Finite Group Databasebeta

← #29692209

Subgroups — #29692209

Order 15616.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic
2#21448463C211yesmaximal, characteristic
3#21446772C211yesmaximal, characteristic, complemented
4#21446585C211yesmaximal, characteristic, complemented
5#21446559C211yesmaximal, characteristic, complemented
6#21446544C211yesmaximal, characteristic, complemented
7#21446515C211yesmaximal, characteristic
8#21446501C211yesmaximal, characteristic
9#21446446C211yesmaximal, characteristic
10#21446439C211yesmaximal, characteristic
11#21446404C211yesmaximal, characteristic, complemented
12#21446397C211yesmaximal, characteristic, complemented
13#21445826C211yesmaximal, characteristic
14#21445676C211yesmaximal, characteristic, complemented
15C61×#2331C211yesmaximal, characteristic, complemented
16C2×C4×D244C2211yescharacteristic
17#16840049C418yescomplemented
18C2×#14519710C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839664C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839651—21no—
35#16839651—21no—
36#16839624—21no—
37#16839621—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4⋊Q8×C61C2211yescharacteristic
50C61×#391—21no—
51C61×#391—21no—
52C61×#389C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C61×#387—21no—
55C61×#387—21no—
56C61×#382C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#376C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C4×D244—22no—
67C4×D244C2×C414yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×C4×Dic61—21no—
79#14519710—41no—
80#14519708—41no—
81#14519708C2×C414yes—
82#14519708C2×C414yes—
83C4⋊#92418—22no—
84C4⋊#92418C2×C414yes—
85#14519697—41no—
86#14519697—41no—
87#14519697C2×C414yes—
88#14519697C2×C414yes—
89#14519688—41no—
90C4×#92418—41no—
91#14519684—21no—
92#14519684—21no—
93#14519684—21no—
94C22×Q8×C61—21no—
95C4⋊Q8×C61—22no—
96C4⋊Q8×C61C2×C414yes—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61C2311yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109#95×C61—21no—
110#95×C61—21no—
111#95×C61—21no—
112#95×C61—21no—
113#95×C61—21no—
114#95×C61—21no—
115#95×C61D411yescharacteristic
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
118#95×C61D411yescharacteristic
119C23×D61—21no—
120C2×Q8×C61—24no—
121C2×Q8×C61—81no—
122C2×Q8×C61—81no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C2×D244—21no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244C22×C412yes—
143C2×C4×D61—24no—
144C2×C4×D61—81no—
145C2×C4×D61—81no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—81no—
154C4⋊C4×C61—81no—
155C4⋊C4×C61C22×C412yes—
156C4⋊C4×C61C22×C412yes—
157C42×C61—21no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—81no—
162C42×C61—81no—
163C42×C61C22×C412yes—
164C42×C61#4112yes—
165#11758198—24no—
166#11758198—81no—
167#11758198—81no—
168#11758198—81no—
169#11758198—81no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61C22×C412yes—
176#11758196—81no—
177#11758196—81no—
178C4×Dic61—81no—
179C2×#92418—21no—
180C2×#92418—21no—
181C2×#92418—21no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418C22×C412yes—
190C2×#92418C22×C412yes—
191C23×C61C22×D411yescharacteristic
192C22×D61—22no—
193C22×D61—41no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199Q8×C61—84no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61C4×D412yes—
229C2×C4×C61C4×D412yes—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×C4×C612−1+411yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—22no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245D244—22no—
246D244—22no—
247D244—41no—
248D244—41no—
249C4×D61—84no—
250#92418—22no—
251#92418—22no—
252#92418—41no—
253#92418—41no—
254#22123—611nomaximal, Sylow
255C22×C61C2×C4×D411yescharacteristic
256C22×C61#53611yescharacteristic
257C22×C61#53711yescharacteristic
258C22×C61#57611yescharacteristic
259C22×C61#57711yescharacteristic
260C22×C61#57711yescharacteristic
261C22×C61#57711yescharacteristic
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—81no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61#51911yescharacteristic
280C4×C61#52011yescharacteristic
281C4×C61#54111yescharacteristic
282C4×C61#54211yescharacteristic
283Dic61—42no—
284Dic61—81no—
285Dic61—81no—
286C2×#476—611no—
287#2331D6111yescharacteristic, complemented
288#2319—611no—
289#1988—611no—
290#1949—611no—
291#1944—611no—
292#1941—611no—
293#1936—611no—
294#1934—611no—
295#1926—611no—
296#1925—611no—
297#1920—611no—
298#1919—611no—
299#1804—611no—
300#1768—611no—
301C2×C61#232311yescharacteristic
302C2×C61#294611yescharacteristic
303C2×C61#294911yescharacteristic
304C2×C61#324111yescharacteristic
305C2×C61#324411yescharacteristic
306C2×C61#325711yescharacteristic
307C2×C61#326311yescharacteristic
308D61—82no—
309C2×C4⋊Q8C2×D6111yescharacteristic
310C2×C4×D4—611no—
311#476—618no—
312C2×#107—611no—
313C2×#106—611no—
314C2×#106—611no—
315C2×#106—611no—
316C2×C4⋊C8—611no—
317C2×C4⋊C8—611no—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—611no—
321C2×#102—611no—
322C2×#97—611no—
323C2×C4×C8—611no—
324#391—21no—
325#391—21no—
326#390—611no—
327#390—611no—
328#389—611no—
329#389—611no—
330#389C2×D6111yescharacteristic
331#389C2×D6111yescharacteristic
332#388—1221no—
333#388—1221no—
334#387—21no—
335#387—21no—
336#382C2×D6111yescharacteristic
337#381—1221no—
338#380—1221no—
339C4×C4⋊C4—611no—
340C4×C4⋊C4C2×D6111yescharacteristic
341C4×C4⋊C4C2×D6111yescharacteristic
342#376C2×D6111yescharacteristic
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#336—611no—
352#336—611no—
353C61#2212311yescharacteristic, complemented, Sylow
354C22×Q8—21no—
355C22×D4—611no—
356C23×C4—1221no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8—611no—
360C22×C8—611no—
361C4⋊Q8—22no—
362C4⋊Q8C4×D6114yes—
363C4×D4—614no—
364C4×D4—1222no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4C22×D6111yescharacteristic
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C22⋊C4—1221no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42—1221no—
385C2×C42C22×D6111yescharacteristic
386C2×C42C22×D6111yescharacteristic
387C2×C42C22×D6111yescharacteristic
388#107—2441no—
389#106—614no—
390#106—614no—
391#106—2441no—
392C4⋊C8—614no—
393C4⋊C8—1222no—
394#102—614no—
395#102—614no—
396#102—2441no—
397#102—2441no—
398#97—2441no—
399C4×C8—2441no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95C22⋊D6111yescharacteristic
410#95C22⋊D6111yescharacteristic
411#95C22⋊D6111yescharacteristic
412#95C22⋊D6111yescharacteristic
413C24—1221no—
414C2×Q8—24no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×D4—612no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—1221no—
428C22×C4—1224no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—4881no—
432C22×C4—4881no—
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—612no—
441C2×C8—612no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—612no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—4881no—
468C4⋊C4—4881no—
469C4⋊C4C2×C4×D6112yes—
470C4⋊C4C2×C4×D6112yes—
471C22⋊C4—1224no—
472C22⋊C4—4881no—
473C22⋊C4—4881no—
474C22⋊C4—4881no—
475C22⋊C4—4881no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—81no—
481C42—81no—
482C42—4881no—
483C42C2×C4×D6112yes—
484C42#1175822212yes—
485C23—1222no—
486C23—2441no—
487C23—2441no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23C22×C22⋊D6111yescharacteristic
493Q8—84no—
494D4—1222no—
495D4—1222no—
496D4—2441no—
497D4—2441no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—1222no—
527C2×C4—4884no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4C4×C22⋊D6112yes—
537C2×C4C4×C22⋊D6112yes—
538C2×C4C22×C4×D6111yescharacteristic
539C2×C4C2×#1175822211yescharacteristic
540C2×C4#1451982311yescharacteristic
541C2×C4#1451982411yescharacteristic
542C2×C4#1451982511yescharacteristic
543C2×C4#1451982511yescharacteristic
544C8—1222no—
545C8—1222no—
546C8—2441no—
547C8—2441no—
548C22—4881no—
549C22—4881no—
550C22—4881no—
551C22—4881no—
552C22—2442no—
553C22—2442no—
554C22—2442no—
555C22C2×C4×C22⋊D6111yescharacteristic
556C22#1684033611yescharacteristic
557C22#1684034011yescharacteristic
558C22#1684056511yescharacteristic
559C22#1684057011yescharacteristic
560C22#1684057011yescharacteristic
561C22#1684057311yescharacteristic
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—4881no—
573C4—4881no—
574C4—2442no—
575C4#1684023811yescharacteristic
576C4#1684024411yescharacteristic
577C4#1684037011yescharacteristic
578C4#1684038411yescharacteristic
579C2—4882no—
580C2#2144849511yescharacteristic
581C2#2145465111yescharacteristic
582C2#2145467811yescharacteristic
583C2#2145707211yescharacteristic
584C2#2145711711yescharacteristic
585C2#2145723011yescharacteristic
586C2#2145728411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2331C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#2331D61
hypocenterC61#22123
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #21446772C2semidirect
4. #21446585C2semidirect
5. #21446559C2semidirect
6. #21446544C2semidirect
11. #21446404C2semidirect
12. #21446397C2semidirect
14. #21445676C2semidirect
15. C61×#2331C2semidirect
17. #16840049C4semidirect
287. #2331D61semidirect
353. C61#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
#214484632
#214467722
#214465852
#214465592
#214465442
#214465152
#214465012
#214464462
#214464392
#214464042
#214463972
#214458262
#214456762
C61×#23312
#2212361