Finite Group Databasebeta

← #29590830

Subgroups — #29590830

Order 15616.

384 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 83 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#21449110C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic
3#21443850C212yesmaximal
4#21443826C212yesmaximal
5C61×#1467C211yesmaximal, characteristic
6C2×C4×Dic122C411yescharacteristic
7#16839786—21no—
8C2×#14519696—21no—
9#16839693—21no—
10#16839692C411yescharacteristic
11#16839692C2211yescharacteristic
12#16839682C411yescharacteristic
13#16839682C2211yescharacteristic
14#16839681—21no—
15C4×#11758196—21no—
16C4×#11758196—21no—
17#16839608C411yescharacteristic
18#16839531C2212yes—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#432—21no—
21C61×#324—21no—
22C61×#324C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C2×#11758194—22no—
25C4×Dic122—41no—
26C4×Dic122—41no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—21no—
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×#11758196—21no—
33C2×C4×Dic61—21no—
34C2×C4×Dic61—21no—
35C4⋊#92418—41no—
36C4⋊#92418—41no—
37#14519696—42no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684C2×C411yescharacteristic
44#14519684C2×C411yescharacteristic
45C22×Q8×C61D411yescharacteristic
46C4×Q8×C61—41no—
47C4×Q8×C61—41no—
48C2×C4⋊C4×C61C2×C411yescharacteristic
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61C2311yescharacteristic
51C2×C42×C61C2×C411yescharacteristic
52C2×C42×C61D411yescharacteristic
53C2×C42×C61D411yescharacteristic
54C4⋊C8×C61—41no—
55C22⋊C8×C61—22no—
56#97×C61—41no—
57C2×Q8×C61—21no—
58C2×Q8×C61—21no—
59C2×Q8×C61—41no—
60C2×Q8×C61—41no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C2×Dic122—41no—
73C2×Dic122—42no—
74C2×C8×C61—42no—
75C4⋊C4×C61—41no—
76C4⋊C4×C61—41no—
77C4⋊C4×C61—41no—
78C42×C61—21no—
79C42×C61—21no—
80C42×C61—21no—
81C42×C61—21no—
82C42×C61—21no—
83C42×C61—21no—
84C4⋊Dic61—41no—
85#11758196—42no—
86#11758196—42no—
87#11758196—42no—
88#11758196—42no—
89#11758196—41no—
90#11758196—41no—
91C4×Dic61—24no—
92C4×Dic61—41no—
93C4×Dic61—41no—
94#11758194—42no—
95#11758194—42no—
96C2×#92418—42no—
97C23×C61C2×C22⋊C411yescharacteristic
98Q8×C61—42no—
99Q8×C61—42no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—22no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61#9911yescharacteristic
116C2×C4×C61#9911yescharacteristic
117C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C61#13711yescharacteristic
119C2×Dic61—22no—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131Dic122—42no—
132Dic122—42no—
133C8×C61—82no—
134#92418—82no—
135#11831—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61C2×#9911yescharacteristic
139C22×C61#41711yescharacteristic
140C22×C61#42111yescharacteristic
141C4×C61—21no—
142C4×C61—21no—
143C4×C61—21no—
144C4×C61—41no—
145C4×C61—41no—
146C4×C61—41no—
147C4×C61—41no—
148C4×C61—41no—
149C4×C61#35612yes—
150Dic61—42no—
151Dic61—42no—
152Dic61—42no—
153Dic61—81no—
154#2399—611no—
155#1801—611no—
156#1467D6111yescharacteristic
157#1354—612no—
158#1346—612no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61#151311yescharacteristic
162C2×C61#212011yescharacteristic
163C2×C61C2×#35611yescharacteristic
164C2×C4×Q8—611no—
165C2×C4×Q8C2×D6111yescharacteristic
166#432—21no—
167#423—1221no—
168C2×#101—1221no—
169#398—611no—
170#398—611no—
171#398—1221no—
172#395—611no—
173#395—611no—
174#395—1221no—
175C4×C4⋊C4—1221no—
176C4×C4⋊C4—1221no—
177#376—611no—
178#343—612no—
179#324—21no—
180#324C2×D6112yes—
181C61#1183111yescharacteristic, complemented, Sylow
182C22×Q8—611no—
183C22×Q8D24411yescharacteristic
184C2×M16—1222no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—2441no—
188C4×Q8—2441no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—1221no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4—1221no—
195C2×C4⋊C4C4×D6111yescharacteristic
196C2×C4⋊C4C22⋊D6111yescharacteristic
197C2×C4⋊C4C22×D6111yescharacteristic
198C2×C42—1221no—
199C2×C42—1221no—
200C2×C42C4×D6111yescharacteristic
201C2×C42D24411yescharacteristic
202C2×C42C22⋊D6111yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—2441no—
205C4⋊C8—2441no—
206#101—2442no—
207C22⋊C8—22no—
208#97—41no—
209#95—611no—
210#95—611no—
211#95—1221no—
212#95—1221no—
213#95—1221no—
214#95—1221no—
215#95—1221no—
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—2441no—
221C2×Q8—2442no—
222C22×C4—1221no—
223C22×C4—1221no—
224C22×C4—1221no—
225C22×C4—1221no—
226C22×C4#1175819811yescharacteristic
227C22×C4#1175819811yescharacteristic
228C22×C4#1175819811yescharacteristic
229C22×C4#1175819811yescharacteristic
230C22×C4C2×C4×D6111yescharacteristic
231C22×C4C2×D24411yescharacteristic
232C22×C4C2×C22⋊D6111yescharacteristic
233M16—2442no—
234M16—2442no—
235C2×C8—42no—
236C2×C8—2442no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—2441no—
241C4⋊C4—2442no—
242C4⋊C4—2442no—
243C4⋊C4—2442no—
244C4⋊C4—2442no—
245C4⋊C4—2441no—
246C4⋊C4—2441no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—1224no—
254C42—2441no—
255C42—2441no—
256C23C2×#1175819811yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—2442no—
260Q8—2442no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—1222no—
277C2×C4—1222no—
278C2×C4—1222no—
279C2×C4—2441no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4#1451969111yescharacteristic
289C2×C4#1451969111yescharacteristic
290C2×C4#1451980511yescharacteristic
291C2×C4#1451981011yescharacteristic
292C8—82no—
293C8—4882no—
294C22—21no—
295C22—21no—
296C22C2×#1451969111yescharacteristic
297C22#1683976411yescharacteristic
298C22#1683977811yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—4881no—
308C4—2442no—
309C4—2442no—
310C4—2442no—
311C4#1683956012yes—
312C2—21no—
313C2—21no—
314C2#2144436311yescharacteristic
315C2#2144745811yescharacteristic
316C2C2×#1683956011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1467C2
socleC22×C61C2×#99
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1467D61
hypocenterC61#11831
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
181. C61#11831semidirect

Maximal subgroups

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

subgroupindex
#214491102
#214458152
#214438502
#214438262
C61×#14672
#1183161