Finite Group Databasebeta

← #29576957

Subgroups — #29576957

Order 15616.

482 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 103 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#21455941C211yesmaximal, characteristic
2#21446737C211yesmaximal, characteristic
3#21444326C211yesmaximal, characteristic
4#21444266C211yesmaximal, characteristic, complemented
5#21443712C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#2320C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840075—21no—
10#16840057C412yescomplemented
11#16839962C412yescomplemented
12#16839901—21no—
13#16839860—22no—
14#16839819C2211yescharacteristic
15#16839819C2211yescharacteristic
16C2×#14519705—22no—
17C4×#11758194C2211yescharacteristic
18#16839495—21no—
19#16839495—21no—
20Q8⋊#92418—22no—
21C2×C4×Q8×C61C2211yescharacteristic
22C61×#391—21no—
23C61×#389—21no—
24C61×#387—22no—
25C61×#383—21no—
26C61×#382—21no—
27C4×C4⋊C4×C61C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C2×#11758222—21no—
30C2×#11758201—22no—
31C2×#11758194—21no—
32#14519777C2×C411yescharacteristic
33#14519774C2×C411yescharacteristic
34#14519765—21no—
35#14519751—21no—
36#14519751—21no—
37C4×D244—21no—
38C4×D244—21no—
39#14519710—22no—
40C4⋊#92418—21no—
41C4⋊#92418C2×C412yes—
42#14519705—44no—
43#14519701—22no—
44#14519697—22no—
45C22⋊#92418—21no—
46C22⋊#92418—21no—
47#14519688—21no—
48#14519688—21no—
49#14519688—21no—
50C4×#92418—21no—
51C22×Q8×C61D411yescharacteristic
52C4×Q8×C61—21no—
53C4×Q8×C61—21no—
54C4×Q8×C61C2×C412yes—
55C4×Q8×C61D412yes—
56C2×C4⋊C4×C61—21no—
57C2×C4⋊C4×C61—21no—
58C2×C4⋊C4×C61—22no—
59C2×C4⋊C4×C61D411yescharacteristic
60C2×C42×C61—21no—
61C2×C42×C61—21no—
62C2×C42×C61D412yes—
63C2×C42×C61C2311yescharacteristic
64#95×C61—22no—
65#95×C61—22no—
66#95×C61—22no—
67#95×C61—21no—
68#95×C61—21no—
69#95×C61D411yescharacteristic
70#95×C61D411yescharacteristic
71C2×Q8×C61—21no—
72C2×Q8×C61—42no—
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×C22⋊D61—41no—
83#11758222—42no—
84#11758222—42no—
85C2×D244—21no—
86C2×D244—21no—
87C2×D244—21no—
88C2×C4×D61—41no—
89C2×Dic122—21no—
90C4⋊C4×C61—22no—
91C4⋊C4×C61—22no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—42no—
94C4⋊C4×C61—82no—
95C4⋊C4×C61C22⋊C412yes—
96C42×C61—22no—
97C42×C61—22no—
98C42×C61—42no—
99C42×C61—42no—
100C42×C61C22×C411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61#4111yescharacteristic
103C42×C61#4111yescharacteristic
104#11758201—44no—
105#11758198—41no—
106#11758198—41no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109#11758194—22no—
110#11758194—41no—
111C2×#92418—22no—
112C2×#92418—41no—
113C2×#92418—41no—
114C23×C61#12011yescharacteristic
115C22×D61—41no—
116Q8×C61—22no—
117Q8×C61—22no—
118Q8×C61—42no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—42no—
131C2×C4×C61—41no—
132C2×C4×C61—42no—
133C2×C4×C61—41no—
134C2×C4×C61—42no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12311yescharacteristic
143C22⋊D61—81no—
144C22⋊D61—81no—
145C2×Dic61—41no—
146D244—42no—
147D244—42no—
148D244—81no—
149C4×D61—82no—
150Dic122—42no—
151#92418—42no—
152#92418—42no—
153#92418—81no—
154#9330—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61#38611yescharacteristic
158C22×C61#45311yescharacteristic
159C22×C61#45411yescharacteristic
160C2×D61—81no—
161C2×D61—81no—
162C2×D61—81no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—42no—
169C4×C61—42no—
170C4×C61—81no—
171C4×C61—82no—
172C4×C61—81no—
173C4×C61#42112yes—
174Dic61—81no—
175#3112—611no—
176#2320D6111yescharacteristic, complemented
177#1980—611no—
178#1505—611no—
179#1491—611no—
180#1300—611no—
181#1285—611no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61#188311yescharacteristic
185C2×C61#189711yescharacteristic
186C2×C61#190511yescharacteristic
187D61—161no—
188#532—611no—
189C2×C4×Q8C2×D6111yescharacteristic
190#480—1221no—
191#477—612no—
192#460—612no—
193#448—1221no—
194#440—1222no—
195#432—611no—
196#432—611no—
197C2×#104—1222no—
198C4×M16—611no—
199#391—21no—
200#389—21no—
201#387—22no—
202#383—21no—
203#382—21no—
204C4×C4⋊C4C2×D6111yescharacteristic
205#376C2×D6111yescharacteristic
206#327—1221no—
207#327—1221no—
208Q8⋊C8—1222no—
209C61#933011yescharacteristic, complemented, Sylow
210C2×#41—1221no—
211C22×Q8C22⋊D6111yescharacteristic
212C2×SD16—1222no—
213C2×M16—1221no—
214C4⋊Q8—611no—
215#127—611no—
216#124—1221no—
217#121—1221no—
218#121—1221no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8C4×D6112yes—
222C4×Q8D24412yes—
223C4×D4—1221no—
224C4×D4—1221no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4C22⋊D6111yescharacteristic
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C22⋊D6112yes—
232C2×C42C22×D6111yescharacteristic
233#107—1222no—
234C4⋊C8—612no—
235C4⋊C8—1221no—
236#104—2444no—
237#103—1222no—
238#102—1222no—
239C22⋊C8—1221no—
240C22⋊C8—1221no—
241#97—1221no—
242#97—1221no—
243#97—1221no—
244C4×C8—1221no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95C22⋊D6111yescharacteristic
251#95C22⋊D6111yescharacteristic
252#41—2442no—
253#41—2442no—
254C2×Q8—21no—
255C2×Q8—42no—
256C2×Q8—1221no—
257C2×Q8#1175819812yes—
258C2×D4—1221no—
259C2×D4—1221no—
260C2×D4—1221no—
261C2×D4—2441no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—2441no—
268C22×C4C2×C22⋊D6111yescharacteristic
269C22×C4C2×C22⋊D6111yescharacteristic
270C22×C4C2×C22⋊D6111yescharacteristic
271SD16—2444no—
272M16—1222no—
273M16—2441no—
274C2×C8—1222no—
275C2×C8—2441no—
276C2×C8—2441no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—42no—
281C4⋊C4—82no—
282C4⋊C4—1221no—
283C4⋊C4—1221no—
284C4⋊C4#1175819812yes—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C42—22no—
288C42—22no—
289C42—42no—
290C42—42no—
291C42C2×C4×D6111yescharacteristic
292C42C2×D24411yescharacteristic
293C42#1175822211yescharacteristic
294C42#1175822211yescharacteristic
295C23—2441no—
296C23#1451974611yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—42no—
300Q8—2442no—
301D4—4881no—
302D4—4881no—
303D4—4881no—
304D4—2442no—
305D4—2442no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—4882no—
325C2×C4—2441no—
326C2×C4C2×#1175819811yescharacteristic
327C2×C4C4×C22⋊D6111yescharacteristic
328C2×C4C4×C22⋊D6111yescharacteristic
329C2×C4#1451975111yescharacteristic
330C2×C4#1451975111yescharacteristic
331C2×C4#1451976011yescharacteristic
332C8—4881no—
333C8—2442no—
334C8—2442no—
335C22—21no—
336C22—21no—
337C22—4881no—
338C22—4881no—
339C22—4881no—
340C22#1683964111yescharacteristic
341C22#1683993011yescharacteristic
342C22#1683993511yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—81no—
351C4—82no—
352C4—81no—
353C4—4881no—
354C4#1683978012yes—
355C2—21no—
356C2—21no—
357C2—9761no—
358C2#2144619511yescharacteristic
359C2#2144626711yescharacteristic
360C2#2144631311yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2320C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#2320D61
hypocenterC61#9330
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2320

Complements

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

NH ≅ G/Ntype
4. #21444266C2semidirect
5. #21443712C2semidirect
6. #21443675C2semidirect
7. C61×#2320C2semidirect
10. #16840057C4semidirect
11. #16839962C4semidirect
176. #2320D61semidirect
209. C61#9330semidirect

Maximal subgroups

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

subgroupindex
#214559412
#214467372
#214443262
#214442662
#214437122
#214436752
C61×#23202
#933061