Finite Group Databasebeta

← #30617763

Subgroups — #30617763

Order 15616.

1230 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462432C211yesmaximal, characteristic, complemented
2#21462420C211yesmaximal, characteristic, complemented
3#21461866C212yesmaximal, complemented
4#21461861C212yesmaximal, complemented
5#21461825C212yesmaximal, complemented
6#21461821C212yesmaximal, complemented
7#21461728C212yesmaximal, complemented
8#21461671C212yesmaximal, complemented
9#21459602C211yesmaximal, characteristic
10#16840842C2212yes—
11#16840830C2212yes—
12#16840761C2214yes—
13#16840755C2214yes—
14#16840748—22no—
15#16840748C2212yes—
16D61⋊#101—22no—
17D61⋊#101C2212yes—
18#16840743—22no—
19#16840743C2212yes—
20#16840741—22no—
21#16840741C2212yes—
22#16840633C2214yes—
23#16840629C2214yes—
24C2×#14519825C2211yescharacteristic
25C2×#14519825C2211yescharacteristic
26#16840427—22no—
27#16840415—22no—
28#16840413—22no—
29#16840413—22no—
30#16840337C411yescharacteristic, complemented
31#16840337C411yescharacteristic, complemented
32#16840337C2211yescharacteristic
33#16840337C2211yescharacteristic
34#16840336C411yescharacteristic, complemented
35#16840336C411yescharacteristic, complemented
36#16840313C411yescharacteristic, complemented
37#16840313C2211yescharacteristic
38#16840312C411yescharacteristic, complemented
39C4⋊Q8×D61C411yescharacteristic, complemented
40C4⋊Q8×D61C2211yescharacteristic
41C4×#11758222C2211yescharacteristic
42C2×C4⋊Q8×C61C411yescharacteristic, complemented
43#14519865—44no—
44D61⋊M16—24no—
45D61⋊M16C2314yes—
46Q8×#6923—44no—
47#14519856—24no—
48#14519856C2314yes—
49#14519842—44no—
50#14519840—44no—
51#14519838—22no—
52#14519838—24no—
53#14519837—22no—
54#14519836—24no—
55#14519836—22no—
56#14519835—22no—
57#14519825—44no—
58#14519825—44no—
59C2×#11758226—21no—
60C2×#11758226C2×C411yescharacteristic
61C2×#11758226C2×C411yescharacteristic
62C2×#11758226D412yes—
63C2×#11758226C2311yescharacteristic
64C2×#11758226C2311yescharacteristic
65C2×#11758222—21no—
66C2×#11758222—21no—
67C2×#11758222C2×C411yescharacteristic
68C2×#11758222C2×C411yescharacteristic
69C2×#11758222D412yes—
70C2×#11758222D412yes—
71C2×#11758222C2311yescharacteristic
72C2×Q8×D61—21no—
73C2×Q8×D61C2×C411yescharacteristic
74C2×Q8×D61C2×C411yescharacteristic
75C2×Q8×D61D412yes—
76C2×Q8×D61C2311yescharacteristic
77C2×Q8×D61C2311yescharacteristic
78#14519777C2×C411yescharacteristic
79#14519776—22no—
80#14519776—22no—
81#14519775C2×C411yescharacteristic
82#14519775C2×C411yescharacteristic
83#14519775C2×C411yescharacteristic
84#14519775C2×C411yescharacteristic
85#14519774C2×C411yescharacteristic
86#14519766C2×C411yescharacteristic
87#14519766C2×C411yescharacteristic
88#14519766C2×C411yescharacteristic
89#14519766C2×C411yescharacteristic
90#14519765C2×C411yescharacteristic
91#14519765C2×C411yescharacteristic
92#14519762—22no—
93#14519762—22no—
94#14519762—22no—
95#14519762—22no—
96#14519760—22no—
97#14519760—22no—
98#14519755—22no—
99#14519755—22no—
100#14519755—22no—
101#14519755—22no—
102#14519753—22no—
103#14519753—22no—
104#14519752—22no—
105#14519752—22no—
106#14519751—22no—
107C4×Dic122—21no—
108C4×D244—21no—
109C4×D244—22no—
110#14519739—22no—
111#14519739—22no—
112C4×C22⋊D61—21no—
113C4×C22⋊D61—21no—
114#14519736C2×C411yescharacteristic
115#14519734C2×C411yescharacteristic
116#14519734C2311yescharacteristic
117C4⋊C4×D61—22no—
118C4⋊C4×D61—22no—
119C42×D61C2×C411yescharacteristic
120C42×D61C2311yescharacteristic
121C22×Q8×C61C2×C411yescharacteristic
122C22×Q8×C61C2×C411yescharacteristic
123C4⋊Q8×C61—22no—
124C4⋊Q8×C61C2×C411yescharacteristic
125C4⋊Q8×C61C2×C411yescharacteristic
126C4⋊Q8×C61C2×C411yescharacteristic
127C4⋊Q8×C61C2×C411yescharacteristic
128C2×C4⋊C4×C61—22no—
129C2×C42×C61C2×C411yescharacteristic
130C2×Q8×C61—21no—
131C2×Q8×C61—42no—
132C2×Q8×C61—42no—
133C2×Q8×C61C22⋊C412yes—
134C2×Q8×C61C22×C411yescharacteristic
135C2×Q8×C61C22×C411yescharacteristic
136C2×Q8×C61C22×C411yescharacteristic
137C2×Q8×C61C22×C411yescharacteristic
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C22⋊C412yes—
141C22×C4×C61C22×C411yescharacteristic
142C2×C22⋊D61—21no—
143C2×C22⋊D61C22⋊C412yes—
144C2×C22⋊D61C2×D411yescharacteristic
145C2×C22⋊D61C2×D411yescharacteristic
146#11758226—24no—
147#11758226—24no—
148#11758226—44no—
149#11758226—42no—
150#11758226—42no—
151#11758226—44no—
152Q8×D61—24no—
153Q8×D61—24no—
154Q8×D61—44no—
155Q8×D61—42no—
156Q8×D61—42no—
157Q8×D61—44no—
158#11758222—24no—
159#11758222—44no—
160#11758222—42no—
161#11758222—42no—
162#11758222—44no—
163#11758222—44no—
164#11758222—44no—
165C2×D244—21no—
166C2×D244—22no—
167C2×D244—21no—
168C2×D244C22⋊C412yes—
169C2×D244C22×C411yescharacteristic
170C2×D244C22×C411yescharacteristic
171C2×D244C2×D412yes—
172C2×D244C2×D411yescharacteristic
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×C4×D61C22⋊C412yes—
181C2×C4×D61C22⋊C412yes—
182C2×C4×D61C22⋊C412yes—
183C2×C4×D61C22×C411yescharacteristic
184C2×C4×D61C2×D412yes—
185C2×C4×D61C2×D412yes—
186C2×C4×D61C2411yescharacteristic
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122C22⋊C412yes—
190C2×Dic122C22×C411yescharacteristic
191C2×Dic122C22×C411yescharacteristic
192C2×Dic122C2×D411yescharacteristic
193C2×Dic122C2×D412yes—
194C22⋊#6923—24no—
195#11758217—24no—
196C4⋊#6923—24no—
197C4⋊#6923—44no—
198C4×#6923—24no—
199C4×#6923—44no—
200#11758213—24no—
201#11758213—44no—
202D61⋊C8—24no—
203D61⋊C8—44no—
204C4⋊C4×C61—22no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—22no—
207C4⋊C4×C61—22no—
208C42×C61—21no—
209C42×C61C22×C411yescharacteristic
210C42×C61C22×C411yescharacteristic
211C22⋊Dic61—21no—
212#11758198—21no—
213#11758198—22no—
214#11758198—22no—
215#11758198—22no—
216#11758198—21no—
217#11758198—22no—
218#11758198—22no—
219#11758198—22no—
220C4⋊Dic61—21no—
221C4⋊Dic61—22no—
222#11758196—22no—
223#11758196—22no—
224#11758196—22no—
225#11758196—22no—
226#11758196—21no—
227#11758196—21no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C22×C411yescharacteristic
232C23×C61C2×C22⋊C411yescharacteristic
233C22×D61—21no—
234C22×D61C2×C22⋊C412yes—
235C22×D61C2×C22⋊C411yescharacteristic
236C22×D61C22×D411yescharacteristic
237C2×#6923—44no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—42no—
243Q8×C61—42no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C23×C411yescharacteristic
256C22⋊D61—22no—
257C22⋊D61—22no—
258C22⋊D61—42no—
259C22⋊D61—42no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C22×D411yescharacteristic
266D244—22no—
267D244—24no—
268D244—22no—
269D244—22no—
270D244—42no—
271D244—44no—
272D244—42no—
273D244—42no—
274C4×D61—24no—
275C4×D61—22no—
276C4×D61—24no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—42no—
280C4×D61—42no—
281C4×D61—42no—
282C4×D61—42no—
283C4×D61—42no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61—42no—
287C4×D61H(5,𝔽2)14yes—
288Dic122—22no—
289Dic122—22no—
290Dic122—24no—
291Dic122—22no—
292Dic122—42no—
293Dic122—42no—
294Dic122—42no—
295#92420—44no—
296#21783—611nomaximal, Sylow
297C22×C61—21no—
298C22×C61—41no—
299C22×C61C22×C22⋊C411yescharacteristic
300C2×D61—21no—
301C2×D61—22no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—41no—
307C2×D61#53412yes—
308#6923—84no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—21no—
312C4×C61—22no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61#51812yes—
317Dic61—22no—
318Dic61—21no—
319Dic61—41no—
320Dic61—41no—
321Dic61—41no—
322Dic61#53412yes—
323#3499—611no—
324#2889—611no—
325#2884—611no—
326#1912—612no—
327#1909—612no—
328#1886—612no—
329#1884—612no—
330#1827—612no—
331#1796—612no—
332C2×C61—21no—
333C2×C61—41no—
334C2×C61#230811yescharacteristic
335D61—22no—
336D61—41no—
337D61—42no—
338D61—81no—
339C2×2−1+4—611no—
340C2×2−1+4—611no—
341#568—612no—
342D4×Q8—1222no—
343#547—1222no—
344#547—1222no—
345#547—1222no—
346#536—611no—
347#536—611no—
348#536—611no—
349#536—611no—
350#536—611no—
351#536—611no—
352#532—611no—
353#532—611no—
354#532—611no—
355C2×C4⋊Q8—611no—
356C2×C4⋊Q8—611no—
357C2×C4⋊Q8#692311yescharacteristic, complemented
358#519—612no—
359C4×#41—611no—
360#421—614no—
361#419—614no—
362#413—612no—
363#413—1222no—
364C2×#101—612no—
365C2×#101—1222no—
366#410—612no—
367#410—612no—
368#410—1222no—
369#410—1222no—
370#339—614no—
371H(3,ℤ/4)—614no—
372C61#2178311yescharacteristic, complemented, Sylow
3732−1+4—2444no—
3742−1+4—2444no—
375C2×#41—612no—
376C2×#41—611no—
377C2×#41—611no—
378C2×#41—612no—
379C2×#41—612no—
380C2×#41—611no—
381C2×#41—611no—
382C2×#41—611no—
383C2×#41—611no—
384C2×#41—611no—
385C2×#41—1221no—
386C2×#41—1221no—
387C2×#41—1221no—
388C22×Q8—611no—
389C22×Q8—612no—
390C22×Q8—611no—
391C22×Q8—611no—
392C22×Q8—611no—
393C22×Q8—1221no—
394C22×Q8C2×#692311yescharacteristic
395C22×Q8C2×#692311yescharacteristic
396#131—2444no—
397C2×M16—614no—
398C2×M16—1224no—
399C4⋊Q8—22no—
400C4⋊Q8—611no—
401C4⋊Q8—611no—
402C4⋊Q8—611no—
403C4⋊Q8—611no—
404C4⋊Q8—611no—
405C4⋊Q8—1222no—
406C4⋊Q8—1222no—
407C4⋊Q8C2×#692311yescharacteristic
408C4⋊Q8C2×#692311yescharacteristic
409C4⋊Q8C2×#692311yescharacteristic
410C4⋊Q8C2×#692311yescharacteristic
411#127—611no—
412#124—611no—
413#124—611no—
414#124—611no—
415#124—611no—
416#124—611no—
417#124—611no—
418#123—1222no—
419#123—1222no—
420#123—1222no—
421#123—1222no—
422#123—1222no—
423#123—1222no—
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426C22⋊Q8—1222no—
427C22⋊Q8—1222no—
428C22⋊Q8—1222no—
429C22⋊Q8—1222no—
430#121—1222no—
431#121—1222no—
432#121—1222no—
433C4×Q8—1221no—
434C4×Q8—2444no—
435C4×D4—1222no—
436C4×D4—1221no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4—1221no—
440C4×D4—1221no—
441#117—611no—
442#117—611no—
443#117—611no—
444#117—614no—
445#117—1224no—
446C2×C4⋊C4—22no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1222no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42C2×#692311yescharacteristic
452#104—2444no—
453#103—2444no—
454#101—1224no—
455#101—1222no—
456#100—1222no—
457#99—1222no—
458#99—1222no—
459#99—1224no—
460#41—1224no—
461#41—1224no—
462#41—1224no—
463#41—2442no—
464#41—2442no—
465#41—2444no—
466#41—2444no—
467#41—2444no—
468#41—2444no—
469#41—2444no—
470#41—2442no—
471#41—2442no—
472#41—2444no—
473C2×Q8—21no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—611no—
477C2×Q8—612no—
478C2×Q8—612no—
479C2×Q8—611no—
480C2×Q8—611no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1224no—
484C2×Q8—1224no—
485C2×Q8—2442no—
486C2×Q8—2442no—
487C2×Q8—2444no—
488C2×Q8—2444no—
489C2×Q8C22⋊#692312yes—
490C2×Q8C22×#692311yescharacteristic
491C2×Q8C22×#692311yescharacteristic
492C2×Q8C22×#692311yescharacteristic
493C2×Q8C22×#692311yescharacteristic
494C2×D4—611no—
495C2×D4—611no—
496C2×D4—612no—
497C2×D4—611no—
498C2×D4—612no—
499C2×D4—612no—
500C2×D4—611no—
501C2×D4—611no—
502C2×D4—1222no—
503C2×D4—1221no—
504C2×D4—1221no—
505C2×D4—1221no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—612no—
511C22×C4—611no—
512C22×C4—612no—
513C22×C4—612no—
514C22×C4—612no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4C22⋊#692312yes—
523C22×C4C22×#692311yescharacteristic
524M16—1224no—
525M16—1224no—
526M16—2444no—
527C2×C8—1224no—
528C2×C8—2444no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—1222no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1224no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1221no—
542C4⋊C4—2444no—
543C22⋊C4—1222no—
544C22⋊C4—1221no—
545C22⋊C4—1221no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—1222no—
549C22⋊C4—1221no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1224no—
553C42—21no—
554C42—611no—
555C42—611no—
556C42—1221no—
557C42—1221no—
558C42—1224no—
559C42—2444no—
560C42C22×#692311yescharacteristic
561C42C22×#692311yescharacteristic
562C23—611no—
563C23—612no—
564C23—611no—
565C23—1221no—
566C23C2×C22⋊#692311yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—42no—
572Q8—42no—
573Q8—1224no—
574Q8—1222no—
575Q8—1222no—
576Q8—1222no—
577Q8—2442no—
578Q8—2442no—
579Q8—2442no—
580D4—1224no—
581D4—1222no—
582D4—1222no—
583D4—1222no—
584D4—1222no—
585D4—1222no—
586D4—2442no—
587D4—2444no—
588D4—2442no—
589D4—2442no—
590D4—2442no—
591D4—2442no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—611no—
601C2×C4—614no—
602C2×C4—611no—
603C2×C4—612no—
604C2×C4—1224no—
605C2×C4—1224no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4—2442no—
613C2×C4—2442no—
614C2×C4—2442no—
615C2×C4—2442no—
616C2×C4—2442no—
617C2×C4—2442no—
618C2×C4—2442no—
619C2×C4—2442no—
620C2×C4—2444no—
621C2×C4C2×C22⋊#692311yescharacteristic
622C2×C4C2×C22⋊#692312yes—
623C2×C4C2×C22⋊#692312yes—
624C2×C4C23×#692311yescharacteristic
625C8—2444no—
626C22—21no—
627C22—41no—
628C22—612no—
629C22—1221no—
630C22—1222no—
631C22—2441no—
632C22—2442no—
633C22—2441no—
634C22—2442no—
635C22—2441no—
636C22C22×C22⋊#692311yescharacteristic
637C4—22no—
638C4—21no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—612no—
645C4—1221no—
646C4—1222no—
647C4—4884no—
648C4—2441no—
649C4—2441no—
650C4—2441no—
651C4#1684082912yes—
652C2—21no—
653C2—41no—
654C2—1222no—
655C2—4881no—
656C2—2441no—
657C2—2442no—
658C2#2146229111yescharacteristic
subgroupG/H
centerC2#21462291
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C4⋊Q8×C61C4
socleC2×C61#2308
prime radical#21459602C2
cosocleC2×C4×D61C24
hypercenterC2×C4⋊Q8#6923
hypocenterC61#21783
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21462432C2semidirect
2. #21462420C2semidirect
3. #21461866C2semidirect
4. #21461861C2semidirect
5. #21461825C2semidirect
6. #21461821C2semidirect
7. #21461728C2semidirect
8. #21461671C2semidirect
30. #16840337C4semidirect
31. #16840337C4semidirect
34. #16840336C4semidirect
35. #16840336C4semidirect
36. #16840313C4semidirect
38. #16840312C4semidirect
39. C4⋊Q8×D61C4semidirect
42. C2×C4⋊Q8×C61C4semidirect
357. C2×C4⋊Q8#6923semidirect
372. C61#21783semidirect

Maximal subgroups

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

subgroupindex
#214624322
#214624202
#214618662
#214618612
#214618252
#214618212
#214617282
#214616712
#214596022
#2178361