Finite Group Databasebeta

← #29592275

Subgroups — #29592275

Order 15616.

398 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 63 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#21446865C211yesmaximal, characteristic
2#21444855C212yesmaximal
3#21443846C212yesmaximal, complemented
4#21443753C211yesmaximal, characteristic, complemented
5C61×#2809C211yesmaximal, characteristic, complemented
6C2×#14519808—21no—
7#16840343C2211yescharacteristic
8#16840080—22no—
9C2×#14519705—21no—
10#16839531C2211yescharacteristic
11#16839495—22no—
12#16839488C2212yes—
13C61×#400C2212yes—
14C61×#398—22no—
15C61×#391—21no—
16C61×#391C2211yescharacteristic
17C61×#389—21no—
18C61×#387—22no—
19C2×#11758222—21no—
20#14519808—42no—
21#14519808—42no—
22C2×#11758202—41no—
23C2×#11758201—41no—
24C2×#11758194—21no—
25#14519757—41no—
26#14519752—41no—
27#14519751—22no—
28#14519708—42no—
29#14519705—42no—
30#14519697—42no—
31C22⋊#92418—22no—
32C22×Q8×C61D411yescharacteristic
33C2×C4⋊C4×C61—22no—
34C2×C4⋊C4×C61—21no—
35C2×C4⋊C4×C61—21no—
36C2×C4⋊C4×C61—21no—
37C2×C4⋊C4×C61D412yes—
38C2×C4⋊C4×C61C2311yescharacteristic
39C2×C42×C61—21no—
40C2×C42×C61D411yescharacteristic
41#95×C61—22no—
42#95×C61—22no—
43#95×C61—22no—
44#95×C61—22no—
45#95×C61D412yes—
46C2×Q8×C61—21no—
47C2×Q8×C61—41no—
48C2×Q8×C61—41no—
49C2×Q8×C61SD1612yes—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—22no—
53C22×C4×C61—22no—
54C22×C4×C61C2×D411yescharacteristic
55C22×C4×C61C2×D412yes—
56C2×C22⋊D61—41no—
57#11758222—41no—
58#11758222—41no—
59#11758222—41no—
60#11758222—41no—
61C2×D244—21no—
62C2×D244—22no—
63C2×C4×D61—41no—
64C2×Dic122—21no—
65C4⋊C4×C61—41no—
66C4⋊C4×C61—44no—
67C4⋊C4×C61—42no—
68C4⋊C4×C61—42no—
69C4⋊C4×C61—81no—
70C4⋊C4×C61—81no—
71C42×C61—41no—
72C42×C61—81no—
73#11758202—44no—
74#11758201—44no—
75#11758198—42no—
76C4⋊Dic61—22no—
77#11758194—22no—
78#11758194—41no—
79C2×#92418—41no—
80C2×#92418—42no—
81C23×C61#12011yescharacteristic
82C22×D61—41no—
83Q8×C61—42no—
84Q8×C61—81no—
85C2×C4×C61—21no—
86C2×C4×C61—21no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61—22no—
90C2×C4×C61—41no—
91C2×C4×C61—41no—
92C2×C4×C61—41no—
93C2×C4×C61—41no—
94C2×C4×C61—42no—
95C2×C4×C61—42no—
96C2×C4×C61—42no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61—42no—
100C2×C4×C61—42no—
101C2×C4×C61C2×SD1611yescharacteristic
102C2×C4×C61#13711yescharacteristic
103C22⋊D61—81no—
104C22⋊D61—81no—
105C2×Dic61—41no—
106D244—41no—
107D244—41no—
108D244—82no—
109C4×D61—81no—
110C4×D61—81no—
111Dic122—41no—
112Dic122—41no—
113#92418—42no—
114#92418—82no—
115#12131—611nomaximal, Sylow
116C22×C61—21no—
117C22×C61—21no—
118C22×C61#44811yescharacteristic
119C22×C61#45311yescharacteristic
120C22×C61B(4,𝔽2)11yescharacteristic
121C2×D61—81no—
122C2×D61—81no—
123C2×D61—81no—
124C4×C61—21no—
125C4×C61—21no—
126C4×C61—42no—
127C4×C61—42no—
128C4×C61—42no—
129C4×C61—82no—
130C4×C61—82no—
131Dic61—81no—
132#2809D6111yescharacteristic, complemented
133#2007—611no—
134#1616—612no—
135#1353—612no—
136#1319—611no—
137C2×C61—21no—
138C2×C61—21no—
139C2×C61#159611yescharacteristic
140C2×C61#220111yescharacteristic
141C2×C61#220411yescharacteristic
142D61—161no—
143C2×#137—1221no—
144#537—611no—
145#481—1222no—
146C2×#104—1221no—
147#400C2×D6112yes—
148#398—22no—
149#391—21no—
150#391C2×D6111yescharacteristic
151#389—21no—
152#387—22no—
153#343—611no—
154#327—1222no—
155#324—612no—
156C61#1213111yescharacteristic, complemented, Sylow
157C2×#41—1221no—
158C22×Q8C22⋊D6111yescharacteristic
159#137—2442no—
160#137—2442no—
161C2×Q16—2441no—
162C2×SD16—2441no—
163C2×M16—1221no—
164C22⋊Q8—2441no—
165#121—1222no—
166#121—2441no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4C22⋊D6112yes—
172C2×C4⋊C4C22×D6111yescharacteristic
173C2×C42—21no—
174C2×C42C22⋊D6111yescharacteristic
175#106—2442no—
176#104—2442no—
177#102—2442no—
178C22⋊C8—1222no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95C22⋊D6112yes—
184#41—2441no—
185#41—2441no—
186#41—2441no—
187#41—2441no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—1221no—
192C2×Q8#1175820112yes—
193C2×D4—1222no—
194C2×D4—1221no—
195C2×D4—2441no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—2441no—
201C22×C4C2×C22⋊D6111yescharacteristic
202C22×C4C2×C22⋊D6112yes—
203Q16—2444no—
204SD16—2444no—
205M16—1222no—
206M16—2441no—
207C2×C8—2442no—
208C2×C8—2441no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C4⋊C4—1222no—
216C22⋊C4—2442no—
217C42—41no—
218C42—81no—
219C23—2441no—
220C23#1451974611yescharacteristic
221Q8—42no—
222Q8—81no—
223Q8—2441no—
224Q8—2441no—
225D4—4882no—
226D4—4881no—
227D4—4881no—
228D4—2441no—
229D4—2441no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—4881no—
247C2×C4—4881no—
248C2×C4—2441no—
249C2×C4C2×#1175820111yescharacteristic
250C2×C4#1451980811yescharacteristic
251C8—4882no—
252C8—2442no—
253C22—21no—
254C22—21no—
255C22—4881no—
256C22—4881no—
257C22—4881no—
258C22#1683990111yescharacteristic
259C22#1683993011yescharacteristic
260C22#1683994911yescharacteristic
261C4—21no—
262C4—21no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—82no—
267C4—82no—
268C4—4881no—
269C2—21no—
270C2—21no—
271C2—9761no—
272C2#2144474211yescharacteristic
273C2#2144783211yescharacteristic
274C2#2144784511yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2809C2
socleC22×C61B(4,𝔽2)
prime radical#16840343C22
hypercenter#2809D61
hypocenterC61#12131
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
3. #21443846C2semidirect
4. #21443753C2semidirect
5. C61×#2809C2semidirect
132. #2809D61semidirect
156. C61#12131semidirect

Maximal subgroups

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

subgroupindex
#214468652
#214448552
#214438462
#214437532
C61×#28092
#1213161