Finite Group Databasebeta

← #29580657

Subgroups — #29580657

Order 15616.

722 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 81 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#21455077C212yesmaximal
2#21445736C211yesmaximal, characteristic
3#21444352C212yesmaximal, complemented
4#21443727C211yesmaximal, characteristic, complemented
5C61×#2817C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C2×#14519808—22no—
8#16839922—24no—
9#16839917—22no—
10#16839912—22no—
11#16839761—22no—
12C2×#14519691—22no—
13#16839726C2212yes—
14#16839625C2211yescharacteristic
15#16839499—24no—
16C2×C22⋊Q8×C61C2212yes—
17C61×#400—22no—
18C61×#397—22no—
19C61×#395—22no—
20C61×#381C2211yescharacteristic
21C2×#11758222—22no—
22C2×#11758222—22no—
23C2×#11758222—44no—
24C22×Dic122D411yescharacteristic
25C22×D244D411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C4×D61—21no—
28#14519808—48no—
29C2×#11758202—24no—
30C2×#11758201—24no—
31C2×#11758194—22no—
32C2×C22⋊Dic61—21no—
33#14519700—24no—
34#14519697—24no—
35#14519691—44no—
36C22⋊#92418—22no—
37C22⋊#92418D414yes—
38#14519684—22no—
39C22×Q8×C61D412yes—
40C23×C4×C61C2311yescharacteristic
41C22⋊Q8×C61—22no—
42C22⋊Q8×C61—22no—
43C22⋊Q8×C61—22no—
44C22⋊Q8×C61—22no—
45C2×C4⋊C4×C61—22no—
46C2×C4⋊C4×C61—22no—
47C2×C4⋊C4×C61D412yes—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61D412yes—
50#95×C61—24no—
51#95×C61—22no—
52#95×C61—22no—
53C24×C61C2×D411yescharacteristic
54C23×D61—21no—
55C2×Q8×C61—22no—
56C2×Q8×C61—22no—
57C2×Q8×C61—44no—
58C22×C4×C61—22no—
59C22×C4×C61—22no—
60C22×C4×C61—22no—
61C22×C4×C61—42no—
62C22×C4×C61C2×D412yes—
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61C2×D412yes—
65C22×C4×C61#4111yescharacteristic
66C2×C22⋊D61—24no—
67C2×C22⋊D61—42no—
68C2×C22⋊D61—41no—
69C2×C22⋊D61—41no—
70C22×Dic61—21no—
71C22×Dic61—41no—
72#11758222—28no—
73#11758222—44no—
74#11758222—44no—
75#11758222—44no—
76C2×D244—22no—
77C2×D244—42no—
78C2×C4×D61—42no—
79C2×C4×D61—42no—
80C2×C4×D61—42no—
81C2×Dic122—22no—
82C2×Dic122—42no—
83C4⋊C4×C61—22no—
84C4⋊C4×C61—22no—
85C4⋊C4×C61—42no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—82no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—42no—
90C22⋊C4×C61—42no—
91C22⋊Dic61—42no—
92#11758202—48no—
93#11758201—48no—
94#11758194—44no—
95C2×#92418—24no—
96C23×C61—22no—
97C23×C61—42no—
98C23×C61#12011yescharacteristic
99C23×C61#12111yescharacteristic
100C23×C61#12711yescharacteristic
101C22×D61—22no—
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—82no—
105Q8×C61—44no—
106Q8×C61—82no—
107C2×C4×C61—24no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—42no—
111C2×C4×C61—44no—
112C2×C4×C61—44no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—44no—
117C2×C4×C61#12012yes—
118C2×C4×C61#12112yes—
119C22⋊D61—44no—
120C22⋊D61—44no—
121C2×Dic61—22no—
122C2×Dic61—41no—
123C2×Dic61—41no—
124C2×Dic61—81no—
125C2×Dic61—82no—
126D244—24no—
127D244—42no—
128C4×D61—48no—
129Dic122—24no—
130Dic122—42no—
131#92418—44no—
132#9982—611nomaximal, Sylow
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—82no—
139C22×C61#39211yescharacteristic
140C22×C61B(4,𝔽2)12yes—
141C2×D61—42no—
142C2×D61—82no—
143C2×D61—82no—
144C2×D61—82no—
145C4×C61—22no—
146C4×C61—22no—
147C4×C61—82no—
148C4×C61—82no—
149C4×C61—82no—
150Dic61—42no—
151Dic61—161no—
152#3007—612no—
153#2817D6111yescharacteristic, complemented
154#1782—611no—
155#1511—612no—
156#1306—611no—
157C2×C61—22no—
158C2×C61—41no—
159C2×C61—41no—
160C2×C61#200912yes—
161C2×C61#202111yescharacteristic
162D61—82no—
163C22×#41—611no—
164C2×#137—1222no—
165C2×C22⋊Q8C2×D6112yes—
166#452—1224no—
167#451—1222no—
168#450—1222no—
169#417—1222no—
170C2×#99—1222no—
171#407—612no—
172#400—22no—
173#397—22no—
174#395—22no—
175#381—611no—
176#381C2×D6111yescharacteristic
177#328—1224no—
178C61#998211yescharacteristic, complemented, Sylow
179C2×#41—1222no—
180C2×#41—1222no—
181C2×#41—2444no—
182C22×Q8—611no—
183C22×Q8C22⋊D6112yes—
184C22×D4—611no—
185C22×D4—1221no—
186C23×C4—1221no—
187C23×C4C22×D6111yescharacteristic
188#137—2448no—
189C2×Q16—1224no—
190C2×SD16—1224no—
191C2×M16—1222no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4C22⋊D6112yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1221no—
201C2×C22⋊C4C22⋊D6112yes—
202#103—1224no—
203#102—1224no—
204#99—2444no—
205C22⋊C8—614no—
206C22⋊C8—1222no—
207#95—24no—
208#95—22no—
209#95—22no—
210#95—1222no—
211C24—1221no—
212C24C2×C22⋊D6111yescharacteristic
213#41—1228no—
214#41—2444no—
215#41—2444no—
216#41—2444no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—44no—
220C2×Q8—1222no—
221C2×Q8—2442no—
222C2×D4—1224no—
223C2×D4—1222no—
224C2×D4—2442no—
225C2×D4—2442no—
226C2×D4—2441no—
227C2×D4—2441no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—1221no—
233C22×C4—2442no—
234C22×C4—2442no—
235C22×C4—2441no—
236C22×C4—2442no—
237C22×C4D4×D6111yescharacteristic
238C22×C4D4×D6112yes—
239C22×C4#1175822411yescharacteristic
240C22×C4C2×C22⋊D6112yes—
241Q16—2448no—
242SD16—2448no—
243M16—2444no—
244C2×C8—1224no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—82no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—2442no—
254C23—22no—
255C23—42no—
256C23—1222no—
257C23—4882no—
258C23—2441no—
259C23—2441no—
260C23#1451974611yescharacteristic
261C23#1451974811yescharacteristic
262C23#1451977311yescharacteristic
263Q8—44no—
264Q8—82no—
265Q8—1224no—
266Q8—2442no—
267D4—1224no—
268D4—2442no—
269D4—2444no—
270D4—2444no—
271C2×C4—24no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—1222no—
282C2×C4—4881no—
283C2×C4—4882no—
284C2×C4—2448no—
285C2×C4—2441no—
286C2×C4—2441no—
287C2×C4#1451974512yes—
288C2×C4#1451974912yes—
289C8—2444no—
290C22—22no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—82no—
296C22—4882no—
297C22—4882no—
298C22—4882no—
299C22—2442no—
300C22#1683967011yescharacteristic
301C22#1683994912yes—
302C4—22no—
303C4—22no—
304C4—82no—
305C4—82no—
306C4—82no—
307C4—9761no—
308C4—2442no—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—4882no—
313C2#2144687912yes—
314C2#2144694311yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#2817C2
socleC22×C61#392
prime radicalC22×#11758222C22
hypercenter#2817D61
hypocenterC61#9982
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
3. #21444352C2semidirect
4. #21443727C2semidirect
5. C61×#2817C2semidirect
153. #2817D61semidirect
178. C61#9982semidirect

Maximal subgroups

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

subgroupindex
#214550772
#214457362
#214443522
#214437272
C61×#28172
#998261