Finite Group Databasebeta

← #29592564

Subgroups — #29592564

Order 15616.

432 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 63 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#21447070C211yesmaximal, characteristic
2#21444770C211yesmaximal, characteristic
3#21444770C211yesmaximal, characteristic
4#21443840C211yesmaximal, characteristic, complemented
5#21443840C211yesmaximal, characteristic, complemented
6#21443767C211yesmaximal, characteristic, complemented
7C61×#2763C211yesmaximal, characteristic, complemented
8#16840324C2211yescharacteristic
9#16839901—21no—
10#16839901—21no—
11C2×#14519691—21no—
12#16839641—21no—
13#16839529C2211yescharacteristic
14#16839495—21no—
15#16839495—21no—
16#16839486C2211yescharacteristic
17#16839486C2211yescharacteristic
18C61×#400—21no—
19C61×#395—21no—
20C61×#393C2211yescharacteristic
21C61×#393C2211yescharacteristic
22C61×#391—21no—
23C61×#388—21no—
24C61×#385—21no—
25C61×#385C2211yescharacteristic
26C61×#382—21no—
27C22×C22⋊D61—21no—
28C2×#11758201—41no—
29C2×#11758201—41no—
30#14519761—41no—
31#14519751—21no—
32#14519751—21no—
33#14519747—41no—
34C2×C22⋊Dic61—21no—
35C2×#11758198—41no—
36#14519701—41no—
37#14519701—41no—
38#14519691—42no—
39C22⋊#92418—21no—
40C22⋊#92418—21no—
41#14519684—41no—
42C22×Q8×C61D411yescharacteristic
43C22×Q8×C61D411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61—21no—
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61D411yescharacteristic
52C2×C22⋊C4×C61C2311yescharacteristic
53C2×C42×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61D411yescharacteristic
62#95×C61D411yescharacteristic
63C24×C61C2×D411yescharacteristic
64C23×D61—21no—
65C2×Q8×C61—42no—
66C2×Q8×C61—41no—
67C2×Q8×C61—41no—
68C2×Q8×C61—81no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—21no—
75C22×C4×C61—41no—
76C22×C4×C61—81no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C2×C22⋊D61—41no—
80C2×C22⋊D61—41no—
81C2×C22⋊D61—41no—
82C2×C22⋊D61—41no—
83C2×C22⋊D61—41no—
84C2×C22⋊D61—41no—
85C22×Dic61—21no—
86C22×Dic61—41no—
87C2×D244—21no—
88C2×D244—21no—
89C4⋊C4×C61—81no—
90C4⋊C4×C61—81no—
91C4⋊C4×C61—81no—
92C4⋊C4×C61—81no—
93C22⋊C4×C61—21no—
94C22⋊C4×C61—41no—
95C22⋊C4×C61—42no—
96C22⋊C4×C61—81no—
97C22⋊C4×C61SD1612yescomplemented
98C42×C61—42no—
99C22⋊Dic61—41no—
100C22⋊Dic61—41no—
101#11758201—82no—
102#11758201—82no—
103#11758198—41no—
104#11758198—41no—
105#11758198—82no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C2×#92418—41no—
109C2×#92418—41no—
110C23×C61—21no—
111C23×C61—21no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61#12011yescharacteristic
115C23×C61C2×SD1611yescharacteristic
116C23×C61#13711yescharacteristic
117C22×D61—41no—
118C22×D61—41no—
119C22×D61—41no—
120C22×D61—81no—
121C22×D61—81no—
122Q8×C61—42no—
123Q8×C61—42no—
124Q8×C61—81no—
125Q8×C61—81no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
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—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—81no—
150C22⋊D61—81no—
151C22⋊D61—81no—
152C22⋊D61—81no—
153C22⋊D61—81no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—81no—
158C2×Dic61—81no—
159C2×Dic61—81no—
160D244—81no—
161D244—81no—
162#92418—81no—
163#92418—81no—
164#12177—611nomaximal, Sylow
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—81no—
174C22×C61#44811yescharacteristic
175C22×C61#45411yescharacteristic
176C22×C61B(4,𝔽2)11yescharacteristic
177C2×D61—82no—
178C2×D61—81no—
179C2×D61—81no—
180C2×D61—81no—
181C2×D61—161no—
182C4×C61—42no—
183C4×C61—41no—
184C4×C61—42no—
185C4×C61—41no—
186C4×C61—81no—
187C4×C61—81no—
188C4×C61—81no—
189C4×C61—81no—
190Dic61—81no—
191Dic61—161no—
192#2763D6111yescharacteristic, complemented
193#2044—611no—
194#1601—611no—
195#1601—611no—
196#1351—611no—
197#1351—611no—
198#1326—611no—
199C2×C61—21no—
200C2×C61—21no—
201C2×C61—41no—
202C2×C61—41no—
203C2×C61#159911yescharacteristic
204C2×C61#219711yescharacteristic
205C2×C61#219911yescharacteristic
206D61—161no—
207#534—611no—
208#448—1221no—
209#448—1221no—
210C2×#99—1221no—
211#400—21no—
212#395—21no—
213#393C2×D6111yescharacteristic
214#393C2×D6111yescharacteristic
215#391—21no—
216#388—21no—
217#386—1221no—
218#385—21no—
219#385C2×D6111yescharacteristic
220#382—21no—
221#342—611no—
222#327—1221no—
223#327—1221no—
224#323—611no—
225#323—611no—
226C61#1217711yescharacteristic, complemented, Sylow
227C22×Q8C22⋊D6111yescharacteristic
228C22×Q8C22⋊D6111yescharacteristic
229C22×D4—1221no—
230C23×C4C22⋊D6111yescharacteristic
231C2×SD16—2441no—
232C2×SD16—2441no—
233#123—2441no—
234#121—1221no—
235#121—1221no—
236#120—2441no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—1221no—
244C2×C22⋊C4—2441no—
245C2×C22⋊C4C22⋊D6111yescharacteristic
246C2×C22⋊C4C22×D6111yescharacteristic
247C2×C42—21no—
248#103—2441no—
249#103—2441no—
250#99—2442no—
251C22⋊C8—1221no—
252C22⋊C8—1221no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—2441no—
261#95C22⋊D6111yescharacteristic
262#95C22⋊D6111yescharacteristic
263C24—1221no—
264C24C2×C22⋊D6111yescharacteristic
265C2×Q8—42no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—81no—
269C2×D4—1221no—
270C2×D4—1221no—
271C2×D4—2441no—
272C2×D4—2441no—
273C2×D4—2441no—
274C2×D4—2441no—
275C2×D4—2441no—
276C2×D4—2441no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—81no—
285C22×C4—1221no—
286C22×C4—2441no—
287C22×C4C2×C22⋊D6111yescharacteristic
288C22×C4C2×C22⋊D6111yescharacteristic
289SD16—4882no—
290SD16—4882no—
291C2×C8—2441no—
292C2×C8—2441no—
293C4⋊C4—81no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—1221no—
298C4⋊C4—1221no—
299C22⋊C4—21no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—81no—
303C22⋊C4—4882no—
304C22⋊C4—2441no—
305C22⋊C4—2441no—
306C22⋊C4—2441no—
307C22⋊C4—2441no—
308C22⋊C4#1175820112yescomplemented
309C42—42no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—4881no—
315C23—4881no—
316C23—2441no—
317C23—2441no—
318C23—2441no—
319C23#1451974611yescharacteristic
320C23C2×#1175820111yescharacteristic
321C23#1451980811yescharacteristic
322Q8—42no—
323Q8—42no—
324Q8—81no—
325Q8—81no—
326D4—4881no—
327D4—4881no—
328D4—4881no—
329D4—4881no—
330D4—4881no—
331D4—4881no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—4881no—
357C2×C4—4881no—
358C2×C4—4881no—
359C2×C4—2441no—
360C2×C4—2441no—
361C2×C4—2441no—
362C8—4881no—
363C8—4881no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—81no—
373C22—9761no—
374C22—4882no—
375C22—4881no—
376C22—4881no—
377C22—4881no—
378C22#1683990111yescharacteristic
379C22#1683993511yescharacteristic
380C22#1683994911yescharacteristic
381C4—42no—
382C4—41no—
383C4—42no—
384C4—41no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—9761no—
390C4—4881no—
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—9761no—
396C2#2144475911yescharacteristic
397C2#2144780911yescharacteristic
398C2#2144782011yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2763C2
socleC22×C61B(4,𝔽2)
prime radical#16840324C22
hypercenter#2763D61
hypocenterC61#12177
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2763

Complements

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

NH ≅ G/Ntype
4. #21443840C2semidirect
5. #21443840C2semidirect
6. #21443767C2semidirect
7. C61×#2763C2semidirect
97. C22⋊C4×C61SD16semidirect
192. #2763D61semidirect
226. C61#12177semidirect
308. C22⋊C4#11758201semidirect

Maximal subgroups

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

subgroupindex
#214470702
#214447702
#214438402
#214437672
C61×#27632
#1217761