Finite Group Databasebeta

← #29654326

Subgroups — #29654326

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457021C212yesmaximal, complemented
2#21451517C211yesmaximal, characteristic
3#21444861C212yesmaximal, complemented
4#21444844C212yesmaximal
5#21444827C212yesmaximal, complemented
6#21444810C212yesmaximal
7#21444331C212yesmaximal, complemented
8#21444153C211yesmaximal, characteristic
9C61×#3458C211yesmaximal, characteristic
10C2×#14519769C2211yescharacteristic
11#16840168—22no—
12#16840130—22no—
13#16840095C2214yes—
14#16840090C2214yes—
15#16840069—24no—
16#16839819C2212yes—
17#16839701C2211yescharacteristic
18#16839698C2211yescharacteristic
19#16839692C2211yescharacteristic
20#16839687C2211yescharacteristic
21#16839657—21no—
22#16839656—21no—
23#16839646—21no—
24#16839635—21no—
25#16839608C2211yescharacteristic
26#16839499C2214yes—
27#16839499C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C61×#544—22no—
31C61×#544—22no—
32C61×#544—21no—
33C61×#544C2212yes—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—24no—
37C61×#533—21no—
38C61×#533C2212yes—
39C2×#117×C61—21no—
40C2×#117×C61C2211yescharacteristic
41#14519769—22no—
42#14519769—22no—
43C2×C4⋊Dic61C2311yescharacteristic
44C2×C4⋊Dic61C2311yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C4×Dic61—21no—
52#14519710—24no—
53#14519708—24no—
54C4⋊#92418—22no—
55#14519701—24no—
56#14519701—24no—
57C22⋊#92418C2314yes—
58#14519688—22no—
59#14519684—21no—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C23×C4×C61D411yescharacteristic
67C4⋊Q8×C61—24no—
68C4⋊Q8×C61—21no—
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61—24no—
71C4⋊Q8×C61D412yes—
72C4⋊Q8×C61D412yes—
73#125×C61—24no—
74#125×C61—22no—
75#125×C61—24no—
76#125×C61—22no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—24no—
79C22⋊Q8×C61—24no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61C2314yes—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61D414yes—
89#117×C61—22no—
90#117×C61—22no—
91#117×C61—22no—
92#117×C61—22no—
93#117×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61D411yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61—22no—
100C2×C22⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61C2311yescharacteristic
102C2×C42×C61—22no—
103C2×C42×C61D411yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D411yescharacteristic
106C2×Q8×C61—22no—
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61C2×D414yes—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—24no—
130C4⋊C4×C61—24no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61C2×D414yes—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—24no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C42×C61—24no—
145C42×C61—24no—
146C42×C61—21no—
147C42×C61—21no—
148C42×C61C2×D412yes—
149C42×C61C2×D412yes—
150C22⋊Dic61—42no—
151C22⋊Dic61—42no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154#11758196—42no—
155#11758196—41no—
156#11758196—41no—
157#11758196—42no—
158#11758196—42no—
159C4×Dic61—42no—
160C2×#92418—24no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—41no—
164C23×C61—41no—
165C23×C61#13712yes—
166C23×C61C22×D411yescharacteristic
167Q8×C61—44no—
168Q8×C61—82no—
169Q8×C61—82no—
170Q8×C61—82no—
171C2×C4×C61—24no—
172C2×C4×C61—24no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—24no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—24no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—82no—
186C2×C4×C61—82no—
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61#12012yes—
190C2×C4×C61#13712yes—
191C2×C4×C61C22×D412yes—
192C2×Dic61—22no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—41no—
204#92418—44no—
205#18340—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—22no—
209C22×C61—42no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—81no—
213C22×C61C2×#12011yescharacteristic
214C22×C61C2×#13711yescharacteristic
215C22×C61C2×#13711yescharacteristic
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—22no—
219C4×C61—22no—
220C4×C61—22no—
221C4×C61—41no—
222C4×C61—44no—
223C4×C61—41no—
224C4×C61—44no—
225Dic61—42no—
226Dic61—81no—
227Dic61—81no—
228Dic61—81no—
229#3458D6111yescharacteristic
230#3236—612no—
231#2611—611no—
232#1617—612no—
233#1614—612no—
234#1611—612no—
235#1608—612no—
236#1506—612no—
237#1464—611no—
238C2×C61—21no—
239C2×C61—21no—
240C2×C61—42no—
241C2×C61#300411yescharacteristic
242C2×C61#301811yescharacteristic
243C2×C61#302811yescharacteristic
244#544—22no—
245#544—22no—
246#544—21no—
247#544C2×D6112yes—
248#543—21no—
249#543C2×D6112yes—
250#533—24no—
251#533—21no—
252#533C2×D6112yes—
253C2×#125—611no—
254C2×#117—21no—
255C2×#117C2×D6111yescharacteristic
256#501—1222no—
257#491—1222no—
258#484—614no—
259#483—614no—
260#479—1224no—
261#432—612no—
262#400—611no—
263#399—611no—
264#398—611no—
265#397—611no—
266#389—1221no—
267#388—1221no—
268#387—1221no—
269#385—1221no—
270#376—611no—
271#328—614no—
272#328—614no—
273#324—612no—
274#323—612no—
275C61#1834011yescharacteristic, complemented, Sylow
276C23×C4C22⋊D6111yescharacteristic
277C4⋊Q8—24no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—24no—
281C4⋊Q8C22⋊D6112yes—
282C4⋊Q8C22⋊D6112yes—
283#125—24no—
284#125—22no—
285#125—24no—
286#125—22no—
287#125—1222no—
288#125—1222no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—24no—
296C22⋊Q8C22×D6114yes—
297C4×Q8—22no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8C22⋊D6114yes—
301#117—22no—
302#117—22no—
303#117—22no—
304#117—22no—
305#117—22no—
306#117—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—611no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4C22⋊D6111yescharacteristic
316C2×C4⋊C4C22×D6111yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4C22⋊D6111yescharacteristic
321C2×C22⋊C4C22×D6111yescharacteristic
322C2×C42—22no—
323C2×C42—1221no—
324C2×C42C22⋊D6111yescharacteristic
325C2×C42C22×D6111yescharacteristic
326#107—1224no—
327#106—1224no—
328C4⋊C8—1222no—
329#103—1224no—
330#103—1224no—
331C22⋊C8—614no—
332#97—1222no—
333#95—611no—
334#95—611no—
335#95—1221no—
336#95—1221no—
337#95—1221no—
338#95—1221no—
339#95—1221no—
340C24C2×C22⋊D6111yescharacteristic
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8C2×C22⋊D6114yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—1221no—
354C22×C4—1221no—
355C22×C4C2×C22⋊D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358C22×C4C2×C22⋊D6111yescharacteristic
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C23×D6111yescharacteristic
361C2×C8—1224no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—1222no—
376C4⋊C4—1222no—
377C4⋊C4—2442no—
378C4⋊C4—2442no—
379C4⋊C4—2441no—
380C4⋊C4—2441no—
381C4⋊C4—2442no—
382C4⋊C4C2×C22⋊D6114yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—2442no—
388C22⋊C4—2442no—
389C42—24no—
390C42—24no—
391C42—21no—
392C42—21no—
393C42—2442no—
394C42C2×C22⋊D6112yes—
395C42C2×C22⋊D6112yes—
396C23—21no—
397C23—21no—
398C23—41no—
399C23—41no—
400C23#1451980812yes—
401C23C22×C22⋊D6111yescharacteristic
402Q8—44no—
403Q8—82no—
404Q8—82no—
405Q8—82no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—1222no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2441no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4#1451974611yescharacteristic
435C2×C4#1451974611yescharacteristic
436C2×C4#1451974612yes—
437C2×C4#1451980812yes—
438C2×C4C22×C22⋊D6112yes—
439C8—2444no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22C2×#1451974611yescharacteristic
448C22C2×#1451980811yescharacteristic
449C22C2×#1451980811yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—21no—
454C4—21no—
455C4—41no—
456C4—44no—
457C4—41no—
458C4—44no—
459C4—4881no—
460C4—4881no—
461C4—4881no—
462C4—2442no—
463C2—21no—
464C2—21no—
465C2—42no—
466C2#2145505311yescharacteristic
467C2#2145516811yescharacteristic
468C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#3458D61
hypocenterC61#18340
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #21457021C2semidirect
3. #21444861C2semidirect
5. #21444827C2semidirect
7. #21444331C2semidirect
275. C61#18340semidirect

Maximal subgroups

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

subgroupindex
#214570212
#214515172
#214448612
#214448442
#214448272
#214448102
#214443312
#214441532
C61×#34582
#1834061