Finite Group Databasebeta

← #29577364

Subgroups — #29577364

Order 15616.

446 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 103 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#21454457C211yesmaximal, characteristic
2#21444421C212yesmaximal, complemented
3#21444153C211yesmaximal, characteristic
4#21443712C212yesmaximal, complemented
5C61×#2611C211yesmaximal, characteristic
6#16839823—24no—
7#16839819C2212yes—
8#16839786—21no—
9#16839726—21no—
10C4×#11758194C2212yes—
11#16839504—24no—
12#16839488—21no—
13#16839486—21no—
14C2×#125×C61C2211yescharacteristic
15C2×#117×C61C2211yescharacteristic
16C61×#400C411yescharacteristic
17C61×#399C411yescharacteristic
18C61×#398C411yescharacteristic
19C61×#397C411yescharacteristic
20C61×#389—21no—
21C61×#388—21no—
22C61×#387—21no—
23C61×#385—21no—
24C61×#376C2211yescharacteristic
25C2×#11758194—22no—
26C4⋊#92418—24no—
27C4⋊#92418—22no—
28C22⋊#92418—24no—
29C22⋊#92418—22no—
30#14519688—22no—
31#14519688D414yes—
32C4×#92418—22no—
33C23×C4×C61D411yescharacteristic
34#125×C61—22no—
35#125×C61—22no—
36#117×C61—22no—
37#117×C61—22no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61—21no—
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61C2×C411yescharacteristic
45C2×C4⋊C4×C61D411yescharacteristic
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—21no—
48C2×C22⋊C4×C61C2×C411yescharacteristic
49C2×C22⋊C4×C61D411yescharacteristic
50C2×C42×C61—21no—
51C2×C42×C61D411yescharacteristic
52C2×C42×C61C2311yescharacteristic
53#95×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61C2×C411yescharacteristic
59#95×C61C2×C411yescharacteristic
60C24×C61C22⋊C411yescharacteristic
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—41no—
68C22×C4×C61—41no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C4⋊C4×C61—22no—
76C4⋊C4×C61—22no—
77C4⋊C4×C61—22no—
78C4⋊C4×C61—22no—
79C4⋊C4×C61—42no—
80C4⋊C4×C61—42no—
81C4⋊C4×C61—41no—
82C4⋊C4×C61—42no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—22no—
85C22⋊C4×C61—22no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—42no—
88C42×C61—21no—
89C42×C61—21no—
90C42×C61—42no—
91C42×C61C2×D412yes—
92C42×C61#4112yes—
93#11758194—42no—
94#11758194—42no—
95C2×#92418—24no—
96C2×#92418—42no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61C2×C22⋊C411yescharacteristic
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—22no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—41no—
112C2×C4×C61—41no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61C4×D412yes—
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61#12112yes—
131C2×C4×C61#12311yescharacteristic
132#92418—44no—
133#92418—82no—
134#9390—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—42no—
138C22×C61—41no—
139C22×C61—41no—
140C22×C61—41no—
141C22×C61—81no—
142C22×C61#38611yescharacteristic
143C22×C61#42111yescharacteristic
144C22×C61#42111yescharacteristic
145C4×C61—22no—
146C4×C61—21no—
147C4×C61—21no—
148C4×C61—22no—
149C4×C61—42no—
150C4×C61—41no—
151C4×C61—41no—
152C4×C61—41no—
153C4×C61—81no—
154C4×C61—81no—
155C4×C61—81no—
156#2921—611no—
157#2611D6111yescharacteristic
158#1528—612no—
159#1464—611no—
160#1300—612no—
161C2×C61—21no—
162C2×C61—21no—
163C2×C61—81no—
164C2×C61#188411yescharacteristic
165C2×C61#189711yescharacteristic
166C2×C61#191511yescharacteristic
167C2×#125C2×D6111yescharacteristic
168C2×#117C2×D6111yescharacteristic
169#433—1224no—
170#432—612no—
171#423—1221no—
172#407—1221no—
173C4×M16—612no—
174#400Dic6111yescharacteristic
175#399Dic6111yescharacteristic
176#398Dic6111yescharacteristic
177#397Dic6111yescharacteristic
178#389—21no—
179#388—21no—
180#387—21no—
181#385—21no—
182#376C2×D6111yescharacteristic
183#330—1224no—
184#324—1221no—
185#323—1221no—
186C61#939011yescharacteristic, complemented, Sylow
187C23×C4C22⋊D6111yescharacteristic
188C2×M16—1222no—
189#125—22no—
190#125—22no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic6111yescharacteristic
198C2×C4⋊C4C2×Dic6111yescharacteristic
199C2×C4⋊C4C2×Dic6111yescharacteristic
200C2×C4⋊C4C22⋊D6111yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C2×Dic6111yescharacteristic
204C2×C22⋊C4C22⋊D6111yescharacteristic
205C2×C42—21no—
206C2×C42C22⋊D6111yescharacteristic
207C2×C42C22×D6111yescharacteristic
208C4⋊C8—1224no—
209C4⋊C8—1222no—
210C22⋊C8—1224no—
211C22⋊C8—1222no—
212#97—614no—
213#97—1222no—
214C4×C8—1222no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95C2×Dic6111yescharacteristic
221#95C2×Dic6111yescharacteristic
222C24C22⋊Dic6111yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4C22⋊Dic6111yescharacteristic
232C22×C4C22⋊Dic6111yescharacteristic
233C22×C4C22⋊Dic6111yescharacteristic
234C22×C4C22×Dic6111yescharacteristic
235C22×C4C2×C22⋊D6111yescharacteristic
236C22×C4C2×C22⋊D6111yescharacteristic
237M16—2442no—
238M16—2442no—
239C2×C8—1224no—
240C2×C8—2442no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C42—21no—
255C42—21no—
256C42—42no—
257C42D4×D6112yes—
258C42#1175822412yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊Dic6111yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4D4×Dic6112yes—
291C2×C4#1451974511yescharacteristic
292C2×C4#1451974912yes—
293C2×C4#1451975811yescharacteristic
294C8—4882no—
295C8—2444no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1683964211yescharacteristic
304C22#1683977911yescharacteristic
305C22#1683977911yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—81no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2144620311yescharacteristic
321C2#2144626811yescharacteristic
322C2#2144637211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2611C2
socleC22×C61#386
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#2611D61
hypocenterC61#9390
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2611

Complements

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

NH ≅ G/Ntype
2. #21444421C2semidirect
4. #21443712C2semidirect
186. C61#9390semidirect

Maximal subgroups

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

subgroupindex
#214544572
#214444212
#214441532
#214437122
C61×#26112
#939061