Finite Group Databasebeta

← #29576915

Subgroups — #29576915

Order 15616.

536 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 111 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#21455949C211yesmaximal, characteristic
2#21446762C211yesmaximal, characteristic
3#21444330C211yesmaximal, characteristic
4#21444266C211yesmaximal, characteristic
5#21443712C211yesmaximal, characteristic
6#21443675C211yesmaximal, characteristic
7C61×#2276C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840092—21no—
10#16840047C412yescomplemented
11#16839972C412yescomplemented
12#16839918—21no—
13#16839867—22no—
14#16839819C2211yescharacteristic
15#16839819C2211yescharacteristic
16C4×#11758194C2211yescharacteristic
17#16839522—22no—
18#16839498—21no—
19#16839498—21no—
20Q8⋊#92418—22no—
21C2×C4×Q8×C61—21no—
22C2×C4×Q8×C61C2211yescharacteristic
23C4×C4⋊C4×C61—21no—
24C4×C4⋊C4×C61—22no—
25C4×C4⋊C4×C61—21no—
26C4×C4⋊C4×C61C2211yescharacteristic
27C61×#376—21no—
28C61×#376C2211yescharacteristic
29C2×#11758202—22no—
30C2×#11758194—21no—
31#14519777C2×C411yescharacteristic
32#14519777C2×C411yescharacteristic
33#14519769—21no—
34#14519756—21no—
35#14519756—21no—
36C4×Dic122—21no—
37C4×Dic122—21no—
38#14519736—21no—
39#14519708—22no—
40C4⋊#92418—21no—
41C4⋊#92418C2×C412yes—
42#14519701—22no—
43#14519700—22no—
44C22⋊#92418—21no—
45C22⋊#92418—21no—
46#14519688—21no—
47#14519688—21no—
48#14519688—21no—
49C4×#92418—21no—
50C22×Q8×C61D411yescharacteristic
51C4×Q8×C61—28no—
52C4×Q8×C61—21no—
53C4×Q8×C61—21no—
54C4×Q8×C61C2×C412yes—
55C4×Q8×C61D412yes—
56C2×C4⋊C4×C61—22no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61D411yescharacteristic
60C2×C42×C61—22no—
61C2×C42×C61—21no—
62C2×C42×C61—21no—
63C2×C42×C61—22no—
64C2×C42×C61D412yes—
65C2×C42×C61C2311yescharacteristic
66#95×C61—22no—
67#95×C61—21no—
68#95×C61—21no—
69#95×C61D411yescharacteristic
70#95×C61D411yescharacteristic
71C2×Q8×C61—24no—
72C2×Q8×C61—21no—
73C2×Q8×C61C22⋊C412yes—
74C22×C4×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—22no—
78C22×C4×C61—21no—
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C2×Dic122—21no—
83C2×Dic122—21no—
84C4⋊C4×C61—28no—
85C4⋊C4×C61—24no—
86C4⋊C4×C61—22no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—21no—
89C4⋊C4×C61C22⋊C412yes—
90C42×C61—28no—
91C42×C61—24no—
92C42×C61—22no—
93C42×C61—22no—
94C42×C61—42no—
95C42×C61—82no—
96C42×C61C22×C411yescharacteristic
97C42×C61C2×D411yescharacteristic
98C42×C61#4111yescharacteristic
99C42×C61#4111yescharacteristic
100C22⋊Dic61—41no—
101#11758202—44no—
102C4⋊Dic61—21no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106#11758196—41no—
107#11758196—41no—
108C4×Dic61—41no—
109#11758194—22no—
110#11758194—41no—
111C2×#92418—22no—
112C2×#92418—41no—
113C2×#92418—41no—
114C23×C61#12011yescharacteristic
115Q8×C61—24no—
116Q8×C61—22no—
117Q8×C61#10414yes—
118C2×C4×C61—24no—
119C2×C4×C61—22no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—24no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—42no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61C2×C22⋊C411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12311yescharacteristic
142C2×Dic61—41no—
143C2×Dic61—41no—
144Dic122—42no—
145Dic122—81no—
146#92418—42no—
147#92418—42no—
148#92418—81no—
149#9324—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61#38611yescharacteristic
153C22×C61#45511yescharacteristic
154C22×C61#45611yescharacteristic
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—24no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—44no—
162C4×C61—42no—
163C4×C61—42no—
164C4×C61—42no—
165C4×C61C2×#10412yes—
166Dic61—81no—
167Dic61—81no—
168#3113—611no—
169#2276D6111yescharacteristic
170#1986—611no—
171#1506—611no—
172#1491—611no—
173#1300—611no—
174#1285—611no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61#188211yescharacteristic
178C2×C61#189611yescharacteristic
179C2×C61#190511yescharacteristic
180#533—611no—
181C2×C4×Q8—21no—
182C2×C4×Q8C2×D6111yescharacteristic
183#483—1221no—
184#475—612no—
185#462—612no—
186#451—1221no—
187#441—1222no—
188#432—611no—
189#432—611no—
190C4×M16—611no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2×D6111yescharacteristic
195#376—21no—
196#376C2×D6111yescharacteristic
197#339—1222no—
198#328—1221no—
199#328—1221no—
200Q8⋊C8—1222no—
201C61#932411yescharacteristic, complemented, Sylow
202C22×Q8C22⋊D6111yescharacteristic
203C2×Q16—1222no—
204C2×M16—1221no—
205C4⋊Q8—611no—
206C4⋊Q8—611no—
207#125—1221no—
208C22⋊Q8—1221no—
209C22⋊Q8—1221no—
210C4×Q8—28no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—1221no—
214C4×Q8—1221no—
215C4×Q8C4×D6112yes—
216C4×Q8D24412yes—
217#117—1221no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22⋊D6111yescharacteristic
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42C22⋊D6112yes—
227C2×C42C22×D6111yescharacteristic
228#106—1222no—
229C4⋊C8—612no—
230C4⋊C8—1221no—
231#103—1222no—
232#103—1222no—
233C22⋊C8—1221no—
234C22⋊C8—1221no—
235#97—1221no—
236#97—1221no—
237#97—1221no—
238C4×C8—1221no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95C22⋊D6111yescharacteristic
243#95C22⋊D6111yescharacteristic
244C2×Q8—24no—
245C2×Q8—21no—
246C2×Q8—1221no—
247C2×Q8—1221no—
248C2×Q8#1175819812yes—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4C2×C22⋊D6111yescharacteristic
255C22×C4C2×C22⋊D6111yescharacteristic
256C22×C4C2×C22⋊D6111yescharacteristic
257Q16—2444no—
258M16—1222no—
259M16—2441no—
260C2×C8—1222no—
261C2×C8—2441no—
262C2×C8—2441no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—1221no—
269C4⋊C4—1221no—
270C4⋊C4—1221no—
271C4⋊C4—1221no—
272C4⋊C4—2441no—
273C4⋊C4—2441no—
274C4⋊C4#1175819812yes—
275C22⋊C4—2441no—
276C42—28no—
277C42—24no—
278C42—22no—
279C42—22no—
280C42—42no—
281C42—82no—
282C42—2441no—
283C42C2×C4×D6111yescharacteristic
284C42C2×D24411yescharacteristic
285C42#1175822211yescharacteristic
286C42#1175822211yescharacteristic
287C23#1451974611yescharacteristic
288Q8—24no—
289Q8—22no—
290Q8—4881no—
291Q8—2442no—
292Q8#1451970514yes—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4C2×#1175819811yescharacteristic
314C2×C4C4×C22⋊D6111yescharacteristic
315C2×C4C4×C22⋊D6111yescharacteristic
316C2×C4#1451975111yescharacteristic
317C2×C4#1451975111yescharacteristic
318C2×C4#1451976011yescharacteristic
319C8—4881no—
320C8—2442no—
321C8—2442no—
322C22—21no—
323C22—21no—
324C22#1683964111yescharacteristic
325C22#1683994011yescharacteristic
326C22#1683994511yescharacteristic
327C4—21no—
328C4—21no—
329C4—24no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—44no—
334C4—42no—
335C4—42no—
336C4—42no—
337C4—4881no—
338C4—4881no—
339C4C2×#1451970512yes—
340C2—21no—
341C2—21no—
342C2#2144618811yescharacteristic
343C2#2144626211yescharacteristic
344C2#2144631311yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2276C2
socleC22×C61#386
prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#2276D61
hypocenterC61#9324
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2276

Complements

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

NH ≅ G/Ntype
10. #16840047C4semidirect
11. #16839972C4semidirect
201. C61#9324semidirect

Maximal subgroups

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

subgroupindex
#214559492
#214467622
#214443302
#214442662
#214437122
#214436752
C61×#22762
#932461