Finite Group Databasebeta

← #29568771

Subgroups — #29568771

Order 15616.

612 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 87 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#21446906C212yesmaximal, complemented
2#21446889C211yesmaximal, characteristic
3#21443645C212yesmaximal, complemented
4#21443640C211yesmaximal, characteristic, complemented
5C61×#2804C211yesmaximal, characteristic, complemented
6C22×#11758201—22no—
7C22×#11758199—21no—
8C2×#14519774C2211yescharacteristic
9C2×#14519697—21no—
10C2×#14519697—22no—
11#16839517C2212yes—
12#16839517C2211yescharacteristic
13C61×#400—21no—
14C61×#393—22no—
15C61×#391C2212yes—
16C61×#390C2211yescharacteristic
17C61×#386—21no—
18C61×#385—22no—
19C22×D244—21no—
20C22×D244—22no—
21C2×#11758201—44no—
22C2×#11758201—44no—
23C2×#11758201—44no—
24C2×#11758199—42no—
25C2×#11758199—42no—
26C2×#11758199—42no—
27C22×#92418—21no—
28C22×#92418—22no—
29#14519774—42no—
30#14519697—42no—
31#14519697—44no—
32C22×Q8×C61D412yes—
33C22×D4×C61D411yescharacteristic
34C23×C4×C61—21no—
35C2×C4⋊C4×C61—21no—
36C2×C4⋊C4×C61D412yes—
37C2×C4⋊C4×C61D411yescharacteristic
38C2×C22⋊C4×C61—22no—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61—21no—
41C2×C42×C61C2311yescharacteristic
42#95×C61—22no—
43#95×C61—21no—
44#95×C61—22no—
45#95×C61—22no—
46C24×C61—21no—
47C23×D61—41no—
48C2×Q8×C61—22no—
49C2×Q8×C61—22no—
50C2×Q8×C61—42no—
51C2×Q8×C61—42no—
52C2×D4×C61—21no—
53C2×D4×C61—41no—
54C2×D4×C61—41no—
55C2×D4×C61SD1612yescomplemented
56C22×C4×C61—22no—
57C22×C4×C61—21no—
58C22×C4×C61—22no—
59C22×C4×C61—82no—
60C22×C4×C61—81no—
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D412yes—
63C2×D244—22no—
64C2×D244—24no—
65C2×D244—41no—
66C2×D244—42no—
67C2×D244—41no—
68C2×D244—42no—
69C2×D244—41no—
70C2×D244—42no—
71C2×D244—42no—
72C2×D244—41no—
73C2×D244—41no—
74C2×D244—42no—
75C4⋊C4×C61—41no—
76C4⋊C4×C61—42no—
77C4⋊C4×C61—81no—
78C22⋊C4×C61—42no—
79C22⋊C4×C61—42no—
80C22⋊C4×C61—82no—
81C22⋊C4×C61—81no—
82C42×C61—41no—
83#11758201—416no—
84#11758199—48no—
85C2×#92418—22no—
86C2×#92418—24no—
87C2×#92418—41no—
88C2×#92418—42no—
89C2×#92418—41no—
90C2×#92418—42no—
91C23×C61—22no—
92C23×C61—42no—
93C23×C61—42no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61#12011yescharacteristic
97C22×D61—42no—
98C22×D61—81no—
99C22×D61—82no—
100C22×D61—81no—
101C22×D61—82no—
102Q8×C61—44no—
103Q8×C61—82no—
104D4×C61—42no—
105D4×C61—81no—
106C2×C4×C61—21no—
107C2×C4×C61—22no—
108C2×C4×C61—21no—
109C2×C4×C61—22no—
110C2×C4×C61—42no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—41no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—84no—
119C2×C4×C61C2×SD1611yescharacteristic
120C2×C4×C61#13512yes—
121C2×C4×C61AGL(1,ℤ/8)12yes—
122C2×C4×C61#13711yescharacteristic
123D244—42no—
124D244—44no—
125D244—42no—
126D244—44no—
127D244—81no—
128D244—81no—
129D244—82no—
130D244—82no—
131#92418—42no—
132#92418—44no—
133#7745—611nomaximal, Sylow
134C22×C61—42no—
135C22×C61—44no—
136C22×C61—82no—
137C22×C61—81no—
138C22×C61—81no—
139C22×C61#44811yescharacteristic
140C22×C61#44912yes—
141C22×C61#45311yescharacteristic
142C22×C61#45312yes—
143C22×C61B(4,𝔽2)11yescharacteristic
144C2×D61—82no—
145C2×D61—84no—
146C2×D61—161no—
147C2×D61—162no—
148C2×D61—161no—
149C4×C61—21no—
150C4×C61—21no—
151C4×C61—22no—
152C4×C61—22no—
153C4×C61—82no—
154C4×C61—81no—
155C4×C61—82no—
156#2804D6111yescharacteristic, complemented
157#2014—612no—
158#2011—611no—
159#1272—612no—
160#1270—611no—
161C2×C61—82no—
162C2×C61#159611yescharacteristic
163C2×C61#160012yes—
164C2×C61#162011yescharacteristic
165C2×C61#162212yes—
166C2×C61#165511yescharacteristic
167D61—162no—
168C22×SD16—1222no—
169C22×D8—1221no—
170C2×#127—611no—
171C2×#102—1221no—
172C2×#102—1222no—
173#400—21no—
174#393—22no—
175#391C2×D6112yes—
176#390C2×D6111yescharacteristic
177#386—21no—
178#385—22no—
179#336—611no—
180#336—612no—
181C61#774511yescharacteristic, complemented, Sylow
182C22×Q8C22⋊D6112yes—
183C22×D4—1221no—
184C22×D4—1222no—
185C22×D4C22⋊D6111yescharacteristic
186C23×C4—21no—
187C2×SD16—2444no—
188C2×SD16—2444no—
189C2×SD16—2444no—
190C2×D8—2442no—
191C2×D8—2442no—
192C2×D8—2442no—
193C22×C8—1221no—
194C22×C8—1222no—
195#127—2442no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22⋊D6112yes—
198C2×C4⋊C4C22⋊D6111yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C42C22×D6111yescharacteristic
203#102—2442no—
204#102—2444no—
205#95—22no—
206#95—21no—
207#95—22no—
208#95—22no—
209C24—21no—
210C24—2441no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×D4—21no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—1222no—
219C2×D4—1224no—
220C2×D4—2441no—
221C2×D4—2442no—
222C2×D4—2441no—
223C2×D4—2442no—
224C2×D4—2441no—
225C2×D4—2442no—
226C2×D4—2441no—
227C2×D4—2441no—
228C2×D4—2442no—
229C2×D4—2442no—
230C2×D4#1175820112yescomplemented
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—82no—
235C22×C4—81no—
236C22×C4C2×C22⋊D6111yescharacteristic
237C22×C4C2×C22⋊D6112yes—
238SD16—24416no—
239D8—2448no—
240C2×C8—1222no—
241C2×C8—1224no—
242C2×C8—2441no—
243C2×C8—2442no—
244C2×C8—2441no—
245C2×C8—2442no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—81no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—81no—
252C22⋊C4—82no—
253C42—41no—
254C23—22no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—41no—
259C23—4881no—
260C23—4882no—
261C23—4881no—
262C23—4882no—
263C23—2442no—
264C23#1451974611yescharacteristic
265Q8—44no—
266Q8—82no—
267D4—42no—
268D4—81no—
269D4—4881no—
270D4—4881no—
271D4—4882no—
272D4—4882no—
273D4—2442no—
274D4—2444no—
275D4—2442no—
276D4—2444no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4C2×#1175820111yescharacteristic
291C2×C4#1451979712yes—
292C2×C4#1451980212yes—
293C2×C4#1451980811yescharacteristic
294C8—2442no—
295C8—2444no—
296C22—44no—
297C22—42no—
298C22—81no—
299C22—82no—
300C22—81no—
301C22—9761no—
302C22—9762no—
303C22—9761no—
304C22—4882no—
305C22—4884no—
306C22#1683990111yescharacteristic
307C22#1683990712yes—
308C22#1683993011yescharacteristic
309C22#1683993012yes—
310C22#1683994911yescharacteristic
311C4—21no—
312C4—21no—
313C4—22no—
314C4—22no—
315C4—81no—
316C4—82no—
317C4—82no—
318C2—82no—
319C2—9762no—
320C2#2144474211yescharacteristic
321C2#2144476512yes—
322C2#2144487811yescharacteristic
323C2#2144489112yes—
324C2#2144510811yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2804C2
socleC23×C61#120
prime radical#21446889C2
hypercenter#2804D61
hypocenterC61#7745
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2804

Complements

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

NH ≅ G/Ntype
1. #21446906C2semidirect
3. #21443645C2semidirect
4. #21443640C2semidirect
5. C61×#2804C2semidirect
55. C2×D4×C61SD16semidirect
156. #2804D61semidirect
181. C61#7745semidirect
230. C2×D4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214469062
#214468892
#214436452
#214436402
C61×#28042
#774561