Finite Group Databasebeta

← #29572550

Subgroups — #29572550

Order 15616.

784 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16840366C211yesmaximal, characteristic
2#21444533C211yesmaximal, characteristic, complemented
3#21444445C212yesmaximal, complemented
4#21444151C212yesmaximal, complemented
5C61×#1555C211yesmaximal, characteristic, complemented
6#16840366—24no—
7#16840366—22no—
8#16840366—22no—
9C2×#14519769C2211yescharacteristic
10C2×#14519758—22no—
11C2×#14519749—22no—
12C2×#14519748C412yes—
13C2×D4×Dic61C412yes—
14C2×#14519734C2211yescharacteristic
15#16839819C2212yes—
16#16839506—22no—
17#16839503—24no—
18#16839487—22no—
19#16839485—22no—
20C2×#127×C61C2211yescharacteristic
21C61×#432C2212yes—
22C61×#331—22no—
23C22×C22⋊D61—22no—
24C23×Dic61—22no—
25C22×C4×D61C2×C411yescharacteristic
26#14519769—21no—
27#14519769—21no—
28#14519769D414yes—
29#14519758—28no—
30#14519758—28no—
31#14519749—28no—
32#14519749—28no—
33#14519748—24no—
34#14519748—24no—
35D4×Dic61—24no—
36D4×Dic61—24no—
37#14519734—21no—
38#14519734—22no—
39#14519734—21no—
40C2×C4⋊Dic61C2×C412yes—
41C2×#11758196—22no—
42C2×#11758196D412yes—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—22no—
45C2×#11758198D412yes—
46C2×C4×Dic61C2×C411yescharacteristic
47C4⋊#92418—22no—
48C22⋊#92418—22no—
49#14519688—22no—
50C22×D4×C61—22no—
51C22×D4×C61C2×C412yes—
52#127×C61—22no—
53#127×C61D412yes—
54#127×C61D412yes—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C22⋊C8×C61—22no—
58#97×C61—22no—
59C24×C61—22no—
60C23×D61C22⋊C411yescharacteristic
61C2×D4×C61—22no—
62C2×D4×C61—24no—
63C2×D4×C61—24no—
64C2×D4×C61—44no—
65C2×D4×C61—84no—
66C2×D4×C61C22⋊C414yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D412yes—
69C2×C22⋊D61—28no—
70C2×C22⋊D61—84no—
71C22×Dic61—22no—
72C22×Dic61—28no—
73C22×Dic61—84no—
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C22⋊C412yes—
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—42no—
79C2×C8×C61—42no—
80C42×C61C2×D411yescharacteristic
81C42×C61C2×D412yes—
82C42×C61C2×D411yescharacteristic
83C22⋊Dic61—28no—
84C22⋊Dic61—28no—
85#11758198—22no—
86#11758198—22no—
87C4⋊Dic61—22no—
88C4⋊Dic61C22⋊C414yes—
89#11758196—24no—
90#11758196—24no—
91#11758196—22no—
92#11758196—22no—
93C4×Dic61—21no—
94C4×Dic61—21no—
95C2×#92418—42no—
96C23×C61—24no—
97C23×C61—44no—
98C23×C61—44no—
99C23×C61—44no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—22no—
103C22×D61—42no—
104D4×C61—44no—
105D4×C61—44no—
106D4×C61—84no—
107D4×C61—84no—
108C2×C4×C61—22no—
109C2×C4×C61—42no—
110C2×C4×C61C2×C22⋊C411yescharacteristic
111C2×C4×C61C2×C22⋊C411yescharacteristic
112C2×C4×C61#12012yes—
113C2×C4×C61#12012yes—
114C22⋊D61—88no—
115C2×Dic61—22no—
116C2×Dic61—24no—
117C2×Dic61—21no—
118C2×Dic61—22no—
119C2×Dic61—44no—
120C2×Dic61—42no—
121C2×Dic61—88no—
122C4×D61—42no—
123C4×D61—42no—
124C8×C61—82no—
125#92418—82no—
126#8526—611nomaximal, Sylow
127C22×C61—22no—
128C22×C61—44no—
129C22×C61—44no—
130C22×C61—44no—
131C22×C61—88no—
132C22×C61#37911yescharacteristic
133C22×C61#41312yes—
134C2×D61—42no—
135C2×D61—41no—
136C2×D61—82no—
137C4×C61—22no—
138C4×C61—22no—
139C4×C61—21no—
140C4×C61—21no—
141Dic61—44no—
142Dic61—42no—
143Dic61—41no—
144C2×#540—611no—
145#1555—611no—
146#1555D6111yescharacteristic, complemented
147#1534—612no—
148#1464—612no—
149C2×C61—22no—
150C2×C61—84no—
151C2×C61#178511yescharacteristic
152C2×C61#179512yes—
153D61—81no—
154#540—1224no—
155#540—1222no—
156#540—1222no—
157C2×#127C2×D6111yescharacteristic
158C2×#125—611no—
159C2×#123—1222no—
160C2×#121—612no—
161C2×#121—1222no—
162C2×C4×D4—612no—
163C2×#117—611no—
164#432—612no—
165#432C2×D6112yes—
166#331—22no—
167#331—1222no—
168#330—1224no—
169#324—1222no—
170#323—1222no—
171C61#852611yescharacteristic, complemented, Sylow
172C22×D4—22no—
173C22×D4—1222no—
174C22×D4C4×D6112yes—
175C23×C4—611no—
176C23×C4—1222no—
177#127—22no—
178#127D24412yes—
179#127C22⋊D6112yes—
180#125—614no—
181#125—1221no—
182#125—1221no—
183#123—1228no—
184#123—1228no—
185#121—1224no—
186#121—1228no—
187#121—1228no—
188#121—1224no—
189C4×D4—1224no—
190C4×D4—1224no—
191#117—1221no—
192#117—1222no—
193#117—1221no—
194C2×C4⋊C4—612no—
195C2×C4⋊C4—612no—
196C2×C4⋊C4—1222no—
197C2×C22⋊C4—612no—
198C2×C22⋊C4—1222no—
199C2×C22⋊C4—1222no—
200C2×C42—611no—
201C2×C42C22×D6111yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—1222no—
204C22⋊C8—22no—
205C22⋊C8—1222no—
206#97—22no—
207#97—1222no—
208C24—22no—
209C24—611no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—44no—
214C2×D4—84no—
215C2×D4—1228no—
216C2×D4—4884no—
217C2×D4#1175819814yes—
218C22×C4—611no—
219C22×C4—612no—
220C22×C4—1228no—
221C22×C4—1222no—
222C22×C4—1221no—
223C22×C4—1221no—
224C22×C4—4884no—
225C22×C4—2442no—
226C22×C4C2×C4×D6111yescharacteristic
227C22×C4D4×D6112yes—
228C2×C8—42no—
229C2×C8—2442no—
230C4⋊C4—614no—
231C4⋊C4—1222no—
232C4⋊C4—1222no—
233C4⋊C4—1224no—
234C4⋊C4—1222no—
235C4⋊C4—1224no—
236C22⋊C4—1228no—
237C22⋊C4—1228no—
238C22⋊C4—1222no—
239C22⋊C4—1222no—
240C42—1221no—
241C42—1221no—
242C42C2×D24411yescharacteristic
243C42D4×D6112yes—
244C42C2×C22⋊D6111yescharacteristic
245C23—24no—
246C23—44no—
247C23—44no—
248C23—44no—
249C23—1222no—
250C23—1221no—
251C23—2442no—
252C23C22⋊C4×D6111yescharacteristic
253D4—44no—
254D4—44no—
255D4—84no—
256D4—84no—
257D4—4888no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—1222no—
261C2×C4—1222no—
262C2×C4—1221no—
263C2×C4—1224no—
264C2×C4—4888no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4—2444no—
268C2×C4—2442no—
269C2×C4C22⋊C4×D6111yescharacteristic
270C2×C4C2×#1175819811yescharacteristic
271C2×C4#1451974512yes—
272C2×C4#1451974712yes—
273C8—82no—
274C8—4882no—
275C22—22no—
276C22—44no—
277C22—44no—
278C22—44no—
279C22—88no—
280C22—4882no—
281C22—2441no—
282C22—2442no—
283C22#1683961711yescharacteristic
284C22#1683974312yes—
285C4—21no—
286C4—21no—
287C4—22no—
288C4—22no—
289C4—2441no—
290C4—2442no—
291C4—2444no—
292C2—22no—
293C2—84no—
294C2—4881no—
295C2#2144574411yescharacteristic
296C2#2144578612yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1555C2
socleC22×C61#379
prime radicalC2×#16840366C2
cosocleC2×C42×C61C23
hypercenter#1555D61
hypocenterC61#8526
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
2. #21444533C2semidirect
3. #21444445C2semidirect
4. #21444151C2semidirect
5. C61×#1555C2semidirect
146. #1555D61semidirect
171. C61#8526semidirect

Maximal subgroups

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

subgroupindex
C2×#168403662
#214445332
#214444452
#214441512
C61×#15552
#852661