Finite Group Databasebeta

← #29568751

Subgroups — #29568751

Order 15616.

536 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#21452851C211yesmaximal, characteristic, complemented
2#21447050C212yesmaximal
3#21443645C211yesmaximal, characteristic, complemented
4#21443643C212yesmaximal, complemented
5C61×#2014C211yesmaximal, characteristic
6C2×#14519775—21no—
7C2×#14519698—22no—
8C2×#14519698—22no—
9#16839692—21no—
10#16839683—22no—
11#16839660C2212yes—
12#16839647—22no—
13#16839517C2212yes—
14C22×SD16×C61—21no—
15C2×#127×C61C2211yescharacteristic
16C2×#102×C61—21no—
17C61×#391C2211yescharacteristic
18C61×#336C2211yescharacteristic
19C22×Dic122—21no—
20C22×#92418—22no—
21#14519775—42no—
22#14519775—42no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61D412yes—
25C2×C4⋊Dic61D412yes—
26C2×#11758196—22no—
27C2×#11758196—22no—
28C2×C4×Dic61—21no—
29#14519698—44no—
30#14519698—44no—
31#14519684—22no—
32#14519684—21no—
33#14519684—22no—
34C22×Q8×C61D411yescharacteristic
35C22×D4×C61—21no—
36C22×D4×C61—22no—
37C2×SD16×C61—42no—
38C2×SD16×C61—42no—
39C2×SD16×C61—42no—
40C22×C8×C61—21no—
41#127×C61—42no—
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C42×C61C2311yescharacteristic
44#102×C61—42no—
45#95×C61—22no—
46C24×C61—41no—
47C2×Q8×C61—21no—
48C2×Q8×C61—41no—
49C2×Q8×C61—41no—
50C2×Q8×C61D812yes—
51C2×D4×C61—22no—
52C2×D4×C61—24no—
53C2×D4×C61—41no—
54C2×D4×C61—41no—
55C2×D4×C61—42no—
56C2×D4×C61—42no—
57C2×D4×C61—42no—
58C2×D4×C61—41no—
59C2×D4×C61—41no—
60C2×D4×C61—42no—
61C2×D4×C61—41no—
62C2×D4×C61—42no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61C2×D411yescharacteristic
66C22×C4×C61C2×D412yes—
67C22×Dic61—22no—
68C22×Dic61—22no—
69C2×Dic122—42no—
70C2×Dic122—81no—
71C2×Dic122—81no—
72SD16×C61—48no—
73C2×C8×C61—22no—
74C2×C8×C61—41no—
75C2×C8×C61—41no—
76C4⋊C4×C61—41no—
77C42×C61—41no—
78C4⋊Dic61—22no—
79C4⋊Dic61—42no—
80C4⋊Dic61—81no—
81C4⋊Dic61SD1614yescomplemented
82#11758196—44no—
83#11758196—42no—
84#11758196—42no—
85C4×Dic61—42no—
86C2×#92418—42no—
87C2×#92418—42no—
88C2×#92418—42no—
89C23×C61—42no—
90C23×C61—81no—
91C23×C61—82no—
92C23×C61—81no—
93C23×C61—82no—
94C23×C61#12011yescharacteristic
95Q8×C61—42no—
96Q8×C61—81no—
97D4×C61—42no—
98D4×C61—42no—
99D4×C61—44no—
100D4×C61—44no—
101D4×C61—81no—
102D4×C61—81no—
103D4×C61—82no—
104D4×C61—82no—
105C2×C4×C61—21no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—21no—
109C2×C4×C61—22no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61C2×D811yescharacteristic
115C2×C4×C61C2×SD1612yes—
116C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C61#13712yes—
118C2×Dic61—24no—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124Dic122—84no—
125C8×C61—42no—
126#92418—82no—
127#7741—611nomaximal, Sylow
128C22×C61—82no—
129C22×C61—84no—
130C22×C61—161no—
131C22×C61—162no—
132C22×C61—161no—
133C22×C61#44711yescharacteristic
134C22×C61#44812yes—
135C22×C61#45311yescharacteristic
136C22×C61#45412yes—
137C22×C61B(4,𝔽2)11yescharacteristic
138C4×C61—21no—
139C4×C61—21no—
140C4×C61—22no—
141C4×C61—22no—
142C4×C61—42no—
143C4×C61—81no—
144Dic61—44no—
145Dic61—82no—
146#2734—611no—
147#2041—612no—
148#2014D6111yescharacteristic
149#1272—611no—
150#1271—612no—
151C2×C61—162no—
152C2×C61#159511yescharacteristic
153C2×C61#159912yes—
154C2×C61#162311yescharacteristic
155C2×C61#162712yes—
156C2×C61#165711yescharacteristic
157C22×SD16—21no—
158C2×C4⋊Q8—1221no—
159C2×#127C2×D6111yescharacteristic
160C2×#102—21no—
161C2×#102—1222no—
162C2×#102—1222no—
163#398—1221no—
164#395—1222no—
165#391C2×D6111yescharacteristic
166#389—612no—
167#387—1222no—
168#336—612no—
169#336C2×D6111yescharacteristic
170C61#774111yescharacteristic, complemented, Sylow
171C22×Q8—1221no—
172C22×Q8C22⋊D6111yescharacteristic
173C22×D4—22no—
174C22×D4—21no—
175C2×SD16—42no—
176C2×SD16—42no—
177C2×SD16—42no—
178C22×C8—21no—
179C22×C8—1222no—
180C4⋊Q8—2442no—
181C4⋊Q8—2442no—
182#127—42no—
183C2×C4⋊C4—612no—
184C2×C4⋊C4—612no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4—1222no—
187C2×C4⋊C4—1222no—
188C2×C4⋊C4C22⋊D6111yescharacteristic
189C2×C42—1221no—
190C2×C42C22×D6111yescharacteristic
191#102—42no—
192#102—2444no—
193#102—2444no—
194#95—22no—
195#95—1221no—
196#95—1222no—
197#95—1222no—
198C24—41no—
199C2×Q8—21no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—4881no—
203C2×Q8—4881no—
204C2×Q8—2442no—
205C2×Q8#1175819912yes—
206C2×D4—24no—
207C2×D4—22no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—42no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—1222no—
221C22×C4—1222no—
222C22×C4D4×D6112yes—
223C22×C4C2×C22⋊D6111yescharacteristic
224SD16—48no—
225C2×C8—22no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—2442no—
229C2×C8—2442no—
230C2×C8—2442no—
231C4⋊C4—41no—
232C4⋊C4—614no—
233C4⋊C4—1222no—
234C4⋊C4—4881no—
235C4⋊C4—2442no—
236C4⋊C4—2444no—
237C4⋊C4—2442no—
238C4⋊C4—2442no—
239C42—41no—
240C42—2442no—
241C23—42no—
242C23—81no—
243C23—82no—
244C23—81no—
245C23—82no—
246C23#1451974511yescharacteristic
247Q8—42no—
248Q8—81no—
249Q8—4884no—
250D4—42no—
251D4—42no—
252D4—44no—
253D4—44no—
254D4—81no—
255D4—81no—
256D4—82no—
257D4—82no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—1224no—
268C2×C4—2442no—
269C2×C4—2442no—
270C2×C4—2442no—
271C2×C4—2442no—
272C2×C4—2442no—
273C2×C4C2×#1175819911yescharacteristic
274C2×C4SD16×D6112yes—
275C2×C4#1451980211yescharacteristic
276C2×C4#1451980612yes—
277C8—42no—
278C8—4882no—
279C22—82no—
280C22—84no—
281C22—161no—
282C22—162no—
283C22—161no—
284C22#1683989511yescharacteristic
285C22#1683989912yes—
286C22#1683992911yescharacteristic
287C22#1683993312yes—
288C22#1683994811yescharacteristic
289C4—21no—
290C4—21no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—81no—
295C4—4882no—
296C4—2444no—
297C2—162no—
298C2#2144473611yescharacteristic
299C2#2144475712yes—
300C2#2144489511yescharacteristic
301C2#2144492012yes—
302C2#2144511511yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2014C2
socleC23×C61#120
prime radicalC2×#127×C61C22
hypercenter#2014D61
hypocenterC61#7741
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2014

Complements

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

NH ≅ G/Ntype
1. #21452851C2semidirect
3. #21443645C2semidirect
4. #21443643C2semidirect
81. C4⋊Dic61SD16semidirect
170. C61#7741semidirect

Maximal subgroups

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

subgroupindex
#214528512
#214470502
#214436452
#214436432
C61×#20142
#774161