Finite Group Databasebeta

← #29723887

Subgroups — #29723887

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839916C211yesmaximal, characteristic
2C2×C4×#11758202C211yesmaximal, characteristic, complemented
3#21449316C211yesmaximal, characteristic
4#21448653C211yesmaximal, characteristic
5#21447369C211yesmaximal, characteristic, complemented
6#21447246C211yesmaximal, characteristic
7#21447139C211yesmaximal, characteristic, complemented
8#21447014C211yesmaximal, characteristic
9#21446498C211yesmaximal, characteristic, complemented
10#21446423C211yesmaximal, characteristic
11#21446157C211yesmaximal, characteristic, complemented
12#21446090C211yesmaximal, characteristic, complemented
13#21445833C211yesmaximal, characteristic, complemented
14#21445752C211yesmaximal, characteristic
15C61×#1878C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519753C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21#16839916—24no—
22C4×#11758202—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839700—21no—
36#16839688—21no—
37#16839669C2211yescharacteristic
38#16839669C2211yescharacteristic
39#16839666—21no—
40#16839666—21no—
41#16839660C2211yescharacteristic
42#16839653—21no—
43#16839646—21no—
44#16839627C2211yescharacteristic
45#16839621—21no—
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48C4×#11758198C2211yescharacteristic
49#16839609—21no—
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—41no—
68C2×Q8×D61—41no—
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C2×#11758202—22no—
72C2×#11758202—22no—
73C2×#11758202D414yes—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519753—22no—
77#14519753—41no—
78C4×Dic122—22no—
79C4×Dic122—41no—
80#14519742—22no—
81#14519742—41no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198D411yescharacteristic
93C2×#11758198D411yescharacteristic
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97#14519710—22no—
98#14519709—41no—
99#14519706—41no—
100#14519702—22no—
101#14519702—22no—
102#14519701—22no—
103#14519701—22no—
104#14519701—22no—
105#14519700—22no—
106#14519689—22no—
107#14519689—41no—
108C4×#92418—22no—
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#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C22×Q8×C61C2311yescharacteristic
122C2×Q16×C61—22no—
123C2×Q16×C61—22no—
124C2×Q16×C61—41no—
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C4×Q8×C61—22no—
128C4×Q8×C61—41no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#103×C61—41no—
135#103×C61—41no—
136#95×C61—21no—
137#95×C61—21no—
138C23×D61C2×D411yescharacteristic
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×C61—21no—
149C2×Q8×C61C2×D412yes—
150C2×Q8×C61#4112yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165Q8×D61—44no—
166Q8×D61—44no—
167Q8×D61—44no—
168Q8×D61—44no—
169C2×C4×D61—24no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—24no—
182C2×Dic122—42no—
183C2×Dic122—42no—
184C2×Dic122C2×D412yes—
185C2×Dic122#4112yes—
186C2×Dic122#4112yes—
187Q16×C61—44no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—42no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—42no—
202#11758202—28no—
203#11758198—22no—
204#11758198—22no—
205#11758198—42no—
206#11758198—42no—
207#11758198—42no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214#11758196—42no—
215#11758196—42no—
216#11758196—42no—
217#11758196—42no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—42no—
221C4×Dic61—42no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418C2×D412yes—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61#13512yes—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—41no—
243Q8×C61—41no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
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×C61#13712yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×Dic61—22no—
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—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C4×D61—48no—
290C4×D61—41no—
291C4×D61—41no—
292C4×D61—41no—
293C4×D61—41no—
294Dic122—22no—
295Dic122—22no—
296Dic122—22no—
297Dic122—48no—
298Dic122#12314yes—
299C8×C61—41no—
300C8×C61—41no—
301#92418—22no—
302#92418—41no—
303#24902—611nomaximal, Sylow
304C22×C61#52511yescharacteristic
305C22×C61D4211yescharacteristic
306C22×C61#54611yescharacteristic
307C22×C61C2×#13511yescharacteristic
308C22×C61C2×#13711yescharacteristic
309C22×C61#57811yescharacteristic
310C22×C61#57811yescharacteristic
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—41no—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61—41no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61C2×#12311yescharacteristic
325C4×C61#53811yescharacteristic
326C4×C61#54611yescharacteristic
327C4×C61#55111yescharacteristic
328Dic61—24no—
329Dic61—41no—
330Dic61—41no—
331Dic61—41no—
332Dic61—81no—
333Dic61—81no—
334Dic61—81no—
335C2×#451—611no—
336C2×C4×Q16—611no—
337#2421—611no—
338Q8×C22⋊C4—611no—
339#2097—611no—
340#2075—611no—
341#2056—611no—
342#2035—611no—
343#1933—611no—
344#1923—611no—
345#1878—611no—
346#1878D6111yescharacteristic, complemented
347#1865—611no—
348#1805—611no—
349#1787—611no—
350C2×C61#241811yescharacteristic
351C2×C61#310811yescharacteristic
352C2×C61#312711yescharacteristic
353C2×C61#327011yescharacteristic
354C2×C61#328511yescharacteristic
355C2×C61#330011yescharacteristic
356C2×C61#332811yescharacteristic
357D61—42no—
358C23×Q8—611no—
359C22×Q16—611no—
360C22×Q16C2×D6111yescharacteristic
361C2×C22⋊Q8—611no—
362C2×C4×Q8—611no—
363C2×C4×Q8—611no—
364C2×C4×Q8C2×D6111yescharacteristic
365#451—1224no—
366C4×Q16—1224no—
367C2×#107—611no—
368C2×#106—611no—
369C2×C4⋊C8—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103C2×D6111yescharacteristic
377C2×#103C2×D6111yescharacteristic
378C2×C22⋊C8—611no—
379C2×C22⋊C8—611no—
380C2×C4×C8—611no—
381#400—1221no—
382#397—1221no—
383#391—611no—
384#391—611no—
385#391—1221no—
386#391—1221no—
387#391C2×D6111yescharacteristic
388#389—611no—
389#388—1221no—
390#387—1221no—
391#382—611no—
392#380—611no—
393#380—611no—
394#380—1221no—
395C4×C22⋊C4—611no—
396C4×C22⋊C4—1221no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340C2×D6111yescharacteristic
401#336—611no—
402#336C2×D6111yescharacteristic
403C61#2490211yescharacteristic, complemented, Sylow
404C22×Q8—611no—
405C22×Q8—1221no—
406C22×Q8—1222no—
407C22×Q8—1222no—
408C22×Q8—1222no—
409C22×Q8—1222no—
410C22×Q8—2441no—
411C22×Q8—2441no—
412C22×Q8C22×D6111yescharacteristic
413C22×Q8C22×D6111yescharacteristic
414C23×C4—611no—
415C23×C4—1221no—
416C2×Q16—22no—
417C2×Q16—22no—
418C2×Q16—41no—
419C2×Q16—614no—
420C2×Q16—1222no—
421C2×Q16—1222no—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8C22×D6111yescharacteristic
425C22×C8C22×D6111yescharacteristic
426C22⋊Q8—1222no—
427C22⋊Q8—2441no—
428C4×Q8—22no—
429C4×Q8—41no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—2441no—
433C4×Q8—2441no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—611no—
445C2×C22⋊C4—611no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C42—21no—
450C2×C42—611no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454#107—1222no—
455#106—2441no—
456C4⋊C8—2441no—
457#103—41no—
458#103—41no—
459#103—1222no—
460#103—1222no—
461#103—1222no—
462#103—1222no—
463#103—1222no—
464#103—1222no—
465C22⋊C8—1222no—
466C22⋊C8—2441no—
467C4×C8—1222no—
468#95—21no—
469#95—21no—
470#95—611no—
471#95—611no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481C24—611no—
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—21no—
491C2×Q8—21no—
492C2×Q8—612no—
493C2×Q8—612no—
494C2×Q8—612no—
495C2×Q8—1224no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—2442no—
500C2×Q8—2442no—
501C2×Q8—2444no—
502C2×Q8—2444no—
503C2×Q8—2444no—
504C2×Q8—2444no—
505C2×Q8#1175822212yes—
506C2×Q8D4×D6112yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—611no—
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—1221no—
524C22×C4—1221no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4D4×D6111yescharacteristic
528C22×C4D4×D6111yescharacteristic
529C22×C4C23×D6111yescharacteristic
530Q16—44no—
531Q16—1228no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—612no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—42no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—2442no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4—2442no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C22⋊C4—2442no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—1221no—
568C42—1221no—
569C42—2442no—
570C42—2442no—
571C23—612no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23C2×D4×D6111yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—41no—
584Q8—41no—
585Q8—41no—
586Q8—41no—
587Q8—614no—
588Q8—1222no—
589Q8—1222no—
590Q8—1222no—
591Q8—2448no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
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—41no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—2448no—
622C2×C4—2441no—
623C2×C4—2441no—
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×C4—2441no—
635C2×C4#1451980912yes—
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4#41×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4C2×#1175822211yescharacteristic
640C2×C4#1451982211yescharacteristic
641C2×C4#1451982211yescharacteristic
642C8—41no—
643C8—41no—
644C8—1222no—
645C8—2441no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—2441no—
650C22—2441no—
651C22—2441no—
652C22—2441no—
653C22#1684027111yescharacteristic
654C22#1684039411yescharacteristic
655C22#1684040911yescharacteristic
656C22#135×D6111yescharacteristic
657C22C2×#1451980911yescharacteristic
658C22#1684057811yescharacteristic
659C22#1684057811yescharacteristic
660C4—22no—
661C4—42no—
662C4—42no—
663C4—41no—
664C4—81no—
665C4—81no—
666C4—1224no—
667C4—4881no—
668C4—4881no—
669C4—4881no—
670C4—2441no—
671C4—2441no—
672C4—2441no—
673C4#123×D6111yescharacteristic
674C4#1684034411yescharacteristic
675C4#1684040211yescharacteristic
676C4#1684044111yescharacteristic
677C2—2442no—
678C2#2144928011yescharacteristic
679C2#2145590311yescharacteristic
680C2#2145611811yescharacteristic
681C2#2145739311yescharacteristic
682C2#2145755811yescharacteristic
683C2#2145771311yescharacteristic
684C2#2145810511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1878C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1878D61
hypocenterC61#24902
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11758202C2semidirect
5. #21447369C2semidirect
7. #21447139C2semidirect
9. #21446498C2semidirect
11. #21446157C2semidirect
12. #21446090C2semidirect
13. #21445833C2semidirect
15. C61×#1878C2semidirect
346. #1878D61semidirect
403. C61#24902semidirect

Maximal subgroups

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

subgroupindex
C2×#168399162
C2×C4×#117582022
#214493162
#214486532
#214473692
#214472462
#214471392
#214470142
#214464982
#214464232
#214461572
#214460902
#214458332
#214457522
C61×#18782
#2490261