Finite Group Databasebeta

← #30611792

Subgroups — #30611792

Order 15616.

1172 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462735C211yesmaximal, characteristic, complemented
2#21462466C211yesmaximal, characteristic, complemented
3#21461331C212yesmaximal, complemented
4#21461330C211yesmaximal, characteristic, complemented
5#21461329C211yesmaximal, characteristic, complemented
6#21461328C212yesmaximal, complemented
7#21461317C212yesmaximal, complemented
8#21461316C211yesmaximal, characteristic, complemented
9D61⋊#332C211yesmaximal, characteristic, complemented
10C2×#16840621C212yesmaximal, complemented
11#21459598C211yesmaximal, characteristic
12C2×#14519864—21no—
13#16840811—21no—
14#16840789—22no—
15D4×#92420—21no—
16#16840780C2211yescharacteristic
17#16840779C2212yes—
18#16840769—21no—
19#16840768—22no—
20#16840767C2211yescharacteristic
21#16840765C2212yes—
22C2×#14519844C2212yes—
23C2×#14519844C2212yes—
24D61⋊(C4⋊C8)C2211yescharacteristic
25D61⋊(C4⋊C8)C2211yescharacteristic
26#16840621C2211yescharacteristic
27#16840621C2212yes—
28#16840621C2212yes—
29#16840621C2212yes—
30#16840621C2212yes—
31#16840621C2211yescharacteristic
32#16840621C2212yes—
33#16840620C2211yescharacteristic
34#16840620C2211yescharacteristic
35#16840620C2212yes—
36C2×#14519824—21no—
37D4×Dic122C411yescharacteristic
38#16840427C411yescharacteristic
39#16840425—21no—
40#16840416C411yescharacteristic
41#16840413C412yes—
42#16840412C412yes—
43#16840411—21no—
44#16840410—22no—
45#16840335—21no—
46#16840334—22no—
47#16840309C2211yescharacteristic
48#16840307C2212yes—
49C2×#14519776C2212yes—
50C4⋊Q8×D61C2211yescharacteristic
51C4×#11758224C2211yescharacteristic
52C61×#547C411yescharacteristic
53#14519864—44no—
54C2×#11758213—21no—
55C2×#11758217—22no—
56C2×D61⋊C8—21no—
57C2×D61⋊C8—21no—
58C22×#92420—22no—
59C22×#92420—22no—
60#14519844—22no—
61#14519844—22no—
62#14519844—21no—
63#14519844C2311yescharacteristic
64#14519844C2312yes—
65#14519844C2311yescharacteristic
66#14519844C2311yescharacteristic
67#14519844C2312yes—
68#14519844C2311yescharacteristic
69C4⋊#92420—21no—
70C4⋊#92420—21no—
71C4⋊#92420—22no—
72#14519834—21no—
73C22⋊#92420—22no—
74C4×#92420—22no—
75C4×#92420—21no—
76#14519824—44no—
77#14519824—44no—
78C2×#11758222—22no—
79C2×#11758222—21no—
80C2×#11758224—21no—
81C2×#11758224—22no—
82C2×#11758224—22no—
83C2×#11758224D411yescharacteristic
84C22×Dic122—22no—
85C22×Dic122—21no—
86C2×Q8×D61—21no—
87C2×Q8×D61—21no—
88#14519777C2×C411yescharacteristic
89#14519776—21no—
90#14519776—22no—
91#14519776—22no—
92#14519776C2×C412yes—
93#14519776C2×C412yes—
94#14519776C2312yes—
95#14519776C2311yescharacteristic
96#14519775—21no—
97#14519773C2311yescharacteristic
98#14519763—22no—
99#14519763—21no—
100#14519762—22no—
101#14519761—22no—
102#14519760—22no—
103#14519759—22no—
104#14519758—22no—
105#14519757—21no—
106#14519756—21no—
107#14519755—22no—
108#14519755—22no—
109#14519754—22no—
110#14519754—22no—
111#14519753—21no—
112#14519752C2×C411yescharacteristic
113#14519751C2×C412yes—
114#14519750—21no—
115#14519749—22no—
116#14519748C2×C411yescharacteristic
117C4×Dic122C2×C411yescharacteristic
118#14519742D411yescharacteristic
119C4×D244C2×C411yescharacteristic
120D4×Dic61C2×C412yes—
121D4×Dic61D411yescharacteristic
122#14519739C2×C412yes—
123C4×C22⋊D61—21no—
124C4×C22⋊D61C2×C412yes—
125#14519737—22no—
126#14519737D412yes—
127#14519735D411yescharacteristic
128#14519733D412yes—
129C2×C4⋊Dic61C2×C412yes—
130C2×#11758196—22no—
131C4⋊C4×D61—21no—
132C4⋊C4×D61C2×C412yes—
133C2×C4×Dic61C2312yes—
134C42×D61C2311yescharacteristic
135C2×#41×C61—21no—
136C4⋊Q8×C61C2×C411yescharacteristic
137#123×C61—22no—
138C22⋊Q8×C61—21no—
139#121×C61C2×C412yes—
140C4×D4×C61C2×C411yescharacteristic
141C4×D4×C61C2×C411yescharacteristic
142C2×C4⋊C4×C61C2×C412yes—
143#41×C61—44no—
144C2×Q8×C61—21no—
145C2×D4×C61—22no—
146C2×D4×C61C22⋊C411yescharacteristic
147C2×D4×C61C22×C411yescharacteristic
148C22×C4×C61—22no—
149C22×C4×C61—21no—
150C22×C4×C61C22×C412yes—
151C2×C22⋊D61—22no—
152C2×C22⋊D61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—21no—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61C2×D412yes—
159C22×Dic61C2×D412yes—
160Q8×D61—42no—
161Q8×D61—42no—
162Q8×D61—44no—
163#11758224—44no—
164#11758224—48no—
165#11758224—44no—
166#11758224—44no—
167#11758224D814yes—
168#11758224SD1614yes—
169#11758222—48no—
170#11758222—42no—
171#11758222—42no—
172C2×D244—21no—
173C2×D244C22×C411yescharacteristic
174C2×C4×D61—22no—
175C2×C4×D61—21no—
176C2×C4×D61—22no—
177C2×C4×D61C22×C411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×Dic122—21no—
181C2×Dic122—22no—
182C2×Dic122—22no—
183C2×Dic122—22no—
184C2×Dic122—21no—
185C2×Dic122—44no—
186C2×Dic122—44no—
187C2×Dic122—44no—
188C2×Dic122C22⋊C411yescharacteristic
189#11758217—44no—
190C2×#92420—21no—
191C2×#92420—22no—
192C2×#92420—21no—
193C2×#92420—21no—
194C2×#92420—21no—
195C2×#92420—22no—
196C2×#92420—44no—
197C2×#92420—82no—
198#11758213—42no—
199D61⋊C8—42no—
200D61⋊C8—81no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61C22⋊C411yescharacteristic
204C4⋊C4×C61C22×C411yescharacteristic
205C4⋊C4×C61C22×C412yes—
206C22⋊C4×C61—21no—
207C22⋊C4×C61—22no—
208C22⋊C4×C61C22⋊C412yes—
209C42×C61C22×C411yescharacteristic
210C22⋊Dic61—21no—
211C22⋊Dic61—22no—
212C22⋊Dic61C22⋊C412yes—
213#11758198—22no—
214#11758198—21no—
215#11758198—21no—
216#11758198—22no—
217#11758198C22⋊C412yes—
218C4⋊Dic61—21no—
219C4⋊Dic61—22no—
220C4⋊Dic61C22⋊C411yescharacteristic
221C4⋊Dic61C22×C412yes—
222C4⋊Dic61C22×C411yescharacteristic
223#11758196—22no—
224#11758196—22no—
225#11758196—22no—
226#11758196—22no—
227#11758196—21no—
228#11758196C22⋊C412yes—
229#11758196C22⋊C412yes—
230C4×Dic61C22×C412yes—
231C4×Dic61C2×D411yescharacteristic
232C4×Dic61C2×D412yes—
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D412yes—
235C4×Dic61C2411yescharacteristic
236C23×C61—21no—
237C23×C61C2×C22⋊C412yes—
238C22×D61—21no—
239C22×D61C2×C22⋊C411yescharacteristic
240Q8×C61—42no—
241D4×C61—41no—
242D4×C61—44no—
243D4×C61#10214yes—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—42no—
247C2×C4×C61—44no—
248C2×C4×C61—42no—
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C23×C411yescharacteristic
253C22⋊D61—44no—
254C22⋊D61—42no—
255C22⋊D61—41no—
256C22⋊D61—41no—
257C22⋊D61—41no—
258C22⋊D61—41no—
259C22⋊D61#10218yes—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—44no—
265C2×Dic61—44no—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C412yes—
269C2×Dic61C2×D814yes—
270C2×Dic61C2×SD1614yes—
271C2×Dic61C22×D411yescharacteristic
272C2×Dic61C22×D411yescharacteristic
273D244—41no—
274D244—42no—
275C4×D61—21no—
276C4×D61—21no—
277C4×D61—41no—
278C4×D61—44no—
279C4×D61—42no—
280C4×D61—44no—
281C4×D61C2×D812yes—
282C4×D61C2×SD1612yes—
283Dic122—42no—
284Dic122—41no—
285Dic122—42no—
286Dic122—41no—
287Dic122—42no—
288Dic122—44no—
289Dic122—44no—
290Dic122#10214yes—
291#92420—41no—
292#92420—41no—
293#92420—42no—
294#92420—42no—
295#92420—41no—
296#92420—41no—
297#17721—611nomaximal, Sylow
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61C2×#10214yes—
304C22×C61C22×C22⋊C411yescharacteristic
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61C2×#10212yes—
311C4×C61—21no—
312C4×C61—22no—
313C4×C61—21no—
314C4×C61—41no—
315C4×C61—42no—
316C4×C61C2×#10212yes—
317Dic61—21no—
318Dic61—22no—
319Dic61—42no—
320Dic61—41no—
321Dic61C2×#10214yes—
322Dic61C22×D811yescharacteristic
323Dic61C22×SD1611yescharacteristic
324Dic61#57711yescharacteristic
325Dic61#57811yescharacteristic
326#3499—611no—
327#2988—611no—
328#2906—611no—
329#1547—611no—
330#1547—612no—
331#1546—612no—
332#1546—611no—
333#1540—611no—
334#1540—612no—
335C2×#332—612no—
336C2×#332—611no—
337C2×C61—24no—
338C2×C61—81no—
339C2×C61#288911yescharacteristic
340C2×C61C22×#10211yescharacteristic
341C2×C61#289811yescharacteristic
342D61—22no—
343D61—81no—
344C2×2−1+4—1221no—
345C2×#131—1221no—
346D4×Q8—611no—
347D4×Q8—611no—
348D4×Q8—1221no—
349#547—612no—
350#547—611no—
351#547—612no—
352#547—1222no—
353#547—1221no—
354#547#692311yescharacteristic
355#536—1221no—
356#536—1222no—
357#532—612no—
358#532—611no—
359C2×C4⋊Q8—611no—
360C2×C4⋊Q8—612no—
361C4×#41—611no—
362C8×Q8—1221no—
363C8×D4—1221no—
364C8×D4—1222no—
365#431—612no—
366#431—611no—
367#424—1222no—
368#424—1221no—
369#423—612no—
370#423—611no—
371C2×C4⋊C8—611no—
372C2×C4⋊C8—612no—
373C2×C4⋊C8—611no—
374C2×C4⋊C8—612no—
375#332—611no—
376#332—612no—
377#332—612no—
378#332—612no—
379#332—612no—
380#332—611no—
381#332—612no—
382#331—611no—
383#331—611no—
384#331—612no—
385C61#1772111yescharacteristic, complemented, Sylow
3862−1+4—2444no—
3872−1+4—2444no—
388C2×#41—21no—
389C2×#41—611no—
390C2×#41—1221no—
391C2×#41—1222no—
392C2×#41—1222no—
393C2×#41—1222no—
394C2×#41—1221no—
395C22×Q8—1222no—
396C22×Q8—1221no—
397C22×Q8—1221no—
398C22×Q8—1221no—
399#131—2444no—
400C2×M16—1222no—
401C2×M16—1221no—
402C22×C8—1222no—
403C22×C8—1221no—
404C22×C8—1222no—
405C22×C8—1221no—
406C4⋊Q8—612no—
407C4⋊Q8—611no—
408C4⋊Q8—612no—
409C4⋊Q8—612no—
410C4⋊Q8—611no—
411C4⋊Q8—1221no—
412C4⋊Q8—1222no—
413C4⋊Q8—1222no—
414C4⋊Q8—1221no—
415C4⋊Q8C2×#692311yescharacteristic
416#127—611no—
417#124—1222no—
418#124—1221no—
419#123—22no—
420#123—1222no—
421#123—1222no—
422#123—1222no—
423#123—1222no—
424#123—1222no—
425C22⋊Q8—21no—
426C22⋊Q8—1222no—
427C22⋊Q8—1222no—
428C22⋊Q8—1222no—
429C22⋊Q8—1222no—
430C22⋊Q8—1221no—
431C22⋊Q8—1221no—
432C22⋊Q8—1221no—
433#121—611no—
434#121—612no—
435#121—611no—
436#121—1221no—
437#121—1222no—
438#121C2×#692312yes—
439C4×Q8—611no—
440C4×Q8—611no—
441C4×D4—611no—
442C4×D4—612no—
443C4×D4—612no—
444C4×D4—612no—
445C4×D4—612no—
446C4×D4—611no—
447C4×D4—1222no—
448C4×D4—1221no—
449C4×D4C2×#692311yescharacteristic
450C4×D4C2×#692311yescharacteristic
451#117—611no—
452#117—612no—
453C2×C4⋊C4—612no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1222no—
457C2×C4⋊C4C2×#692312yes—
458C2×C42—612no—
459C2×C42—611no—
460C4⋊C8—611no—
461C4⋊C8—612no—
462C4⋊C8—611no—
463C4⋊C8—611no—
464C4⋊C8—612no—
465C4⋊C8—611no—
466C4⋊C8—1221no—
467C4⋊C8—1221no—
468C4⋊C8—1222no—
469C4⋊C8—1222no—
470C4⋊C8—1222no—
471C4⋊C8—1221no—
472C22⋊C8—1222no—
473C22⋊C8—1221no—
474C4×C8—1222no—
475C4×C8—1221no—
476#41—44no—
477#41—614no—
478#41—614no—
479#41—2442no—
480#41—2442no—
481#41—2448no—
482#41—2444no—
483#41—2444no—
484#41—2448no—
485#41—2444no—
486C2×Q8—21no—
487C2×Q8—611no—
488C2×Q8—1222no—
489C2×Q8—1222no—
490C2×Q8—1221no—
491C2×Q8—1222no—
492C2×Q8—1221no—
493C2×Q8—2444no—
494C2×Q8—2444no—
495C2×Q8—2444no—
496C2×Q8—2442no—
497C2×Q8—2442no—
498C2×Q8—2444no—
499C2×D4—22no—
500C2×D4—612no—
501C2×D4—611no—
502C2×D4—1221no—
503C2×D4—1221no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1222no—
507C2×D4C22⋊#692311yescharacteristic
508C2×D4C22×#692311yescharacteristic
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—612no—
512C22×C4—611no—
513C22×C4—612no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1221no—
521C22×C4C22×#692312yes—
522M16—2442no—
523M16—2444no—
524C2×C8—1221no—
525C2×C8—1222no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1222no—
530C2×C8—4881no—
531C2×C8—4882no—
532C2×C8—2442no—
533C2×C8—2444no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—612no—
537C4⋊C4—612no—
538C4⋊C4—611no—
539C4⋊C4—611no—
540C4⋊C4—612no—
541C4⋊C4—1222no—
542C4⋊C4—1221no—
543C4⋊C4—1222no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—1221no—
548C4⋊C4C22⋊#692311yescharacteristic
549C4⋊C4C22×#692312yes—
550C4⋊C4C22×#692311yescharacteristic
551C22⋊C4—22no—
552C22⋊C4—21no—
553C22⋊C4—612no—
554C22⋊C4—612no—
555C22⋊C4—1222no—
556C22⋊C4—1221no—
557C22⋊C4—1222no—
558C22⋊C4—1221no—
559C22⋊C4—1221no—
560C22⋊C4—1222no—
561C22⋊C4C22⋊#692312yes—
562C42—612no—
563C42—611no—
564C42—612no—
565C42—611no—
566C42—611no—
567C42—612no—
568C42C22×#692311yescharacteristic
569C23—21no—
570C23—611no—
571C23—1221no—
572C23C2×C22⋊#692312yes—
573Q8—42no—
574Q8—614no—
575Q8—2441no—
576Q8—2441no—
577Q8—2444no—
578Q8—2442no—
579Q8—2442no—
580Q8—2444no—
581Q8—2442no—
582D4—41no—
583D4—44no—
584D4—618no—
585D4—2441no—
586D4—2442no—
587D4—2441no—
588D4—2441no—
589D4—2441no—
590D4—2441no—
591D4—2442no—
592D4—2444no—
593D4#1451983914yes—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—614no—
600C2×C4—614no—
601C2×C4—612no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—611no—
607C2×C4—612no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1221no—
614C2×C4—2441no—
615C2×C4—2444no—
616C2×C4—2444no—
617C2×C4—2444no—
618C2×C4—2444no—
619C2×C4—2442no—
620C2×C4C2×C22⋊#692312yes—
621C2×C4C2×C22⋊#692311yescharacteristic
622C2×C4C2×C22⋊#692311yescharacteristic
623C2×C4C23×#692311yescharacteristic
624C8—2441no—
625C8—2442no—
626C8—2441no—
627C8—2441no—
628C8—2441no—
629C8—2442no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—612no—
636C22—1221no—
637C22—1221no—
638C22—2441no—
639C22—2441no—
640C22—2441no—
641C22C2×#1451983914yes—
642C22C22×C22⋊#692311yescharacteristic
643C4—21no—
644C4—22no—
645C4—21no—
646C4—41no—
647C4—42no—
648C4—611no—
649C4—611no—
650C4—614no—
651C4—611no—
652C4—611no—
653C4—1221no—
654C4—1222no—
655C4—2442no—
656C4—2441no—
657C4C2×#1451983912yes—
658C2—24no—
659C2—81no—
660C2—1222no—
661C2—4881no—
662C2#2146243111yescharacteristic
663C2C22×#1451983911yescharacteristic
664C2#2146244111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#547C4
socleC22×C61C22×C22⋊C4
prime radical#21459598C2
cosocleC4×Dic61C24
hypercenter#547#6923
hypocenterC61#17721
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #21462735C2semidirect
2. #21462466C2semidirect
3. #21461331C2semidirect
4. #21461330C2semidirect
5. #21461329C2semidirect
6. #21461328C2semidirect
7. #21461317C2semidirect
8. #21461316C2semidirect
9. D61⋊#332C2semidirect
10. C2×#16840621C2semidirect
385. C61#17721semidirect

Maximal subgroups

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

subgroupindex
#214627352
#214624662
#214613312
#214613302
#214613292
#214613282
#214613172
#214613162
D61⋊#3322
C2×#168406212
#214595982
#1772161