Finite Group Databasebeta

← #29580623

Subgroups — #29580623

Order 15616.

716 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 85 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#21445730C211yesmaximal, characteristic
3#21444183C212yesmaximal, complemented
4#21443696C211yesmaximal, characteristic, complemented
5C61×#3189C211yesmaximal, characteristic, complemented
6C22×#11758222C2211yescharacteristic
7C2×#14519808—22no—
8#16839922—24no—
9#16839917—22no—
10#16839912—22no—
11#16839761—22no—
12#16839726C2212yes—
13C2×C22⋊#92418C2211yescharacteristic
14C2×#14519689—22no—
15#16839488—24no—
16C2×C22⋊Q8×C61C2212yes—
17C61×#484—22no—
18C61×#467—22no—
19C61×#451—22no—
20C2×C22⋊C8×C61C2211yescharacteristic
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—
32C22×#92418—21no—
33#14519700—24no—
34#14519697—24no—
35C22⋊#92418—22no—
36C22⋊#92418—22no—
37C22⋊#92418D414yes—
38#14519689—44no—
39C22×Q8×C61D412yes—
40C23×C4×C61C2311yescharacteristic
41C2×Q16×C61—22no—
42C22×C8×C61D412yes—
43C22⋊Q8×C61—22no—
44C22⋊Q8×C61—22no—
45C22⋊Q8×C61—22no—
46C22⋊Q8×C61—22no—
47C2×C4⋊C4×C61—22no—
48C2×C4⋊C4×C61D412yes—
49C2×C22⋊C4×C61—22no—
50#107×C61—22no—
51#103×C61—24no—
52C22⋊C8×C61—22no—
53C24×C61#4111yescharacteristic
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—42no—
61C22×C4×C61C2×D412yes—
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×C4×C61C2×D412yes—
65C2×C22⋊D61—42no—
66C2×C22⋊D61—42no—
67C2×C22⋊D61—42no—
68C22×Dic61—21no—
69#11758222—28no—
70#11758222—44no—
71#11758222—44no—
72#11758222—44no—
73C2×D244—22no—
74C2×D244—42no—
75C2×C4×D61—24no—
76C2×C4×D61—42no—
77C2×C4×D61—41no—
78C2×C4×D61—41no—
79C2×Dic122—22no—
80C2×Dic122—42no—
81Q16×C61—82no—
82C2×C8×C61—22no—
83C2×C8×C61—42no—
84C4⋊C4×C61—22no—
85C4⋊C4×C61—22no—
86C4⋊C4×C61—42no—
87C4⋊C4×C61—42no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—42no—
90#11758202—48no—
91#11758201—48no—
92#11758194—44no—
93C2×#92418—24no—
94C2×#92418—42no—
95C2×#92418—41no—
96C23×C61—22no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61#12112yes—
100C23×C61#12111yescharacteristic
101C22×D61—22no—
102C22×D61—41no—
103C22×D61—42no—
104C22×D61—41no—
105C22×D61—42no—
106Q8×C61—44no—
107Q8×C61—44no—
108C2×C4×C61—22no—
109C2×C4×C61—24no—
110C2×C4×C61—41no—
111C2×C4×C61—42no—
112C2×C4×C61—44no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—82no—
116C2×C4×C61#12012yes—
117C2×C4×C61#12011yescharacteristic
118C2×C4×C61#12711yescharacteristic
119C22⋊D61—48no—
120C2×Dic61—22no—
121C2×Dic61—41no—
122C2×Dic61—41no—
123D244—24no—
124D244—42no—
125C4×D61—44no—
126C4×D61—44no—
127Dic122—24no—
128Dic122—42no—
129C8×C61—42no—
130#92418—44no—
131#92418—81no—
132#9976—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—22no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—82no—
139C22×C61#39211yescharacteristic
140C22×C61#47312yes—
141C2×D61—44no—
142C2×D61—42no—
143C2×D61—82no—
144C2×D61—81no—
145C2×D61—81no—
146C4×C61—22no—
147C4×C61—21no—
148C4×C61—82no—
149C4×C61—82no—
150C4×C61B(4,𝔽2)12yes—
151Dic61—42no—
152#3189D6111yescharacteristic, complemented
153#3007—612no—
154#1780—611no—
155#1472—612no—
156#1293—611no—
157C2×C61—22no—
158C2×C61—42no—
159C2×C61#201711yescharacteristic
160C2×C61#202011yescharacteristic
161C2×C61#202111yescharacteristic
162D61—82no—
163C22×#41—611no—
164C2×#137—1222no—
165C2×C22⋊Q8C2×D6112yes—
166#484—22no—
167#467—22no—
168#452—1224no—
169#451—22no—
170#451—1222no—
171#450—1222no—
172#417—1222no—
173#407—612no—
174C2×C22⋊C8—611no—
175C2×C22⋊C8—1222no—
176C2×C22⋊C8C2×D6111yescharacteristic
177#324—1224no—
178C61#997611yescharacteristic, 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—22no—
190C2×Q16—1224no—
191C2×SD16—1224no—
192C2×M16—1222no—
193C22×C8—1221no—
194C22×C8C22⋊D6112yes—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C22⋊D6112yes—
201C2×C22⋊C4—22no—
202#107—22no—
203#103—24no—
204#103—1224no—
205#102—1224no—
206C22⋊C8—22no—
207C22⋊C8—614no—
208C22⋊C8—1222no—
209C22⋊C8—1222no—
210C22⋊C8—2444no—
211C24—1221no—
212C24#1175822411yescharacteristic
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—1222no—
223C2×D4—2442no—
224C2×D4—2442no—
225C2×D4—2442no—
226C2×D4—2442no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—1224no—
231C22×C4—1221no—
232C22×C4—2442no—
233C22×C4—2441no—
234C22×C4—2441no—
235C22×C4D4×D6111yescharacteristic
236C22×C4D4×D6112yes—
237C22×C4C2×C22⋊D6112yes—
238C22×C4C2×C22⋊D6111yescharacteristic
239Q16—82no—
240Q16—2448no—
241SD16—2448no—
242M16—2444no—
243C2×C8—22no—
244C2×C8—42no—
245C2×C8—1224no—
246C2×C8—2442no—
247C2×C8—2441no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C23—22no—
255C23—41no—
256C23—41no—
257C23—1222no—
258C23—2441no—
259C23—2442no—
260C23—2442no—
261C23—2441no—
262C23#1451974811yescharacteristic
263C23#1451974912yes—
264Q8—44no—
265Q8—44no—
266Q8—1224no—
267Q8—2442no—
268D4—1224no—
269D4—2442no—
270D4—2448no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—1222no—
280C2×C4—2444no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2444no—
284C2×C4#1451974512yes—
285C2×C4#1451974611yescharacteristic
286C2×C4#1451977311yescharacteristic
287C8—42no—
288C8—4881no—
289C8—2444no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—82no—
296C22—4882no—
297C22—4881no—
298C22—4881no—
299C22—2444no—
300C22—2442no—
301C22#1683967011yescharacteristic
302C22#1684003412yes—
303C4—21no—
304C4—22no—
305C4—82no—
306C4—82no—
307C4—2442no—
308C4#1683994912yes—
309C2—22no—
310C2—42no—
311C2—4882no—
312C2#2144692211yescharacteristic
313C2#2144693811yescharacteristic
314C2#2144694311yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC61×#3189C2
socleC22×C61#392
prime radicalC22×#11758222C22
hypercenter#3189D61
hypocenterC61#9976
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
3. #21444183C2semidirect
4. #21443696C2semidirect
5. C61×#3189C2semidirect
152. #3189D61semidirect
178. C61#9976semidirect

Maximal subgroups

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

subgroupindex
#214550772
#214457302
#214441832
#214436962
C61×#31892
#997661