Finite Group Databasebeta

← #29573406

Subgroups — #29573406

Order 15616.

588 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448226C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic
3#21443698C214yesmaximal
4C61×#1434C211yesmaximal, characteristic
5C2×#14519736—21no—
6C2×C22⋊#92418C2212yes—
7#16839632C411yescharacteristic, complemented
8#16839632C411yescharacteristic, complemented
9#16839628—21no—
10#16839622—21no—
11#16839622C2212yes—
12C4×C4⋊Dic61—21no—
13C4×C4⋊Dic61C412yescomplemented
14#16839517—21no—
15#16839517—21no—
16C22×C42×C61C2211yescharacteristic
17C2×#95×C61—21no—
18C2×#95×C61—21no—
19C2×#95×C61C2212yes—
20C22×#92418—22no—
21#14519736—44no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61C2×C412yes—
27C2×C4⋊Dic61C2×C412yes—
28C2×C22⋊Dic61—22no—
29C2×C4×Dic61—22no—
30C22⋊#92418—24no—
31#14519684—22no—
32#14519684—22no—
33C23×C4×C61—22no—
34C23×C4×C61D411yescharacteristic
35C23×C4×C61D411yescharacteristic
36C23×C4×C61C2311yescharacteristic
37C2×C42×C61—24no—
38C2×C42×C61C2×C411yescharacteristic
39C2×C42×C61C2×C411yescharacteristic
40C2×C42×C61D411yescharacteristic
41C2×C42×C61D411yescharacteristic
42#95×C61—24no—
43#95×C61—24no—
44#95×C61—42no—
45C24×C61C2×D411yescharacteristic
46C22×C4×C61—22no—
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—24no—
51C22×C4×C61—41no—
52C22×C4×C61—41no—
53C22×C4×C61—41no—
54C22×C4×C61—42no—
55C22×C4×C61—42no—
56C22×C4×C61—42no—
57C22×C4×C61—42no—
58C22×C4×C61—42no—
59C22×C4×C61—42no—
60C22×C4×C61C22⋊C411yescharacteristic
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61SD1612yes—
65C22×C4×C61Q1612yes—
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61#4114yes—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C42×C61—22no—
72C42×C61—22no—
73C42×C61—22no—
74C42×C61—22no—
75C22⋊Dic61—44no—
76C4⋊Dic61—42no—
77C4⋊Dic61—42no—
78C4⋊Dic61—42no—
79C4⋊Dic61—42no—
80C4⋊Dic61—42no—
81C4⋊Dic61—42no—
82C4×Dic61—44no—
83C2×#92418—42no—
84C2×#92418—42no—
85C2×#92418—42no—
86C23×C61—21no—
87C23×C61—21no—
88C23×C61—21no—
89C23×C61—21no—
90C23×C61C2×C22⋊C411yescharacteristic
91C23×C61#12312yes—
92C23×C61C2×SD1611yescharacteristic
93C23×C61C2×Q1611yescharacteristic
94C23×C61AGL(1,ℤ/8)12yes—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
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—24no—
106C2×C4×C61—24no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—44no—
111C2×C4×C61—44no—
112C2×C4×C61—44no—
113C2×C4×C61—44no—
114C2×C4×C61—44no—
115C2×C4×C61—82no—
116C2×C4×C61—82no—
117C2×C4×C61—82no—
118C2×C4×C61—82no—
119C2×C4×C61#10312yes—
120C2×C4×C61#10312yes—
121C2×C4×C61#10412yes—
122C2×C4×C61#11712yes—
123C2×C4×C61#12312yes—
124C2×Dic61—42no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130#92418—82no—
131#8717—611nomaximal, Sylow
132C22×C61—22no—
133C22×C61—22no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—22no—
137C22×C61—22no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61#38111yescharacteristic
141C22×C61#41011yescharacteristic
142C22×C61#41011yescharacteristic
143C22×C61C2×#10311yescharacteristic
144C22×C61#41811yescharacteristic
145C22×C61C2×#10411yescharacteristic
146C22×C61#42111yescharacteristic
147C22×C61#48112yes—
148C22×C61#48412yes—
149C4×C61—22no—
150C4×C61—22no—
151C4×C61—22no—
152C4×C61—22no—
153C4×C61—22no—
154C4×C61—41no—
155C4×C61—84no—
156Dic61—82no—
157Dic61—82no—
158#2285—611no—
159#1817—611no—
160#1434D6111yescharacteristic
161#1294—614no—
162C2×C61—22no—
163C2×C61—22no—
164C2×C61#150311yescharacteristic
165C2×C61#150511yescharacteristic
166C2×C61#151411yescharacteristic
167C2×C61#151811yescharacteristic
168C2×C61#182011yescharacteristic
169C2×C61#182411yescharacteristic
170C2×C61#182611yescharacteristic
171C2×#117—1221no—
172C22×C42C2×D6111yescharacteristic
173C2×C22⋊C8—612no—
174#384—611no—
175#384—611no—
176#382—1221no—
177#380—612no—
178#380—1221no—
179C4×C4⋊C4—612no—
180C4×C4⋊C4—1221no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95C2×D6112yes—
184#336—1221no—
185#336—1221no—
186C61#871711yescharacteristic, complemented, Sylow
187C23×C4—22no—
188C23×C4C22⋊D6111yescharacteristic
189C23×C4C22⋊D6111yescharacteristic
190C23×C4C22×D6111yescharacteristic
191C22×C8—1222no—
192#117—2444no—
193C2×C4⋊C4—612no—
194C2×C4⋊C4—612no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4—1221no—
197C2×C4⋊C4—1221no—
198C2×C4⋊C4—1221no—
199C2×C22⋊C4—1222no—
200C2×C42—24no—
201C2×C42—1222no—
202C2×C42C4×D6111yescharacteristic
203C2×C42C4×D6111yescharacteristic
204C2×C42D24411yescharacteristic
205C2×C42D24411yescharacteristic
206C22⋊C8—1224no—
207#95—24no—
208#95—24no—
209#95—42no—
210#95—1222no—
211#95—1222no—
212C24C2×C22⋊D6111yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—1222no—
228C22×C4—1222no—
229C22×C4#1175819811yescharacteristic
230C22×C4#1175819811yescharacteristic
231C22×C4#1175819811yescharacteristic
232C22×C4#1175819811yescharacteristic
233C22×C4#1175820012yes—
234C22×C4#1175820212yes—
235C22×C4C2×C4×D6111yescharacteristic
236C22×C4C2×D24411yescharacteristic
237C22×C4#1175822414yes—
238C2×C8—2442no—
239C2×C8—2442no—
240C2×C8—2442no—
241C4⋊C4—2442no—
242C4⋊C4—2442no—
243C4⋊C4—2442no—
244C4⋊C4—2442no—
245C4⋊C4—2442no—
246C4⋊C4—2442no—
247C22⋊C4—2444no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—2444no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23C2×#1175819811yescharacteristic
258C23#1451975812yes—
259C23C2×#1175820011yescharacteristic
260C23C2×#1175820211yescharacteristic
261C23#1451980212yes—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—2442no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4#1451970012yes—
293C2×C4#1451970012yes—
294C2×C4#1451970512yes—
295C2×C4#1451973312yes—
296C2×C4#1451976112yes—
297C8—4882no—
298C22—22no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22#1683962311yescharacteristic
307C22#1683973511yescharacteristic
308C22#1683973511yescharacteristic
309C22C2×#1451970011yescharacteristic
310C22#1683976611yescharacteristic
311C22C2×#1451970511yescharacteristic
312C22#1683978011yescharacteristic
313C22#1684007812yes—
314C22#1684009512yes—
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—84no—
322C4—4882no—
323C4—4882no—
324C2—22no—
325C2—22no—
326C2#2144431711yescharacteristic
327C2#2144432611yescharacteristic
328C2#2144436511yescharacteristic
329C2#2144438211yescharacteristic
330C2#2144589411yescharacteristic
331C2#2144590711yescharacteristic
332C2#2144591911yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1434C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1434D61
hypocenterC61#8717
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
7. #16839632C4semidirect
8. #16839632C4semidirect
13. C4×C4⋊Dic61C4semidirect
186. C61#8717semidirect

Maximal subgroups

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

subgroupindex
#214482262
#214458872
#214436982
C61×#14342
#871761