Finite Group Databasebeta

← #29691323

Subgroups — #29691323

Order 15616.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839916C211yesmaximal, characteristic
2#21448495C211yesmaximal, characteristic
3#21448402C211yesmaximal, characteristic
4#21446687C211yesmaximal, characteristic, complemented
5#21446681C211yesmaximal, characteristic, complemented
6#21446506C211yesmaximal, characteristic, complemented
7#21446493C211yesmaximal, characteristic, complemented
8#21446461C211yesmaximal, characteristic, complemented
9#21446450C211yesmaximal, characteristic, complemented
10#21446160C211yesmaximal, characteristic, complemented
11#21446128C211yesmaximal, characteristic
12#21446021C211yesmaximal, characteristic
13#21445752C211yesmaximal, characteristic
14#21445584C211yesmaximal, characteristic
15C61×#1878C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519753C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21#16839916C418yescomplemented
22C2×#14519711C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839669C2211yescharacteristic
35#16839669C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839666—21no—
39#16839666—21no—
40#16839658C2211yescharacteristic
41#16839657—21no—
42#16839653—21no—
43#16839652—21no—
44#16839646—21no—
45#16839624—21no—
46#16839624—21no—
47#16839619C2211yescharacteristic
48C4×C4⋊Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53C22×Q16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C22×Dic122C2311yescharacteristic
62C2×Q8×D61—22no—
63C2×Q8×D61—41no—
64C2×Q8×D61—41no—
65C2×Q8×D61—41no—
66C2×Q8×D61—41no—
67C2×Q8×D61C2×C414yes—
68C22×C4×D61—21no—
69C22×C4×D61C2311yescharacteristic
70C2×#11758202—41no—
71C2×#11758202—41no—
72C2×#11758202C2×C414yes—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519753—22no—
76#14519753C2×C414yes—
77#14519735—41no—
78#14519735—41no—
79#14519734—41no—
80#14519734—41no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196D411yescharacteristic
87C2×#11758196D411yescharacteristic
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198D411yescharacteristic
92C2×#11758198D411yescharacteristic
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96#14519711—41no—
97#14519708—41no—
98#14519702—41no—
99#14519702C2×C414yes—
100#14519701—41no—
101#14519701—41no—
102#14519701C2×C414yes—
103#14519700—41no—
104#14519689—22no—
105#14519689C2×C414yes—
106#14519687—41no—
107C8×Dic61—41no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×Q8×C61C2311yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C2×Q16×C61—41no—
122C2×Q16×C61—41no—
123C2×Q16×C61C2×C414yes—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C4×Q8×C61—41no—
127C4×Q8×C61—41no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#103×C61—41no—
134#103×C61—41no—
135#95×C61—21no—
136#95×C61—21no—
137C23×D61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C22×C412yes—
149C2×Q8×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164Q8×D61—44no—
165Q8×D61—44no—
166Q8×D61—44no—
167Q8×D61—44no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—24no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61C22×C412yes—
176C2×C4×D61#4112yes—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—24no—
183C2×Dic122—42no—
184C2×Dic122—42no—
185C2×Dic122C22×C412yes—
186Q16×C61—44no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61C22×C412yes—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—42no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—42no—
201#11758202—44no—
202#11758198—24no—
203#11758198—41no—
204#11758198—41no—
205#11758198—42no—
206#11758198—81no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61C22×C412yes—
213#11758196—24no—
214#11758196—41no—
215#11758196—41no—
216#11758196—81no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—42no—
220C4×Dic61—81no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418C22×C412yes—
227C23×C61C22×D411yescharacteristic
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61C4×D412yes—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—41no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—41no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61H(5,𝔽2)11yescharacteristic
264C2×C4×C61H(5,𝔽2)11yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61C4×D412yes—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—48no—
293Dic122—22no—
294Dic122—22no—
295Dic122—48no—
296Dic122—41no—
297Dic122—41no—
298C8×C61—22no—
299C8×C61—41no—
300#92418—22no—
301#92418—41no—
302#22054—611nomaximal, Sylow
303C22×C61C2×C4×D411yescharacteristic
304C22×C61#53411yescharacteristic
305C22×C61#53511yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57811yescharacteristic
308C22×C61#57811yescharacteristic
309C22×C61#57811yescharacteristic
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61#51811yescharacteristic
324C4×C61#51811yescharacteristic
325C4×C61#53811yescharacteristic
326C4×C61#53811yescharacteristic
327Dic61—22no—
328Dic61—42no—
329Dic61—41no—
330Dic61—41no—
331Dic61—42no—
332Dic61—81no—
333Dic61—81no—
334C2×#451—611no—
335#2323—611no—
336#2310—611no—
337#1971—611no—
338#1969—611no—
339#1935—611no—
340#1933—611no—
341#1928—611no—
342#1927—611no—
343#1878—611no—
344#1878D6111yescharacteristic, complemented
345#1873—611no—
346#1854—611no—
347#1787—611no—
348#1750—611no—
349C2×C61#232111yescharacteristic
350C2×C61#294711yescharacteristic
351C2×C61#295011yescharacteristic
352C2×C61#320111yescharacteristic
353C2×C61#320211yescharacteristic
354C2×C61#322311yescharacteristic
355C2×C61#322511yescharacteristic
356D61—42no—
357C23×Q8—611no—
358C22×Q16—611no—
359C22×Q16C2×D6111yescharacteristic
360C2×C22⋊Q8—611no—
361C2×C4×Q8C2×D6111yescharacteristic
362C2×#117—611no—
363C2×#117—611no—
364#451—618no—
365C2×#107—611no—
366C2×#106—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103C2×D6111yescharacteristic
374C2×#103C2×D6111yescharacteristic
375C2×C22⋊C8—611no—
376C2×C22⋊C8—611no—
377C2×#97—611no—
378C2×C4×C8—611no—
379#391—611no—
380#391—611no—
381#391—611no—
382#391—611no—
383#391—1221no—
384#391—1221no—
385#391C2×D6111yescharacteristic
386#389—611no—
387#389—1221no—
388#388—1221no—
389#388—1221no—
390#387—1221no—
391#381—1221no—
392#381—1221no—
393#380—611no—
394C4×C4⋊C4—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#336—611no—
400#336C2×D6111yescharacteristic
401C61#2205411yescharacteristic, complemented, Sylow
402C22×Q8—614no—
403C22×Q8—611no—
404C22×Q8—1221no—
405C22×Q8—1222no—
406C22×Q8—2441no—
407C22×Q8—2441no—
408C22×Q8—2441no—
409C22×Q8—2441no—
410C22×Q8C22×D6111yescharacteristic
411C22×Q8C22×D6111yescharacteristic
412C23×C4—611no—
413C23×C4—1221no—
414C2×Q16—41no—
415C2×Q16—41no—
416C2×Q16—614no—
417C2×Q16—2441no—
418C2×Q16—2441no—
419C2×Q16C4×D6114yes—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C22⋊Q8—614no—
425C22⋊Q8—1222no—
426C4×Q8—41no—
427C4×Q8—41no—
428#117—2441no—
429#117—2441no—
430#117—2441no—
431#117—2441no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C22⋊C4—611no—
443C2×C22⋊C4—611no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—21no—
448C2×C42—611no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42C22×D6111yescharacteristic
452#107—2441no—
453#106—2441no—
454#103—41no—
455#103—41no—
456#103—614no—
457#103—614no—
458#103—2441no—
459#103—2441no—
460#103—2441no—
461#103—2441no—
462C22⋊C8—614no—
463C22⋊C8—1222no—
464#97—2441no—
465C4×C8—2441no—
466#95—21no—
467#95—21no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—611no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—612no—
491C2×Q8—1224no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—2442no—
498C2×Q8—2442no—
499C2×Q8—2444no—
500C2×Q8—2444no—
501C2×Q8—2444no—
502C2×Q8—2444no—
503C2×Q8C2×C4×D6112yes—
504C2×Q8C2×C4×D6112yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—611no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—1224no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4D4×D6111yescharacteristic
526C22×C4D4×D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528Q16—44no—
529Q16—2444no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—612no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8C2×C4×D6112yes—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—42no—
547C4⋊C4—612no—
548C4⋊C4—1224no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—4881no—
555C4⋊C4—2441no—
556C4⋊C4—2441no—
557C22⋊C4—1224no—
558C22⋊C4—4881no—
559C22⋊C4—2441no—
560C22⋊C4—2441no—
561C22⋊C4—2442no—
562C42—21no—
563C42—21no—
564C42—42no—
565C42—1221no—
566C42—1221no—
567C42—4881no—
568C42—2442no—
569C23—612no—
570C23—1221no—
571C23—1221no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23C2×D4×D6111yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—41no—
582Q8—41no—
583Q8—41no—
584Q8—41no—
585Q8—1222no—
586Q8—1222no—
587Q8—2448no—
588Q8—2441no—
589Q8—2441no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—612no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—2448no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4C22×C4×D6111yescharacteristic
635C2×C4#41×D6111yescharacteristic
636C2×C4#1451982211yescharacteristic
637C2×C4#1451982211yescharacteristic
638C2×C4#1451982211yescharacteristic
639C2×C4#1451982311yescharacteristic
640C8—22no—
641C8—41no—
642C8—1222no—
643C8—2441no—
644C22—1222no—
645C22—1222no—
646C22—1222no—
647C22—2441no—
648C22—2441no—
649C22—2441no—
650C22—2441no—
651C22C4×D4×D6111yescharacteristic
652C22#1684032211yescharacteristic
653C22#1684032811yescharacteristic
654C22#1684057111yescharacteristic
655C22#1684057511yescharacteristic
656C22#1684057611yescharacteristic
657C22#1684057811yescharacteristic
658C4—42no—
659C4—42no—
660C4—41no—
661C4—41no—
662C4—81no—
663C4—81no—
664C4—1222no—
665C4—4881no—
666C4—4881no—
667C4—2442no—
668C4—2441no—
669C4—2442no—
670C4—2441no—
671C4#1684023411yescharacteristic
672C4#1684023511yescharacteristic
673C4#1684034411yescharacteristic
674C4#1684034811yescharacteristic
675C2—2442no—
676C2#2144847511yescharacteristic
677C2#2145466111yescharacteristic
678C2#2145468211yescharacteristic
679C2#2145670011yescharacteristic
680C2#2145671211yescharacteristic
681C2#2145689911yescharacteristic
682C2#2145693511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1878C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1878D61
hypocenterC61#22054
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
4. #21446687C2semidirect
5. #21446681C2semidirect
6. #21446506C2semidirect
7. #21446493C2semidirect
8. #21446461C2semidirect
9. #21446450C2semidirect
10. #21446160C2semidirect
15. C61×#1878C2semidirect
21. #16839916C4semidirect
344. #1878D61semidirect
401. C61#22054semidirect

Maximal subgroups

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

subgroupindex
C2×#168399162
#214484952
#214484022
#214466872
#214466812
#214465062
#214464932
#214464612
#214464502
#214461602
#214461282
#214460212
#214457522
#214455842
C61×#18782
#2205461