Finite Group Databasebeta

← #30598776

Subgroups — #30598776

Order 15616.

910 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 100 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#21461714C212yesmaximal
2#21461714C212yesmaximal
3#21461714C212yesmaximal
4C22×#14519772C211yesmaximal, characteristic
5#16840703C2214yes—
6#16840640—24no—
7#16840640—24no—
8#16840640—24no—
9C2×#14519772—24no—
10C2×#14519772—24no—
11C2×#14519772—24no—
12C22×C4⋊Dic61C411yescharacteristic, complemented
13C22×#11758196C411yescharacteristic, complemented
14C22×#11758198C2211yescharacteristic
15C22×#11758198C2211yescharacteristic
16C22×#11758198C2211yescharacteristic
17C22×C4×Dic61C411yescharacteristic, complemented
18C22×C4⋊C4×C61C411yescharacteristic, complemented
19C2×C22⋊#6923—24no—
20C2×C22⋊#6923—24no—
21C2×C22⋊#6923—24no—
22C24×D61C2311yescharacteristic
23C23×Dic61C2×C411yescharacteristic
24C23×Dic61C2×C411yescharacteristic
25C23×Dic61C2×C411yescharacteristic
26#14519772—232no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61—22no—
30C2×#11758196—22no—
31C2×#11758196—22no—
32C2×#11758196—22no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C2×#11758198—22no—
37C2×#11758198—22no—
38C2×#11758198—22no—
39C2×#11758198D414yes—
40C2×#11758198D414yes—
41C2×#11758198D414yes—
42C2×C4×Dic61—22no—
43C2×C4×Dic61—22no—
44C2×C4×Dic61—22no—
45C23×C4×C61C2×C411yescharacteristic
46C23×C4×C61C2×C411yescharacteristic
47C23×C4×C61C2×C411yescharacteristic
48C2×C4⋊C4×C61—22no—
49C2×C4⋊C4×C61—22no—
50C2×C4⋊C4×C61—22no—
51C24×C61C22×C411yescharacteristic
52C23×D61—42no—
53C23×D61—42no—
54C23×D61—42no—
55C23×D61C2×D412yes—
56C23×D61C2×D412yes—
57C23×D61C2×D412yes—
58C22×#6923—44no—
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—42no—
66C22×C4×C61—42no—
67C22×C4×C61—42no—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22⋊C412yes—
70C22×C4×C61C22⋊C412yes—
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—42no—
78C22×Dic61—42no—
79C22×Dic61—42no—
80C22×Dic61C22⋊C412yes—
81C22×Dic61C22⋊C412yes—
82C22×Dic61C22⋊C412yes—
83C22⋊#6923—48no—
84C22⋊#6923—48no—
85C22⋊#6923—48no—
86C4⋊C4×C61—28no—
87#11758198—28no—
88#11758198—28no—
89#11758198—28no—
90C4⋊Dic61—28no—
91#11758196—28no—
92C4×Dic61—28no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61C2×C22⋊C411yescharacteristic
97C23×C61C2×C22⋊C411yescharacteristic
98C23×C61C2×C22⋊C411yescharacteristic
99C22×D61—24no—
100C22×D61—24no—
101C22×D61—24no—
102C22×D61—42no—
103C22×D61—42no—
104C22×D61—42no—
105C22×D61—42no—
106C22×D61—42no—
107C22×D61—42no—
108C22×D61—88no—
109C22×D61#12014yes—
110C2×#6923—84no—
111C2×#6923—84no—
112C2×#6923—84no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—22no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×Dic61—22no—
126C2×Dic61—22no—
127C2×Dic61—22no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—42no—
132C2×Dic61—42no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137#8545—611nomaximal, Sylow
138C22×C61—28no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—21no—
143C22×C61—21no—
144C22×C61—21no—
145C22×C61#37911yescharacteristic
146C22×C61#41012yes—
147C22×C61#41012yes—
148C22×C61#41012yes—
149C2×D61—24no—
150C2×D61—24no—
151C2×D61—24no—
152C2×D61—84no—
153C2×D61—84no—
154C2×D61—84no—
155#6923—164no—
156C4×C61—42no—
157C4×C61—42no—
158C4×C61—42no—
159Dic61—42no—
160Dic61—42no—
161Dic61—42no—
162C22×#126—611no—
163#1819—612no—
164#1819—612no—
165#1819—612no—
166C2×C61—22no—
167C2×C61—22no—
168C2×C61—22no—
169C2×C61#178211yescharacteristic
170C2×C61#178211yescharacteristic
171C2×C61#178211yescharacteristic
172D61—44no—
173C2×#126—1224no—
174C2×#126—1224no—
175C2×#126—1224no—
176C22×C4⋊C4—611no—
177C22×C4⋊C4—611no—
178C22×C4⋊C4#692311yescharacteristic, complemented
179C22×C22⋊C4—611no—
180C22×C22⋊C4—611no—
181C22×C22⋊C4—611no—
182C22×C42—611no—
183#379—614no—
184#342—1224no—
185#342—1224no—
186#342—1224no—
187C61#854511yescharacteristic, complemented, Sylow
188C25—611no—
189C23×C4—611no—
190C23×C4—611no—
191C23×C4—611no—
192C23×C4C2×#692311yescharacteristic
193C23×C4C2×#692311yescharacteristic
194C23×C4C2×#692311yescharacteristic
195#126—12232no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—1222no—
200C2×C4⋊C4—1222no—
201C2×C4⋊C4—1222no—
202C2×C4⋊C4—1222no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1222no—
205C2×C22⋊C4—614no—
206C2×C22⋊C4—614no—
207C2×C22⋊C4—614no—
208C2×C22⋊C4—1222no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4—1222no—
211C2×C22⋊C4—1222no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1224no—
215C2×C22⋊C4—1224no—
216C2×C22⋊C4—1224no—
217C2×C42—1222no—
218C2×C42—1222no—
219C2×C42—1222no—
220C24—612no—
221C24—612no—
222C24—612no—
223C24—2442no—
224C24—2442no—
225C24—2442no—
226C24C22×#692311yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—612no—
237C22×C4—612no—
238C22×C4—612no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4—1221no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4—2444no—
246C22×C4—2442no—
247C22×C4—2442no—
248C22×C4—2442no—
249C22×C4C22⋊#692312yes—
250C22×C4C22⋊#692312yes—
251C22×C4C22⋊#692312yes—
252C4⋊C4—28no—
253C4⋊C4—1228no—
254C4⋊C4—1228no—
255C22⋊C4—1228no—
256C22⋊C4—1228no—
257C22⋊C4—1228no—
258C22⋊C4—2448no—
259C22⋊C4—2448no—
260C22⋊C4—2448no—
261C42—1228no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—614no—
266C23—1224no—
267C23—1224no—
268C23—1224no—
269C23—4888no—
270C23—2442no—
271C23—2442no—
272C23—2442no—
273C23—2442no—
274C23—2442no—
275C23—2442no—
276C23C2×C22⋊#692311yescharacteristic
277C23C2×C22⋊#692311yescharacteristic
278C23C2×C22⋊#692311yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—4884no—
295C2×C4—4884no—
296C2×C4—4884no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4—2442no—
305C2×C4—2442no—
306C22—28no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—1224no—
314C22—1224no—
315C22—1224no—
316C22—4884no—
317C22—4884no—
318C22—4884no—
319C22#1684070311yescharacteristic
320C22#1684074112yes—
321C22#1684074212yes—
322C22#1684074312yes—
323C4—42no—
324C4—42no—
325C4—42no—
326C4—9764no—
327C4—2442no—
328C4—2442no—
329C4—2442no—
330C2—22no—
331C2—22no—
332C2—22no—
333C2—2444no—
334C2#2146164811yescharacteristic
335C2#2146164911yescharacteristic
336C2#2146165011yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC22×C61#379
prime radical, cosocleC24×D61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8545
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. C22×C4⋊Dic61C4semidirect
13. C22×#11758196C4semidirect
17. C22×C4×Dic61C4semidirect
18. C22×C4⋊C4×C61C4semidirect
178. C22×C4⋊C4#6923semidirect
187. C61#8545semidirect

Maximal subgroups

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

subgroupindex
#214617142
C22×#145197722
#854561