Finite Group Databasebeta

← #29593000

Subgroups — #29593000

Order 15616.

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

normal subgroups only

This group has 458 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#21447172C211yesmaximal, characteristic, complemented
2#21444809C211yesmaximal, characteristic, complemented
3#21444741C211yesmaximal, characteristic
4#21443819C211yesmaximal, characteristic, complemented
5#21443816C211yesmaximal, characteristic
6#21443767C211yesmaximal, characteristic, complemented
7C61×#2712C211yesmaximal, characteristic
8#16840388C2211yescharacteristic
9#16840092—21no—
10#16839913—21no—
11#16839635—21no—
12#16839529C2211yescharacteristic
13#16839528—21no—
14#16839498—21no—
15#16839498—21no—
16#16839486C2211yescharacteristic
17#16839486C2211yescharacteristic
18C2×#124×C61—21no—
19C61×#397C2211yescharacteristic
20C61×#394—21no—
21C61×#393—21no—
22C61×#392C2211yescharacteristic
23C61×#391—21no—
24C61×#388—21no—
25C61×#386—21no—
26C61×#385C2211yescharacteristic
27C2×#11758200—41no—
28#14519756—21no—
29#14519756—21no—
30#14519754—41no—
31#14519754—41no—
32C2×#11758196—41no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—21no—
35#14519708—41no—
36#14519700—41no—
37#14519698—41no—
38C22⋊#92418—21no—
39C22⋊#92418—21no—
40#14519684—41no—
41C22×Q8×C61—21no—
42C22×D4×C61—21no—
43C22×D4×C61D411yescharacteristic
44C23×C4×C61D411yescharacteristic
45#124×C61—42no—
46#124×C61—42no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C22⋊C4×C61—21no—
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2311yescharacteristic
57C2×C42×C61—21no—
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61—21no—
62#95×C61D411yescharacteristic
63#95×C61D411yescharacteristic
64C24×C61—21no—
65C24×C61C2×D411yescharacteristic
66C2×Q8×C61—42no—
67C2×Q8×C61—42no—
68C2×Q8×C61—81no—
69C2×D4×C61—41no—
70C2×D4×C61—42no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—81no—
74C2×D4×C61—81no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—41no—
81C22×C4×C61—81no—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—41no—
87C2×Dic122—21no—
88C2×Dic122—21no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—81no—
91C22⋊C4×C61—21no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—42no—
94C22⋊C4×C61—42no—
95C22⋊C4×C61—42no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C22⋊C4×C61—81no—
99C22⋊C4×C61—81no—
100C22⋊C4×C61SD1612yescomplemented
101C42×C61—42no—
102C22⋊Dic61—22no—
103C22⋊Dic61—41no—
104C22⋊Dic61—41no—
105C22⋊Dic61—41no—
106#11758200—82no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109#11758196—41no—
110#11758196—41no—
111#11758196—82no—
112C2×#92418—41no—
113C2×#92418—41no—
114C23×C61—21no—
115C23×C61—21no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61—41no—
122C23×C61—41no—
123C23×C61—41no—
124C23×C61—41no—
125C23×C61#12011yescharacteristic
126C23×C61C2×SD1611yescharacteristic
127C23×C61AGL(1,ℤ/8)11yescharacteristic
128Q8×C61—84no—
129D4×C61—42no—
130D4×C61—82no—
131D4×C61—82no—
132D4×C61—81no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
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—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—81no—
154C2×Dic61—42no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61—81no—
158C2×Dic61—81no—
159C2×Dic61—81no—
160C2×Dic61—81no—
161Dic122—81no—
162Dic122—81no—
163#92418—81no—
164#92418—81no—
165#12245—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—41no—
173C22×C61—41no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61—81no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61#45011yescharacteristic
183C22×C61#45611yescharacteristic
184C22×C61B(4,𝔽2)11yescharacteristic
185C4×C61—42no—
186C4×C61—42no—
187C4×C61—41no—
188C4×C61—41no—
189C4×C61—81no—
190C4×C61—81no—
191C4×C61—81no—
192Dic61—82no—
193Dic61—81no—
194Dic61—81no—
195#2712D6111yescharacteristic
196#2062—611no—
197#1608—611no—
198#1596—611no—
199#1344—611no—
200#1343—611no—
201#1326—611no—
202C2×C61—21no—
203C2×C61—21no—
204C2×C61—41no—
205C2×C61—41no—
206C2×C61—161no—
207C2×C61#160411yescharacteristic
208C2×C61#219811yescharacteristic
209C2×C61#219911yescharacteristic
210#543—611no—
211C2×#124—21no—
212#483—1221no—
213#450—1221no—
214#397C2×D6111yescharacteristic
215#394—21no—
216#393—21no—
217#392C2×D6111yescharacteristic
218#391—21no—
219#388—21no—
220#386—21no—
221#385—1221no—
222#385C2×D6111yescharacteristic
223#342—611no—
224#342—1221no—
225#328—1221no—
226#328—1221no—
227#323—611no—
228#323—611no—
229C61#1224511yescharacteristic, complemented, Sylow
230C22×Q8—21no—
231C22×D4—21no—
232C22×D4C22⋊D6111yescharacteristic
233C23×C4C22⋊D6111yescharacteristic
234C2×SD16—2441no—
235#124—42no—
236#124—42no—
237C22⋊Q8—1221no—
238C22⋊Q8—1221no—
239C22⋊Q8—2441no—
240C22⋊Q8—2441no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—2441no—
243C2×C4⋊C4C22⋊D6111yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—1221no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4C22⋊D6111yescharacteristic
253C2×C22⋊C4C22×D6111yescharacteristic
254C2×C42—21no—
255#106—2441no—
256#103—2441no—
257#102—2441no—
258C22⋊C8—1221no—
259C22⋊C8—1221no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—2441no—
265#95C22⋊D6111yescharacteristic
266#95C22⋊D6111yescharacteristic
267C24—21no—
268C24C2×C22⋊D6111yescharacteristic
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—81no—
272C2×Q8—1221no—
273C2×Q8—1221no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—81no—
279C2×D4—81no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—41no—
286C22×C4—81no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—2441no—
290C22×C4C2×C22⋊D6111yescharacteristic
291C22×C4C2×C22⋊D6111yescharacteristic
292SD16—4882no—
293C2×C8—2441no—
294C2×C8—2441no—
295C4⋊C4—41no—
296C4⋊C4—81no—
297C4⋊C4—1221no—
298C4⋊C4—1221no—
299C4⋊C4—4882no—
300C4⋊C4—2441no—
301C4⋊C4—2441no—
302C22⋊C4—21no—
303C22⋊C4—41no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C22⋊C4—1222no—
312C22⋊C4—2441no—
313C22⋊C4—2441no—
314C22⋊C4—2441no—
315C22⋊C4#1175820012yescomplemented
316C42—42no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#1451974611yescharacteristic
329C23C2×#1175820011yescharacteristic
330C23#1451980311yescharacteristic
331Q8—84no—
332Q8—4881no—
333Q8—4881no—
334D4—42no—
335D4—82no—
336D4—82no—
337D4—81no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
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—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—4881no—
360C2×C4—4881no—
361C2×C4—4881no—
362C2×C4—4881no—
363C2×C4—2442no—
364C2×C4—2441no—
365C2×C4—2441no—
366C8—4881no—
367C8—4881no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#1683991311yescharacteristic
385C22#1683994511yescharacteristic
386C22#1683994911yescharacteristic
387C4—42no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—4882no—
395C4—4881no—
396C4—4881no—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—161no—
402C2#2144478811yescharacteristic
403C2#2144781511yescharacteristic
404C2#2144782011yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C22⋊C4×C61C23
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2712C2
socleC22×C61B(4,𝔽2)
prime radicalC61×#392C22
hypercenter#2712D61
hypocenterC61#12245
LCSG ▷ C2×C22⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2712

Complements

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

NH ≅ G/Ntype
1. #21447172C2semidirect
2. #21444809C2semidirect
4. #21443819C2semidirect
6. #21443767C2semidirect
100. C22⋊C4×C61SD16semidirect
229. C61#12245semidirect
315. C22⋊C4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214471722
#214448092
#214447412
#214438192
#214438162
#214437672
C61×#27122
#1224561