Finite Group Databasebeta

← #30601636

Subgroups — #30601636

Order 15616.

610 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 128 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#21461752C211yesmaximal, characteristic, complemented
2C4×#14519832C211yesmaximal, characteristic, complemented
3C2×Q8⋊#92420C212yesmaximal, complemented
4C2×D4⋊#92420C212yesmaximal
5#124×Dic61C211yesmaximal, characteristic
6C2×#14519844—22no—
7C2×C4⋊#92420C2212yes—
8C2×#14519832—22no—
9C2×C4×#92420C2212yes—
10Q8⋊#92420—24no—
11D4⋊#92420—24no—
12C2×Q8×Dic61C2211yescharacteristic
13C2×D4×Dic61C2211yescharacteristic
14#16839669C411yescharacteristic
15#16839665C411yescharacteristic
16#16839655—22no—
17#16839628C411yescharacteristic
18C22⋊C4×Dic61—22no—
19C42×Dic61C2211yescharacteristic
20C2×#124×C61C411yescharacteristic
21C22×#92420—22no—
22C22×#92420—22no—
23#14519844—44no—
24C4⋊#92420—22no—
25C4⋊#92420—22no—
26#14519832—28no—
27C4×#92420—22no—
28C4×#92420D414yes—
29C23×Dic61—21no—
30Q8×Dic61—22no—
31Q8×Dic61—22no—
32D4×Dic61—22no—
33D4×Dic61—22no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—21no—
39C2×C4×Dic61—21no—
40C2×C4×Dic61C2×C411yescharacteristic
41C2×C4×Dic61D412yes—
42C2×C4×Dic61D412yes—
43C2×C4×Dic61C2311yescharacteristic
44#14519684—22no—
45#14519684—22no—
46C22×Q8×C61C2×C411yescharacteristic
47C22×D4×C61C2×C411yescharacteristic
48#124×C61—22no—
49#124×C61—22no—
50C2×C22⋊C4×C61—22no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61—21no—
53C2×Q8×C61—21no—
54C2×Q8×C61—22no—
55C2×Q8×C61—22no—
56C2×Q8×C61M1612yes—
57C2×D4×C61—22no—
58C2×D4×C61—21no—
59C2×D4×C61—22no—
60C2×D4×C61M1612yes—
61C22×C4×C61—21no—
62C22×C4×C61C22⋊C412yes—
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—21no—
65C22×Dic61—21no—
66C22×Dic61—41no—
67C22×Dic61—41no—
68C22×Dic61—42no—
69C22×Dic61—42no—
70C22×Dic61C22⋊C412yes—
71C22×Dic61C2×D411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C2×#92420—28no—
74C2×#92420—24no—
75C2×#92420—42no—
76C2×#92420—44no—
77C22⋊C4×C61—28no—
78C42×C61—21no—
79C42×C61—21no—
80C22⋊Dic61—44no—
81C22⋊Dic61—42no—
82C4⋊Dic61—22no—
83C4⋊Dic61—22no—
84C4⋊Dic61—42no—
85C4×Dic61—22no—
86C4×Dic61—24no—
87C4×Dic61—22no—
88C4×Dic61—22no—
89C4×Dic61—41no—
90C4×Dic61—41no—
91C4×Dic61C2×D412yes—
92C4×Dic61#4112yes—
93C23×C61—22no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61C2×C22⋊C411yescharacteristic
99Q8×C61—24no—
100Q8×C61—42no—
101D4×C61—24no—
102D4×C61—42no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61C4×D412yes—
110C2×C4×C61C2×M1611yescharacteristic
111C2×C4×C61C2×M1611yescharacteristic
112C2×C4×C61#13112yes—
113C2×Dic61—22no—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—82no—
121C2×Dic61—81no—
122C2×Dic61—81no—
123C2×Dic61#10414yes—
124C2×Dic61#12112yes—
125C2×Dic61#12411yescharacteristic
126C2×Dic61#12711yescharacteristic
127C2×Dic61AGL(1,ℤ/8)12yes—
128C2×Dic61#13712yes—
129#92420—28no—
130#92420—82no—
131#9646—611nomaximal, Sylow
132C22×C61—42no—
133C22×C61—44no—
134C22×C61—81no—
135C22×C61—81no—
136C22×C61—82no—
137C22×C61#39011yescharacteristic
138C22×C61#40711yescharacteristic
139C22×C61#40811yescharacteristic
140C22×C61#41712yes—
141C22×C61C2×#10412yes—
142C4×C61—42no—
143C4×C61—42no—
144C4×C61#43512yes—
145C4×C61#43512yes—
146Dic61—22no—
147Dic61—24no—
148Dic61—81no—
149Dic61—81no—
150Dic61#48912yes—
151Dic61#49611yescharacteristic
152Dic61#49711yescharacteristic
153C4×#124—611no—
154#1842—611no—
155C4×#97—611no—
156C2×Q8⋊C8—612no—
157C2×D4⋊C8—612no—
158C2×C61—82no—
159C2×C61#148312yes—
160C2×C61#148912yes—
161C2×C61#195511yescharacteristic
162C2×C61#195911yescharacteristic
163C2×C61#196211yescharacteristic
164C2×#124#692311yescharacteristic
165C2×C4×Q8—611no—
166C2×C4×D4—611no—
167C2×C4⋊C8—612no—
168C2×C4⋊C8—1222no—
169C2×#97—1222no—
170C2×C4×C8—612no—
171#391—611no—
172#390—611no—
173#388—1222no—
174#382—611no—
175C4×C22⋊C4—1222no—
176C43—611no—
177Q8⋊C8—1224no—
178D4⋊C8—1224no—
179C61#964611yescharacteristic, complemented, Sylow
180C22×Q8C2×#692311yescharacteristic
181C22×D4C2×#692311yescharacteristic
182C23×C4—1221no—
183C22×C8—1222no—
184C22×C8—1222no—
185#124—22no—
186#124—22no—
187C4×Q8—1222no—
188C4×Q8—1222no—
189C4×D4—1222no—
190C4×D4—1222no—
191C2×C4⋊C4—611no—
192C2×C4⋊C4—611no—
193C2×C4⋊C4—1221no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—1221no—
196C2×C22⋊C4—1222no—
197C2×C42—611no—
198C2×C42—612no—
199C2×C42—612no—
200C2×C42—611no—
201C2×C42—1221no—
202C2×C42C2×#692311yescharacteristic
203C4⋊C8—1222no—
204C4⋊C8—1222no—
205C4⋊C8—2444no—
206#97—1228no—
207C4×C8—614no—
208C4×C8—1222no—
209#95—1222no—
210#95—1222no—
211C24—21no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8#1175821712yes—
216C2×D4—22no—
217C2×D4—21no—
218C2×D4—22no—
219C2×D4#1175821712yes—
220C22×C4—21no—
221C22×C4—611no—
222C22×C4—612no—
223C22×C4—611no—
224C22×C4—1221no—
225C22×C4—1221no—
226C22×C4—2441no—
227C22×C4—2441no—
228C22×C4—2442no—
229C22×C4—2442no—
230C22×C4C22⋊#692312yes—
231C22×C4C22×#692311yescharacteristic
232C2×C8—1224no—
233C2×C8—1228no—
234C2×C8—2442no—
235C2×C8—2444no—
236C4⋊C4—1222no—
237C4⋊C4—1222no—
238C4⋊C4—2442no—
239C22⋊C4—28no—
240C22⋊C4—2442no—
241C22⋊C4—2444no—
242C42—21no—
243C42—21no—
244C42—612no—
245C42—612no—
246C42—1224no—
247C42—1222no—
248C42—1222no—
249C42—1222no—
250C42—2441no—
251C42—2441no—
252C23—22no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—42no—
257C23C2×C22⋊#692311yescharacteristic
258Q8—24no—
259Q8—42no—
260D4—24no—
261D4—42no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—611no—
269C2×C4—611no—
270C2×C4—612no—
271C2×C4—612no—
272C2×C4—612no—
273C2×C4—614no—
274C2×C4—1222no—
275C2×C4—1222no—
276C2×C4—1222no—
277C2×C4—4882no—
278C2×C4—4881no—
279C2×C4—4881no—
280C2×C4—2441no—
281C2×C4—2441no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4D4×#692312yes—
285C2×C4C2×#1175821711yescharacteristic
286C2×C4C2×#1175821711yescharacteristic
287C2×C4#1451986412yes—
288C8—1228no—
289C8—4882no—
290C22—44no—
291C22—42no—
292C22—82no—
293C22—81no—
294C22—81no—
295C22#1684071411yescharacteristic
296C22#1684073211yescharacteristic
297C22#1684073611yescharacteristic
298C22#1684075312yes—
299C22D61⋊#10412yes—
300C4—42no—
301C4—42no—
302C4—612no—
303C4—611no—
304C4—611no—
305C4—1224no—
306C4—1222no—
307C4—4881no—
308C4—4881no—
309C4#1684079312yes—
310C4#1684079312yes—
311C2—82no—
312C2#2146113512yes—
313C2#2146115412yes—
314C2#2146194311yescharacteristic
315C2#2146194911yescharacteristic
316C2#2146195411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenterC2×#124#6923
hypocenterC61#9646
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21461752C2semidirect
2. C4×#14519832C2semidirect
3. C2×Q8⋊#92420C2semidirect
179. C61#9646semidirect

Maximal subgroups

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

subgroupindex
#214617522
C4×#145198322
C2×Q8⋊#924202
C2×D4⋊#924202
#124×Dic612
#964661