Finite Group Databasebeta

← #29656093

Subgroups — #29656093

Order 15616.

1068 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 163 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×#16840498C211yesmaximal, characteristic, complemented
2#21455937C212yesmaximal, complemented
3#21444938C214yesmaximal, complemented
4#21444876C214yesmaximal, complemented
5#21444533C211yesmaximal, characteristic, complemented
6#21444267C212yesmaximal, complemented
7C61×#3151C211yesmaximal, characteristic
8#16840498—24no—
9#16840498—22no—
10#16840498C2214yes—
11C2×#14519764—22no—
12C2×#14519758—22no—
13C2×C4×Dic122C2212yes—
14C2×D4×Dic61—22no—
15#16840073—22no—
16#16840054C2214yescomplemented
17#16839959C2214yescomplemented
18#16839900—22no—
19#16839862—24no—
20#16839819C2212yes—
21#16839506C2214yes—
22#16839494C2218yescomplemented
23C2×AGL(1,ℤ/8)×C61—22no—
24C61×#532C2211yescharacteristic
25C2×#127×C61C2211yescharacteristic
26C61×#496—22no—
27C61×#493C2212yes—
28C61×#492C2212yes—
29C4×M16×C61C2211yescharacteristic
30C22×Dic122D412yes—
31C23×Dic61—22no—
32C2×#11758200—24no—
33#14519764—28no—
34#14519764—28no—
35#14519758—28no—
36#14519758—28no—
37C4×Dic122—22no—
38C4×Dic122—22no—
39C4×Dic122D414yes—
40C4×Dic122C2314yes—
41D4×Dic61—28no—
42D4×Dic61—28no—
43C2×C4⋊Dic61D412yes—
44C2×#11758196—22no—
45C2×C22⋊Dic61—24no—
46C2×C22⋊Dic61—22no—
47C2×C4×Dic61—22no—
48#14519710—24no—
49C4⋊#92418C2314yes—
50#14519700—24no—
51#14519698—24no—
52C22⋊#92418—22no—
53#14519688—22no—
54C2×#41×C61—21no—
55C22×D4×C61—22no—
56C22×D4×C61D412yes—
57AGL(1,ℤ/8)×C61—48no—
58C2×SD16×C61—24no—
59C2×D8×C61—24no—
60C2×M16×C61—21no—
61C4⋊Q8×C61C2311yescharacteristic
62#127×C61—22no—
63#127×C61D412yes—
64#127×C61C2311yescharacteristic
65#127×C61C2312yes—
66#124×C61—21no—
67#121×C61—22no—
68C4×D4×C61—22no—
69C2×C42×C61C2311yescharacteristic
70#97×C61—21no—
71C4×C8×C61C2312yes—
72C24×C61—22no—
73#41×C61—44no—
74C2×Q8×C61—21no—
75C2×D4×C61—24no—
76C2×D4×C61—22no—
77C2×D4×C61—24no—
78C2×D4×C61—22no—
79C2×D4×C61—21no—
80C2×D4×C61—48no—
81C2×D4×C61—44no—
82C2×D4×C61—41no—
83C2×D4×C61C2×D414yes—
84C22×C4×C61—41no—
85C22×C4×C61C2×D412yes—
86C22×C4×C61C2×D411yescharacteristic
87C22×Dic61—22no—
88C22×Dic61—22no—
89C22×Dic61—28no—
90C22×Dic61—84no—
91C2×Dic122—22no—
92C2×Dic122—44no—
93C2×Dic122C2×D414yes—
94SD16×C61—48no—
95D8×C61—48no—
96M16×C61—24no—
97C2×C8×C61—22no—
98C4⋊C4×C61—22no—
99C22⋊C4×C61—42no—
100C42×C61C2×D412yes—
101C42×C61C2×D411yescharacteristic
102C42×C61C2411yescharacteristic
103C22⋊Dic61—216no—
104C22⋊Dic61—28no—
105#11758200—48no—
106C4⋊Dic61—22no—
107C4⋊Dic61C2×D414yes—
108#11758196—24no—
109#11758196—24no—
110C4×Dic61—24no—
111C4×Dic61—24no—
112C2×#92418—24no—
113C23×C61—24no—
114C23×C61—48no—
115C23×C61—44no—
116C23×C61—41no—
117C23×C61#12011yescharacteristic
118Q8×C61—42no—
119D4×C61—44no—
120D4×C61—48no—
121D4×C61—42no—
122D4×C61—48no—
123D4×C61—44no—
124D4×C61—82no—
125D4×C61—82no—
126C2×C4×C61—22no—
127C2×C4×C61—24no—
128C2×C4×C61—41no—
129C2×C4×C61—82no—
130C2×C4×C61#12012yes—
131C2×C4×C61#12011yescharacteristic
132C2×C4×C61C22×D411yescharacteristic
133C2×C4×C61C22×D412yes—
134C2×Dic61—24no—
135C2×Dic61—24no—
136C2×Dic61—44no—
137C2×Dic61—44no—
138C2×Dic61—88no—
139Dic122—24no—
140Dic122—24no—
141Dic122—44no—
142C8×C61—24no—
143#92418—44no—
144#18526—611nomaximal, Sylow
145C22×C61—22no—
146C22×C61—48no—
147C22×C61—44no—
148C22×C61—88no—
149C22×C61—82no—
150C22×C61—81no—
151C22×C61#45312yes—
152C22×C61C2×#12011yescharacteristic
153C4×C61—22no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—81no—
157C4×C61#57514yes—
158Dic61—44no—
159Dic61—44no—
160C2×#559—611no—
161#3151D6111yescharacteristic
162#3112—612no—
163#1630—614no—
164#1620—614no—
165#1555—611no—
166#1491—612no—
167C2×C61—22no—
168C2×C61—84no—
169C2×C61—161no—
170C2×C61#300512yes—
171C2×C61C2×#45311yescharacteristic
172C2×AGL(1,ℤ/8)—22no—
173#559—614no—
174#559—1224no—
175#559—1222no—
176#532C2×D6111yescharacteristic
177C2×#127C2×D6111yescharacteristic
178C2×#124—1222no—
179C2×#123—1222no—
180C2×C4×Q8—612no—
181C2×C4×D4—1222no—
182#496—22no—
183#493C2×D6112yes—
184#492C2×D6112yes—
185#480—1222no—
186#477—614no—
187#460—614no—
188#448—1222no—
189#440—1224no—
190#432—612no—
191C4×M16C2×D6111yescharacteristic
192#331—614no—
193Q8⋊C8—618no—
194C61#1852611yescharacteristic, complemented, Sylow
195C2×#41—21no—
196C22×Q8—612no—
197C22×D4—22no—
198C22×D4C22⋊D6112yes—
199C23×C4—1222no—
200AGL(1,ℤ/8)—48no—
201C2×SD16—24no—
202C2×SD16—1224no—
203C2×D8—24no—
204C2×M16—21no—
205C4⋊Q8C22×D6111yescharacteristic
206#127—22no—
207#127C22⋊D6112yes—
208#127C22×D6111yescharacteristic
209#127C22×D6112yes—
210#124—21no—
211#124—1228no—
212#124—1228no—
213#123—1228no—
214#123—1228no—
215#121—22no—
216C4×Q8—614no—
217C4×Q8—614no—
218C4×Q8—1222no—
219C4×Q8—1222no—
220C4×D4—22no—
221C4×D4—1228no—
222C4×D4—1228no—
223C2×C4⋊C4—612no—
224C2×C4⋊C4—1222no—
225C2×C22⋊C4—1224no—
226C2×C22⋊C4—1222no—
227C2×C42—1222no—
228C2×C42C22×D6111yescharacteristic
229#107—1224no—
230C4⋊C8—614no—
231#103—1224no—
232#102—1224no—
233C22⋊C8—1222no—
234#97—21no—
235#97—1222no—
236C4×C8C22×D6112yes—
237C24—22no—
238#41—44no—
239C2×Q8—21no—
240C2×Q8—614no—
241C2×Q8—1222no—
242C2×Q8—2444no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—48no—
249C2×D4—44no—
250C2×D4—41no—
251C2×D4C2×C22⋊D6114yes—
252C22×C4—41no—
253C22×C4—1228no—
254C22×C4—1222no—
255C22×C4—1222no—
256C22×C4—4884no—
257C22×C4D4×D6112yes—
258C22×C4C2×C22⋊D6111yescharacteristic
259SD16—48no—
260SD16—2448no—
261D8—48no—
262M16—24no—
263C2×C8—22no—
264C2×C8—1224no—
265C4⋊C4—22no—
266C4⋊C4—614no—
267C4⋊C4—1224no—
268C4⋊C4—1222no—
269C4⋊C4—1224no—
270C22⋊C4—42no—
271C22⋊C4—12216no—
272C22⋊C4—1228no—
273C42—1224no—
274C42—1224no—
275C42D4×D6112yes—
276C42C2×C22⋊D6111yescharacteristic
277C42C23×D6111yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—41no—
282C23#1451974511yescharacteristic
283Q8—42no—
284Q8—1224no—
285Q8—1224no—
286Q8—2444no—
287D4—44no—
288D4—48no—
289D4—42no—
290D4—48no—
291D4—44no—
292D4—82no—
293D4—82no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—41no—
297C2×C4—82no—
298C2×C4—1224no—
299C2×C4—1224no—
300C2×C4—4888no—
301C2×C4—2444no—
302C2×C4—2444no—
303C2×C4#1451974512yes—
304C2×C4#1451974511yescharacteristic
305C2×C4C2×D4×D6112yes—
306C2×C4C22×C22⋊D6111yescharacteristic
307C8—24no—
308C8—2444no—
309C22—22no—
310C22—44no—
311C22—48no—
312C22—88no—
313C22—81no—
314C22—82no—
315C22#1683992912yes—
316C22C2×#1451974511yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—81no—
321C4—2444no—
322C4—2444no—
323C4#1684055814yes—
324C2—22no—
325C2—84no—
326C2—161no—
327C2#2145505812yes—
328C2C2×#1683992911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3151C2
socleC22×C61C2×#120
hypercenter#3151D61
hypocenterC61#18526
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
1. C2×#16840498C2semidirect
2. #21455937C2semidirect
3. #21444938C2semidirect
4. #21444876C2semidirect
5. #21444533C2semidirect
6. #21444267C2semidirect
16. #16840054C22semidirect
17. #16839959C22semidirect
22. #16839494C22semidirect
194. C61#18526semidirect

Maximal subgroups

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

subgroupindex
C2×#168404982
#214559372
#214449382
#214448762
#214445332
#214442672
C61×#31512
#1852661