Finite Group Databasebeta

← #29575798

Subgroups — #29575798

Order 15616.

648 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 131 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#21454150C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic, complemented
3C2×#16839495C212yesmaximal
4#21443698C212yesmaximal, complemented
5C2×C61×#383C211yesmaximal, characteristic, complemented
6C2×#14519760—21no—
7C2×#14519751C2211yescharacteristic
8C2×C22⋊#92418C2212yes—
9#16839664C412yescomplemented
10#16839660C412yescomplemented
11#16839653—22no—
12#16839641—21no—
13#16839622C2211yescharacteristic
14#16839517—22no—
15#16839495—24no—
16C22×C42×C61C2211yescharacteristic
17C61×#383—24no—
18C2×#95×C61—21no—
19C2×#95×C61—21no—
20C2×#95×C61C2212yes—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—21no—
23C22×#92418—22no—
24#14519760—44no—
25#14519751—22no—
26#14519751—22no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×C4⋊Dic61C2×C412yes—
29C2×#11758196—22no—
30C2×C22⋊Dic61—21no—
31C2×#11758198—22no—
32C2×#11758198—22no—
33C2×#11758198—21no—
34C22⋊#92418—22no—
35C22⋊#92418—22no—
36#14519684—22no—
37#14519684—21no—
38C23×C4×C61—22no—
39C23×C4×C61D411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C2×C42×C61—22no—
43C2×C42×C61—22no—
44C2×C42×C61C2×C412yes—
45C2×C42×C61D412yes—
46#95×C61—24no—
47#95×C61—24no—
48#95×C61—22no—
49#95×C61D414yes—
50C24×C61C2×D411yescharacteristic
51C23×D61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61—24no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61—44no—
62C22×C4×C61—44no—
63C22×C4×C61—42no—
64C22×C4×C61—42no—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61C22×C411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D412yes—
70C22×C4×C61#4112yes—
71C2×C22⋊D61—41no—
72C2×C22⋊D61—41no—
73C2×C22⋊D61—42no—
74C2×C22⋊D61—42no—
75C22×Dic61—21no—
76C22×Dic61—22no—
77C2×D244—22no—
78C2×D244—42no—
79C42×C61—24no—
80C42×C61—24no—
81C22⋊Dic61—42no—
82#11758198—44no—
83#11758198—44no—
84#11758198—42no—
85C4⋊Dic61—22no—
86C4⋊Dic61—42no—
87#11758196—44no—
88C2×#92418—44no—
89C2×#92418—42no—
90C23×C61—22no—
91C23×C61—22no—
92C23×C61C2×C22⋊C411yescharacteristic
93C23×C61#12011yescharacteristic
94C23×C61#12311yescharacteristic
95C23×C61#13512yes—
96C23×C61AGL(1,ℤ/8)12yes—
97C22×D61—41no—
98C22×D61—42no—
99C22×D61—41no—
100C22×D61—41no—
101C22×D61—42no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—44no—
117C2×C4×C61—44no—
118C2×C4×C61—44no—
119C2×C4×C61—84no—
120C2×C4×C61—84no—
121C2×C4×C61C4×D412yes—
122C2×C4×C61#12112yes—
123C22⋊D61—82no—
124C22⋊D61—82no—
125C2×Dic61—41no—
126C2×Dic61—42no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131D244—82no—
132#92418—82no—
133#9157—611nomaximal, Sylow
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—24no—
139C22×C61—24no—
140C22×C61#38611yescharacteristic
141C22×C61#41012yes—
142C22×C61#41611yescharacteristic
143C22×C61#41811yescharacteristic
144C22×C61#42112yes—
145C22×C61#44912yes—
146C22×C61#48212yes—
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—81no—
150C2×D61—82no—
151C2×D61—82no—
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—42no—
155C4×C61—42no—
156C4×C61—84no—
157Dic61—81no—
158Dic61—82no—
159#2854—611no—
160C2×#383D6111yescharacteristic, complemented
161#1817—611no—
162C2×#327—612no—
163#1294—612no—
164C2×C61—22no—
165C2×C61—22no—
166C2×C61#151412yes—
167C2×C61#152112yes—
168C2×C61#187511yescharacteristic
169C2×C61#189711yescharacteristic
170C2×C61#189911yescharacteristic
171D61—161no—
172C2×#123—1221no—
173C2×#121—611no—
174C22×C42C2×D6111yescharacteristic
175C2×C22⋊C8—612no—
176#390—612no—
177#389—612no—
178#388—1222no—
179#386—1221no—
180#383—24no—
181#380—611no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95C2×D6112yes—
185#336—1222no—
186#327—1224no—
187C61#915711yescharacteristic, complemented, Sylow
188C22×D4—611no—
189C22×D4—1221no—
190C23×C4—22no—
191C23×C4C22⋊D6111yescharacteristic
192C23×C4C22⋊D6111yescharacteristic
193C23×C4C22×D6111yescharacteristic
194C22×C8—1222no—
195#123—2444no—
196#121—1222no—
197#121—1222no—
198C2×C4⋊C4—611no—
199C2×C4⋊C4—612no—
200C2×C4⋊C4—1222no—
201C2×C22⋊C4—1221no—
202C2×C22⋊C4—1222no—
203C2×C22⋊C4—1222no—
204C2×C22⋊C4—1221no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D6112yes—
208C2×C42D24412yes—
209C22⋊C8—1222no—
210C22⋊C8—1222no—
211#95—24no—
212#95—24no—
213#95—22no—
214#95—1222no—
215#95—1221no—
216#95C22⋊D6114yes—
217C24—1221no—
218C24C2×C22⋊D6111yescharacteristic
219C2×D4—1222no—
220C2×D4—2442no—
221C2×D4—2441no—
222C2×D4—2441no—
223C2×D4—2442no—
224C2×D4—2442no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—1222no—
239C22×C4—1221no—
240C22×C4#1175819812yes—
241C22×C4#1175819812yes—
242C22×C4C2×C4×D6111yescharacteristic
243C22×C4C2×D24411yescharacteristic
244C22×C4#1175822212yes—
245C22×C4C2×C22⋊D6112yes—
246C2×C8—2444no—
247C2×C8—2442no—
248C4⋊C4—1222no—
249C4⋊C4—2442no—
250C4⋊C4—2444no—
251C22⋊C4—2444no—
252C22⋊C4—2444no—
253C22⋊C4—2442no—
254C22⋊C4—2442no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—2442no—
260C23—2441no—
261C23—2441no—
262C23—2442no—
263C23—2441no—
264C23C2×#1175819811yescharacteristic
265C23#1451974611yescharacteristic
266C23#1451976011yescharacteristic
267C23#1451979712yes—
268C23#1451980212yes—
269D4—4882no—
270D4—4882no—
271D4—4882no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4—2442no—
292C2×C4—2441no—
293C2×C4—2442no—
294C2×C4—2441no—
295C2×C4—2442no—
296C2×C4—2441no—
297C2×C4C4×C22⋊D6112yes—
298C2×C4#1451975112yes—
299C8—4882no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—4882no—
307C22—4882no—
308C22—4881no—
309C22—4881no—
310C22—4881no—
311C22#1683964111yescharacteristic
312C22#1683973512yes—
313C22#1683975611yescharacteristic
314C22#1683976611yescharacteristic
315C22#1683978012yes—
316C22#1683990712yes—
317C22#1684008612yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—84no—
323C4—4882no—
324C4—4881no—
325C2—22no—
326C2—22no—
327C2—9761no—
328C2#2144436512yes—
329C2#2144439412yes—
330C2#2144614111yescharacteristic
331C2#2144626711yescharacteristic
332C2#2144627711yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#383C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenterC2×#383D61
hypocenterC61#9157
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#383

Complements

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

NH ≅ G/Ntype
2. #21445887C2semidirect
4. #21443698C2semidirect
5. C2×C61×#383C2semidirect
9. #16839664C4semidirect
10. #16839660C4semidirect
160. C2×#383D61semidirect
187. C61#9157semidirect

Maximal subgroups

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

subgroupindex
#214541502
#214458872
C2×#168394952
#214436982
C2×C61×#3832
#915761