Finite Group Databasebeta

← #29567100

Subgroups — #29567100

Order 15616.

372 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 87 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#21446112C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic
3C61×#1799C211yesmaximal, characteristic
4C23×#92418C411yescharacteristic, complemented
5#16839786—21no—
6C2×#14519694—21no—
7C2×C22⋊#92418—21no—
8C2×C22⋊#92418C411yescharacteristic, complemented
9#16839531C412yescomplemented
10#16839517—21no—
11C2×C4×D4×C61C2211yescharacteristic
12C2×#99×C61—21no—
13C61×#380—21no—
14C61×#342C412yes—
15C2×#11758194—22no—
16C22×#92418—21no—
17C22×#92418—21no—
18C22×#92418—42no—
19C22×#92418—41no—
20C4⋊#92418—41no—
21C4⋊#92418—41no—
22#14519694—41no—
23#14519694—41no—
24C22⋊#92418—42no—
25C22⋊#92418—41no—
26C22×D4×C61D411yescharacteristic
27C23×C4×C61C2×C411yescharacteristic
28C23×C4×C61D411yescharacteristic
29C4×D4×C61—41no—
30C4×D4×C61—41no—
31C2×C4⋊C4×C61—41no—
32C2×C4⋊C4×C61C2×C411yescharacteristic
33C2×C22⋊C4×C61—22no—
34C2×C22⋊C4×C61C2×C411yescharacteristic
35C2×C22⋊C4×C61Q811yescharacteristic
36C2×C42×C61D411yescharacteristic
37#99×C61—41no—
38#99×C61—41no—
39#95×C61—41no—
40C24×C61C22⋊C411yescharacteristic
41C24×C61C4⋊C411yescharacteristic
42C2×D4×C61—41no—
43C2×D4×C61—41no—
44C2×D4×C61D811yescharacteristic
45C2×D4×C61SD1611yescharacteristic
46C2×D4×C61SD1611yescharacteristic
47C2×D4×C61Q1611yescharacteristic
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—41no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61—41no—
56C22×C4×C61—42no—
57C22×C4×C61C4211yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C4⋊C411yescharacteristic
61C22×C4×C61C4⋊C411yescharacteristic
62C4⋊C4×C61—41no—
63C4⋊C4×C61—81no—
64C4⋊C4×C61—81no—
65C22⋊C4×C61—41no—
66C22⋊C4×C61—41no—
67C22⋊C4×C61—42no—
68C22⋊C4×C61—42no—
69C42×C61—21no—
70C42×C61—21no—
71#11758194—42no—
72#11758194—42no—
73C2×#92418—22no—
74C2×#92418—41no—
75C2×#92418—42no—
76C2×#92418—41no—
77C2×#92418—42no—
78C2×#92418—41no—
79C2×#92418—42no—
80C2×#92418—41no—
81C2×#92418—41no—
82C23×C61—21no—
83C23×C61—21no—
84C23×C61—21no—
85C23×C61—21no—
86C23×C61—41no—
87C23×C61—41no—
88C23×C61—41no—
89C23×C61—41no—
90C23×C61#9511yescharacteristic
91C23×C61#10211yescharacteristic
92C23×C61#10311yescharacteristic
93C23×C61#10611yescharacteristic
94C23×C61#10711yescharacteristic
95D4×C61—41no—
96D4×C61—41no—
97D4×C61—41no—
98D4×C61—41no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—42no—
107C2×C4×C61—41no—
108C2×C4×C61—41no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—81no—
115C2×C4×C61—82no—
116C2×C4×C61—82no—
117C2×C4×C61—82no—
118C2×C4×C61#9911yescharacteristic
119C2×C4×C61#9911yescharacteristic
120C2×C4×C61#10211yescharacteristic
121C2×C4×C61#10311yescharacteristic
122#92418—42no—
123#92418—81no—
124#92418—82no—
125#7236—611nomaximal, Sylow
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—21no—
129C22×C61—21no—
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—41no—
133C22×C61—41no—
134C22×C61—41no—
135C22×C61—41no—
136C22×C61—42no—
137C22×C61—41no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—81no—
142C22×C61H(3,ℤ/4)11yescharacteristic
143C22×C61#34011yescharacteristic
144C22×C61#34211yescharacteristic
145C4×C61—21no—
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—41no—
149C4×C61—41no—
150C4×C61—41no—
151C4×C61—162no—
152C4×C61AΣL(1,𝔽16)11yescharacteristic
153C4×C61#35211yescharacteristic
154#1869—611no—
155#1800—611no—
156#1799D6111yescharacteristic
157C2×C61—21no—
158C2×C61—21no—
159C2×C61—41no—
160C2×C61—41no—
161C2×C61—41no—
162C2×C61—41no—
163C2×C61#128911yescharacteristic
164C2×C61#139011yescharacteristic
165C2×C61#139511yescharacteristic
166C23×C8—611no—
167C2×C4×D4C2×D6111yescharacteristic
168#423—1221no—
169C2×#100—1221no—
170C2×#99—21no—
171C2×C22⋊C8—611no—
172C2×C22⋊C8—1221no—
173#380—21no—
174#343—612no—
175#342Dic6112yes—
176#336—1221no—
177C61#723611yescharacteristic, complemented, Sylow
178C22×D4C22⋊D6111yescharacteristic
179C23×C4C4×D6111yescharacteristic
180C23×C4C22⋊D6111yescharacteristic
181C2×M16—1222no—
182C22×C8—1221no—
183C22×C8—1221no—
184C22×C8—2442no—
185C22×C8—2441no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C4×D6111yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4Dic12211yescharacteristic
192C2×C22⋊C4C2×Dic6111yescharacteristic
193C2×C42D24411yescharacteristic
194C4⋊C8—2441no—
195C4⋊C8—2441no—
196#100—2441no—
197#100—2441no—
198#99—41no—
199#99—41no—
200C22⋊C8—2442no—
201C22⋊C8—2441no—
202#95—41no—
203C24#1175819611yescharacteristic
204C24C22⋊Dic6111yescharacteristic
205C2×D4—41no—
206C2×D4—41no—
207C2×D4#1175819911yescharacteristic
208C2×D4#1175820011yescharacteristic
209C2×D4#1175820111yescharacteristic
210C2×D4#1175820211yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4C4×Dic6111yescharacteristic
221C22×C4#1175819611yescharacteristic
222C22×C4C4⋊Dic6111yescharacteristic
223C22×C4#1175819811yescharacteristic
224C22×C4#1175819811yescharacteristic
225M16—2442no—
226M16—2442no—
227C2×C8—1222no—
228C2×C8—2441no—
229C2×C8—2442no—
230C2×C8—2441no—
231C2×C8—2441no—
232C2×C8—2441no—
233C2×C8—2442no—
234C2×C8—2441no—
235C2×C8—2442no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—21no—
244C42—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1451968411yescharacteristic
254C23#1451969811yescharacteristic
255C23#1451970111yescharacteristic
256C23#1451970811yescharacteristic
257C23#1451971011yescharacteristic
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#1451969111yescharacteristic
282C2×C4#1451969111yescharacteristic
283C2×C4#1451969711yescharacteristic
284C2×C4#1451970011yescharacteristic
285C8—4881no—
286C8—4882no—
287C8—2442no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22#1683951811yescharacteristic
305C22#1683952411yescharacteristic
306C22#1683952811yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—162no—
314C4#1683954811yescharacteristic
315C4#1683955111yescharacteristic
316C2—21no—
317C2—21no—
318C2—41no—
319C2—41no—
320C2—41no—
321C2—41no—
322C2#2144368611yescharacteristic
323C2#2144393611yescharacteristic
324C2#2144394711yescharacteristic
subgroupG/H
centerC22#16839528
commutatorC22×C4×C61C42
FrattiniC2×C4×D4C2×D61
FittingC61×#1799C2
socleC22×C61#342
prime radicalC22×D4×C61D4
cosocleC2×C4×D4×C61C22
hypercenter#1799D61
hypocenterC61#7236
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. C23×#92418C4semidirect
8. C2×C22⋊#92418C4semidirect
9. #16839531C4semidirect
177. C61#7236semidirect

Maximal subgroups

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

subgroupindex
#214461122
#214458112
C61×#17992
#723661