Finite Group Databasebeta

← #29577885

Subgroups — #29577885

Order 15616.

522 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 111 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#21454904C211yesmaximal, characteristic, complemented
2#21450724C211yesmaximal, characteristic
3#21444243C212yesmaximal
4#21443679C212yesmaximal, complemented
5C61×#1801C211yesmaximal, characteristic, complemented
6C2×#14519753—21no—
7C2×C4×D244C2211yescharacteristic
8#16839893—24no—
9#16839819—22no—
10#16839815C2212yes—
11#16839692C412yes—
12#16839679C412yes—
13#16839653—22no—
14#16839628C2211yescharacteristic
15#16839619—21no—
16#16839489—24no—
17C2×C4×Q8×C61C2211yescharacteristic
18C61×#423C2212yes—
19C2×#101×C61—21no—
20C61×#343—21no—
21C22×D244C2×C411yescharacteristic
22C22×C4×D61—21no—
23#14519753—44no—
24C4×D244—22no—
25C4×D244—22no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—22no—
29C2×#11758198—22no—
30C2×#11758198—22no—
31C2×#11758198—21no—
32C4⋊#92418—24no—
33C4⋊#92418—22no—
34C22⋊#92418—22no—
35#14519688—24no—
36C4×#92418D414yes—
37#14519684—22no—
38#14519684—21no—
39#14519684C2×C412yes—
40C22×Q8×C61D411yescharacteristic
41C2×M16×C61—22no—
42C4×Q8×C61—22no—
43C4×Q8×C61—22no—
44C2×C4⋊C4×C61C2×C412yes—
45C2×C4⋊C4×C61D411yescharacteristic
46C2×C42×C61D412yes—
47C2×C42×C61C2311yescharacteristic
48C4⋊C8×C61—22no—
49C4⋊C8×C61—22no—
50#101×C61—42no—
51C23×D61—21no—
52C2×Q8×C61—22no—
53C2×Q8×C61—21no—
54C2×Q8×C61—22no—
55C2×Q8×C61—21no—
56C22×C4×C61C22⋊C412yes—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22×C411yescharacteristic
59C22×C4×C61C2×D411yescharacteristic
60C22×C4×C61C2×D411yescharacteristic
61C22×Dic61—21no—
62C22×Dic61—22no—
63C2×D244—22no—
64C2×D244—42no—
65C2×C4×D61—42no—
66C2×C4×D61—41no—
67C2×C4×D61—42no—
68C2×C4×D61—41no—
69M16×C61—42no—
70M16×C61—42no—
71C2×C8×C61—42no—
72C4⋊C4×C61—24no—
73C4⋊C4×C61—22no—
74C42×C61—22no—
75C42×C61—22no—
76C42×C61C2×D412yes—
77C42×C61#4112yes—
78#11758198—44no—
79#11758198—44no—
80#11758198—41no—
81#11758198—41no—
82C4⋊Dic61—22no—
83C4⋊Dic61—42no—
84#11758196—44no—
85C2×#92418—24no—
86C23×C61C2×C22⋊C411yescharacteristic
87C22×D61—22no—
88C22×D61—41no—
89C22×D61—41no—
90C22×D61—41no—
91C22×D61—41no—
92C22×D61—41no—
93C22×D61—41no—
94Q8×C61—42no—
95Q8×C61—42no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61—42no—
103C2×C4×C61C4×D412yes—
104C2×C4×C61#12011yescharacteristic
105C2×C4×C61#12112yes—
106C2×C4×C61#12311yescharacteristic
107C2×C4×C61AGL(1,ℤ/8)12yes—
108C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
109C2×C4×C61#13711yescharacteristic
110C2×Dic61—42no—
111C2×Dic61—42no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—42no—
116D244—24no—
117D244—42no—
118C4×D61—82no—
119C4×D61—82no—
120C8×C61—82no—
121#92418—44no—
122#9490—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61#38611yescharacteristic
126C22×C61#41711yescharacteristic
127C22×C61#41811yescharacteristic
128C2×D61—42no—
129C2×D61—42no—
130C2×D61—42no—
131C2×D61—81no—
132C2×D61—81no—
133C2×D61—81no—
134C2×D61—81no—
135C4×C61—22no—
136C4×C61—21no—
137C4×C61—21no—
138C4×C61—42no—
139C4×C61—42no—
140C4×C61#46812yes—
141C4×C61#46912yes—
142Dic61—82no—
143Dic61—81no—
144#2983—611no—
145#2536—611no—
146#1801D6111yescharacteristic, complemented
147#1486—612no—
148#1286—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61#189111yescharacteristic
152C2×C61#189511yescharacteristic
153C2×C61#191211yescharacteristic
154D61—82no—
155C2×C22⋊Q8—1221no—
156C2×C4×Q8C2×D6111yescharacteristic
157C2×C4×D4—611no—
158#446—1224no—
159#432—1222no—
160#431—612no—
161#423C2×D6112yes—
162C2×#101—21no—
163#398—612no—
164#394—612no—
165#388—1222no—
166#382—611no—
167#380—1221no—
168#343—21no—
169D4⋊C8—1224no—
170C61#949011yescharacteristic, complemented, Sylow
171C22×Q8C22⋊D6111yescharacteristic
172C22×D4—611no—
173C23×C4—1221no—
174C2×M16—22no—
175C22⋊Q8—2444no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1222no—
179C4×D4—1222no—
180C2×C4⋊C4—611no—
181C2×C4⋊C4—1222no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4C4×D6112yes—
184C2×C4⋊C4C22⋊D6111yescharacteristic
185C2×C22⋊C4—1222no—
186C2×C22⋊C4—1222no—
187C2×C22⋊C4—1221no—
188C2×C42D24412yes—
189C2×C42C22×D6111yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—1224no—
193C4⋊C8—1222no—
194#101—42no—
195C22⋊C8—1222no—
196#97—1224no—
197C4×C8—614no—
198#95—612no—
199#95—1222no—
200#95—1221no—
201C24—1221no—
202C2×Q8—21no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—21no—
206C2×D4—1222no—
207C2×D4—2442no—
208C22×C4—1222no—
209C22×C4—1221no—
210C22×C4—2441no—
211C22×C4—2442no—
212C22×C4—2442no—
213C22×C4—2441no—
214C22×C4#1175819812yes—
215C22×C4#1175819812yes—
216C22×C4C2×C4×D6111yescharacteristic
217C22×C4C2×D24411yescharacteristic
218C22×C4C2×C22⋊D6111yescharacteristic
219M16—42no—
220M16—42no—
221C2×C8—42no—
222C2×C8—1224no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—1222no—
226C4⋊C4—2444no—
227C4⋊C4—2442no—
228C22⋊C4—2444no—
229C22⋊C4—2444no—
230C22⋊C4—2441no—
231C22⋊C4—2441no—
232C42—22no—
233C42—22no—
234C42D4×D6112yes—
235C42#1175822412yes—
236C23—1222no—
237C23—2441no—
238C23—2441no—
239C23—2441no—
240C23—2441no—
241C23—2441no—
242C23—2441no—
243C23C2×#1175819811yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—1224no—
247D4—2442no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—4882no—
256C2×C4—4882no—
257C2×C4—2442no—
258C2×C4—2442no—
259C2×C4—2441no—
260C2×C4—2442no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4#1451973712yes—
264C2×C4#1451974711yescharacteristic
265C2×C4#1451974912yes—
266C2×C4#1451976111yescharacteristic
267C2×C4#1451980212yes—
268C2×C4#1451980311yescharacteristic
269C2×C4#1451980811yescharacteristic
270C8—82no—
271C8—2444no—
272C22—21no—
273C22—21no—
274C22—4881no—
275C22—4881no—
276C22—4881no—
277C22—4881no—
278C22—2442no—
279C22—2442no—
280C22—2442no—
281C22#1683963911yescharacteristic
282C22#1683976111yescharacteristic
283C22#1683976611yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—4882no—
290C4—4881no—
291C4#1684001012yes—
292C4#1684001312yes—
293C2—21no—
294C2—21no—
295C2—4882no—
296C2#2144623511yescharacteristic
297C2#2144625511yescharacteristic
298C2#2144635411yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1801C2
socleC22×C61#386
prime radicalC22×D244C2×C4
cosocleC2×C42×C61C23
hypercenter#1801D61
hypocenterC61#9490
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #21454904C2semidirect
4. #21443679C2semidirect
5. C61×#1801C2semidirect
146. #1801D61semidirect
170. C61#9490semidirect

Maximal subgroups

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

subgroupindex
#214549042
#214507242
#214442432
#214436792
C61×#18012
#949061