61C2311yescharacteristic
70C2×C42×C61C2311yescharacteristic
71C22⋊C8×C61—24no—
72C24×C61C2×D411yescharacteristic
73C23×D61—22no—
74C23×D61—24no—
75C23×D61—42no—
76C23×D61—42no—
77C23×D61—84no—
78C22×C4×C61—22no—
79C22×C4×C61—22no—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D414yes—
84C22×C4×C61C2411yescharacteristic
85C22×Dic61—22no—
86C22×Dic61—22no—
87C22×Dic61—22no—
88C2×D244—24no—
89C2×D244—24no—
90C2×D244—24no—
91C2×D244—22no—
92C2×D244—416no—
93C2×D244—42no—
94C2×D244—42no—
95C2×D244#4114yes—
96C2×Dic122—24no—
97C2×Dic122—24no—
98C2×Dic122—22no—
99C2×Dic122C2×D414yes—
100C2×C8×C61—24no—
101C2×C8×C61—24no—
102C2×C8×C61—24no—
103C4⋊C4×C61—24no—
104C22⋊C4×C61—24no—
105C42×C61—22no—
106C42×C61—22no—
107C22⋊Dic61—28no—
108C22⋊Dic61—42no—
109#11758198—84no—
110#11758198—84no—
111C4⋊Dic61—28no—
112C4⋊Dic61—22no—
113C4⋊Dic61—24no—
114C4⋊Dic61C2×D414yes—
115#11758196—28no—
116#11758196—28no—
117#11758196—28no—
118C4×Dic61—28no—
119C4×Dic61—42no—
120#11758190—416no—
121C23×C61—21no—
122C23×C61—21no—
123C23×C61—24no—
124C23×C61#12012yes—
125C23×C61C22×D411yescharacteristic
126C22×D61—24no—
127C22×D61—48no—
128C22×D61—42no—
129C22×D61—42no—
130C22×D61—44no—
131C22×D61—44no—
132C22×D61—88no—
133C22×D61—88no—
134C22×D61—88no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—28no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—24no—
141C2×C4×C61—24no—
142C2×C4×C61#12012yes—
143C2×C4×C61C22×D412yes—
144C2×C4×C61C2×#4112yes—
145C2×C4×C61H(5,𝔽2)12yes—
146C2×Dic61—24no—
147C2×Dic61—24no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—44no—
151C2×Dic61—42no—
152C2×Dic61—44no—
153D244—28no—
154D244—48no—
155D244—44no—
156Dic122—28no—
157Dic122—44no—
158C8×C61—44no—
159#23942—611nomaximal, Sylow
160C22×C61—22no—
161C22×C61—24no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61C2×#12011yescharacteristic
165C22×C61#52511yescharacteristic
166C22×C61#53511yescharacteristic
167C22×C61#57714yes—
168C2×D61—48no—
169C2×D61—44no—
170C2×D61—816no—
171C2×D61—82no—
172C2×D61—82no—
173C2×D61—84no—
174C4×C61—22no—
175C4×C61—42no—
176C4×C61—42no—
177C4×C61#54614yes—
178Dic61—44no—
179Dic61—44no—
180Dic61—82no—
181C2×#541—611no—
182C2×#450—612no—
183#2309—611no—
184#2025—612no—
185#2013—612no—
186#1880—614no—
187#1873—612no—
188#1818D6111yescharacteristic, complemented
189C2×C61—42no—
190C2×C61#240411yescharacteristic
191C2×C61#301012yes—
192C2×C61#312312yes—
193C2×C61#322212yes—
194D61—84no—
195C23×D4—611no—
196C22×SD16—614no—
197#541—1228no—
198C2×C4⋊Q8—611no—
199C2×#126—1222no—
200C2×#125—611no—
201C2×#125—1222no—
202C2×C22⋊Q8—612no—
203C2×C4×Q8—612no—
204C2×#117—611no—
205C2×#117C2×D6111yescharacteristic
206#450—1228no—
207C2×#103—614no—
208C2×#102—614no—
209C2×C22⋊C8C2×D6112yes—
210#390—612no—
211#390—612no—
212#390—1222no—
213#379—1222no—
214#340—614no—
215#336C2×D6112yescomplemented
216#336C2×D6112yescomplemented
217C61#2394211yescharacteristic, complemented, Sylow
218C25—1221no—
219C22×Q8—612no—
220C22×D4—612no—
221C22×D4—1224no—
222C22×D4—1221no—
223C22×D4—1224no—
224C22×D4—2442no—
225C23×C4C22×D6111yescharacteristic
226C2×SD16—1228no—
227C2×SD16—1228no—
228C2×SD16—2444no—
229C22×C8C22×D6114yes—
230C4⋊Q8—1224no—
231#126—12216no—
232#125—1224no—
233#125—12216no—
234C22⋊Q8—618no—
235C22⋊Q8—1224no—
236C4×Q8—1224no—
237C4×Q8—1224no—
238#117—24no—
239#117—2442no—
240C2×C4⋊C4—612no—
241C2×C4⋊C4—612no—
242C2×C4⋊C4—1222no—
243C2×C4⋊C4—1222no—
244C2×C4⋊C4—1222no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4D24412yes—
247C2×C22⋊C4—1222no—
248C2×C22⋊C4—1224no—
249C2×C22⋊C4—1224no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4D24412yes—
252C2×C42—1222no—
253C2×C42—1221no—
254C2×C42C22×D6111yescharacteristic
255C2×C42C22×D6111yescharacteristic
256#103—2444no—
257#102—1228no—
258C22⋊C8—24no—
259C24—1224no—
260C24—1222no—
261C24—4884no—
262C24—2442no—
263C24—2442no—
264C24C2×D24411yescharacteristic
265C2×Q8—614no—
266C2×Q8—1222no—
267C2×Q8—1224no—
268C2×Q8—1224no—
269C2×D4—614no—
270C2×D4—1224no—
271C2×D4—1224no—
272C2×D4—1222no—
273C2×D4—1224no—
274C2×D4—24416no—
275C2×D4—2442no—
276C2×D4—2442no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—1222no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4C2×D24412yes—
283C22×C4C2×D24412yes—
284C22×C4C2×D24411yescharacteristic
285C22×C4D4×D6114yes—
286C22×C4C23×D6111yescharacteristic
287SD16—24416no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C4⋊C4—24no—
292C4⋊C4—614no—
293C4⋊C4—1228no—
294C4⋊C4—1228no—
295C4⋊C4—1228no—
296C4⋊C4—1222no—
297C4⋊C4—1228no—
298C4⋊C4—1224no—
299C22⋊C4—24no—
300C22⋊C4—1228no—
301C22⋊C4—4884no—
302C22⋊C4—4884no—
303C22⋊C4—2442no—
304C42—22no—
305C42—22no—
306C42—1228no—
307C42—2442no—
308C23—21no—
309C23—24no—
310C23—21no—
311C23—1224no—
312C23—4888no—
313C23—4888no—
314C23—4888no—
315C23—2448no—
316C23—2442no—
317C23—2444no—
318C23—2444no—
319C23—2442no—
320C23#1451974712yes—
321C23C22×D24411yescharacteristic
322Q8—1228no—
323Q8—2444no—
324D4—1228no—
325D4—2448no—
326D4—2444no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—28no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—1224no—
335C2×C4—1224no—
336C2×C4—2444no—
337C2×C4—2442no—
338C2×C4—2442no—
339C2×C4—2444no—
340C2×C4—2442no—
341C2×C4#1451974712yes—
342C2×C4C2×D4×D6112yes—
343C2×C4#41×D6112yes—
344C2×C4#1451982312yes—
345C8—44no—
346C22—24no—
347C22—22no—
348C22—42no—
349C22—42no—
350C22—48816no—
351C22—4884no—
352C22—4882no—
353C22—4882no—
354C22—2444no—
355C22—2448no—
356C22C2×#1451974711yescharacteristic
357C22#1684027411yescharacteristic
358C22#1684033111yescharacteristic
359C22#1684057414yes—
360C4—22no—
361C4—42no—
362C4—42no—
363C4—4882no—
364C4—2444no—
365C4—2444no—
366C4#1684040414yes—
367C2—42no—
368C2—4884no—
369C2#2144914011yescharacteristic
370C2#2145510712yes—
371C2#2145607212yes—
372C2#2145689312yes—
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1818C2
socleC23×C61C22×D4
prime radicalC23×D244C22
hypercenter#1818D61
hypocenterC61#23942
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
1. C2×#16840374C2semidirect
4. #21446966C2semidirect
5. #21446904C2semidirect
6. #21446173C2semidirect
7. #21446129C2semidirect
8. C61×#1818C2semidirect
12. C2×#14519777C22semidirect
14. C2×#14519769C22semidirect
17. C2×C4×Dic122C22semidirect
20. C2×#14519702C22semidirect
26. #16839525C22semidirect
29. C61×#336C22semidirect
30. C61×#336C22semidirect
188. #1818D61semidirect
215. #336C2×D61semidirect
216. #336C2×D61semidirect
217. C61#23942semidirect

Maximal subgroups

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

subgroupindex
C2×#168403742
C2×#168399152
#214483992
#214469662
#214469042
#214461732
#214461292
C61×#18182
#2394261