Finite Group Databasebeta

← #29686901

Subgroups — #29686901

Order 15616.

882 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840035C211yesmaximal, characteristic
2#21454643C211yesmaximal, characteristic
3#137×Dic61C211yesmaximal, characteristic
4C2×#16839854C211yesmaximal, characteristic, complemented
5#131×Dic61C211yesmaximal, characteristic
6#21446731C211yesmaximal, characteristic, complemented
7#21446603C211yesmaximal, characteristic, complemented
8#21446475C211yesmaximal, characteristic, complemented
9#21446471C211yesmaximal, characteristic, complemented
10#21446334C211yesmaximal, characteristic
11#21445881C211yesmaximal, characteristic
12#21445720C211yesmaximal, characteristic, complemented
13#101×Dic61C211yesmaximal, characteristic, complemented
14#21445614C211yesmaximal, characteristic, complemented
15C61×#1908C211yesmaximal, characteristic
16C2×#14519807C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#41×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22#16840035C418yescomplemented
23#16839869—21no—
24#16839866—21no—
25#16839863—21no—
26#16839859—21no—
27Q16×Dic61—21no—
28#16839854—21no—
29#16839854—21no—
30#16839854—21no—
31#16839854—21no—
32#16839854—21no—
33#16839853—21no—
34SD16×Dic61—21no—
35C2×#14519712C2211yescharacteristic
36C2×#14519712C2211yescharacteristic
37#16839804C2211yescharacteristic
38#16839802C2211yescharacteristic
39C2×#14519710C2211yescharacteristic
40#16839772C2211yescharacteristic
41#16839771C2211yescharacteristic
42#16839763C2211yescharacteristic
43C2×#14519702C2211yescharacteristic
44C2×#14519696C2211yescharacteristic
45C2×#14519695C2211yescharacteristic
46C2×#14519695C2211yescharacteristic
47#16839720—21no—
48#16839720C2211yescharacteristic
49#16839717—21no—
50#16839717C2211yescharacteristic
51M16×Dic61—21no—
52M16×Dic61C2211yescharacteristic
53M16×Dic61C2211yescharacteristic
54M16×Dic61C2211yescharacteristic
55C2×C8×Dic61—21no—
56C2×C8×Dic61C2211yescharacteristic
57#16839531C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60#16839527C2211yescharacteristic
61C2×#137×C61C2211yescharacteristic
62C2×#131×C61C2211yescharacteristic
63C61×#417C2211yescharacteristic
64C2×#103×C61C2211yescharacteristic
65C2×#101×C61C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#341C2211yescharacteristic
68C22×Dic122C2311yescharacteristic
69#14519807—22no—
70#14519807C2×C414yes—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758202—21no—
74C2×#11758202—22no—
75C2×#11758202C2×C414yescomplemented
76C2×#11758200—21no—
77#14519784—22no—
78#14519784C2×C414yes—
79C2×#11758194—21no—
80C2×#11758194C2311yescharacteristic
81C2×#11758194C2311yescharacteristic
82C2×#11758194C2311yescharacteristic
83C22×#92418—21no—
84C22×#92418C2311yescharacteristic
85Q8×Dic61—21no—
86Q8×Dic61—21no—
87Q8×Dic61—21no—
88Q8×Dic61—21no—
89Q8×Dic61D411yescharacteristic
90Q8×Dic61D411yescharacteristic
91#14519742—21no—
92#14519742—21no—
93#14519742—21no—
94#14519742—21no—
95D4×Dic61—21no—
96D4×Dic61—21no—
97D4×Dic61D411yescharacteristic
98D4×Dic61D411yescharacteristic
99#14519736—21no—
100#14519736C2311yescharacteristic
101C2×C4⋊Dic61—21no—
102C2×C4⋊Dic61C2311yescharacteristic
103C2×#11758196—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108#14519712—22no—
109#14519712C2×C414yescomplemented
110#14519711—21no—
111#14519710—21no—
112#14519710—21no—
113#14519710—21no—
114#14519709—21no—
115#14519708—21no—
116#14519702—21no—
117#14519702—21no—
118#14519702—21no—
119#14519702—21no—
120#14519701—21no—
121#14519701—21no—
122#14519701—21no—
123#14519698—21no—
124#14519696C2×C414yescomplemented
125#14519695—22no—
126#14519695C2×C414yescomplemented
127#14519687—21no—
128#14519687—21no—
129#14519687—21no—
130#14519687—21no—
131#14519687—21no—
132C8×Dic61—21no—
133C8×Dic61—21no—
134C8×Dic61—21no—
135C8×Dic61—21no—
136C8×Dic61—21no—
137C8×Dic61D411yescharacteristic
138C8×Dic61D411yescharacteristic
139C8×Dic61D411yescharacteristic
140C8×Dic61D411yescharacteristic
141C2×#41×C61C2311yescharacteristic
142C22×Q8×C61C2311yescharacteristic
143#137×C61—22no—
144#137×C61C2×C414yes—
145C2×Q16×C61—21no—
146C2×SD16×C61—21no—
147#131×C61—22no—
148#131×C61—22no—
149C2×M16×C61—21no—
150C2×M16×C61C2311yescharacteristic
151C2×M16×C61C2311yescharacteristic
152C2×M16×C61C2311yescharacteristic
153C22×C8×C61—21no—
154C22×C8×C61C2311yescharacteristic
155C2×C4⋊C4×C61C2311yescharacteristic
156#103×C61—21no—
157#103×C61—21no—
158#103×C61—21no—
159#102×C61—21no—
160#101×C61—22no—
161#41×C61—21no—
162#41×C61—21no—
163#41×C61C22×C412yes—
164#41×C61#4112yes—
165C2×Q8×C61—21no—
166C2×Q8×C61—21no—
167C2×Q8×C61—21no—
168C2×Q8×C61—21no—
169C2×Q8×C61—21no—
170C2×Q8×C61C22×C412yes—
171C2×Q8×C61C2×D411yescharacteristic
172C2×D4×C61—21no—
173C2×D4×C61C2×D411yescharacteristic
174C22×C4×C61—21no—
175C22×C4×C61—21no—
176C22×C4×C61—21no—
177C22×C4×C61C2411yescharacteristic
178C22×Dic61—21no—
179C22×Dic61—21no—
180C22×Dic61—21no—
181C22×Dic61—21no—
182C22×Dic61#4111yescharacteristic
183C22×Dic61#4111yescharacteristic
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122C22×C412yes—
190Q16×C61—24no—
191SD16×C61—24no—
192M16×C61—24no—
193M16×C61—21no—
194M16×C61—21no—
195M16×C61—21no—
196M16×C61—21no—
197M16×C61C22×C412yes—
198M16×C61#4112yes—
199C2×C8×C61—24no—
200C2×C8×C61—21no—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C2×C8×C61—21no—
204C2×C8×C61—21no—
205C2×C8×C61C2×D411yescharacteristic
206C2×C8×C61C2×D411yescharacteristic
207C4⋊C4×C61—21no—
208C4⋊C4×C61—21no—
209C22⋊Dic61—21no—
210C22⋊Dic61—21no—
211C22⋊Dic61—21no—
212#11758202—24no—
213#11758202—24no—
214#11758202—24no—
215#11758200—24no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—41no—
221C4⋊Dic61—41no—
222C4⋊Dic61C2×D411yescharacteristic
223C4⋊Dic61C2×D411yescharacteristic
224#11758196—41no—
225#11758196—41no—
226C4×Dic61—21no—
227C4×Dic61—21no—
228C4×Dic61—21no—
229C4×Dic61—41no—
230C4×Dic61—41no—
231C4×Dic61—41no—
232C4×Dic61—41no—
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237#11758194—21no—
238#11758194—21no—
239#11758194—24no—
240#11758194—21no—
241#11758194—21no—
242#11758194C22×C412yes—
243#11758194C22×C412yes—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—24no—
247C2×#92418—21no—
248C2×#92418C22×C412yes—
249C2×#92418C2×D411yescharacteristic
250C2×#92418C2×D411yescharacteristic
251C2×#92418#4112yes—
252C23×C61—21no—
253C23×C61C2×#4111yescharacteristic
254Q8×C61—21no—
255Q8×C61—22no—
256Q8×C61—22no—
257Q8×C61—41no—
258Q8×C61—41no—
259Q8×C61C4×D412yes—
260D4×C61—22no—
261D4×C61—22no—
262D4×C61—21no—
263D4×C61C4×D412yes—
264C2×C4×C61—21no—
265C2×C4×C61—22no—
266C2×C4×C61—22no—
267C2×C4×C61—22no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—81no—
272C2×C4×C61C23×C411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—41no—
283C2×Dic61—42no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—42no—
287C2×Dic61—41no—
288C2×Dic61—81no—
289C2×Dic61#12111yescharacteristic
290C2×Dic61#12111yescharacteristic
291C2×Dic61#12111yescharacteristic
292C2×Dic61#12111yescharacteristic
293Dic122—22no—
294Dic122—22no—
295Dic122—41no—
296Dic122—41no—
297C8×C61—22no—
298C8×C61—21no—
299C8×C61—22no—
300C8×C61—21no—
301C8×C61—41no—
302#92418—22no—
303#92418—22no—
304#92418—41no—
305#92418C4×D412yes—
306#92418C4×D412yes—
307C4×#473—611nomaximal, Sylow
308C22×C61—21no—
309C22×C61—21no—
310C22×C61—22no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61C4×#4111yescharacteristic
314C22×C61C2×#12111yescharacteristic
315C22×C61#52911yescharacteristic
316C4×C61—22no—
317C4×C61—42no—
318C4×C61—81no—
319C4×C61C2×C4×D411yescharacteristic
320C4×C61C2×C4×D411yescharacteristic
321C4×C61#52511yescharacteristic
322C4×C61#53211yescharacteristic
323Dic61—21no—
324Dic61—21no—
325Dic61—21no—
326Dic61—41no—
327Dic61—41no—
328Dic61—42no—
329Dic61—81no—
330Dic61#47312yescomplemented
331C2×#473—611no—
332C4×#137—611no—
333C4×#137—611no—
334C2×C4×Q16—611no—
335C4×#131—611no—
336#1979—611no—
337#1952—611no—
338#1930—611no—
339#1930—611no—
340#1908—611no—
341#1908D6111yescharacteristic
342#1816—611no—
343C4×#108—611no—
344C4×#101—611no—
345C4×#101—611no—
346C2×C61—21no—
347C2×C61—21no—
348C2×C61—42no—
349C2×C61C4×#12111yescharacteristic
350C2×C61C2×#47311yescharacteristic
351C2×C61#306711yescharacteristic
352C2×#137—611no—
353C2×#137C2×D6111yescharacteristic
354C22×Q16—611no—
355C2×#131—611no—
356C2×#131C2×D6111yescharacteristic
357C4×#41—611no—
358C2×C4×Q8—611no—
359C2×C4×Q8—611no—
360#473—618no—
361#441—1221no—
362#441—1221no—
363#440—1221no—
364#440—1221no—
365C4×Q16—1221no—
366C4×Q16—1221no—
367C4×Q16—1221no—
368C4×Q16—1221no—
369C4×Q16—1221no—
370C4×Q16—1221no—
371C4×SD16—1221no—
372C4×SD16—1221no—
373C2×#108—611no—
374C2×#108—611no—
375#428—611no—
376#428—611no—
377C2×#107—611no—
378#419—611no—
379#419—611no—
380#417—611no—
381#417C2×D6111yescharacteristic
382C2×#103—611no—
383C2×#103C2×D6111yescharacteristic
384C2×#101—611no—
385C2×#101—611no—
386C2×#101—611no—
387C2×#101C2×D6111yescharacteristic
388#405—611no—
389#405—611no—
390#405—1221no—
391#405—1221no—
392C4×M16—611no—
393C4×M16—611no—
394C4×M16—611no—
395C4×M16—1221no—
396C2×C4×C8—611no—
397C2×C4×C8—1221no—
398#343—611no—
399#343—611no—
400#343—611no—
401#343C2×D6111yescharacteristic
402#341—611no—
403#341C2×D6111yescharacteristic
404C61C4×#47311yescharacteristic, complemented, Sylow
405C2×#41C22×D6111yescharacteristic
406C22×Q8—611no—
407C22×Q8C22×D6111yescharacteristic
408#137—22no—
409#137—614no—
410#137—1222no—
411#137C4×D6114yes—
412C2×Q16—21no—
413C2×Q16—614no—
414C2×Q16—1221no—
415C2×Q16—1221no—
416C2×Q16—1222no—
417C2×Q16—1221no—
418C2×SD16—21no—
419C2×SD16—1221no—
420#131—22no—
421#131—22no—
422#131—614no—
423#131—1222no—
424C2×M16—21no—
425C2×M16—611no—
426C2×M16—611no—
427C2×M16—611no—
428C2×M16—1221no—
429C2×M16C22×D6111yescharacteristic
430C2×M16C22×D6111yescharacteristic
431C2×M16C22×D6111yescharacteristic
432C22×C8—21no—
433C22×C8—611no—
434C22×C8—1221no—
435C22×C8C22×D6111yescharacteristic
436C4×Q8—611no—
437C4×Q8—611no—
438C4×Q8—1221no—
439C4×Q8—1221no—
440C4×Q8—1221no—
441C4×Q8—1221no—
442C4×Q8—1221no—
443C4×Q8—1221no—
444C4×Q8—1221no—
445C4×Q8—1221no—
446C4×D4—611no—
447C4×D4—611no—
448C4×D4—1221no—
449C4×D4—1221no—
450#117—611no—
451#117—1221no—
452C2×C4⋊C4—611no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C42—611no—
457C2×C42—1221no—
458C2×C42—1221no—
459C2×C42—1221no—
460#108—614no—
461#108—1222no—
462#107—1221no—
463#107—1221no—
464#107—1221no—
465#107—1221no—
466#106—1221no—
467#106—1221no—
468#103—21no—
469#103—21no—
470#103—21no—
471#103—1221no—
472#103—1221no—
473#103—1221no—
474#103—1221no—
475#103—1221no—
476#103—1221no—
477#103—1221no—
478#102—21no—
479#102—1221no—
480#101—22no—
481#101—614no—
482#101—614no—
483#101—1222no—
484#97—1221no—
485#97—1221no—
486#97—1221no—
487#97—1221no—
488#97—1221no—
489C4×C8—611no—
490C4×C8—611no—
491C4×C8—611no—
492C4×C8—611no—
493C4×C8—1221no—
494C4×C8—1221no—
495C4×C8—1221no—
496C4×C8—1221no—
497C4×C8—1221no—
498#41—21no—
499#41—21no—
500#41C2×C4×D6112yes—
501#41#1175822412yes—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—612no—
508C2×Q8—1221no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8C2×C4×D6112yes—
514C2×Q8D4×D6111yescharacteristic
515C2×D4—21no—
516C2×D4D4×D6111yescharacteristic
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4C23×D6111yescharacteristic
527Q16—24no—
528Q16—1224no—
529Q16—1224no—
530Q16—1224no—
531SD16—24no—
532SD16—1224no—
533M16—24no—
534M16—21no—
535M16—21no—
536M16—21no—
537M16—21no—
538M16—612no—
539M16—612no—
540M16—1224no—
541M16—1221no—
542M16—1221no—
543M16—1221no—
544M16—1221no—
545M16C2×C4×D6112yes—
546M16#1175822412yes—
547C2×C8—24no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—612no—
554C2×C8—611no—
555C2×C8—611no—
556C2×C8—612no—
557C2×C8—1224no—
558C2×C8—1221no—
559C2×C8—1221no—
560C2×C8—1221no—
561C2×C8D4×D6111yescharacteristic
562C2×C8D4×D6111yescharacteristic
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—611no—
566C4⋊C4—611no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4—1221no—
571C4⋊C4—2441no—
572C4⋊C4—2441no—
573C4⋊C4—2441no—
574C4⋊C4—2441no—
575C22⋊C4—1221no—
576C22⋊C4—1221no—
577C22⋊C4—1221no—
578C42—611no—
579C42—611no—
580C42—611no—
581C42—611no—
582C42—1221no—
583C42—1221no—
584C42—1221no—
585C42—2441no—
586C42—2441no—
587C42—2441no—
588C42—2441no—
589C23—21no—
590C23#41×D6111yescharacteristic
591Q8—21no—
592Q8—22no—
593Q8—22no—
594Q8—41no—
595Q8—41no—
596Q8—1222no—
597Q8—1222no—
598Q8—2441no—
599Q8—2441no—
600Q8#1451973712yes—
601D4—22no—
602D4—22no—
603D4—21no—
604D4#1451973712yes—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—81no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—611no—
616C2×C4—611no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—4881no—
622C2×C4—2442no—
623C2×C4—2441no—
624C2×C4—2442no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4C22×C4×D6111yescharacteristic
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C2×D4×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4C2×#1175822411yescharacteristic
633C2×C4C2×#1175822411yescharacteristic
634C8—22no—
635C8—21no—
636C8—22no—
637C8—21no—
638C8—41no—
639C8—612no—
640C8—612no—
641C8—1222no—
642C8—1222no—
643C8—2441no—
644C22—21no—
645C22—21no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22C4×#1175822411yescharacteristic
650C22#121×D6111yescharacteristic
651C22#1684028911yescharacteristic
652C4—22no—
653C4—42no—
654C4—81no—
655C4—612no—
656C4—1221no—
657C4—1221no—
658C4—1221no—
659C4—4881no—
660C4—2442no—
661C4—2441no—
662C4—2441no—
663C4C4×D4×D6111yescharacteristic
664C4C2×#1451973711yescharacteristic
665C4#1684026811yescharacteristic
666C4#1684030711yescharacteristic
667C2—21no—
668C2—21no—
669C2—42no—
670C2#2144832911yescharacteristic
671C2#473×D6111yescharacteristic
672C2#2145553411yescharacteristic
subgroupG/H
centerC22#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1908C2
socleC22×C61C2×#121
prime radicalC2×#41×C61C23
cosocleC22×C4×C61C24
hypercenter#1908D61
hypocenterC61C4×#473
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1908

Complements

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

NH ≅ G/Ntype
4. C2×#16839854C2semidirect
6. #21446731C2semidirect
7. #21446603C2semidirect
8. #21446475C2semidirect
9. #21446471C2semidirect
12. #21445720C2semidirect
13. #101×Dic61C2semidirect
14. #21445614C2semidirect
22. #16840035C4semidirect
75. C2×#11758202C2×C4semidirect
109. #14519712C2×C4semidirect
124. #14519696C2×C4semidirect
126. #14519695C2×C4semidirect
330. Dic61#473semidirect
404. C61C4×#473semidirect

Maximal subgroups

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

subgroupindex
C2×#168400352
#214546432
#137×Dic612
C2×#168398542
#131×Dic612
#214467312
#214466032
#214464752
#214464712
#214463342
#214458812
#214457202
#101×Dic612
#214456142
C61×#19082
C4×#47361