Finite Group Databasebeta

← #29577734

Subgroups — #29577734

Order 15616.

512 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 93 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#21450305C211yesmaximal, characteristic
2#21448420C211yesmaximal, characteristic
3#21445897C211yesmaximal, characteristic
4#21444352C211yesmaximal, characteristic
5#21443673C211yesmaximal, characteristic
6#21443671C211yesmaximal, characteristic
7C61×#1511C211yesmaximal, characteristic
8C22×C4⋊Dic61C2211yescharacteristic
9#16839726C2211yescharacteristic
10#16839701—22no—
11#16839669C412yescomplemented
12#16839647—22no—
13#16839645—22no—
14#16839627C412yescomplemented
15#16839625C2211yescharacteristic
16#16839623—21no—
17#16839623—21no—
18#16839620—21no—
19C22⋊C4×Dic61—21no—
20C4×C22⋊Dic61C2211yescharacteristic
21#16839530—22no—
22#16839499—22no—
23C2×C22⋊Q8×C61C2211yescharacteristic
24C61×#407C2211yescharacteristic
25C61×#381C2211yescharacteristic
26C61×#328—22no—
27C23×Dic61D411yescharacteristic
28C23×Dic61D411yescharacteristic
29C2×#11758194—21no—
30C2×C4⋊Dic61—21no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61C2×C412yes—
35C2×C4⋊Dic61D412yes—
36C2×#11758196—22no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C4×Dic61—22no—
40C2×C4×Dic61C2×C412yes—
41C22⋊#92418—21no—
42C22⋊#92418—21no—
43#14519684—22no—
44#14519684—22no—
45#14519684—21no—
46#14519684—21no—
47#14519684—22no—
48#14519684—22no—
49#14519684—22no—
50#14519684—22no—
51#14519684D412yes—
52C22×Q8×C61C2×C411yescharacteristic
53C23×C4×C61C2311yescharacteristic
54C2×M16×C61—21no—
55C22⋊Q8×C61—21no—
56C22⋊Q8×C61—21no—
57C22⋊Q8×C61—21no—
58C22⋊Q8×C61—21no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C22⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C22⋊C8×C61—21no—
64C22⋊C8×C61—21no—
65#95×C61—21no—
66#95×C61D412yes—
67C24×C61C2×D411yescharacteristic
68C2×Q8×C61—21no—
69C2×Q8×C61—21no—
70C2×Q8×C61—42no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—22no—
74C22×C4×C61—42no—
75C22×C4×C61C22×C411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61#4111yescharacteristic
80C22×C4×C61#4111yescharacteristic
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—22no—
86C22×Dic61—22no—
87C22×Dic61—42no—
88C22×Dic61—42no—
89C22×Dic61C22⋊C412yes—
90C22×Dic61C22⋊C412yes—
91M16×C61—42no—
92C2×C8×C61—41no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—41no—
96C4⋊C4×C61—41no—
97C22⋊C4×C61—42no—
98C22⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C22⋊Dic61—42no—
101C22⋊Dic61—42no—
102C4⋊Dic61—22no—
103C4⋊Dic61—22no—
104C4⋊Dic61—22no—
105C4⋊Dic61—22no—
106#11758196—82no—
107C4×Dic61—42no—
108C4×Dic61—82no—
109#11758194—42no—
110C2×#92418—41no—
111C23×C61—22no—
112C23×C61—22no—
113C23×C61—41no—
114C23×C61C2×C22⋊C411yescharacteristic
115C23×C61#12011yescharacteristic
116C23×C61#12311yescharacteristic
117Q8×C61—81no—
118Q8×C61—81no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61C4×D411yescharacteristic
133C2×C4×C61C4×D411yescharacteristic
134C2×C4×C61#12111yescharacteristic
135C2×C4×C61#12111yescharacteristic
136C2×Dic61—22no—
137C2×Dic61—22no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—81no—
149C2×Dic61—81no—
150C8×C61—81no—
151#92418—81no—
152#9460—611nomaximal, Sylow
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—44no—
161C22×C61#38611yescharacteristic
162C22×C61#41012yes—
163C22×C61#45512yes—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—82no—
169C4×C61—81no—
170C4×C61—81no—
171Dic61—42no—
172Dic61—42no—
173Dic61—82no—
174Dic61—82no—
175#2503—611no—
176#2313—611no—
177#1821—611no—
178#1511—611no—
179#1511D6111yescharacteristic
180#1284—611no—
181#1284—611no—
182C2×C61—22no—
183C2×C61—22no—
184C2×C61—41no—
185C2×C61#188511yescharacteristic
186C2×C61#189911yescharacteristic
187C2×C61#191211yescharacteristic
188C2×C22⋊Q8C2×D6111yescharacteristic
189C22×C4⋊C4—611no—
190#407—611no—
191#407C2×D6111yescharacteristic
192#400—1222no—
193#391—612no—
194#387—1222no—
195#387—1222no—
196#382—612no—
197#381—611no—
198#381—1221no—
199#381—1221no—
200#381C2×D6111yescharacteristic
201#380—1221no—
202C4×C22⋊C4—611no—
203C4×C22⋊C4—1221no—
204#343—1222no—
205#328—22no—
206#328—1222no—
207C61#946011yescharacteristic, complemented, Sylow
208C22×Q8C4×D6111yescharacteristic
209C23×C4—611no—
210C23×C4—611no—
211C23×C4C22×D6111yescharacteristic
212C2×M16—21no—
213C2×M16—1221no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—612no—
220C2×C4⋊C4—612no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—1221no—
223C2×C4⋊C4—1221no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4—1222no—
226C2×C4⋊C4C4×D6111yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—1221no—
230C2×C22⋊C4—1221no—
231C2×C42—612no—
232C2×C42—1222no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235C22⋊C8—1221no—
236C22⋊C8—1221no—
237#95—21no—
238#95—612no—
239#95—1222no—
240#95—1222no—
241#95—1222no—
242#95—1222no—
243#95—1222no—
244#95—1222no—
245#95—1221no—
246#95—1221no—
247#95C22⋊D6112yes—
248C24D4×D6111yescharacteristic
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—42no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—612no—
257C22×C4—612no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1222no—
263C22×C4—1222no—
264C22×C4—2442no—
265C22×C4—2442no—
266C22×C4C2×C4×D6111yescharacteristic
267C22×C4#1175822211yescharacteristic
268C22×C4D4×D6111yescharacteristic
269C22×C4D4×D6111yescharacteristic
270C22×C4#1175822411yescharacteristic
271C22×C4C2×C22⋊D6111yescharacteristic
272M16—42no—
273M16—2442no—
274C2×C8—41no—
275C2×C8—2441no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—1222no—
281C4⋊C4—1222no—
282C4⋊C4—1222no—
283C4⋊C4—1222no—
284C4⋊C4—4882no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—2442no—
289C22⋊C4—2442no—
290C42—4882no—
291C42—2442no—
292C23—22no—
293C23—22no—
294C23—41no—
295C23C22⋊C4×D6111yescharacteristic
296C23#1451974511yescharacteristic
297C23#1451975911yescharacteristic
298Q8—81no—
299Q8—81no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—1222no—
314C2×C4—1222no—
315C2×C4—4881no—
316C2×C4—4881no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4—2442no—
320C2×C4—2442no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4#1451973711yescharacteristic
328C2×C4C4×C22⋊D6111yescharacteristic
329C2×C4#1451974811yescharacteristic
330C2×C4#1451975011yescharacteristic
331C8—81no—
332C8—4881no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—44no—
341C22#1683963811yescharacteristic
342C22#1683973412yes—
343C22#1683993812yes—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—82no—
349C4—81no—
350C4—81no—
351C4—4882no—
352C4—4882no—
353C4—2442no—
354C4—2442no—
355C2—22no—
356C2—22no—
357C2—41no—
358C2#2144620411yescharacteristic
359C2#2144627411yescharacteristic
360C2#2144635311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1511C2
socleC22×C61#386
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1511D61
hypocenterC61#9460
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1511

Complements

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

NH ≅ G/Ntype
11. #16839669C4semidirect
14. #16839627C4semidirect
207. C61#9460semidirect

Maximal subgroups

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

subgroupindex
#214503052
#214484202
#214458972
#214443522
#214436732
#214436712
C61×#15112
#946061