Finite Group Databasebeta

← #29577426

Subgroups — #29577426

Order 15616.

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

normal subgroups only

This group has 570 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
2#21450772C211yesmaximal, characteristic
3#21444505C212yesmaximal, complemented
4#21443720C212yesmaximal, complemented
5C61×#1800C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7C2×#14519756—22no—
8#16839840—24no—
9#16839815C2212yes—
10#16839726—22no—
11#16839698C414yes—
12#16839665—21no—
13#16839631—21no—
14C4×C4⋊Dic61C2211yescharacteristic
15#16839507—24no—
16C2×C4×D4×C61C2211yescharacteristic
17C61×#423C2212yes—
18C2×#100×C61—21no—
19C61×#343—21no—
20C22×Dic122—21no—
21C2×#11758194—24no—
22#14519777—22no—
23#14519777—22no—
24#14519756—48no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61C2×C412yes—
28C2×#11758196—24no—
29C2×C22⋊Dic61—24no—
30C2×C4×Dic61—21no—
31C4⋊#92418D414yes—
32C22⋊#92418—24no—
33C22⋊#92418—22no—
34C4×#92418—22no—
35#14519684C2×C412yes—
36C22×D4×C61D411yescharacteristic
37C23×C4×C61D412yes—
38C2×M16×C61—22no—
39C4×D4×C61—22no—
40C4×D4×C61—22no—
41C2×C4⋊C4×C61D411yescharacteristic
42C2×C22⋊C4×C61C2×C412yes—
43C2×C42×C61C2311yescharacteristic
44C4⋊C8×C61—22no—
45C4⋊C8×C61—22no—
46#100×C61—42no—
47C24×C61C22⋊C412yes—
48C2×D4×C61—22no—
49C2×D4×C61—21no—
50C2×D4×C61—22no—
51C2×D4×C61—21no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61—44no—
55C22×C4×C61C22⋊C412yes—
56C22×C4×C61C22×C411yescharacteristic
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D411yescharacteristic
59C22×Dic61—22no—
60C22×Dic61—22no—
61C2×Dic122—44no—
62C2×Dic122—41no—
63C2×Dic122—41no—
64M16×C61—42no—
65M16×C61—42no—
66C2×C8×C61—42no—
67C4⋊C4×C61—22no—
68C22⋊C4×C61—24no—
69C42×C61C2×D412yes—
70C42×C61#4112yes—
71C22⋊Dic61—48no—
72C4⋊Dic61—24no—
73C4⋊Dic61—24no—
74C4⋊Dic61—42no—
75#11758196—48no—
76C4×Dic61—41no—
77C4×Dic61—41no—
78#11758194—84no—
79C2×#92418—24no—
80C23×C61—22no—
81C23×C61—22no—
82C23×C61—22no—
83C23×C61—42no—
84C23×C61—42no—
85C23×C61C2×C22⋊C411yescharacteristic
86D4×C61—42no—
87D4×C61—42no—
88C2×C4×C61—22no—
89C2×C4×C61—22no—
90C2×C4×C61—41no—
91C2×C4×C61—44no—
92C2×C4×C61—42no—
93C2×C4×C61—82no—
94C2×C4×C61C4×D412yes—
95C2×C4×C61#12011yescharacteristic
96C2×C4×C61#12112yes—
97C2×C4×C61#12311yescharacteristic
98C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
99C2×C4×C61#13712yes—
100C2×C4×C61#13711yescharacteristic
101C2×Dic61—24no—
102C2×Dic61—42no—
103C2×Dic61—42no—
104C2×Dic61—44no—
105C2×Dic61—42no—
106Dic122—82no—
107Dic122—82no—
108C8×C61—82no—
109#92418—44no—
110#9403—611nomaximal, Sylow
111C22×C61—21no—
112C22×C61—21no—
113C22×C61—44no—
114C22×C61—42no—
115C22×C61—42no—
116C22×C61—42no—
117C22×C61—82no—
118C22×C61#38611yescharacteristic
119C22×C61#41711yescharacteristic
120C22×C61#41911yescharacteristic
121C4×C61—21no—
122C4×C61—22no—
123C4×C61—42no—
124C4×C61#45312yes—
125C4×C61#46812yes—
126C4×C61#47012yes—
127Dic61—44no—
128Dic61—82no—
129#2975—611no—
130#2542—611no—
131#1800D6111yescharacteristic
132#1549—612no—
133#1303—612no—
134C2×C61—21no—
135C2×C61—21no—
136C2×C61—82no—
137C2×C61#188711yescharacteristic
138C2×C61#189411yescharacteristic
139C2×C61#191511yescharacteristic
140C2×C4⋊Q8—611no—
141C2×C22⋊Q8—1222no—
142C2×C4×D4C2×D6111yescharacteristic
143#436—1224no—
144#431—612no—
145#423C2×D6112yes—
146C2×#100—21no—
147#407—1222no—
148#399—614no—
149#390—1221no—
150#384—1221no—
151C4×C4⋊C4—611no—
152#343—21no—
153#332—1224no—
154C61#940311yescharacteristic, complemented, Sylow
155C22×Q8—1221no—
156C22×D4C22⋊D6111yescharacteristic
157C23×C4D24412yes—
158C2×M16—22no—
159C2×M16—1224no—
160C4⋊Q8—1222no—
161C4⋊Q8—1222no—
162C22⋊Q8—2448no—
163C4×D4—22no—
164C4×D4—22no—
165C2×C4⋊C4—612no—
166C2×C4⋊C4—1221no—
167C2×C4⋊C4—1224no—
168C2×C4⋊C4—1221no—
169C2×C4⋊C4C22⋊D6111yescharacteristic
170C2×C22⋊C4—1224no—
171C2×C22⋊C4C4×D6112yes—
172C2×C42—1221no—
173C2×C42C22×D6111yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—614no—
177#100—42no—
178C22⋊C8—1224no—
179C22⋊C8—1222no—
180C4×C8—1222no—
181#95—612no—
182C24#1175819812yes—
183C2×Q8—2444no—
184C2×Q8—2441no—
185C2×Q8—2441no—
186C2×D4—21no—
187C2×D4—22no—
188C2×D4—21no—
189C2×D4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—1222no—
194C22×C4—1222no—
195C22×C4#1175819812yes—
196C22×C4C2×C4×D6111yescharacteristic
197C22×C4C2×D24411yescharacteristic
198C22×C4C2×C22⋊D6111yescharacteristic
199M16—42no—
200M16—42no—
201M16—4884no—
202C2×C8—42no—
203C2×C8—1224no—
204C4⋊C4—22no—
205C4⋊C4—1224no—
206C4⋊C4—1224no—
207C4⋊C4—2448no—
208C4⋊C4—2442no—
209C22⋊C4—24no—
210C22⋊C4—2448no—
211C42—2441no—
212C42—2441no—
213C42D4×D6112yes—
214C42#1175822412yes—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23C2×#1175819811yescharacteristic
221Q8—4882no—
222Q8—4882no—
223D4—42no—
224D4—42no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—1224no—
232C2×C4—2444no—
233C2×C4—2442no—
234C2×C4—2442no—
235C2×C4—2442no—
236C2×C4#1451973712yes—
237C2×C4#1451974711yescharacteristic
238C2×C4#1451974912yes—
239C2×C4#1451976111yescharacteristic
240C2×C4#1451980311yescharacteristic
241C2×C4#1451980712yes—
242C2×C4#1451980811yescharacteristic
243C8—82no—
244C8—2444no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22#1683963911yescharacteristic
253C22#1683976111yescharacteristic
254C22#1683977011yescharacteristic
255C4—21no—
256C4—22no—
257C4—42no—
258C4—4882no—
259C4—2444no—
260C4#1683993212yes—
261C4#1684000612yes—
262C4#1684001812yes—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2144621511yescharacteristic
267C2#2144625011yescharacteristic
268C2#2144636911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1800C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC2×C42×C61C23
hypercenter#1800D61
hypocenterC61#9403
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
3. #21444505C2semidirect
4. #21443720C2semidirect
154. C61#9403semidirect

Maximal subgroups

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

subgroupindex
#214548602
#214507722
#214445052
#214437202
C61×#18002
#940361