Finite Group Databasebeta

← #30595920

Subgroups — #30595920

Order 15616.

400 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 84 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#21461747C211yesmaximal, characteristic, complemented
2#21461746C211yesmaximal, characteristic, complemented
3#21460638C212yesmaximal, complemented
4#21460635C212yesmaximal
5#21454183C211yesmaximal, characteristic
6#16840628—21no—
7#16840628—21no—
8#16840628—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×#14519770—21no—
13#16839665C411yescharacteristic
14#16839665C2211yescharacteristic
15#16839661C411yescharacteristic
16#16839660C411yescharacteristic
17#16839658C2211yescharacteristic
18#16839656—21no—
19#16839655—21no—
20#16839650—21no—
21C42×Dic61C2211yescharacteristic
22C61×#390C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25#14519770—44no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35C2×C4×Dic61C2×C411yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37C2×C4×Dic61D411yescharacteristic
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61C2311yescharacteristic
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44C22×D4×C61C2×C411yescharacteristic
45C2×C4⋊C4×C61C2×C411yescharacteristic
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—21no—
48C2×C42×C61C2×C411yescharacteristic
49C24×C61—21no—
50C2×D4×C61—22no—
51C2×D4×C61—41no—
52C2×D4×C61—41no—
53C22×C4×C61—21no—
54C22×C4×C61C22⋊C411yescharacteristic
55C22×C4×C61C22⋊C411yescharacteristic
56C22×C4×C61C22×C411yescharacteristic
57C22×Dic61—21no—
58C22×Dic61—21no—
59C22×Dic61C22⋊C411yescharacteristic
60C22×Dic61C22⋊C411yescharacteristic
61C22×Dic61C2×D411yescharacteristic
62C22×Dic61C2×D411yescharacteristic
63C2×#92420—42no—
64C2×#92420—42no—
65C2×#92420—42no—
66C2×#92420—42no—
67C2×#92420—42no—
68C2×#92420—42no—
69C4⋊C4×C61—41no—
70C22⋊C4×C61—42no—
71C22⋊C4×C61—42no—
72C42×C61—22no—
73C22⋊Dic61—42no—
74C22⋊Dic61—42no—
75C22⋊Dic61—42no—
76C22⋊Dic61—42no—
77C4⋊Dic61—22no—
78C4⋊Dic61—42no—
79C4⋊Dic61—41no—
80#11758196—42no—
81#11758196—42no—
82C4×Dic61—22no—
83C4×Dic61—22no—
84C4×Dic61—41no—
85C4×Dic61—41no—
86C4×Dic61—41no—
87C4×Dic61—41no—
88C23×C61—41no—
89C23×C61—41no—
90C23×C61—41no—
91C23×C61—41no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61—41no—
95C23×C61C2×C22⋊C411yescharacteristic
96D4×C61—82no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—41no—
106C2×C4×C61—41no—
107C2×C4×C61—41no—
108C2×C4×C61#13112yes—
109C2×Dic61—21no—
110C2×Dic61—21no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114C2×Dic61—21no—
115C2×Dic61—21no—
116C2×Dic61—21no—
117C2×Dic61—21no—
118C2×Dic61—21no—
119C2×Dic61—41no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61#13512yes—
126C2×Dic61AGL(1,ℤ/8)12yes—
127#92420—82no—
128#92420—82no—
129#7726—611nomaximal, Sylow
130C22×C61—81no—
131C22×C61—81no—
132C22×C61—81no—
133C22×C61—81no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61—81no—
137C22×C61#40811yescharacteristic
138C22×C61#41011yescharacteristic
139C22×C61#41311yescharacteristic
140C22×C61#41611yescharacteristic
141C22×C61#41811yescharacteristic
142C22×C61#42111yescharacteristic
143C22×C61#42111yescharacteristic
144C4×C61—22no—
145C4×C61—41no—
146C4×C61—41no—
147C4×C61—81no—
148Dic61—22no—
149Dic61—22no—
150Dic61—41no—
151Dic61—41no—
152Dic61—81no—
153Dic61—81no—
154#2860—611no—
155#1839—611no—
156#1839—611no—
157#1271—612no—
158#1270—612no—
159C2×C61—161no—
160C2×C61#147311yescharacteristic
161C2×C61#148211yescharacteristic
162C2×C61#149611yescharacteristic
163C2×C61#151411yescharacteristic
164C2×C61#152111yescharacteristic
165C2×C61#153511yescharacteristic
166C2×C61#155311yescharacteristic
167C2×#126—1221no—
168#390—611no—
169#390—611no—
170#390#692311yescharacteristic
171#389—611no—
172#389—611no—
173#389—611no—
174#388—1221no—
175#388—1221no—
176#388—1221no—
177C43—611no—
178#336—612no—
179#336—612no—
180#336—1221no—
181#336—1221no—
182#336—1221no—
183#336—1221no—
184C61#772611yescharacteristic, complemented, Sylow
185C22×D4C2×#692311yescharacteristic
186C22×C8—1222no—
187C22×C8—1222no—
188#126—2444no—
189C2×C4⋊C4—611no—
190C2×C4⋊C4—611no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4C2×#692311yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—1221no—
198C2×C22⋊C4—1221no—
199C2×C22⋊C4—1221no—
200C2×C22⋊C4—1221no—
201C2×C42—611no—
202C2×C42—611no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42—611no—
206C2×C42—611no—
207C2×C42C2×#692311yescharacteristic
208#95—1221no—
209#95—1221no—
210#95—1221no—
211C24—21no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C22×C4—21no—
216C22×C4—611no—
217C22×C4—611no—
218C22×C4—611no—
219C22×C4—611no—
220C22×C4—1221no—
221C22×C4—1221no—
222C22×C4C22⋊#692311yescharacteristic
223C22×C4C22⋊#692311yescharacteristic
224C22×C4C22×#692311yescharacteristic
225C2×C8—2442no—
226C2×C8—2442no—
227C2×C8—2442no—
228C2×C8—2442no—
229C2×C8—2442no—
230C2×C8—2442no—
231C4⋊C4—41no—
232C4⋊C4—1222no—
233C4⋊C4—2441no—
234C4⋊C4—2442no—
235C4⋊C4—2442no—
236C4⋊C4—2442no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—2442no—
240C22⋊C4—2442no—
241C22⋊C4—2442no—
242C22⋊C4—2442no—
243C42—22no—
244C42—1222no—
245C42—1222no—
246C42—2441no—
247C42—2441no—
248C42—2441no—
249C42—2441no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23C2×C22⋊#692311yescharacteristic
258D4—82no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—612no—
271C2×C4—612no—
272C2×C4—1221no—
273C2×C4—1221no—
274C2×C4—1221no—
275C2×C4—1221no—
276C2×C4—1221no—
277C2×C4—1221no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4—2441no—
288C2×C4#1451986412yes—
289C8—4882no—
290C8—4882no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22#1684073511yescharacteristic
299C22#1684074211yescharacteristic
300C22#1684074811yescharacteristic
301C22#1684075211yescharacteristic
302C22#1684075411yescharacteristic
303C22#1684075911yescharacteristic
304C22#1684076011yescharacteristic
305C4—22no—
306C4—41no—
307C4—41no—
308C4—81no—
309C4—1222no—
310C4—1222no—
311C4—4881no—
312C4—4881no—
313C4—2441no—
314C4—2441no—
315C2—161no—
316C2#2146110111yescharacteristic
317C2#2146113011yescharacteristic
318C2#2146117811yescharacteristic
319C2#2146123911yescharacteristic
320C2#2146126011yescharacteristic
321C2#2146130311yescharacteristic
322C2#2146134211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7726
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461747C2semidirect
2. #21461746C2semidirect
3. #21460638C2semidirect
184. C61#7726semidirect

Maximal subgroups

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

subgroupindex
#214617472
#214617462
#214606382
#214606352
#214541832
#772661