Finite Group Databasebeta

← #29584983

Subgroups — #29584983

Order 15616.

356 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 93 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#21446617C211yesmaximal, characteristic
2#21445960C211yesmaximal, characteristic, complemented
3#21444594C211yesmaximal, characteristic, complemented
4#21444590C211yesmaximal, characteristic, complemented
5#21444245C211yesmaximal, characteristic
6#21444245C211yesmaximal, characteristic
7C61×#1275C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic
9#16839786C2211yescharacteristic
10#16839786C2211yescharacteristic
11C2×C4⋊#92418C2211yescharacteristic
12C2×#14519689—22no—
13#16839527—22no—
14#16839515—22no—
15#16839513—22no—
16#16839491—22no—
17#16839491—22no—
18C61×#431C2211yescharacteristic
19C61×#431C2211yescharacteristic
20C2×#97×C61C2211yescharacteristic
21C22×D244Q811yescharacteristic
22C22×C4×D61—21no—
23C2×#11758194—21no—
24C2×#11758194—21no—
25C22×#92418—21no—
26C4×D244—21no—
27C4×D244—21no—
28C4×D244—21no—
29C4×D244—21no—
30C2×C4⋊Dic61Q811yescharacteristic
31C2×#11758198—21no—
32C4⋊#92418—21no—
33C4⋊#92418—21no—
34C4⋊#92418—21no—
35C4⋊#92418—21no—
36C4⋊#92418—21no—
37C4⋊#92418—21no—
38#14519689—44no—
39C22×C8×C61D412yes—
40C2×C42×C61C2311yescharacteristic
41C4⋊C8×C61—21no—
42C4⋊C8×C61—21no—
43C22⋊C8×C61—21no—
44C22⋊C8×C61—21no—
45#97×C61—21no—
46#97×C61—21no—
47C4×C8×C61D412yes—
48C4×C8×C61D412yes—
49C23×D61—21no—
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61C2×Q811yescharacteristic
52C22×C4×C61#4111yescharacteristic
53C22×Dic61—21no—
54C2×D244—21no—
55C2×D244—21no—
56C2×D244—42no—
57C2×C4×D61—41no—
58C2×C4×D61—41no—
59C2×C4×D61—42no—
60C2×C4×D61—42no—
61C2×C8×C61—22no—
62C2×C8×C61—22no—
63C2×C8×C61—22no—
64C2×C8×C61—22no—
65C2×C8×C61—22no—
66C42×C61C2×D411yescharacteristic
67C42×C61C2×D411yescharacteristic
68C42×C61#4111yescharacteristic
69C42×C61#4111yescharacteristic
70#11758198—41no—
71#11758198—41no—
72C4⋊Dic61—21no—
73C4⋊Dic61—21no—
74#11758194—42no—
75#11758194—42no—
76C2×#92418—41no—
77C2×#92418—41no—
78C2×#92418—41no—
79C2×#92418—42no—
80C23×C61C22⋊Q811yescharacteristic
81C22×D61—22no—
82C22×D61—41no—
83C22×D61—42no—
84C22×D61—41no—
85C22×D61—42no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61C22⋊Q811yescharacteristic
89C2×C4×C61C22⋊Q811yescharacteristic
90C2×C4×C61#12311yescharacteristic
91C2×C4×C61#12311yescharacteristic
92C2×C4×C61#12311yescharacteristic
93C2×C4×C61#12711yescharacteristic
94C2×C4×C61#13512yes—
95C2×C4×C61AGL(1,ℤ/8)12yes—
96C2×Dic61—41no—
97C2×Dic61—42no—
98D244—22no—
99D244—22no—
100D244—41no—
101D244—41no—
102C4×D61—82no—
103C4×D61—81no—
104C4×D61—81no—
105C8×C61—42no—
106C8×C61—42no—
107C8×C61—42no—
108#92418—81no—
109#92418—81no—
110#92418—81no—
111#10659—611nomaximal, Sylow
112C22×C61—21no—
113C22×C61—21no—
114C22×C61#39911yescharacteristic
115C22×C61#47812yes—
116C2×D61—44no—
117C2×D61—42no—
118C2×D61—82no—
119C2×D61—81no—
120C2×D61—81no—
121C4×C61—21no—
122C4×C61—21no—
123C4×C61—21no—
124C4×C61—21no—
125C4×C61#48212yes—
126C4×C61#49511yescharacteristic
127C4×C61#49611yescharacteristic
128Dic61—81no—
129#1955—611no—
130#1838—611no—
131#1571—611no—
132#1570—611no—
133#1486—611no—
134#1486—611no—
135#1275D6111yescharacteristic, complemented
136C2×C61—22no—
137C2×C61#209111yescharacteristic
138C2×C61#209211yescharacteristic
139C2×C61#209811yescharacteristic
140D61—82no—
141C2×C4×D4—611no—
142#431C2×D6111yescharacteristic
143#431C2×D6111yescharacteristic
144#423—611no—
145#423—611no—
146C2×C4⋊C8—611no—
147C2×C22⋊C8—1222no—
148C2×#97C2×D6111yescharacteristic
149#341—1222no—
150#335—1222no—
151#334—1222no—
152D4⋊C8—1222no—
153D4⋊C8—1222no—
154C61#1065911yescharacteristic, complemented, Sylow
155C22×D4—611no—
156C23×C4—1221no—
157C2×M16—1221no—
158C2×M16—1221no—
159C22×C8—1221no—
160C22×C8D24412yes—
161C4×D4—1221no—
162C4×D4—1221no—
163C4×D4—1221no—
164C4×D4—1221no—
165C2×C4⋊C4—611no—
166C2×C22⋊C4—1221no—
167C2×C42C22×D6111yescharacteristic
168C4⋊C8—21no—
169C4⋊C8—21no—
170C4⋊C8—1221no—
171C4⋊C8—1221no—
172C4⋊C8—1221no—
173C4⋊C8—1221no—
174C4⋊C8—1221no—
175C4⋊C8—1221no—
176C22⋊C8—21no—
177C22⋊C8—21no—
178C22⋊C8—2444no—
179#97—21no—
180#97—21no—
181C4×C8D24412yes—
182C4×C8D24412yes—
183C24—1221no—
184C2×D4—1221no—
185C2×D4—1221no—
186C2×D4—2442no—
187C22×C4—1221no—
188C22×C4—2441no—
189C22×C4—2441no—
190C22×C4—2442no—
191C22×C4—2442no—
192C22×C4C2×D24411yescharacteristic
193C22×C4#1175822411yescharacteristic
194C22×C4Q8×D6111yescharacteristic
195M16—2442no—
196M16—2442no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—2441no—
203C2×C8—2441no—
204C2×C8—2441no—
205C2×C8—2442no—
206C4⋊C4—1221no—
207C4⋊C4—1221no—
208C22⋊C4—2441no—
209C22⋊C4—2441no—
210C42C2×D24411yescharacteristic
211C42C2×D24411yescharacteristic
212C42#1175822411yescharacteristic
213C42#1175822411yescharacteristic
214C23—1222no—
215C23—2441no—
216C23—2442no—
217C23—2441no—
218C23—2442no—
219C23#1451975711yescharacteristic
220D4—1222no—
221D4—1222no—
222D4—2441no—
223D4—2441no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—4882no—
227C2×C4—4881no—
228C2×C4—4881no—
229C2×C4—2441no—
230C2×C4—2442no—
231C2×C4#1451975711yescharacteristic
232C2×C4#1451975711yescharacteristic
233C2×C4#1451976111yescharacteristic
234C2×C4#1451976111yescharacteristic
235C2×C4#1451976111yescharacteristic
236C2×C4#1451977411yescharacteristic
237C2×C4#1451979912yes—
238C2×C4#1451980512yes—
239C8—42no—
240C8—42no—
241C8—42no—
242C8—4881no—
243C8—4881no—
244C8—4881no—
245C22—21no—
246C22—21no—
247C22—4882no—
248C22—4881no—
249C22—4881no—
250C22—2444no—
251C22—2442no—
252C22#1683969911yescharacteristic
253C22#1684006312yes—
254C4—21no—
255C4—21no—
256C4—21no—
257C4—21no—
258C4—4881no—
259C4#1684008712yes—
260C4#1684014611yescharacteristic
261C4#1684015111yescharacteristic
262C2—22no—
263C2—4882no—
264C2#2144733611yescharacteristic
265C2#2144733911yescharacteristic
266C2#2144737311yescharacteristic
subgroupG/H
centerC2×C4#14519774
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1275C2
socleC22×C61#399
prime radicalC22×D244Q8
hypercenter#1275D61
hypocenterC61#10659
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
2. #21445960C2semidirect
3. #21444594C2semidirect
4. #21444590C2semidirect
7. C61×#1275C2semidirect
135. #1275D61semidirect
154. C61#10659semidirect

Maximal subgroups

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

subgroupindex
#214466172
#214459602
#214445942
#214445902
#214442452
C61×#12752
#1065961