Finite Group Databasebeta

← #29567331

Subgroups — #29567331

Order 15616.

482 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 71 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#21444146C211yesmaximal, characteristic
2#21444068C211yesmaximal, characteristic
3C61×#1462C211yesmaximal, characteristic
4C22×C22⋊Dic61C411yescharacteristic
5C22×C22⋊Dic61C411yescharacteristic
6#16839726—21no—
7C2×#14519684—21no—
8C2×#14519684—21no—
9#16839486—21no—
10#16839486C412yescomplemented
11C22×C22⋊C4×C61C2211yescharacteristic
12C61×#407—21no—
13C61×#323—21no—
14C61×#323C412yescomplemented
15C23×Dic61—21no—
16C23×Dic61—21no—
17C2×#11758194—41no—
18C2×C22⋊Dic61—41no—
19C2×C22⋊Dic61—41no—
20C2×C22⋊Dic61—42no—
21C2×C22⋊Dic61—42no—
22C22⋊#92418—22no—
23C22⋊#92418—41no—
24#14519684—42no—
25#14519684—42no—
26C25×C61C2×C411yescharacteristic
27C23×C4×C61D411yescharacteristic
28C23×C4×C61D411yescharacteristic
29C2×M16×C61—41no—
30C2×C22⋊C4×C61—41no—
31C2×C22⋊C4×C61—41no—
32C2×C22⋊C4×C61C2×C411yescharacteristic
33C2×C22⋊C4×C61C2×C411yescharacteristic
34C2×C22⋊C4×C61D411yescharacteristic
35C2×C22⋊C4×C61Q811yescharacteristic
36C22⋊C8×C61—22no—
37C22⋊C8×C61—41no—
38C24×C61—21no—
39C24×C61—21no—
40C24×C61—41no—
41C24×C61—41no—
42C24×C61—41no—
43C24×C61—41no—
44C24×C61—41no—
45C24×C61—41no—
46C24×C61C4211yescharacteristic
47C24×C61C22⋊C411yescharacteristic
48C24×C61C4⋊C411yescharacteristic
49C22×C4×C61—21no—
50C22×C4×C61—21no—
51C22×C4×C61—21no—
52C22×C4×C61—21no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61—41no—
56C22×C4×C61—41no—
57C22×C4×C61C22⋊C411yescharacteristic
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C4⋊C411yescharacteristic
60C22×C4×C61C4⋊C411yescharacteristic
61C22×Dic61—22no—
62C22×Dic61—22no—
63C22×Dic61—41no—
64C22×Dic61—41no—
65C22×Dic61—41no—
66C22×Dic61—41no—
67C22×Dic61—41no—
68C22×Dic61—41no—
69C22×Dic61—41no—
70C22×Dic61—41no—
71C22×Dic61—41no—
72C22×Dic61—41no—
73C22×Dic61—41no—
74C22×Dic61—41no—
75M16×C61—81no—
76M16×C61—81no—
77C2×C8×C61—42no—
78C22⋊C4×C61—41no—
79C22⋊C4×C61—41no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C22⋊Dic61—42no—
83C22⋊Dic61—42no—
84C22⋊Dic61—42no—
85C22⋊Dic61—42no—
86#11758194—81no—
87#11758194—81no—
88C2×#92418—42no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61—42no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—81no—
114C23×C61—81no—
115C23×C61—81no—
116C23×C61—81no—
117C23×C61—81no—
118C23×C61—81no—
119C23×C61—81no—
120C23×C61—81no—
121C23×C61#9511yescharacteristic
122C23×C61#9911yescharacteristic
123C23×C61#9911yescharacteristic
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—81no—
137C2×C4×C61—81no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—81no—
145C2×Dic61—81no—
146C2×Dic61—81no—
147C2×Dic61—81no—
148C2×Dic61—81no—
149C2×Dic61—81no—
150C2×Dic61—81no—
151C2×Dic61—81no—
152C8×C61—82no—
153#92418—82no—
154#7324—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—21no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—41no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—82no—
180C22×C61—82no—
181C22×C61—82no—
182C22×C61—82no—
183C22×C61#34211yescharacteristic
184C22×C61#34411yescharacteristic
185C22×C61#34411yescharacteristic
186C22×C61AΣL(1,𝔽16)11yescharacteristic
187C22×C61AΣL(1,𝔽16)11yescharacteristic
188C22×C61#35211yescharacteristic
189C22×C61#35211yescharacteristic
190C4×C61—41no—
191C4×C61—41no—
192C4×C61—41no—
193C4×C61—41no—
194Dic61—82no—
195Dic61—82no—
196#1462—611no—
197#1462D6111yescharacteristic
198#1440—611no—
199C2×C61—21no—
200C2×C61—21no—
201C2×C61—21no—
202C2×C61—21no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—81no—
206C2×C61—81no—
207C2×C61#130611yescharacteristic
208C2×C61#139011yescharacteristic
209C2×C61#139011yescharacteristic
210C22×C22⋊C4—611no—
211C22×C22⋊C4—611no—
212C22×C22⋊C4C2×D6111yescharacteristic
213#407—21no—
214#407—1221no—
215C2×#95—1221no—
216C2×#95—1221no—
217#323—21no—
218#323—612no—
219#323—1221no—
220#323Dic6112yescomplemented
221C61#732411yescharacteristic, complemented, Sylow
222C25C4×D6111yescharacteristic
223C23×C4—1221no—
224C23×C4—1221no—
225C23×C4D24411yescharacteristic
226C23×C4C22⋊D6111yescharacteristic
227C2×M16—41no—
228C2×M16—2441no—
229C2×C22⋊C4—41no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—2441no—
232C2×C22⋊C4—2441no—
233C2×C22⋊C4—2442no—
234C2×C22⋊C4—2442no—
235C2×C22⋊C4Dic12211yescharacteristic
236C2×C22⋊C4C4×D6111yescharacteristic
237C2×C22⋊C4C2×Dic6111yescharacteristic
238C2×C22⋊C4C22⋊D6111yescharacteristic
239C22⋊C8—22no—
240C22⋊C8—41no—
241C22⋊C8—1222no—
242C22⋊C8—2441no—
243#95—2442no—
244#95—2442no—
245C24—21no—
246C24—21no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—41no—
252C24—41no—
253C24C4×Dic6111yescharacteristic
254C24#1175819611yescharacteristic
255C24#1175819811yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—1222no—
265C22×C4—1222no—
266C22×C4—2441no—
267C22×C4—2441no—
268C22×C4—2441no—
269C22×C4—2441no—
270C22×C4—2441no—
271C22×C4—2441no—
272C22×C4—2441no—
273C22×C4—2441no—
274C22×C4—2441no—
275C22×C4—2441no—
276C22×C4—2441no—
277C22×C4—2441no—
278C22×C4#1175819611yescharacteristic
279C22×C4C4⋊Dic6111yescharacteristic
280C22×C4#1175819811yescharacteristic
281C22×C4C22⋊Dic6111yescharacteristic
282M16—81no—
283M16—81no—
284M16—4881no—
285M16—4881no—
286C2×C8—42no—
287C2×C8—2442no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—2442no—
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C22⋊C4—2442no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23#1451968411yescharacteristic
329C23#1451969111yescharacteristic
330C23#1451969111yescharacteristic
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—4881no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—4881no—
349C2×C4—4881no—
350C2×C4—4881no—
351C2×C4—4881no—
352C2×C4—4881no—
353C2×C4—2442no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4—2442no—
357C2×C4—2442no—
358C2×C4—2442no—
359C8—82no—
360C8—4882no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22#1683952811yescharacteristic
390C22#1683953311yescharacteristic
391C22#1683953411yescharacteristic
392C22#1683954811yescharacteristic
393C22#1683955011yescharacteristic
394C22#1683955111yescharacteristic
395C22#1683955311yescharacteristic
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—4882no—
401C4—4882no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—81no—
409C2—81no—
410C2#2144372811yescharacteristic
411C2#2144393611yescharacteristic
412C2#2144393811yescharacteristic
subgroupG/H
centerC22#16839528
commutatorC24×C61C42
FrattiniC22×C22⋊C4C2×D61
FittingC61×#1462C2
socleC22×C61#342
prime radicalC25×C61C2×C4
cosocleC22×C22⋊C4×C61C22
hypercenter#1462D61
hypocenterC61#7324
LCSG ▷ C24×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
10. #16839486C4semidirect
14. C61×#323C4semidirect
220. #323Dic61semidirect
221. C61#7324semidirect

Maximal subgroups

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

subgroupindex
#214441462
#214440682
C61×#14622
#732461