Finite Group Databasebeta

← #29589552

Subgroups — #29589552

Order 15616.

460 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 79 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#21450089C211yesmaximal, characteristic
2#21444153C211yesmaximal, characteristic, complemented
3#21443766C212yesmaximal, complemented
4#21443746C212yesmaximal
5C61×#1458C211yesmaximal, characteristic, complemented
6C22×#11758198C411yescharacteristic
7#16839819—21no—
8#16839697—21no—
9#16839688—21no—
10#16839653—21no—
11#16839653—21no—
12#16839634C411yescharacteristic
13#16839634C2211yescharacteristic
14#16839625C411yescharacteristic
15#16839617C411yescharacteristic
16#16839617C2211yescharacteristic
17#16839488—21no—
18#16839486C2212yes—
19C2×#117×C61C2211yescharacteristic
20C61×#431—21no—
21C61×#324—21no—
22C61×#323C2212yes—
23C24×D61C2×C411yescharacteristic
24C23×Dic61C2×C411yescharacteristic
25C2×#11758196—21no—
26C2×#11758196—21no—
27C2×C22⋊Dic61C2×C411yescharacteristic
28C2×C22⋊Dic61C2×C411yescharacteristic
29C2×#11758198—21no—
30C2×#11758198—21no—
31C2×#11758198—21no—
32C2×#11758198—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—41no—
36C2×#11758198—41no—
37C4⋊#92418—41no—
38C22⋊#92418—22no—
39#14519688—41no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43#14519684—21no—
44C23×C4×C61C2×C411yescharacteristic
45#117×C61—41no—
46#117×C61—41no—
47C2×C4⋊C4×C61D411yescharacteristic
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C22⋊C4×C61C2×C411yescharacteristic
50C2×C22⋊C4×C61C2311yescharacteristic
51C2×C42×C61D411yescharacteristic
52C2×C42×C61D411yescharacteristic
53C4⋊C8×C61—41no—
54C22⋊C8×C61—22no—
55C4×C8×C61—41no—
56C24×C61C22×C411yescharacteristic
57C23×D61—21no—
58C23×D61—22no—
59C23×D61—42no—
60C23×D61—41no—
61C23×D61—41no—
62C23×D61—81no—
63C22×C4×C61—22no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—41no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61C2×D411yescharacteristic
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—41no—
77C22×Dic61—81no—
78C2×C8×C61—42no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—41no—
83C42×C61—21no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—21no—
87C22⋊Dic61—22no—
88C22⋊Dic61—41no—
89#11758198—42no—
90#11758198—42no—
91#11758198—42no—
92#11758198—41no—
93#11758198—41no—
94#11758198—81no—
95#11758198—81no—
96#11758196—81no—
97#11758196—81no—
98C2×#92418—42no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61C2×C22⋊C411yescharacteristic
104C23×C61#13112yes—
105C22×D61—22no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—41no—
109C22×D61—42no—
110C22×D61—41no—
111C22×D61—41no—
112C22×D61—41no—
113C22×D61—42no—
114C22×D61—84no—
115C22×D61—84no—
116C22×D61—82no—
117C22×D61—82no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61#9911yescharacteristic
134C2×C4×C61#9911yescharacteristic
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—82no—
145C8×C61—82no—
146#92418—82no—
147#11546—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—22no—
151C22×C61—42no—
152C22×C61—42no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61#40811yescharacteristic
156C22×C61C2×#9911yescharacteristic
157C22×C61#41311yescharacteristic
158C2×D61—42no—
159C2×D61—42no—
160C2×D61—42no—
161C2×D61—81no—
162C2×D61—81no—
163C2×D61—84no—
164C2×D61—81no—
165C2×D61—81no—
166C2×D61—84no—
167C4×C61—21no—
168C4×C61—21no—
169C4×C61—41no—
170C4×C61—41no—
171C4×C61—41no—
172C4×C61—41no—
173C4×C61—41no—
174Dic61—81no—
175Dic61—81no—
176Dic61—81no—
177#2482—611no—
178#1464—611no—
179#1458D6111yescharacteristic, complemented
180#1326—612no—
181#1316—612no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61—42no—
185C2×C61#146811yescharacteristic
186C2×C61#212011yescharacteristic
187C2×C61#212211yescharacteristic
188D61—82no—
189C2×#117C2×D6111yescharacteristic
190C22×C22⋊C4—611no—
191#432—1221no—
192#431—21no—
193#399—1221no—
194#397—1221no—
195#388—1221no—
196#388—1221no—
197#385—611no—
198#385—611no—
199#381—611no—
200#379—611no—
201#379—611no—
202#324—21no—
203#324—1221no—
204#323—612no—
205#323C2×D6112yes—
206C61#1154611yescharacteristic, complemented, Sylow
207C25—611no—
208C23×C4—611no—
209C23×C4C4×D6111yescharacteristic
210#117—41no—
211#117—41no—
212C2×C4⋊C4—1221no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4D24411yescharacteristic
215C2×C4⋊C4C22⋊D6111yescharacteristic
216C2×C22⋊C4—611no—
217C2×C22⋊C4—611no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C22⋊C4—1221no—
221C2×C22⋊C4—1221no—
222C2×C22⋊C4—1221no—
223C2×C22⋊C4—1221no—
224C2×C22⋊C4—2441no—
225C2×C22⋊C4—2441no—
226C2×C22⋊C4C4×D6111yescharacteristic
227C2×C22⋊C4C22×D6111yescharacteristic
228C2×C42D24411yescharacteristic
229C2×C42C22⋊D6111yescharacteristic
230C4⋊C8—41no—
231C4⋊C8—2441no—
232C22⋊C8—22no—
233C22⋊C8—1222no—
234#97—2441no—
235C4×C8—41no—
236#95—1221no—
237#95—1221no—
238#95—1221no—
239#95—1221no—
240C24—1221no—
241C24—1222no—
242C24—4881no—
243C24—2441no—
244C24—2441no—
245C24—2442no—
246C24C2×C4×D6111yescharacteristic
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—1221no—
254C22×C4—4881no—
255C22×C4—2441no—
256C22×C4#1175819811yescharacteristic
257C22×C4#1175819811yescharacteristic
258C22×C4#1175819811yescharacteristic
259C22×C4#1175819811yescharacteristic
260C22×C4C2×D24411yescharacteristic
261C22×C4C2×C22⋊D6111yescharacteristic
262C2×C8—42no—
263C2×C8—2442no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—4881no—
267C4⋊C4—4881no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—1222no—
271C22⋊C4—4881no—
272C22⋊C4—4881no—
273C22⋊C4—2442no—
274C22⋊C4—2442no—
275C22⋊C4—2442no—
276C22⋊C4—2441no—
277C22⋊C4—2441no—
278C22⋊C4—2441no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23—1222no—
288C23—4882no—
289C23—4882no—
290C23—4884no—
291C23—4884no—
292C23—2441no—
293C23—2441no—
294C23—2441no—
295C23—2442no—
296C23—2442no—
297C23—2441no—
298C23—2441no—
299C23—2441no—
300C23C2×#1175819811yescharacteristic
301C23#1451978312yes—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—4882no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4#1451969111yescharacteristic
328C2×C4#1451969111yescharacteristic
329C8—82no—
330C8—4882no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—4881no—
339C22—4881no—
340C22—4884no—
341C22—4884no—
342C22—4881no—
343C22—4881no—
344C22—2442no—
345C22—2442no—
346C22—2442no—
347C22#1683972811yescharacteristic
348C22C2×#1451969111yescharacteristic
349C22#1683974411yescharacteristic
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—4881no—
358C4—4881no—
359C4—4881no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—4882no—
364C2#2144416511yescharacteristic
365C2#2144746111yescharacteristic
366C2#2144746811yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC24×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#1458C2
socleC22×C61C2×#99
prime radicalC24×D61C2×C4
cosocleC2×C22⋊C4×C61C23
hypercenter#1458D61
hypocenterC61#11546
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #21444153C2semidirect
3. #21443766C2semidirect
5. C61×#1458C2semidirect
179. #1458D61semidirect
206. C61#11546semidirect

Maximal subgroups

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

subgroupindex
#214500892
#214441532
#214437662
#214437462
C61×#14582
#1154661