Finite Group Databasebeta

← #29580575

Subgroups — #29580575

Order 15616.

804 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#16839920C212yesmaximal
2#21451933C211yesmaximal, characteristic
3C2×#16839500C212yesmaximal
4#21444089C211yesmaximal, characteristic
5C61×#1280C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C22×Dic244—22no—
8C2×#14519756C2212yes—
9#16839920—24no—
10#16839920—24no—
11C2×#14519702—22no—
12#16839694—22no—
13#16839683—22no—
14#16839622—22no—
15C2×#14519684—22no—
16C2×#14519684C2211yescharacteristic
17#16839500—24no—
18C2×C22⋊C8×C61C2212yes—
19C2×#95×C61C2211yescharacteristic
20C22×Dic122—22no—
21C22×Dic122—22no—
22C22×Dic122—21no—
23C22×Dic122—44no—
24C22×Dic122D412yes—
25C22×Dic122D412yes—
26C23×Dic61—21no—
27C23×Dic61—21no—
28C2×Dic244—28no—
29C2×Dic244—48no—
30#14519756—22no—
31#14519756—22no—
32#14519756—22no—
33#14519756—22no—
34C2×C4⋊Dic61—22no—
35C2×C4⋊Dic61—22no—
36C2×C4⋊Dic61D412yes—
37C2×#11758196—22no—
38C2×C22⋊Dic61—22no—
39#14519702—28no—
40#14519684—24no—
41#14519684—22no—
42#14519684—44no—
43C23×C4×C61D412yes—
44C23×C4×C61C2311yescharacteristic
45C22×C8×C61—22no—
46C22⋊C8×C61—22no—
47C22⋊C8×C61D414yes—
48#95×C61—22no—
49C24×C61C2×D411yescharacteristic
50C22×C4×C61—22no—
51C22×C4×C61—22no—
52C22×C4×C61—42no—
53C22×C4×C61—42no—
54C22×C4×C61—42no—
55C22×C4×C61Q1614yescomplemented
56C22×C4×C61C2×D412yes—
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D412yes—
59C22×C4×C61#4111yescharacteristic
60C22×Dic61—24no—
61C22×Dic61—22no—
62C22×Dic61—22no—
63C22×Dic61—22no—
64C22×Dic61—22no—
65C22×Dic61—42no—
66C22×Dic61—41no—
67C22×Dic61—41no—
68C22×Dic61—41no—
69C22×Dic61—42no—
70C22×Dic61—41no—
71C22×Dic61—42no—
72C2×Dic122—28no—
73C2×Dic122—24no—
74C2×Dic122—22no—
75C2×Dic122—22no—
76C2×Dic122—44no—
77C2×Dic122—48no—
78C2×Dic122—44no—
79C2×Dic122—44no—
80C2×Dic122—42no—
81C2×Dic122—44no—
82C2×C8×C61—24no—
83C2×C8×C61—44no—
84C22⋊Dic61—42no—
85C22⋊Dic61—42no—
86C4⋊Dic61—22no—
87C4⋊Dic61—22no—
88C4⋊Dic61—82no—
89C4⋊Dic61—82no—
90#11758196—42no—
91#11758196—42no—
92Dic244—416no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—22no—
96C23×C61#12011yescharacteristic
97C23×C61#12111yescharacteristic
98C23×C61#12711yescharacteristic
99C23×C61C2×Q1612yes—
100C23×C61#13712yes—
101C2×C4×C61—24no—
102C2×C4×C61—24no—
103C2×C4×C61—41no—
104C2×C4×C61—42no—
105C2×C4×C61—41no—
106C2×C4×C61—42no—
107C2×C4×C61—42no—
108C2×C4×C61—44no—
109C2×C4×C61—82no—
110C2×C4×C61#12012yes—
111C2×C4×C61#12112yes—
112C2×Dic61—24no—
113C2×Dic61—42no—
114C2×Dic61—44no—
115C2×Dic61—42no—
116C2×Dic61—44no—
117C2×Dic61—44no—
118C2×Dic61—42no—
119C2×Dic61—42no—
120C2×Dic61—42no—
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—44no—
124C2×Dic61—82no—
125C2×Dic61—81no—
126C2×Dic61—81no—
127Dic122—28no—
128Dic122—44no—
129Dic122—48no—
130Dic122—44no—
131Dic122—82no—
132C8×C61—44no—
133#9967—611nomaximal, Sylow
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61#39211yescharacteristic
143C22×C61#45112yes—
144C22×C61#45511yescharacteristic
145C22×C61#45611yescharacteristic
146C22×C61B(4,𝔽2)12yes—
147C22×C61#46212yes—
148C22×C61#49411yescharacteristic
149C22×C61#49711yescharacteristic
150C4×C61—22no—
151C4×C61—22no—
152C4×C61—44no—
153Dic61—44no—
154Dic61—82no—
155Dic61—82no—
156Dic61—82no—
157C2×#451—612no—
158#2659—611no—
159C2×#328—612no—
160#1446—611no—
161#1280D6111yescharacteristic
162C2×C61—21no—
163C2×C61—22no—
164C2×C61—21no—
165C2×C61#160212yes—
166C2×C61#160512yes—
167C2×C61#200711yescharacteristic
168C2×C61#201611yescharacteristic
169C2×C61#202111yescharacteristic
170C23×Q8—611no—
171C22×Q16—1222no—
172C2×C22⋊Q8—612no—
173#451—1224no—
174#451—1224no—
175C2×#103—1222no—
176C2×C22⋊C8C2×D6112yes—
177#398—1222no—
178#395—1222no—
179#380—1222no—
180C2×#95—611no—
181C2×#95—1222no—
182C2×#95C2×D6111yescharacteristic
183#328—1224no—
184C61#996711yescharacteristic, complemented, Sylow
185C22×Q8—612no—
186C22×Q8—612no—
187C22×Q8—1221no—
188C22×Q8—1222no—
189C22×Q8—1222no—
190C22×Q8—2444no—
191C23×C4—1221no—
192C23×C4—1221no—
193C23×C4D24412yes—
194C23×C4C22×D6111yescharacteristic
195C2×Q16—1228no—
196C2×Q16—2448no—
197C22×C8—22no—
198C22⋊Q8—1222no—
199C22⋊Q8—1222no—
200C22⋊Q8—1222no—
201C22⋊Q8—1222no—
202C2×C4⋊C4—612no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—1222no—
206C2×C22⋊C4—1222no—
207#103—1228no—
208C22⋊C8—22no—
209C22⋊C8D24414yes—
210#95—22no—
211#95—1224no—
212#95—1222no—
213#95—2444no—
214C24C2×D24411yescharacteristic
215C2×Q8—1228no—
216C2×Q8—1224no—
217C2×Q8—1222no—
218C2×Q8—1222no—
219C2×Q8—2444no—
220C2×Q8—2444no—
221C2×Q8—2444no—
222C2×Q8—2444no—
223C2×Q8—2448no—
224C2×Q8—2442no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—1224no—
231C22×C4—1222no—
232C22×C4—1222no—
233C22×C4—1222no—
234C22×C4—1222no—
235C22×C4—2442no—
236C22×C4—2441no—
237C22×C4—2442no—
238C22×C4—2441no—
239C22×C4—2442no—
240C22×C4—2441no—
241C22×C4—2441no—
242C22×C4Dic24414yescomplemented
243C22×C4C2×D24412yes—
244C22×C4D4×D6112yes—
245C22×C4D4×D6111yescharacteristic
246C22×C4#1175822611yescharacteristic
247Q16—24416no—
248C2×C8—24no—
249C2×C8—44no—
250C4⋊C4—1222no—
251C4⋊C4—1222no—
252C4⋊C4—4882no—
253C4⋊C4—4882no—
254C4⋊C4—2442no—
255C4⋊C4—2442no—
256C22⋊C4—2442no—
257C22⋊C4—2442no—
258C23—21no—
259C23—21no—
260C23—22no—
261C23#1451974711yescharacteristic
262C23#1451975211yescharacteristic
263C23#1451977411yescharacteristic
264C23C2×Dic24412yes—
265C23#1451981012yes—
266Q8—1228no—
267Q8—4882no—
268Q8—2444no—
269Q8—2444no—
270Q8—2448no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—82no—
280C2×C4—1224no—
281C2×C4—4881no—
282C2×C4—4882no—
283C2×C4—4881no—
284C2×C4—2444no—
285C2×C4—2442no—
286C2×C4—2442no—
287C2×C4—2442no—
288C2×C4—2444no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2444no—
293C2×C4—2444no—
294C2×C4—2442no—
295C2×C4#1451974712yes—
296C2×C4#1451975212yes—
297C8—44no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22#1683967211yescharacteristic
307C22#1683992012yes—
308C22#1683994211yescharacteristic
309C22#1683994711yescharacteristic
310C22#1683995012yes—
311C22#1683997312yes—
312C22#1684014211yescharacteristic
313C22#1684015611yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—4882no—
318C4—4882no—
319C4—4882no—
320C4—2444no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2#2144477812yes—
325C2#2144479512yes—
326C2#2144686811yescharacteristic
327C2#2144692111yescharacteristic
328C2#2144694611yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1280C2
socleC23×C61#120
prime radicalC24×C61C2×D4
hypercenter#1280D61
hypocenterC61#9967
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
55. C22×C4×C61Q16semidirect
184. C61#9967semidirect
242. C22×C4Dic244semidirect

Maximal subgroups

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

subgroupindex
C2×#168399202
#214519332
C2×#168395002
#214440892
C61×#12802
#996761