Finite Group Databasebeta

← #29577977

Subgroups — #29577977

Order 15616.

542 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 107 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#21450127C211yesmaximal, characteristic
2#21445897C212yesmaximal, complemented
3#21444157C211yesmaximal, characteristic
4#21443724C212yesmaximal, complemented
5C61×#2975C211yesmaximal, characteristic
6C2×#14519758C412yes—
7C22×C4⋊Dic61C2211yescharacteristic
8#16839726C2212yes—
9#16839692—21no—
10#16839687—21no—
11#16839660—21no—
12#16839623C412yes—
13#16839622—21no—
14#16839622C2211yescharacteristic
15#16839530—24no—
16#16839488—21no—
17#16839486—21no—
18C2×C4×D4×C61C2211yescharacteristic
19C61×#436—24no—
20C61×#431—21no—
21C61×#407—21no—
22C61×#407C2212yes—
23C23×Dic61C2×C412yes—
24C2×#11758194—22no—
25#14519758—44no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—22no—
28C2×C4⋊Dic61—21no—
29C2×C4⋊Dic61—21no—
30C2×C4⋊Dic61—22no—
31C2×#11758196—22no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61C2×C412yes—
34C22⋊#92418—22no—
35C22⋊#92418—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684—21no—
39C22×D4×C61C2×C411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C2×M16×C61—22no—
43C2×M16×C61D414yes—
44C4×D4×C61—22no—
45C4×D4×C61—22no—
46C2×C4⋊C4×C61D411yescharacteristic
47C2×C22⋊C4×C61C2×C411yescharacteristic
48C2×C22⋊C4×C61D411yescharacteristic
49C2×C42×C61D411yescharacteristic
50C4⋊C8×C61—22no—
51C22⋊C8×C61—22no—
52C22⋊C8×C61—22no—
53C22⋊C8×C61—22no—
54C4×C8×C61—22no—
55C24×C61C22⋊C411yescharacteristic
56C24×C61C22×C411yescharacteristic
57C2×D4×C61—22no—
58C2×D4×C61—22no—
59C2×D4×C61—41no—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—42no—
63C22×C4×C61—42no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C2×D412yes—
68C22×C4×C61C2×D411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61#4112yes—
71C22×Dic61—22no—
72C22×Dic61—22no—
73C22×Dic61—24no—
74C22×Dic61—42no—
75C22×Dic61—42no—
76M16×C61—44no—
77M16×C61—82no—
78C2×C8×C61—24no—
79C4⋊C4×C61—22no—
80C22⋊C4×C61—22no—
81C22⋊C4×C61—22no—
82C42×C61—21no—
83C42×C61—21no—
84C22⋊Dic61—44no—
85C22⋊Dic61—42no—
86C4⋊Dic61—42no—
87C4⋊Dic61—41no—
88C4⋊Dic61—42no—
89C4⋊Dic61—41no—
90#11758196—44no—
91#11758194—44no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61C2×C22⋊C411yescharacteristic
103C23×C61C4×D412yes—
104D4×C61—44no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—24no—
108C2×C4×C61—22no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—81no—
117C2×C4×C61#10012yes—
118C2×C4×C61#12011yescharacteristic
119C2×C4×C61#12112yes—
120C2×C4×C61#12311yescharacteristic
121C2×Dic61—42no—
122C2×Dic61—42no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—84no—
128C8×C61—44no—
129#92418—82no—
130#9508—611nomaximal, Sylow
131C22×C61—21no—
132C22×C61—21no—
133C22×C61—22no—
134C22×C61—22no—
135C22×C61—22no—
136C22×C61—42no—
137C22×C61—42no—
138C22×C61—42no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—41no—
142C22×C61—81no—
143C22×C61#38611yescharacteristic
144C22×C61C2×#10011yescharacteristic
145C22×C61#41311yescharacteristic
146C4×C61—22no—
147C4×C61—21no—
148C4×C61—22no—
149C4×C61—41no—
150C4×C61—41no—
151C4×C61#45812yes—
152Dic61—82no—
153Dic61—82no—
154#2975D6111yescharacteristic
155#2486—611no—
156#1821—612no—
157#1465—611no—
158#1305—612no—
159C2×C61—21no—
160C2×C61—21no—
161C2×C61—42no—
162C2×C61—81no—
163C2×C61#189011yescharacteristic
164C2×C61#190011yescharacteristic
165C2×C61#191511yescharacteristic
166C2×#123—612no—
167C2×C4×D4C2×D6111yescharacteristic
168C22×C4⋊C4—611no—
169#436—24no—
170#431—21no—
171#407—21no—
172#407—612no—
173#407C2×D6112yes—
174#398—1221no—
175#397—1221no—
176#389—1221no—
177#381—612no—
178#380—611no—
179#380—1221no—
180#343—1224no—
181#324—1221no—
182#323—1221no—
183C61#950811yescharacteristic, complemented, Sylow
184C22×D4C4×D6111yescharacteristic
185C23×C4—612no—
186C23×C4C22⋊D6111yescharacteristic
187C23×C4C22×D6111yescharacteristic
188C2×M16—22no—
189C2×M16—1222no—
190C2×M16C22⋊D6114yes—
191#123—2444no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—1222no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4—1222no—
197C2×C4⋊C4—1222no—
198C2×C4⋊C4—1221no—
199C2×C4⋊C4—1221no—
200C2×C4⋊C4D24411yescharacteristic
201C2×C22⋊C4—612no—
202C2×C22⋊C4—1222no—
203C2×C22⋊C4C4×D6111yescharacteristic
204C2×C22⋊C4D24411yescharacteristic
205C2×C42C22⋊D6111yescharacteristic
206C4⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—1222no—
211C22⋊C8—1222no—
212C4×C8—22no—
213#95—1222no—
214#95—1222no—
215#95—1221no—
216C24#1175819811yescharacteristic
217C24C2×C4×D6111yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1224no—
226C22×C4—1222no—
227C22×C4—1222no—
228C22×C4—2442no—
229C22×C4—2442no—
230C22×C4#1175819811yescharacteristic
231C22×C4#1175819811yescharacteristic
232C22×C4#1175819811yescharacteristic
233C22×C4C2×D24411yescharacteristic
234C22×C4#1175822212yes—
235C22×C4C2×C22⋊D6112yes—
236C22×C4C2×C22⋊D6111yescharacteristic
237M16—44no—
238M16—82no—
239M16—2444no—
240C2×C8—24no—
241C2×C8—2442no—
242C4⋊C4—22no—
243C4⋊C4—2442no—
244C4⋊C4—2444no—
245C4⋊C4—2441no—
246C4⋊C4—2442no—
247C4⋊C4—2441no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—2444no—
251C22⋊C4—2442no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×#1175819811yescharacteristic
264C23C4×C22⋊D6112yes—
265D4—44no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—4884no—
279C2×C4—2442no—
280C2×C4—2442no—
281C2×C4—2442no—
282C2×C4—2442no—
283C2×C4—2442no—
284C2×C4—2442no—
285C2×C4#1451969312yes—
286C2×C4#1451974611yescharacteristic
287C2×C4#1451975112yes—
288C2×C4#1451976011yescharacteristic
289C8—44no—
290C8—4882no—
291C22—21no—
292C22—21no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1683964111yescharacteristic
304C22C2×#1451969311yescharacteristic
305C22#1683974411yescharacteristic
306C4—22no—
307C4—21no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—4882no—
312C4—4882no—
313C4#1683995212yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—81no—
318C2#2144623211yescharacteristic
319C2#2144628211yescharacteristic
320C2#2144637111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2975C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#2975D61
hypocenterC61#9508
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #21445897C2semidirect
4. #21443724C2semidirect
183. C61#9508semidirect

Maximal subgroups

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

subgroupindex
#214501272
#214458972
#214441572
#214437242
C61×#29752
#950861