Finite Group Databasebeta

← #29723895

Subgroups — #29723895

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×#16839919C211yesmaximal, characteristic
2C2×C4×Dic244C211yesmaximal, characteristic, complemented
3#21449323C211yesmaximal, characteristic
4#21448653C211yesmaximal, characteristic
5#21447365C211yesmaximal, characteristic, complemented
6#21447244C211yesmaximal, characteristic
7#21447140C211yesmaximal, characteristic, complemented
8#21447018C211yesmaximal, characteristic
9#21446425C211yesmaximal, characteristic
10#21446163C211yesmaximal, characteristic, complemented
11#21446161C211yesmaximal, characteristic, complemented
12#21446086C211yesmaximal, characteristic, complemented
13#21445833C211yesmaximal, characteristic, complemented
14#21445752C211yesmaximal, characteristic
15C61×#1933C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23#16839919—24no—
24C4×Dic244—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519689C2211yescharacteristic
35C2×#14519689C2211yescharacteristic
36#16839704—21no—
37#16839691—21no—
38#16839669C2211yescharacteristic
39#16839667C2211yescharacteristic
40#16839666—21no—
41#16839666—21no—
42#16839652—21no—
43#16839648—21no—
44#16839627C2211yescharacteristic
45#16839621—21no—
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48C4×#11758198C2211yescharacteristic
49#16839609—21no—
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—22no—
68C2×Q8×D61—41no—
69C2×Q8×D61—41no—
70C22×C4×D61—21no—
71C22×C4×D61C2311yescharacteristic
72C2×Dic244—22no—
73C2×Dic244—22no—
74C2×Dic244D414yes—
75C2×#11758202—22no—
76C2×#11758202—22no—
77C2×#11758202—41no—
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80#14519757—22no—
81#14519757—41no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122—41no—
85C4×Dic122—41no—
86#14519742—22no—
87#14519742—41no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198D411yescharacteristic
98C2×#11758198D411yescharacteristic
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61C2311yescharacteristic
103#14519711—22no—
104#14519708—41no—
105#14519706—41no—
106#14519702—22no—
107#14519702—22no—
108#14519701—41no—
109#14519700—22no—
110#14519700—22no—
111#14519700—41no—
112#14519689—22no—
113#14519689—41no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684D411yescharacteristic
123#14519684D411yescharacteristic
124C22×Q8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132#103×C61—22no—
133#103×C61—22no—
134C4×C8×C61—22no—
135#95×C61—21no—
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#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159Q8×D61—44no—
160Q8×D61—44no—
161Q8×D61—44no—
162Q8×D61—44no—
163C2×C4×D61—24no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—24no—
180C2×Dic122—21no—
181C2×Dic122—42no—
182C2×Dic122—42no—
183C2×Dic122C2×D412yes—
184C2×Dic122C2×D412yes—
185C2×Dic122#4112yes—
186C2×Dic122#4112yes—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61C2×D412yes—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—42no—
200C42×C61—21no—
201C42×C61—21no—
202#11758202—44no—
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—42no—
213#11758196—42no—
214#11758196—42no—
215#11758196—42no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—42no—
219C4×Dic61—42no—
220C4×Dic61—42no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227Dic244—28no—
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—41no—
239Q8×C61—41no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
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#13712yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C61H(5,𝔽2)11yescharacteristic
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
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—
285C4×D61—48no—
286C4×D61—41no—
287C4×D61—41no—
288C4×D61—41no—
289C4×D61—41no—
290Dic122—22no—
291Dic122—22no—
292Dic122—22no—
293Dic122—22no—
294Dic122—22no—
295Dic122—48no—
296Dic122—41no—
297Dic122—41no—
298Dic122#12314yes—
299C8×C61—22no—
300C8×C61—41no—
301#92418—41no—
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—41no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61C2×#12311yescharacteristic
325C4×C61#53811yescharacteristic
326C4×C61#54611yescharacteristic
327C4×C61#55111yescharacteristic
328Dic61—24no—
329Dic61—41no—
330Dic61—42no—
331Dic61—41no—
332Dic61—41no—
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#1933D6111yescharacteristic, complemented
344#1923—611no—
345#1878—611no—
346#1878—611no—
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×Q16—611no—
361C2×C22⋊Q8—611no—
362C2×C4×Q8—611no—
363C2×C4×Q8—611no—
364C2×C4×Q8—611no—
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×C8C2×D6111yescharacteristic
381#400—1221no—
382#397—1221no—
383#391—611no—
384#391—611no—
385#391—1221no—
386#391—1221no—
387#391C2×D6111yescharacteristic
388#389C2×D6111yescharacteristic
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#340C2×D6111yescharacteristic
400#340C2×D6111yescharacteristic
401#336—611no—
402#336—611no—
403C61#2490211yescharacteristic, complemented, Sylow
404C22×Q8—611no—
405C22×Q8—611no—
406C22×Q8—1221no—
407C22×Q8—1222no—
408C22×Q8—1222no—
409C22×Q8—1222no—
410C22×Q8—1222no—
411C22×Q8—2441no—
412C22×Q8—2441no—
413C22×Q8C22×D6111yescharacteristic
414C23×C4—611no—
415C23×C4—1221no—
416C2×Q16—614no—
417C2×Q16—1222no—
418C2×Q16—1222no—
419C2×Q16—1222no—
420C2×Q16—1222no—
421C2×Q16—2441no—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8C22×D6111yescharacteristic
425C22×C8C22×D6111yescharacteristic
426C22⋊Q8—1222no—
427C22⋊Q8—2441no—
428C4×Q8—1222no—
429C4×Q8—1222no—
430C4×Q8—1222no—
431C4×Q8—2441no—
432C4×Q8—2441no—
433C4×Q8—2441no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4C22×D6111yescharacteristic
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—611no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454#107—1222no—
455#106—2441no—
456C4⋊C8—2441no—
457#103—22no—
458#103—22no—
459#103—1222no—
460#103—1222no—
461#103—1222no—
462#103—1222no—
463#103—2441no—
464#103—2441no—
465C22⋊C8—1222no—
466C22⋊C8—2441no—
467C4×C8—22no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—611no—
472#95—611no—
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—612no—
488C2×Q8—612no—
489C2×Q8—612no—
490C2×Q8—612no—
491C2×Q8—1224no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—2442no—
501C2×Q8—2442no—
502C2×Q8—2444no—
503C2×Q8—2444no—
504C2×Q8—2444no—
505C2×Q8—2444no—
506C2×Q8#1175822212yes—
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—1228no—
531Q16—2444no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8D4×D6112yes—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—42no—
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—1221no—
567C42—1221no—
568C42—2442no—
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—41no—
582Q8—41no—
583Q8—614no—
584Q8—1222no—
585Q8—1222no—
586Q8—1222no—
587Q8—1222no—
588Q8—1222no—
589Q8—2448no—
590Q8—2441no—
591Q8—2441no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
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—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—1222no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—2448no—
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#1451980612yes—
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4#41×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4C2×#1175822211yescharacteristic
640C2×C4#1451982311yescharacteristic
641C2×C4#1451982311yescharacteristic
642C8—22no—
643C8—41no—
644C8—2441no—
645C8—2441no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—2441no—
650C22—2441no—
651C22—2441no—
652C22—2441no—
653C22#1684027111yescharacteristic
654C22#1684039611yescharacteristic
655C22#1684040511yescharacteristic
656C22#135×D6111yescharacteristic
657C22C2×#1451980611yescharacteristic
658C22#1684057511yescharacteristic
659C22#1684057611yescharacteristic
660C4—22no—
661C4—42no—
662C4—41no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—1224no—
667C4—4881no—
668C4—4881no—
669C4—2441no—
670C4—2441no—
671C4—2442no—
672C4—2441no—
673C4#123×D6111yescharacteristic
674C4#1684035111yescharacteristic
675C4#1684040811yescharacteristic
676C4#1684043611yescharacteristic
677C2—2442no—
678C2#2144928811yescharacteristic
679C2#2145589711yescharacteristic
680C2#2145611611yescharacteristic
681C2#2145738611yescharacteristic
682C2#2145756311yescharacteristic
683C2#2145770511yescharacteristic
684C2#2145810111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1933C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1933D61
hypocenterC61#24902
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1933

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic244C2semidirect
5. #21447365C2semidirect
7. #21447140C2semidirect
10. #21446163C2semidirect
11. #21446161C2semidirect
12. #21446086C2semidirect
13. #21445833C2semidirect
15. C61×#1933C2semidirect
343. #1933D61semidirect
403. C61#24902semidirect

Maximal subgroups

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

subgroupindex
C2×#168399192
C2×C4×Dic2442
#214493232
#214486532
#214473652
#214472442
#214471402
#214470182
#214464252
#214461632
#214461612
#214460862
#214458332
#214457522
C61×#19332
#2490261