Finite Group Databasebeta

← #29733706

Subgroups — #29733706

Order 15616.

1240 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456272C211yesmaximal, characteristic, complemented
2#21455299C211yesmaximal, characteristic, complemented
3#21454177C211yesmaximal, characteristic, complemented
4#21446861C214yesmaximal, complemented
5#21446809C211yesmaximal, characteristic, complemented
6#21446648C211yesmaximal, characteristic, complemented
7#21446195C214yesmaximal, complemented
8C4×#14519704C211yesmaximal, characteristic, complemented
9C61×#3511C211yesmaximal, characteristic, complemented
10C2×#14519802C2214yes—
11C2×#14519802C2212yes—
12C2×#14519774C2211yescharacteristic
13C2×#14519766—22no—
14#16840152—22no—
15#16840138—21no—
16#16840136—21no—
17#16839962—21no—
18#16839962—21no—
19#16839957—21no—
20#16839957—21no—
21#16839786C2211yescharacteristic
22C2×#14519704C2212yes—
23C2×#14519704C2214yes—
24#16839766C2212yes—
25C4×#11758194C2211yescharacteristic
26#16839663—22no—
27#16839663C2214yes—
28C42×Dic61C2211yescharacteristic
29#16839531C2214yes—
30#16839521C2212yes—
31C61×#556—21no—
32C61×#555—21no—
33C61×#554—21no—
34C61×#553—22no—
35C61×#548—22no—
36C61×#547—21no—
37C61×#542—22no—
38C61×#537C2212yes—
39C61×#533—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520C2212yes—
44C4×#41×C61C2211yescharacteristic
45C22×D244—21no—
46C22×D244—21no—
47C22×D244C2312yes—
48#14519802—28no—
49#14519802—44no—
50#14519802—44no—
51C2×#11758201—22no—
52C2×#11758201—24no—
53C2×#11758199—22no—
54C2×#11758199—24no—
55C2×#11758194C2312yes—
56C2×#11758194C2312yes—
57#14519774—21no—
58#14519774—21no—
59#14519774—21no—
60#14519774—21no—
61#14519766—48no—
62C2×#11758198—28no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2312yes—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68#14519704—24no—
69#14519704—44no—
70#14519697—22no—
71#14519697—22no—
72#14519688—21no—
73C4×#92418—21no—
74C2×#41×C61—22no—
75C2×#41×C61C2312yes—
76C2×#41×C61C2311yescharacteristic
77C22×Q8×C61D412yes—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61D411yescharacteristic
83#127×C61D411yescharacteristic
84#126×C61—24no—
85#125×C61—22no—
86#124×C61—22no—
87#124×C61—24no—
88#124×C61D412yes—
89#123×C61—24no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61—24no—
92C22⋊Q8×C61—22no—
93C22⋊Q8×C61—22no—
94#121×C61—22no—
95#121×C61—22no—
96#121×C61—22no—
97#121×C61—21no—
98#121×C61—21no—
99#121×C61—24no—
100C4×Q8×C61—22no—
101C4×Q8×C61—22no—
102C4×Q8×C61D411yescharacteristic
103C4×Q8×C61D411yescharacteristic
104C4×D4×C61—22no—
105C4×D4×C61—22no—
106C4×D4×C61—21no—
107C4×D4×C61—24no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110C4×D4×C61—21no—
111C4×D4×C61D411yescharacteristic
112C4×D4×C61D411yescharacteristic
113#117×C61—22no—
114#117×C61—22no—
115#117×C61—21no—
116#117×C61C2311yescharacteristic
117C2×C4⋊C4×C61—22no—
118C2×C4⋊C4×C61D412yes—
119C2×C4⋊C4×C61C2312yes—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C23×D61—22no—
123#41×C61—22no—
124#41×C61—24no—
125#41×C61—24no—
126#41×C61—22no—
127#41×C61—84no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—44no—
131C2×Q8×C61C2×D412yes—
132C2×Q8×C61C2×D411yescharacteristic
133C2×D4×C61—21no—
134C2×D4×C61—22no—
135C2×D4×C61—24no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C2×D412yes—
139C2×D4×C61C2×D411yescharacteristic
140C22×C4×C61—24no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—22no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D414yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—22no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—24no—
153C2×D244—24no—
154C2×D244—44no—
155C2×D244—44no—
156C2×D244—41no—
157C2×D244—41no—
158C2×D244—41no—
159C2×D244—41no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C42×C61—24no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188#11758201—48no—
189#11758201—44no—
190#11758199—48no—
191#11758199—44no—
192#11758198—416no—
193C4×Dic61—24no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196C4×Dic61—42no—
197#11758194—22no—
198#11758194—24no—
199#11758194#4114yescomplemented
200C2×#92418—22no—
201C2×#92418—22no—
202C23×C61—21no—
203C23×C61—22no—
204C23×C61C22×D411yescharacteristic
205C22×D61—44no—
206C22×D61—44no—
207C22×D61—42no—
208C22×D61—44no—
209Q8×C61—21no—
210Q8×C61—21no—
211Q8×C61—24no—
212Q8×C61—44no—
213Q8×C61—82no—
214D4×C61—21no—
215D4×C61—21no—
216D4×C61—24no—
217D4×C61—42no—
218D4×C61—44no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—84no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—21no—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—24no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—44no—
235C2×C4×C61—84no—
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61#12011yescharacteristic
238C2×C4×C61#12011yescharacteristic
239C2×C4×C61#12011yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C2×#4112yes—
243C2×C4×C61H(5,𝔽2)12yes—
244C2×Dic61—24no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247D244—42no—
248D244—42no—
249D244—44no—
250D244—82no—
251D244—82no—
252D244—82no—
253D244—82no—
254#92418—24no—
255#92418—42no—
256#25616—611nomaximal, Sylow
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—42no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—42no—
263C22×C61—41no—
264C22×C61—42no—
265C22×C61C2×#12011yescharacteristic
266C22×C61#55012yes—
267C2×D61—88no—
268C2×D61—82no—
269C2×D61—82no—
270C2×D61—82no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—42no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—44no—
277C4×C61—41no—
278C4×C61#45312yes—
279C4×C61#45412yes—
280C4×C61#53211yescharacteristic
281C4×C61#53511yescharacteristic
282C4×C61#54612yes—
283Dic61—44no—
284#3511D6111yescharacteristic, complemented
285#3151—611no—
286#3038—611no—
287#2858—611no—
288#2006—614no—
289#1997—611no—
290#1962—611no—
291#1883—614no—
292C4×#104—611no—
293C2×C61—22no—
294C2×C61—81no—
295C2×C61—82no—
296C2×C61#243111yescharacteristic
297C2×C61C2×#45311yescharacteristic
298C2×C61C2×#45411yescharacteristic
299D61—162no—
300C2×AGL(1,ℤ/8)—614no—
301C2×AGL(1,ℤ/8)—612no—
302#556—21no—
303#555—21no—
304#554—21no—
305#553—22no—
306#548—22no—
307#547—21no—
308#542—22no—
309#537C2×D6112yes—
310#533—21no—
311#532—21no—
312#532C2×D6111yescharacteristic
313C2×C4⋊Q8C2×D6111yescharacteristic
314C2×#127—611no—
315C2×#124—1222no—
316#520C2×D6112yes—
317C4×#41C2×D6111yescharacteristic
318#496—1222no—
319#493—1221no—
320#492—1221no—
321#460—1221no—
322#460—1221no—
323#459—1221no—
324#459—1221no—
325#423—611no—
326C2×#104—612no—
327C2×#104—614no—
328#418—612no—
329C4×M16—611no—
330#390—614no—
331#390—1222no—
332C43—611no—
333#343—614no—
334#339—612no—
335C61#2561611yescharacteristic, complemented, Sylow
336C2×#41—22no—
337C2×#41C22×D6111yescharacteristic
338C2×#41C22×D6112yes—
339C22×Q8C22⋊D6112yes—
340C22×D4—612no—
341C22×D4—1221no—
342C22×D4—1221no—
343AGL(1,ℤ/8)—1228no—
344AGL(1,ℤ/8)—2444no—
345AGL(1,ℤ/8)—2444no—
346C2×SD16—1222no—
347C2×SD16—1224no—
348C2×D8—1222no—
349C2×D8—1224no—
350C2×M16—612no—
351C2×M16—612no—
352C4⋊Q8—21no—
353C4⋊Q8—21no—
354C4⋊Q8—21no—
355C4⋊Q8—21no—
356C4⋊Q8C22⋊D6111yescharacteristic
357#127—1221no—
358#127—1221no—
359#127—1221no—
360#127—1221no—
361#127C22⋊D6111yescharacteristic
362#126—24no—
363#125—22no—
364#124—22no—
365#124—24no—
366#124—2448no—
367#124C22⋊D6112yes—
368#123—24no—
369C22⋊Q8—22no—
370C22⋊Q8—24no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373#121—22no—
374#121—22no—
375#121—22no—
376#121—21no—
377#121—21no—
378#121—24no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×Q8C22⋊D6111yescharacteristic
382C4×Q8C22⋊D6111yescharacteristic
383C4×D4—22no—
384C4×D4—22no—
385C4×D4—21no—
386C4×D4—21no—
387C4×D4—24no—
388C4×D4—21no—
389C4×D4—21no—
390C4×D4C22⋊D6111yescharacteristic
391C4×D4C22⋊D6111yescharacteristic
392#117—22no—
393#117—22no—
394#117—21no—
395#117C22×D6111yescharacteristic
396C2×C4⋊C4—22no—
397C2×C4⋊C4C22⋊D6112yes—
398C2×C4⋊C4C22×D6112yes—
399C2×C22⋊C4—1228no—
400C2×C42—21no—
401C2×C42—612no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42C22×D6111yescharacteristic
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407#104—1224no—
408#104—2444no—
409#102—1222no—
410#102—1222no—
411#97—1221no—
412C4×C8—1221no—
413C24—1222no—
414#41—22no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—84no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—44no—
422C2×Q8C2×C22⋊D6112yes—
423C2×Q8C2×C22⋊D6111yescharacteristic
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1224no—
432C2×D4—1224no—
433C2×D4—2444no—
434C2×D4—2444no—
435C2×D4—2441no—
436C2×D4—2441no—
437C2×D4—2441no—
438C2×D4—2441no—
439C2×D4C2×C22⋊D6112yes—
440C2×D4C2×C22⋊D6111yescharacteristic
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1222no—
447C22×C4C2×C22⋊D6111yescharacteristic
448C22×C4C2×C22⋊D6111yescharacteristic
449C22×C4C2×C22⋊D6114yes—
450C22×C4C23×D6111yescharacteristic
451SD16—2448no—
452SD16—2444no—
453D8—2448no—
454D8—2444no—
455M16—614no—
456M16—1222no—
457M16—1224no—
458C2×C8—1222no—
459C2×C8—1222no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4C2×C22⋊D6111yescharacteristic
471C4⋊C4C2×C22⋊D6111yescharacteristic
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—24416no—
481C42—24no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—1224no—
486C42—1222no—
487C42—1222no—
488C42—2442no—
489C42C2×C22⋊D6111yescharacteristic
490C42C2×C22⋊D6111yescharacteristic
491C42C2×C22⋊D6111yescharacteristic
492C42C2×C22⋊D6111yescharacteristic
493C23—21no—
494C23—22no—
495C23—2444no—
496C23—2444no—
497C23—2442no—
498C23—2444no—
499C23C22×C22⋊D6111yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—24no—
503Q8—44no—
504Q8—82no—
505D4—21no—
506D4—21no—
507D4—24no—
508D4—42no—
509D4—44no—
510D4—42no—
511D4—42no—
512D4—84no—
513D4—4882no—
514D4—4882no—
515D4—4882no—
516D4—4882no—
517D4—2442no—
518D4—2442no—
519D4—2444no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—84no—
534C2×C4—1224no—
535C2×C4—2442no—
536C2×C4—2442no—
537C2×C4#1451974611yescharacteristic
538C2×C4#1451974611yescharacteristic
539C2×C4#1451974611yescharacteristic
540C2×C4#1451974611yescharacteristic
541C2×C4C22×C22⋊D6111yescharacteristic
542C2×C4C22×C22⋊D6111yescharacteristic
543C2×C4#41×D6112yes—
544C2×C4#1451982312yes—
545C8—1224no—
546C8—2442no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—42no—
554C22—41no—
555C22—4888no—
556C22—4882no—
557C22—4882no—
558C22—4882no—
559C22C2×#1451974611yescharacteristic
560C22#1684043112yes—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—44no—
565C4—41no—
566C4—42no—
567C4—41no—
568C4—2444no—
569C4#1683993012yes—
570C4#1683993512yes—
571C4#1684030811yescharacteristic
572C4#1684032611yescharacteristic
573C4#1684040012yes—
574C2—22no—
575C2—81no—
576C2—82no—
577C2—9762no—
578C2#2144943411yescharacteristic
579C2C2×#1683993011yescharacteristic
580C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#25616
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21456272C2semidirect
2. #21455299C2semidirect
3. #21454177C2semidirect
4. #21446861C2semidirect
5. #21446809C2semidirect
6. #21446648C2semidirect
7. #21446195C2semidirect
8. C4×#14519704C2semidirect
9. C61×#3511C2semidirect
199. #11758194#41semidirect
284. #3511D61semidirect
335. C61#25616semidirect

Maximal subgroups

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

subgroupindex
#214562722
#214552992
#214541772
#214468612
#214468092
#214466482
#214461952
C4×#145197042
C61×#35112
#2561661