Finite Group Databasebeta

← #29646296

Subgroups — #29646296

Order 15616.

922 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840450C211yesmaximal, characteristic
2#21454909C211yesmaximal, characteristic, complemented
3#21454906C211yesmaximal, characteristic, complemented
4#21444187C212yesmaximal, complemented
5#21444187C212yesmaximal, complemented
6#21444175C211yesmaximal, characteristic, complemented
7#21444159C211yesmaximal, characteristic
8#21444152C212yesmaximal, complemented
9#21444139C211yesmaximal, characteristic
10#21444132C212yesmaximal, complemented
11C61×#1470C211yesmaximal, characteristic, complemented
12#16840450—21no—
13#16840450—21no—
14#16840450—21no—
15#16840450—22no—
16#16840450—22no—
17#16840450—21no—
18C2×#14519772C2212yes—
19C2×#14519768C412yes—
20C2×#14519762C412yes—
21C2×#14519757C411yescharacteristic
22C2×#14519757C411yescharacteristic
23C2×C4×Dic122C2211yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×#14519735C411yescharacteristic
26C2×#14519735C2212yes—
27C2×C4⋊C4×D61C411yescharacteristic
28#16839892—21no—
29#16839891—22no—
30#16839890—22no—
31#16839889—21no—
32C8×Dic122—21no—
33#16839886—21no—
34#16839819C2211yescharacteristic
35#16839816C2212yes—
36#16839816C2212yes—
37#16839815C2211yescharacteristic
38#16839812C2212yes—
39#16839812C2212yes—
40#16839488C2212yes—
41#16839488C2212yes—
42#16839485C2211yescharacteristic
43#16839485C2212yes—
44#16839485C2211yescharacteristic
45#16839485C2211yescharacteristic
46#16839485C2211yescharacteristic
47#16839485C2212yes—
48C2×#125×C61C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C61×#431C2211yescharacteristic
51C61×#324C2212yes—
52C61×#324C2212yes—
53C22×Dic122C2×C411yescharacteristic
54C22×D244C2×C411yescharacteristic
55C22×C4×D61C2×C412yes—
56C22×C4×D61C2×C411yescharacteristic
57C22×C4×D61C2×C411yescharacteristic
58#14519772—22no—
59#14519772—22no—
60#14519772—22no—
61#14519772—22no—
62#14519768—22no—
63#14519768—22no—
64#14519768—22no—
65#14519768—22no—
66#14519762—22no—
67#14519762—22no—
68#14519762—22no—
69#14519762—22no—
70#14519757—21no—
71#14519757—21no—
72#14519757—22no—
73#14519757—21no—
74#14519757—21no—
75#14519757—22no—
76C4×Dic122D411yescharacteristic
77C4×Dic122D411yescharacteristic
78C4×Dic122D411yescharacteristic
79C4×Dic122D412yes—
80C4×Dic122D412yes—
81C4×Dic122D411yescharacteristic
82C4×D244—21no—
83C4×D244—21no—
84C4×D244—21no—
85C4×D244—21no—
86#14519735—22no—
87#14519735—21no—
88#14519735—21no—
89#14519735—22no—
90#14519735—22no—
91#14519735—22no—
92#14519735—22no—
93C2×C4⋊Dic61C2×C412yes—
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×C4⋊Dic61C2×C412yes—
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758196C2×C412yes—
98C2×#11758196C2×C412yes—
99C4⋊C4×D61—22no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61—21no—
102C2×#11758198C2×C412yes—
103C2×#11758198C2×C412yes—
104C2×#11758198C2×C412yes—
105C2×#11758198C2311yescharacteristic
106C2×#11758198C2311yescharacteristic
107C2×C4×Dic61C2×C412yes—
108C2×C4×Dic61C2312yes—
109C4⋊#92418—21no—
110C4⋊#92418—21no—
111#14519706—22no—
112#14519706—22no—
113#14519706—22no—
114#14519706—22no—
115C22⋊#92418C2311yescharacteristic
116C22⋊#92418C2312yes—
117C22⋊#92418C2311yescharacteristic
118#14519688—21no—
119#14519687—22no—
120#14519687—22no—
121C4×#92418—21no—
122C8×Dic61—22no—
123C8×Dic61—22no—
124#125×C61—21no—
125#125×C61—21no—
126#125×C61—21no—
127#125×C61—21no—
128C2×C4⋊C4×C61C2×C411yescharacteristic
129C2×C4⋊C4×C61C2×C411yescharacteristic
130C2×C4⋊C4×C61C2×C412yes—
131C2×C4⋊C4×C61C2312yes—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61—21no—
134C4⋊C8×C61—21no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2312yes—
137C22⋊C8×C61C2311yescharacteristic
138#97×C61—21no—
139C4×C8×C61—21no—
140C23×D61C22×C411yescharacteristic
141C23×D61C22×C411yescharacteristic
142C22×C4×C61C22×C412yes—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C412yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61C22×C412yes—
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C412yes—
151C2×D244—22no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—41no—
155C2×C4×D61—24no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×Dic122—42no—
166C2×Dic122C22⋊C411yescharacteristic
167C2×Dic122C22⋊C411yescharacteristic
168C2×Dic122C22⋊C412yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184#11758198—21no—
185#11758198—21no—
186#11758198—21no—
187#11758198—21no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61C22⋊C412yes—
201C4⋊Dic61C22⋊C411yescharacteristic
202C4⋊Dic61C22⋊C411yescharacteristic
203#11758196—22no—
204#11758196—22no—
205#11758196C22⋊C412yes—
206#11758196C22⋊C412yes—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61C2×D412yes—
212C4×Dic61C2×D412yes—
213C4×Dic61C2×D412yes—
214C4×Dic61C2×D412yes—
215C2×#92418—21no—
216C2×#92418—22no—
217C2×#92418—21no—
218C23×C61C23×C411yescharacteristic
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61#13112yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—42no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254C2×Dic61C2×C22⋊C412yes—
255D244—42no—
256D244—42no—
257C4×D61—44no—
258C4×D61—44no—
259C4×D61—42no—
260C4×D61—42no—
261C4×D61—42no—
262C4×D61—42no—
263Dic122—22no—
264Dic122—22no—
265Dic122—42no—
266C8×C61—41no—
267C8×C61—41no—
268C8×C61—42no—
269#92418—41no—
270#92418—42no—
271#92418—41no—
272#17423—611nomaximal, Sylow
273C22×C61—21no—
274C22×C61—21no—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61C2×#13111yescharacteristic
277C22×C61#56811yescharacteristic
278C2×D61—22no—
279C2×D61—42no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—42no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—81no—
287C2×D61—81no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61#41312yes—
296Dic61—22no—
297Dic61—22no—
298Dic61—22no—
299Dic61—22no—
300Dic61—41no—
301Dic61—41no—
302C2×#552—611no—
303#2984—611no—
304#2984—611no—
305#1473—612no—
306#1473—612no—
307#1470—611no—
308#1470D6111yescharacteristic, complemented
309#1466—611no—
310#1464—612no—
311#1460—611no—
312#1458—612no—
313C2×C61—21no—
314C2×C61—21no—
315C2×C61#288011yescharacteristic
316C2×C61#288411yescharacteristic
317C2×C61C2×#41311yescharacteristic
318D61—42no—
319D61—81no—
320#552—1221no—
321#552—1221no—
322#552—1222no—
323#552—1222no—
324#552—1221no—
325#552—1221no—
326C2×#126—612no—
327C2×#125—612no—
328C2×#125C2×D6111yescharacteristic
329C2×#123—612no—
330C2×C22⋊Q8—611no—
331C2×C22⋊Q8—611no—
332C2×C4×Q8—611no—
333C2×C4×D4—611no—
334C2×#117—611no—
335C2×#117—612no—
336C22×C4⋊C4—611no—
337#446—1221no—
338#446—1222no—
339#446—1221no—
340#446—1222no—
341C8×Q8—1221no—
342C8×Q8—1221no—
343#432—611no—
344#432—612no—
345#432—612no—
346#432C2×D6111yescharacteristic
347#431—611no—
348#431—612no—
349#431—612no—
350#431C2×D6111yescharacteristic
351#324—612no—
352#324—612no—
353#324C2×D6112yes—
354#324C2×D6112yes—
355#323—611no—
356#323—612no—
357#323—611no—
358#323—611no—
359#323—611no—
360#323—612no—
361C61#1742311yescharacteristic, complemented, Sylow
362C22×Q8—611no—
363C22×D4—611no—
364C23×C4—612no—
365C23×C4—611no—
366C23×C4—611no—
367#126—1222no—
368#126—1222no—
369#126—1222no—
370#126—1222no—
371#125—21no—
372#125—21no—
373#125—21no—
374#125—21no—
375#125—1222no—
376#125—1222no—
377#125—1222no—
378#125—1222no—
379#123—1222no—
380#123—1222no—
381#123—1222no—
382#123—1222no—
383C22⋊Q8—1221no—
384C22⋊Q8—1221no—
385C22⋊Q8—1222no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8—1222no—
389C4×Q8—611no—
390C4×Q8—611no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—611no—
394C4×Q8—611no—
395C4×D4—1221no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4—1221no—
399#117—1222no—
400#117—1222no—
401#117—1222no—
402#117—1222no—
403#117—1222no—
404#117—1221no—
405#117—1221no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—612no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C4×D6111yescharacteristic
416C2×C4⋊C4C4×D6111yescharacteristic
417C2×C4⋊C4C4×D6112yes—
418C2×C4⋊C4C22×D6112yes—
419C2×C22⋊C4—612no—
420C2×C22⋊C4—612no—
421C2×C22⋊C4—612no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—611no—
424C2×C42—612no—
425C2×C42—612no—
426C2×C42C22×D6111yescharacteristic
427C4⋊C8—21no—
428C4⋊C8—21no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431C4⋊C8—1222no—
432C4⋊C8—1222no—
433C4⋊C8—1221no—
434C4⋊C8—1221no—
435C22⋊C8—611no—
436C22⋊C8—612no—
437C22⋊C8—611no—
438C22⋊C8C22×D6111yescharacteristic
439C22⋊C8C22×D6112yes—
440C22⋊C8C22×D6111yescharacteristic
441#97—21no—
442#97—1222no—
443#97—1222no—
444#97—1221no—
445C4×C8—21no—
446C4×C8—1222no—
447C4×C8—1222no—
448C4×C8—1221no—
449C24—611no—
450C24—611no—
451C2×Q8—611no—
452C2×Q8—611no—
453C2×Q8—612no—
454C2×Q8—2442no—
455C2×D4—1222no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—2441no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1224no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4C2×C4×D6112yes—
474C22×C4C2×C4×D6111yescharacteristic
475C22×C4C2×C4×D6111yescharacteristic
476C22×C4C2×C4×D6112yes—
477C22×C4C23×D6111yescharacteristic
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—22no—
481C2×C8—1221no—
482C2×C8—1222no—
483C2×C8—1221no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—612no—
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—612no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—612no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—1221no—
517C42—612no—
518C42—612no—
519C42—612no—
520C42—612no—
521C42—1222no—
522C42—1222no—
523C42D4×D6111yescharacteristic
524C42D4×D6111yescharacteristic
525C42D4×D6111yescharacteristic
526C42D4×D6111yescharacteristic
527C23—612no—
528C23—1221no—
529C23—1221no—
530C23—1221no—
531C23—1221no—
532C23—1221no—
533C23—2441no—
534C23—2441no—
535C23—2441no—
536C23—2441no—
537C23C22×C4×D6111yescharacteristic
538Q8—1222no—
539Q8—1222no—
540Q8—2442no—
541D4—2442no—
542D4—2442no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2444no—
563C2×C4—2444no—
564C2×C4—2442no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4C22⋊C4×D6111yescharacteristic
570C2×C4C22⋊C4×D6111yescharacteristic
571C2×C4C22⋊C4×D6111yescharacteristic
572C2×C4C22⋊C4×D6111yescharacteristic
573C2×C4C2×D4×D6111yescharacteristic
574C2×C4C2×D4×D6111yescharacteristic
575C8—41no—
576C8—41no—
577C8—42no—
578C8—2441no—
579C8—2442no—
580C8—2441no—
581C22—21no—
582C22—21no—
583C22—1222no—
584C22—4881no—
585C22—4881no—
586C22—2442no—
587C22—2442no—
588C22—2441no—
589C22—2441no—
590C22—2441no—
591C22—2441no—
592C22—2441no—
593C22C2×C22⋊C4×D6111yescharacteristic
594C22#131×D6111yescharacteristic
595C22#1684052711yescharacteristic
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—42no—
602C4—42no—
603C4—1222no—
604C4—1222no—
605C4—1222no—
606C4—1222no—
607C4—2441no—
608C4—2441no—
609C4#1683974512yes—
610C2—21no—
611C2—21no—
612C2—4881no—
613C2—2442no—
614C2#2145426711yescharacteristic
615C2#2145428111yescharacteristic
616C2C2×#1683974511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1470C2
socleC22×C61C22×C22⋊C4
prime radicalC22×D244C2×C4
cosocleC22×C4×C61C24
hypercenter#1470D61
hypocenterC61#17423
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1470

Complements

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

NH ≅ G/Ntype
2. #21454909C2semidirect
3. #21454906C2semidirect
4. #21444187C2semidirect
5. #21444187C2semidirect
6. #21444175C2semidirect
8. #21444152C2semidirect
10. #21444132C2semidirect
11. C61×#1470C2semidirect
308. #1470D61semidirect
361. C61#17423semidirect

Maximal subgroups

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

subgroupindex
C2×#168404502
#214549092
#214549062
#214441872
#214441752
#214441592
#214441522
#214441392
#214441322
C61×#14702
#1742361