Finite Group Databasebeta

← #29663200

Subgroups — #29663200

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457135C211yesmaximal, characteristic, complemented
2#21457126C211yesmaximal, characteristic
3#21456142C211yesmaximal, characteristic, complemented
4#21445366C212yesmaximal, complemented
5#21445352C212yesmaximal, complemented
6#21445270C212yesmaximal, complemented
7#21445256C212yesmaximal, complemented
8C2×#16839515C211yesmaximal, characteristic, complemented
9#21444510C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic, complemented
11C61×#2931C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840312C2211yescharacteristic
14#16840117—21no—
15#16840117—21no—
16#16840113C2211yescharacteristic
17#16840113C2211yescharacteristic
18#16840109C2211yescharacteristic
19#16840109C2211yescharacteristic
20#16840052C2212yes—
21#16840044—21no—
22#16840042C2212yes—
23#16840039—21no—
24#16839996—22no—
25#16839989—22no—
26#16839967C2212yes—
27#16839959—21no—
28#16839957C2212yes—
29#16839954—21no—
30C2×C4⋊#92418C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32#16839759—21no—
33#16839759—21no—
34#16839515C2212yes—
35#16839515C2212yes—
36#16839510C2212yes—
37#16839509C2212yes—
38#16839507C2212yes—
39#16839505C2212yes—
40C2×#131×C61—21no—
41C61×#532C2211yescharacteristic
42C61×#436—21no—
43C61×#436C2212yes—
44C8×D4×C61—21no—
45C8×D4×C61C2212yes—
46C61×#423C2211yescharacteristic
47C61×#408—21no—
48C2×C4×C8×C61C2211yescharacteristic
49C2×#11758222—21no—
50C2×#11758200—22no—
51C2×#11758199—22no—
52C22×#92418—21no—
53C22×#92418—21no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519777C2311yescharacteristic
57#14519774C2311yescharacteristic
58#14519769—21no—
59#14519765—21no—
60#14519756—21no—
61#14519756—21no—
62#14519751—21no—
63#14519751—21no—
64C4×Dic122—21no—
65C4×Dic122—21no—
66C4×D244—21no—
67C4×D244—21no—
68#14519736—21no—
69#14519710—22no—
70#14519708—22no—
71C4⋊#92418—21no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74C4⋊#92418C2312yes—
75#14519702—21no—
76#14519702—21no—
77#14519702—21no—
78#14519702—21no—
79#14519700—22no—
80#14519699—21no—
81#14519699—21no—
82#14519699—21no—
83#14519699—21no—
84#14519698—22no—
85#14519698—22no—
86#14519697—22no—
87C2×#41×C61—21no—
88#131×C61—44no—
89C2×M16×C61—21no—
90C2×M16×C61—22no—
91C22×C8×C61—22no—
92C22×C8×C61D411yescharacteristic
93C22×C8×C61D411yescharacteristic
94C4⋊Q8×C61D411yescharacteristic
95#127×C61D411yescharacteristic
96#124×C61—21no—
97#121×C61—21no—
98#121×C61D412yes—
99C4×D4×C61—21no—
100C4×D4×C61C2312yes—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—21no—
103C4⋊C8×C61C2312yes—
104C22⋊C8×C61—22no—
105C22⋊C8×C61—22no—
106C4×C8×C61D411yescharacteristic
107C4×C8×C61D411yescharacteristic
108C4×C8×C61C2311yescharacteristic
109C4×C8×C61C2311yescharacteristic
110#41×C61—42no—
111#41×C61—42no—
112C2×Q8×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61C2×D412yes—
117C22×C4×C61—22no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61#4111yescharacteristic
121C2×C22⋊D61—41no—
122#11758222—42no—
123#11758222—42no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244—21no—
127C2×C4×D61—41no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122—21no—
131M16×C61—44no—
132M16×C61—42no—
133C2×C8×C61—22no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C2×C8×C61—44no—
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2×D411yescharacteristic
140C2×C8×C61C2×D411yescharacteristic
141C4⋊C4×C61—21no—
142C4⋊C4×C61C2×D412yes—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61#4111yescharacteristic
148C42×C61C2411yescharacteristic
149C22⋊Dic61—41no—
150#11758200—82no—
151#11758199—82no—
152#11758198—41no—
153#11758198—41no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—81no—
166C2×#92418—81no—
167C23×C61—22no—
168C23×C61#12111yescharacteristic
169C22×D61—41no—
170Q8×C61—42no—
171D4×C61—22no—
172D4×C61—22no—
173D4×C61—42no—
174D4×C61—44no—
175D4×C61—42no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—22no—
179C2×C4×C61—41no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#12111yescharacteristic
184C2×C4×C61#12111yescharacteristic
185C2×C4×C61#13512yes—
186C2×C4×C61AGL(1,ℤ/8)12yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C22⋊D61—82no—
191C2×Dic61—41no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194D244—41no—
195D244—41no—
196D244—41no—
197D244—41no—
198D244—81no—
199C4×D61—82no—
200Dic122—41no—
201Dic122—41no—
202Dic122—41no—
203Dic122—41no—
204Dic122—81no—
205C8×C61—21no—
206C8×C61—21no—
207C8×C61—21no—
208C8×C61—21no—
209C8×C61—42no—
210#92418—42no—
211#92418—42no—
212#19201—611nomaximal, Sylow
213C22×C61—21no—
214C22×C61—21no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61#46312yes—
219C22×C61C2×#12111yescharacteristic
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61#46312yes—
228C4×C61C2×#13511yescharacteristic
229C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C4×C61#57611yescharacteristic
231C4×C61#57811yescharacteristic
232Dic61—81no—
233Dic61—81no—
234Dic61—81no—
235#3246—611no—
236#3245—611no—
237#3131—611no—
238#2931D6111yescharacteristic, complemented
239#1699—612no—
240#1697—612no—
241#1683—612no—
242#1681—612no—
243C2×#335—611no—
244#1550—611no—
245#1547—611no—
246C2×C61—22no—
247C2×C61—82no—
248C2×C61C2×#46311yescharacteristic
249C2×C61#306111yescharacteristic
250C2×C61#306711yescharacteristic
251D61—161no—
252C2×#131—21no—
253#533—611no—
254#532—611no—
255#532C2×D6111yescharacteristic
256#488—1221no—
257#488—1221no—
258#487—611no—
259#487—611no—
260#486—611no—
261#486—611no—
262#476—612no—
263#475—1221no—
264#474—612no—
265#474—1221no—
266#466—1222no—
267#465—1222no—
268#461—612no—
269#460—1221no—
270#459—612no—
271#459—1221no—
272#436—21no—
273#436C2×D6112yes—
274C8×D4—21no—
275C8×D4C2×D6112yes—
276#423C2×D6111yescharacteristic
277C2×C4⋊C8—611no—
278C2×C4⋊C8—611no—
279#416—1221no—
280#416—1221no—
281#408—21no—
282C2×C4×C8C2×D6111yescharacteristic
283#335—612no—
284#335—612no—
285#333—612no—
286#332—612no—
287#332—612no—
288#331—612no—
289C61#1920111yescharacteristic, complemented, Sylow
290C2×#41—21no—
291C2×#41—1221no—
292C2×SD16—1222no—
293C2×D8—1222no—
294#131—44no—
295C2×M16—21no—
296C2×M16—22no—
297C22×C8—22no—
298C22×C8—1221no—
299C22×C8—1221no—
300C22×C8C22⋊D6111yescharacteristic
301C22×C8C22⋊D6111yescharacteristic
302C4⋊Q8—611no—
303C4⋊Q8—611no—
304C4⋊Q8—611no—
305C4⋊Q8D24411yescharacteristic
306#127—611no—
307#127D24411yescharacteristic
308#125—1221no—
309#124—21no—
310#124—1221no—
311C22⋊Q8—1221no—
312C22⋊Q8—1221no—
313#121—21no—
314#121—1221no—
315#121—1221no—
316#121D24412yes—
317C4×Q8—1221no—
318C4×Q8—1221no—
319C4×D4—21no—
320C4×D4—1221no—
321C4×D4—1221no—
322C4×D4C22×D6112yes—
323#117—1221no—
324C2×C42C22×D6111yescharacteristic
325#107—1222no—
326#106—1222no—
327C4⋊C8—21no—
328C4⋊C8—612no—
329C4⋊C8—612no—
330C4⋊C8—1221no—
331C4⋊C8—1221no—
332C4⋊C8C22×D6112yes—
333#103—1221no—
334#103—1221no—
335#103—1221no—
336#103—1221no—
337#103—1222no—
338#102—1222no—
339#102—1222no—
340#102—1221no—
341#102—1221no—
342#102—1221no—
343#102—1221no—
344#102—1222no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C4×C8C22⋊D6111yescharacteristic
348C4×C8C22⋊D6111yescharacteristic
349C4×C8C22×D6111yescharacteristic
350C4×C8C22×D6111yescharacteristic
351#41—42no—
352#41—42no—
353#41—2442no—
354#41—2442no—
355C2×Q8—21no—
356C2×Q8—1221no—
357C2×Q8—1221no—
358C2×Q8—1221no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—1221no—
363C2×D4—1221no—
364C2×D4—1221no—
365C2×D4—2441no—
366C2×D4C2×D24412yes—
367C22×C4—22no—
368C22×C4—2441no—
369C22×C4C2×D24411yescharacteristic
370C22×C4#1175822211yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372SD16—4882no—
373D8—4882no—
374M16—44no—
375M16—42no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—44no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—4881no—
383C2×C8—4881no—
384C2×C8C2×C22⋊D6111yescharacteristic
385C2×C8C2×C22⋊D6111yescharacteristic
386C2×C8C2×C22⋊D6111yescharacteristic
387C2×C8C2×C22⋊D6111yescharacteristic
388C4⋊C4—21no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—1221no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4C2×D24412yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C22⋊C4—2441no—
403C42—2441no—
404C42C2×D24411yescharacteristic
405C42#1175822211yescharacteristic
406C42C2×C22⋊D6111yescharacteristic
407C42C23×D6111yescharacteristic
408C23—22no—
409C23—2441no—
410C23#1451975111yescharacteristic
411Q8—42no—
412Q8—4881no—
413Q8—2441no—
414Q8—2441no—
415Q8—2441no—
416Q8—2441no—
417D4—22no—
418D4—22no—
419D4—42no—
420D4—44no—
421D4—42no—
422D4—4881no—
423D4—4882no—
424D4—2441no—
425D4—2441no—
426D4—2441no—
427D4—2441no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—4882no—
435C2×C4—2441no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4#1451975111yescharacteristic
439C2×C4#1451975111yescharacteristic
440C2×C4#1451975111yescharacteristic
441C2×C4#1451979712yes—
442C2×C4#1451980312yes—
443C2×C4C22×C22⋊D6111yescharacteristic
444C2×C4C22×D24411yescharacteristic
445C2×C4C2×#1175822211yescharacteristic
446C8—21no—
447C8—21no—
448C8—21no—
449C8—21no—
450C8—42no—
451C8—2442no—
452C8—2442no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—4881no—
459C22—4881no—
460C22—4881no—
461C22#1683997712yes—
462C22C2×#1451975111yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—4881no—
468C4—4881no—
469C4—4881no—
470C4#1683997712yes—
471C4C2×#1451979711yescharacteristic
472C4C2×#1451980311yescharacteristic
473C4#1684056711yescharacteristic
474C4#1684057911yescharacteristic
475C2—22no—
476C2—82no—
477C2—9761no—
478C2C2×#1683997711yescharacteristic
479C2#2145548911yescharacteristic
480C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2931C2
socleC22×C61C2×#121
prime radical#21457126C2
hypercenter#2931D61
hypocenterC61#19201
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931

Complements

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

NH ≅ G/Ntype
1. #21457135C2semidirect
3. #21456142C2semidirect
4. #21445366C2semidirect
5. #21445352C2semidirect
6. #21445270C2semidirect
7. #21445256C2semidirect
8. C2×#16839515C2semidirect
9. #21444510C2semidirect
10. #21444497C2semidirect
11. C61×#2931C2semidirect
238. #2931D61semidirect
289. C61#19201semidirect

Maximal subgroups

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

subgroupindex
#214571352
#214571262
#214561422
#214453662
#214453522
#214452702
#214452562
C2×#168395152
#214445102
#214444972
C61×#29312
#1920161