Finite Group Databasebeta

← #29580659

Subgroups — #29580659

Order 15616.

722 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 79 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#21455079C212yesmaximal
2#21453856C211yesmaximal, characteristic, complemented
3#21444351C212yesmaximal, complemented
4#21443727C211yesmaximal, characteristic, complemented
5C61×#1782C211yesmaximal, characteristic
6C2×#14519807—22no—
7C2×#14519756C2212yes—
8#16839924—24no—
9#16839918—22no—
10#16839913—22no—
11#16839763—22no—
12#16839726C2212yes—
13#16839702—22no—
14#16839687—22no—
15#16839683—22no—
16#16839498—24no—
17C22×#41×C61C2211yescharacteristic
18C2×#99×C61—22no—
19C61×#381C2211yescharacteristic
20C61×#381C2211yescharacteristic
21C22×Dic122D412yes—
22#14519807—48no—
23C2×#11758202—24no—
24C2×#11758200—24no—
25C2×#11758194—22no—
26#14519756—22no—
27#14519756—22no—
28#14519756—22no—
29#14519756—22no—
30C2×C4⋊Dic61—22no—
31C2×C4⋊Dic61D412yes—
32C2×#11758196—22no—
33C2×C22⋊Dic61—22no—
34#14519701—24no—
35#14519698—24no—
36C22⋊#92418—22no—
37C22⋊#92418D414yes—
38#14519684—24no—
39#14519684—22no—
40C2×#41×C61—22no—
41C2×#41×C61—22no—
42C2×#41×C61—44no—
43C22×Q8×C61D411yescharacteristic
44C22×D4×C61—21no—
45C22×D4×C61D411yescharacteristic
46C23×C4×C61—21no—
47C23×C4×C61C2311yescharacteristic
48C2×C22⋊C4×C61—21no—
49C2×C22⋊C4×C61D412yes—
50#99×C61—44no—
51#95×C61—22no—
52#95×C61—22no—
53C24×C61—21no—
54C24×C61C2×D411yescharacteristic
55#41×C61—28no—
56#41×C61—44no—
57#41×C61—44no—
58#41×C61—44no—
59C2×Q8×C61—22no—
60C2×Q8×C61—42no—
61C2×D4×C61—24no—
62C2×D4×C61—22no—
63C2×D4×C61—42no—
64C2×D4×C61—42no—
65C2×D4×C61—41no—
66C2×D4×C61—41no—
67C22×C4×C61—21no—
68C22×C4×C61—22no—
69C22×C4×C61—42no—
70C22×C4×C61—42no—
71C22×C4×C61—42no—
72C22×C4×C61—42no—
73C22×C4×C61—41no—
74C22×C4×C61C2×D412yes—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D412yes—
77C22×C4×C61#4111yescharacteristic
78C22×Dic61—22no—
79C22×Dic61—22no—
80C2×Dic122—22no—
81C2×Dic122—22no—
82C2×Dic122—44no—
83C22⋊C4×C61—42no—
84C22⋊C4×C61—42no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87#11758202—48no—
88#11758200—48no—
89C4⋊Dic61—22no—
90C4⋊Dic61—22no—
91C4⋊Dic61—82no—
92#11758196—42no—
93#11758196—42no—
94#11758194—44no—
95C2×#92418—24no—
96C23×C61—22no—
97C23×C61—22no—
98C23×C61—42no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—82no—
102C23×C61#12011yescharacteristic
103C23×C61#12111yescharacteristic
104C23×C61#12711yescharacteristic
105Q8×C61—24no—
106Q8×C61—42no—
107D4×C61—24no—
108D4×C61—42no—
109D4×C61—44no—
110D4×C61—44no—
111C2×C4×C61—24no—
112C2×C4×C61—22no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61—48no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—44no—
120C2×C4×C61—42no—
121C2×C4×C61—81no—
122C2×C4×C61—82no—
123C2×C4×C61#12012yes—
124C2×C4×C61#12112yes—
125C2×Dic61—44no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—44no—
129Dic122—44no—
130Dic122—82no—
131#92418—44no—
132#9982—611nomaximal, Sylow
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—42no—
139C22×C61—82no—
140C22×C61—82no—
141C22×C61—82no—
142C22×C61—82no—
143C22×C61#39211yescharacteristic
144C22×C61B(4,𝔽2)12yes—
145C4×C61—22no—
146C4×C61—22no—
147C4×C61—42no—
148C4×C61—82no—
149C4×C61—161no—
150Dic61—82no—
151Dic61—82no—
152#3007—612no—
153#2817—611no—
154#1782D6111yescharacteristic
155#1511—612no—
156#1306—611no—
157C2×C61—22no—
158C2×C61—41no—
159C2×C61—41no—
160C2×C61—82no—
161C2×C61#200912yes—
162C2×C61#202111yescharacteristic
163C22×#41C2×D6111yescharacteristic
164C2×#137—1222no—
165C2×C22⋊Q8—612no—
166#452—1224no—
167#451—1222no—
168#450—1222no—
169#417—1222no—
170C2×#99—22no—
171#407—612no—
172#400—1222no—
173#397—1222no—
174#395—1222no—
175#381C2×D6111yescharacteristic
176#381C2×D6111yescharacteristic
177#328—1224no—
178C61#998211yescharacteristic, complemented, Sylow
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—44no—
182C22×Q8—612no—
183C22×Q8D24411yescharacteristic
184C22×D4—21no—
185C22×D4D24411yescharacteristic
186C23×C4—21no—
187C23×C4C22×D6111yescharacteristic
188#137—2448no—
189C2×Q16—1224no—
190C2×SD16—1224no—
191C2×M16—1222no—
192C22⋊Q8—1222no—
193C22⋊Q8—1222no—
194C22⋊Q8—1222no—
195C22⋊Q8—1222no—
196C2×C4⋊C4—612no—
197C2×C4⋊C4—1222no—
198C2×C4⋊C4—1222no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—1222no—
201C2×C22⋊C4C22⋊D6112yes—
202#103—1224no—
203#102—1224no—
204#99—44no—
205C22⋊C8—614no—
206C22⋊C8—1222no—
207#95—22no—
208#95—22no—
209#95—1224no—
210#95—1222no—
211C24—21no—
212C24C2×C22⋊D6111yescharacteristic
213#41—28no—
214#41—44no—
215#41—44no—
216#41—44no—
217C2×Q8—22no—
218C2×Q8—42no—
219C2×Q8—1222no—
220C2×Q8—1222no—
221C2×Q8—2444no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—41no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—1222no—
236C22×C4—1222no—
237C22×C4C2×D24411yescharacteristic
238C22×C4#1175822211yescharacteristic
239C22×C4D4×D6112yes—
240C22×C4D4×D6112yes—
241Q16—2448no—
242SD16—2448no—
243M16—2444no—
244C2×C8—1224no—
245C4⋊C4—1222no—
246C4⋊C4—1222no—
247C4⋊C4—4882no—
248C4⋊C4—2442no—
249C4⋊C4—2442no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—2442no—
253C22⋊C4—2442no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23—82no—
260C23#1451974511yescharacteristic
261C23#1451975111yescharacteristic
262C23#1451977311yescharacteristic
263Q8—24no—
264Q8—42no—
265Q8—4882no—
266Q8—2444no—
267D4—24no—
268D4—42no—
269D4—44no—
270D4—44no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—48no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—2444no—
284C2×C4—2444no—
285C2×C4—2442no—
286C2×C4—2442no—
287C2×C4#1451974712yes—
288C2×C4#1451975012yes—
289C8—2444no—
290C22—22no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—42no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22#1683967111yescharacteristic
301C22#1683994812yes—
302C4—22no—
303C4—22no—
304C4—42no—
305C4—82no—
306C4—161no—
307C4—4882no—
308C4—4882no—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—82no—
313C2#2144687812yes—
314C2#2144694511yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#1782C2
socleC22×C61#392
prime radicalC22×#41×C61C22
hypercenter#1782D61
hypocenterC61#9982
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
2. #21453856C2semidirect
3. #21444351C2semidirect
4. #21443727C2semidirect
178. C61#9982semidirect

Maximal subgroups

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

subgroupindex
#214550792
#214538562
#214443512
#214437272
C61×#17822
#998261