Finite Group Databasebeta

← #29589641

Subgroups — #29589641

Order 15616.

412 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 83 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#21450795C211yesmaximal, characteristic
2#21444163C211yesmaximal, characteristic, complemented
3#21443752C212yesmaximal, complemented
4#21443750C212yesmaximal
5C61×#1461C211yesmaximal, characteristic, complemented
6C2×#14519775C411yescharacteristic
7C2×#14519766C411yescharacteristic
8C2×#14519765C411yescharacteristic
9#16839819—21no—
10#16839684—21no—
11#16839684—21no—
12#16839667C2211yescharacteristic
13#16839663C2211yescharacteristic
14#16839653—21no—
15#16839653—21no—
16C4×C4⋊Dic61C411yescharacteristic
17#16839488—21no—
18#16839488C2212yes—
19C2×C4×Q8×C61C2211yescharacteristic
20C61×#431—21no—
21C61×#324—21no—
22C61×#324C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C22×D244C2×C411yescharacteristic
25#14519775—42no—
26#14519766—42no—
27#14519765—41no—
28#14519765—41no—
29C2×C4⋊Dic61—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×#11758198—21no—
33C2×#11758198—21no—
34C2×#11758198—21no—
35C2×#11758198—21no—
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×C4×Dic61C2×C411yescharacteristic
39C2×C4×Dic61C2×C411yescharacteristic
40C4⋊#92418—41no—
41C22⋊#92418—22no—
42#14519688—41no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46C22×Q8×C61C2×C411yescharacteristic
47C4×Q8×C61—41no—
48C4×Q8×C61—41no—
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61D411yescharacteristic
51C2×C4⋊C4×C61C2311yescharacteristic
52C2×C42×C61C2×C411yescharacteristic
53C2×C42×C61D411yescharacteristic
54C2×C42×C61D411yescharacteristic
55C4⋊C8×C61—41no—
56C22⋊C8×C61—22no—
57C4×C8×C61—41no—
58C23×D61—21no—
59C2×Q8×C61—22no—
60C2×Q8×C61—22no—
61C2×Q8×C61—41no—
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22⋊C411yescharacteristic
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×C4×C61C2×D411yescharacteristic
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C2×D244—22no—
73C2×D244—41no—
74C2×D244—41no—
75C2×Dic122—22no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×C8×C61—42no—
79C4⋊C4×C61—41no—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—41no—
82C42×C61—21no—
83C42×C61—21no—
84C42×C61—21no—
85C42×C61—21no—
86C42×C61—21no—
87C42×C61—21no—
88#11758198—42no—
89#11758198—42no—
90#11758198—42no—
91#11758198—42no—
92#11758198—41no—
93#11758198—41no—
94#11758198—42no—
95C4⋊Dic61—42no—
96#11758196—42no—
97#11758196—42no—
98C4×Dic61—41no—
99C4×Dic61—41no—
100C2×#92418—42no—
101C23×C61C2×C22⋊C411yescharacteristic
102C22×D61—41no—
103C22×D61—41no—
104C22×D61—41no—
105C22×D61—41no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61—41no—
109Q8×C61—44no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61#9911yescharacteristic
125C2×C4×C61#9911yescharacteristic
126C2×C4×C61#10012yes—
127C2×C4×C61#13112yes—
128C2×Dic61—22no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137D244—82no—
138Dic122—82no—
139C8×C61—82no—
140#92418—82no—
141#11565—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#40811yescharacteristic
145C22×C61C2×#9911yescharacteristic
146C22×C61C2×#10011yescharacteristic
147C2×D61—81no—
148C2×D61—81no—
149C2×D61—81no—
150C2×D61—81no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61—81no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—41no—
159C4×C61—41no—
160C4×C61—41no—
161C4×C61—41no—
162Dic61—42no—
163Dic61—81no—
164Dic61—81no—
165#2545—611no—
166#1467—611no—
167#1461D6111yescharacteristic, complemented
168#1319—612no—
169#1318—612no—
170C2×C61—21no—
171C2×C61—21no—
172C2×C61#147111yescharacteristic
173C2×C61#212611yescharacteristic
174C2×C61#213311yescharacteristic
175D61—161no—
176C2×C4⋊Q8—611no—
177C2×#124—611no—
178C2×#124—611no—
179C2×C4×Q8C2×D6111yescharacteristic
180#432—1221no—
181#431—21no—
182#396—1221no—
183#396—1221no—
184#391—611no—
185#390—611no—
186#388—1221no—
187#388—1221no—
188C4×C4⋊C4—611no—
189#324—21no—
190#324—612no—
191#324—1221no—
192#324C2×D6112yes—
193C61#1156511yescharacteristic, complemented, Sylow
194C22×Q8—611no—
195C22×Q8C4×D6111yescharacteristic
196C22×D4—611no—
197C4⋊Q8—2442no—
198#124—2441no—
199#124—2441no—
200#124—2442no—
201C4×Q8—41no—
202C4×Q8—41no—
203C2×C4⋊C4—1221no—
204C2×C4⋊C4—1221no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4D24411yescharacteristic
207C2×C4⋊C4C22⋊D6111yescharacteristic
208C2×C4⋊C4C22×D6111yescharacteristic
209C2×C22⋊C4—1221no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1221no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42C4×D6111yescharacteristic
218C2×C42D24411yescharacteristic
219C2×C42C22⋊D6111yescharacteristic
220C4⋊C8—41no—
221C4⋊C8—2441no—
222C22⋊C8—22no—
223C22⋊C8—1222no—
224#97—2441no—
225C4×C8—41no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229C24—1221no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—1222no—
234C2×Q8—2441no—
235C2×Q8—2441no—
236C2×D4—1222no—
237C2×D4—2441no—
238C2×D4—2441no—
239C22×C4—1221no—
240C22×C4—1221no—
241C22×C4—1221no—
242C22×C4#1175819811yescharacteristic
243C22×C4#1175819811yescharacteristic
244C22×C4#1175819811yescharacteristic
245C22×C4#1175819811yescharacteristic
246C22×C4C2×C4×D6111yescharacteristic
247C22×C4C2×D24411yescharacteristic
248C22×C4C2×C22⋊D6111yescharacteristic
249C2×C8—42no—
250C2×C8—2442no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—2442no—
255C4⋊C4—2442no—
256C4⋊C4—2442no—
257C22⋊C4—2442no—
258C22⋊C4—2442no—
259C22⋊C4—2442no—
260C22⋊C4—2442no—
261C22⋊C4—2441no—
262C22⋊C4—2441no—
263C22⋊C4—2442no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—2441no—
271C42—2441no—
272C23—2441no—
273C23—2441no—
274C23—2441no—
275C23—2441no—
276C23—2441no—
277C23—2441no—
278C23—2441no—
279C23C2×#1175819811yescharacteristic
280Q8—44no—
281Q8—4882no—
282D4—4882no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—1222no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4#1451969111yescharacteristic
307C2×C4#1451969111yescharacteristic
308C2×C4#1451969312yes—
309C2×C4#1451978312yes—
310C8—82no—
311C8—4882no—
312C22—21no—
313C22—21no—
314C22—4881no—
315C22—4881no—
316C22—4881no—
317C22—4881no—
318C22—4881no—
319C22—4881no—
320C22—4881no—
321C22#1683972811yescharacteristic
322C22C2×#1451969111yescharacteristic
323C22C2×#1451969311yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—4881no—
333C4—4881no—
334C4—2442no—
335C2—21no—
336C2—21no—
337C2—9761no—
338C2#2144417811yescharacteristic
339C2#2144748211yescharacteristic
340C2#2144750611yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1461C2
socleC22×C61C2×#99
prime radicalC22×D244C2×C4
cosocleC2×C4⋊C4×C61C23
hypercenter#1461D61
hypocenterC61#11565
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #21444163C2semidirect
3. #21443752C2semidirect
5. C61×#1461C2semidirect
167. #1461D61semidirect
193. C61#11565semidirect

Maximal subgroups

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

subgroupindex
#214507952
#214441632
#214437522
#214437502
C61×#14612
#1156561