Finite Group Databasebeta

← #29592129

Subgroups — #29592129

Order 15616.

476 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 67 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#21456570C211yesmaximal, characteristic
2#21447738C212yesmaximal, complemented
3#21446829C211yesmaximal, characteristic, complemented
4#21443829C212yesmaximal, complemented
5C61×#1314C211yesmaximal, characteristic, complemented
6C22×Dic244C2211yescharacteristic, complemented
7C2×#14519756—21no—
8C2×#14519751—21no—
9C2×#14519722—22no—
10#16840076—22no—
11#16839998—21no—
12#16839998C2212yes—
13#16839898—22no—
14C2×#14519702—21no—
15#16839567—22no—
16#16839525C2211yescharacteristic, complemented
17C2×C22⋊C8×C61C2211yescharacteristic
18C61×C22⋊C16C2212yescomplemented
19C22×Dic122—21no—
20C22×D244—21no—
21C22×C22⋊D61—41no—
22C2×Dic244—22no—
23C2×Dic244—42no—
24C2×D488—21no—
25C2×D488D412yes—
26#14519756—42no—
27#14519756—42no—
28#14519751—22no—
29#14519751—22no—
30#14519751—41no—
31#14519751—41no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—21no—
34C2×#11758196—41no—
35C2×C22⋊Dic61—41no—
36C2×#11758198—41no—
37#14519722—44no—
38#14519711—21no—
39#14519711D412yes—
40#14519702—42no—
41#14519699—22no—
42#14519699—22no—
43C23×C4×C61D411yescharacteristic
44C22×C8×C61D411yescharacteristic
45C22×C8×C61C2311yescharacteristic
46C2×C16×C61—22no—
47C22⋊C8×C61—22no—
48C24×C61D811yescharacteristic
49C23×D61—41no—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—41no—
53C22×C4×C61—41no—
54C22×C4×C61D811yescharacteristic
55C22×C4×C61C2×D411yescharacteristic
56C2×C22⋊D61—42no—
57C2×C22⋊D61—42no—
58C2×C22⋊D61—82no—
59C2×C22⋊D61—82no—
60C22×Dic61—41no—
61C22×Dic61—41no—
62C22×Dic61—41no—
63C2×D244—22no—
64C2×D244—41no—
65C2×D244—81no—
66C2×D244—81no—
67C2×Dic122—22no—
68C2×Dic122—41no—
69C2×Dic122—41no—
70C2×Dic122—41no—
71C2×Dic122—41no—
72C2×Dic122—41no—
73C2×C8×C61—21no—
74C2×C8×C61—41no—
75C2×C8×C61—41no—
76C2×C8×C61C2×D412yes—
77C22⋊Dic61—44no—
78#11758198—42no—
79#11758198—42no—
80C4⋊Dic61—22no—
81C4⋊Dic61—22no—
82C4⋊Dic61—41no—
83C4⋊Dic61—41no—
84#11758196—44no—
85Dic244—24no—
86Dic244—42no—
87D488—42no—
88D488—81no—
89C16×C61—42no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—41no—
93C23×C61—41no—
94C23×C61SD3212yescomplemented
95C23×C61C2×D811yescharacteristic
96C22×D61—42no—
97C22×D61—82no—
98C22×D61—82no—
99C22×D61—82no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—41no—
104C2×C4×C61—41no—
105C2×C4×C61—41no—
106C2×C4×C61—81no—
107C2×C4×C61#12011yescharacteristic
108C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
109C22⋊D61—84no—
110C22⋊D61—82no—
111C22⋊D61—82no—
112C2×Dic61—42no—
113C2×Dic61—42no—
114C2×Dic61—42no—
115C2×Dic61—81no—
116C2×Dic61—81no—
117C2×Dic61—82no—
118C2×Dic61—81no—
119C2×Dic61—81no—
120D244—42no—
121D244—42no—
122D244—82no—
123Dic122—42no—
124Dic122—42no—
125Dic122—81no—
126Dic122—81no—
127C8×C61—22no—
128C8×C61—41no—
129#12105—611nomaximal, Sylow
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—22no—
133C22×C61—42no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—81no—
137C22×C61#44711yescharacteristic
138C22×C61C2×SD3211yescharacteristic
139C22×C61#50911yescharacteristic
140C2×D61—82no—
141C2×D61—82no—
142C2×D61—82no—
143C2×D61—164no—
144C4×C61—21no—
145C4×C61—41no—
146C4×C61#45311yescharacteristic
147C4×C61B(4,𝔽2)11yescharacteristic
148Dic61—82no—
149Dic61—82no—
150Dic61—82no—
151#3186—611no—
152#2185—612no—
153#2001—611no—
154#1347—612no—
155#1314D6111yescharacteristic, complemented
156C2×C61—21no—
157C2×C61—21no—
158C2×C61—42no—
159C2×C61#159511yescharacteristic
160C2×C61#218711yescharacteristic
161C2×C61#219111yescharacteristic
162D61—162no—
163C22×Q16—611no—
164C2×C22⋊Q8—1221no—
165C2×#121—1221no—
166C2×SD32—1222no—
167#480—1222no—
168#466—612no—
169#466—1221no—
170#447—1222no—
171C2×#103—1221no—
172C2×C22⋊C8C2×D6111yescharacteristic
173#358—1222no—
174C22⋊C16C2×D6112yescomplemented
175#340—611no—
176C61#1210511yescharacteristic, complemented, Sylow
177C22×Q8—1221no—
178C22×D4—1221no—
179C22×D4—2441no—
180C23×C4D24411yescharacteristic
181C2×Q16—1222no—
182C2×Q16—2442no—
183C2×D8—612no—
184C2×D8—1221no—
185C22×C8D24411yescharacteristic
186C22×C8C22×D6111yescharacteristic
187C22⋊Q8—2442no—
188C22⋊Q8—2442no—
189#121—1222no—
190#121—1222no—
191#121—2441no—
192#121—2441no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4—1221no—
195C2×C4⋊C4—2441no—
196C2×C22⋊C4—2441no—
197C2×C22⋊C4—2441no—
198SD32—2444no—
199C2×C16—22no—
200#107—612no—
201#107—1221no—
202#103—2442no—
203#102—1222no—
204#102—1222no—
205C22⋊C8—22no—
206C24—2441no—
207C24D48811yescharacteristic
208C2×Q8—1222no—
209C2×Q8—2441no—
210C2×Q8—2441no—
211C2×Q8—2441no—
212C2×Q8—2441no—
213C2×Q8—2441no—
214C2×D4—1222no—
215C2×D4—4881no—
216C2×D4—4881no—
217C2×D4—4882no—
218C2×D4—4882no—
219C2×D4—2441no—
220C2×D4—2442no—
221C2×D4—2442no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—2441no—
227C22×C4—2441no—
228C22×C4—2441no—
229C22×C4D48811yescharacteristic
230C22×C4C2×D24411yescharacteristic
231Q16—1224no—
232Q16—2442no—
233D8—4881no—
234D8—2442no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8D4×D6112yes—
239C4⋊C4—1222no—
240C4⋊C4—1222no—
241C4⋊C4—2441no—
242C4⋊C4—2441no—
243C4⋊C4—2444no—
244C22⋊C4—2442no—
245C22⋊C4—2444no—
246C22⋊C4—2442no—
247C16—42no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—4882no—
253C23—4882no—
254C23—4882no—
255C23—2442no—
256C23#1451972212yescomplemented
257C23C2×D48811yescharacteristic
258Q8—4881no—
259Q8—4881no—
260Q8—2442no—
261Q8—2442no—
262D4—4882no—
263D4—4884no—
264D4—4882no—
265D4—4882no—
266D4—2442no—
267D4—2442no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—81no—
275C2×C4—4881no—
276C2×C4—4882no—
277C2×C4—4881no—
278C2×C4—4881no—
279C2×C4—4881no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4#1451974711yescharacteristic
284C2×C4#1451980511yescharacteristic
285C8—22no—
286C8—41no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—9764no—
295C22—4882no—
296C22—4882no—
297C22—4882no—
298C22#1683989811yescharacteristic
299C22C2×#1451972211yescharacteristic
300C22#1684020011yescharacteristic
301C4—21no—
302C4—41no—
303C4—4882no—
304C4—4882no—
305C4—4882no—
306C4#1683993211yescharacteristic
307C4#1683995011yescharacteristic
308C2—21no—
309C2—21no—
310C2—42no—
311C2—9762no—
312C2#2144473911yescharacteristic
313C2#2144775011yescharacteristic
314C2#2144777211yescharacteristic
subgroupG/H
centerC22#16839898
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1314C2
socleC22×C61#447
prime radical#21456570C2
hypercenter#1314D61
hypocenterC61#12105
LCSG ▷ C22×C8×C61 ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #21447738C2semidirect
3. #21446829C2semidirect
4. #21443829C2semidirect
5. C61×#1314C2semidirect
6. C22×Dic244C22semidirect
16. #16839525C22semidirect
18. C61×C22⋊C16C22semidirect
94. C23×C61SD32semidirect
155. #1314D61semidirect
174. C22⋊C16C2×D61semidirect
176. C61#12105semidirect
256. C23#14519722semidirect

Maximal subgroups

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

subgroupindex
#214565702
#214477382
#214468292
#214438292
C61×#13142
#1210561