Finite Group Databasebeta

← #29577436

Subgroups — #29577436

Order 15616.

706 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 115 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#21454860C211yesmaximal, characteristic, complemented
2#21450765C211yesmaximal, characteristic
3#21444488C212yesmaximal, complemented
4#21443720C212yesmaximal, complemented
5C61×#1800C211yesmaximal, characteristic, complemented
6C2×#14519774C2211yescharacteristic
7C2×#14519773—21no—
8C2×#14519751—22no—
9#16839840—24no—
10#16839815C2212yes—
11#16839726—22no—
12#16839671C414yes—
13#16839663—21no—
14C4×C4⋊Dic61C2211yescharacteristic
15#16839505—24no—
16C2×C4×D4×C61C2211yescharacteristic
17C61×#423C2212yes—
18C2×#100×C61—21no—
19C61×#343—21no—
20C22×D244—21no—
21C22×D244—21no—
22C22×D244C2×C412yes—
23C22×C22⋊D61—24no—
24C2×#11758194—24no—
25#14519774—22no—
26#14519774—22no—
27#14519773—44no—
28#14519751—48no—
29C2×C4⋊Dic61—21no—
30C2×#11758198—24no—
31C2×C4×Dic61—21no—
32C4⋊#92418D414yescomplemented
33C22⋊#92418—24no—
34C22⋊#92418—22no—
35C4×#92418—22no—
36#14519684C2×C412yes—
37C22×D4×C61D411yescharacteristic
38C23×C4×C61D412yes—
39C2×M16×C61—22no—
40C4×D4×C61—22no—
41C4×D4×C61—22no—
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61C2×C412yes—
44C2×C42×C61C2311yescharacteristic
45C4⋊C8×C61—22no—
46C4⋊C8×C61—22no—
47#100×C61—42no—
48C24×C61C22⋊C412yes—
49C23×D61—22no—
50C2×D4×C61—22no—
51C2×D4×C61—21no—
52C2×D4×C61—22no—
53C2×D4×C61—21no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—44no—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C2×C22⋊D61—48no—
62C2×C22⋊D61—48no—
63C2×C22⋊D61—84no—
64C22×Dic61—22no—
65C2×D244—24no—
66C2×D244—44no—
67C2×D244—41no—
68C2×D244—42no—
69C2×D244—41no—
70C2×D244—41no—
71C2×D244—42no—
72C2×D244—41no—
73C2×D244—82no—
74M16×C61—42no—
75M16×C61—42no—
76C2×C8×C61—42no—
77C4⋊C4×C61—22no—
78C22⋊C4×C61—24no—
79C42×C61C2×D412yes—
80C42×C61#4112yes—
81#11758198—48no—
82C4⋊Dic61—24no—
83C4×Dic61—42no—
84#11758194—84no—
85C2×#92418—24no—
86C23×C61—22no—
87C23×C61—22no—
88C23×C61—22no—
89C23×C61—42no—
90C23×C61—42no—
91C23×C61C2×C22⋊C411yescharacteristic
92C22×D61—44no—
93C22×D61—44no—
94C22×D61—42no—
95C22×D61—42no—
96C22×D61—42no—
97D4×C61—42no—
98D4×C61—42no—
99C2×C4×C61—22no—
100C2×C4×C61—22no—
101C2×C4×C61—41no—
102C2×C4×C61—44no—
103C2×C4×C61—42no—
104C2×C4×C61—82no—
105C2×C4×C61C4×D412yes—
106C2×C4×C61#12011yescharacteristic
107C2×C4×C61#12112yes—
108C2×C4×C61#12311yescharacteristic
109C2×C4×C61AGL(1,ℤ/8)12yes—
110C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C61#13711yescharacteristic
112C22⋊D61—816no—
113C2×Dic61—24no—
114C2×Dic61—42no—
115C2×Dic61—42no—
116D244—84no—
117D244—82no—
118D244—82no—
119D244—82no—
120D244—82no—
121C8×C61—82no—
122#92418—44no—
123#9405—611nomaximal, Sylow
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—44no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61—82no—
131C22×C61#38611yescharacteristic
132C22×C61#41711yescharacteristic
133C22×C61#41811yescharacteristic
134C2×D61—84no—
135C2×D61—84no—
136C2×D61—82no—
137C2×D61—82no—
138C2×D61—82no—
139C4×C61—21no—
140C4×C61—22no—
141C4×C61—42no—
142C4×C61#45312yes—
143C4×C61#46812yes—
144C4×C61#46912yes—
145Dic61—44no—
146#2975—611no—
147#2541—611no—
148#1800D6111yescharacteristic, complemented
149#1545—612no—
150#1303—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61—82no—
154C2×C61#188711yescharacteristic
155C2×C61#189511yescharacteristic
156C2×C61#191411yescharacteristic
157D61—162no—
158C2×#127—611no—
159C2×#127—1221no—
160C2×#121—1222no—
161C2×C4×D4C2×D6111yescharacteristic
162#436—1224no—
163#431—612no—
164#423C2×D6112yes—
165C2×#100—21no—
166#407—1222no—
167#392—614no—
168#390—1221no—
169C4×C4⋊C4—611no—
170#343—21no—
171#331—1224no—
172C61#940511yescharacteristic, complemented, Sylow
173C22×D4—612no—
174C22×D4—1224no—
175C22×D4—1221no—
176C22×D4—1221no—
177C22×D4C22⋊D6111yescharacteristic
178C23×C4D24412yes—
179C2×M16—22no—
180C2×M16—1224no—
181#127—1222no—
182#127—1222no—
183#127—2444no—
184#121—2448no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—1221no—
188C2×C4⋊C4C22⋊D6111yescharacteristic
189C2×C22⋊C4—1224no—
190C2×C22⋊C4C4×D6112yes—
191C2×C42—1221no—
192C2×C42C22×D6111yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—614no—
196#100—42no—
197C22⋊C8—1224no—
198C22⋊C8—1222no—
199C4×C8—1222no—
200#95—612no—
201C24—1222no—
202C24#1175819812yes—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—1224no—
208C2×D4—4882no—
209C2×D4—4884no—
210C2×D4—2444no—
211C2×D4—2448no—
212C2×D4—2448no—
213C2×D4—2441no—
214C2×D4—2441no—
215C2×D4—2442no—
216C2×D4—2441no—
217C2×D4—2442no—
218C2×D4—2441no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—44no—
222C22×C4—1222no—
223C22×C4#1175819812yes—
224C22×C4C2×C4×D6111yescharacteristic
225C22×C4C2×D24411yescharacteristic
226C22×C4C2×C22⋊D6111yescharacteristic
227M16—42no—
228M16—42no—
229M16—4884no—
230C2×C8—42no—
231C2×C8—1224no—
232C4⋊C4—22no—
233C4⋊C4—1224no—
234C22⋊C4—24no—
235C22⋊C4—2448no—
236C42—2442no—
237C42D4×D6112yes—
238C42#1175822412yes—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—2444no—
245C23—2444no—
246C23—2442no—
247C23—2442no—
248C23—2442no—
249C23C2×#1175819811yescharacteristic
250D4—42no—
251D4—42no—
252D4—4884no—
253D4—4882no—
254D4—48816no—
255D4—4882no—
256D4—4882no—
257D4—4882no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—1224no—
265C2×C4—2442no—
266C2×C4—2442no—
267C2×C4#1451973712yes—
268C2×C4#1451974711yescharacteristic
269C2×C4#1451974912yes—
270C2×C4#1451976111yescharacteristic
271C2×C4#1451980212yes—
272C2×C4#1451980311yescharacteristic
273C2×C4#1451980811yescharacteristic
274C8—82no—
275C8—2444no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—82no—
283C22—4884no—
284C22—4884no—
285C22—4882no—
286C22—4882no—
287C22—4882no—
288C22#1683963911yescharacteristic
289C22#1683976111yescharacteristic
290C22#1683976611yescharacteristic
291C4—21no—
292C4—22no—
293C4—42no—
294C4—2444no—
295C4#1683993212yes—
296C4#1684001012yes—
297C4#1684001312yes—
298C2—21no—
299C2—21no—
300C2—82no—
301C2—9762no—
302C2#2144621511yescharacteristic
303C2#2144625511yescharacteristic
304C2#2144636411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1800C2
socleC22×C61#386
prime radical#21450765C2
cosocleC2×C42×C61C23
hypercenter#1800D61
hypocenterC61#9405
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
1. #21454860C2semidirect
3. #21444488C2semidirect
4. #21443720C2semidirect
5. C61×#1800C2semidirect
32. C4⋊#92418D4semidirect
148. #1800D61semidirect
172. C61#9405semidirect

Maximal subgroups

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

subgroupindex
#214548602
#214507652
#214444882
#214437202
C61×#18002
#940561