Finite Group Databasebeta

← #29718821

Subgroups — #29718821

Order 15616.

896 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839916C211yesmaximal, characteristic
2C2×#16839822C211yesmaximal, characteristic
3#21451619C211yesmaximal, characteristic
4#21448402C211yesmaximal, characteristic
5#21447213C211yesmaximal, characteristic, complemented
6#21447200C211yesmaximal, characteristic, complemented
7#21447141C211yesmaximal, characteristic, complemented
8#21446687C211yesmaximal, characteristic, complemented
9#21446681C211yesmaximal, characteristic, complemented
10#21446511C211yesmaximal, characteristic, complemented
11#21446496C211yesmaximal, characteristic, complemented
12#21446128C211yesmaximal, characteristic
13#21446021C211yesmaximal, characteristic
14#21445752C211yesmaximal, characteristic
15C61×#2056C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519753C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21#16839916—24no—
22#16839822—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839703—21no—
37#16839696—21no—
38#16839691—21no—
39#16839689—21no—
40#16839669C2211yescharacteristic
41#16839669C2211yescharacteristic
42#16839668C2211yescharacteristic
43#16839667C2211yescharacteristic
44#16839666—21no—
45#16839666—21no—
46#16839661C2211yescharacteristic
47#16839660C2211yescharacteristic
48#16839624—21no—
49#16839624—21no—
50#16839619C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#391C2211yescharacteristic
60C61×#340C2211yescharacteristic
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×#11758202—41no—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519753—22no—
77#14519753—22no—
78#14519735—41no—
79#14519735—41no—
80#14519734—22no—
81#14519734—22no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196D411yescharacteristic
89C2×#11758196D411yescharacteristic
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198D411yescharacteristic
94C2×#11758198D411yescharacteristic
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97#14519711—41no—
98#14519708—41no—
99C4⋊#92418—22no—
100#14519702—22no—
101#14519702—41no—
102#14519701—22no—
103#14519701—41no—
104#14519701—41no—
105#14519700—22no—
106#14519689—22no—
107#14519689D414yes—
108#14519687—22no—
109C8×Dic61—22no—
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—
119#14519684—21no—
120#14519684—21no—
121C22×Q8×C61C2311yescharacteristic
122C22×Q8×C61C2311yescharacteristic
123C2×Q16×C61—22no—
124C2×Q16×C61—22no—
125C2×Q16×C61—41no—
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—22no—
132#103×C61—22no—
133#103×C61—41no—
134#95×C61—21no—
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—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
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—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—24no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61C2×D412yes—
177C2×C4×D61#4112yes—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—24no—
183C2×Dic122—21no—
184C2×Dic122—42no—
185C2×Dic122—42no—
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×C61C2×D412yes—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200#11758202—44no—
201#11758198—22no—
202#11758198—22no—
203#11758198—42no—
204#11758198—42no—
205#11758198—81no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—81no—
213#11758196—22no—
214#11758196—22no—
215#11758196—42no—
216#11758196—81no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—42no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418C2×D412yes—
226C23×C61C22×D411yescharacteristic
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61#12112yes—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61—41no—
241Q8×C61—41no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
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—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61H(5,𝔽2)11yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
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#12112yes—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—48no—
292Dic122—22no—
293Dic122—22no—
294Dic122—48no—
295Dic122—41no—
296Dic122—41no—
297C8×C61—22no—
298C8×C61—41no—
299#92418—22no—
300#92418—41no—
301#24514—611nomaximal, Sylow
302C22×C61C2×#12111yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61#53511yescharacteristic
305C22×C61#57711yescharacteristic
306C22×C61#57811yescharacteristic
307C22×C61#57811yescharacteristic
308C22×C61#57811yescharacteristic
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—41no—
316C4×C61—42no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—81no—
320C4×C61—81no—
321C4×C61—81no—
322C4×C61#54611yescharacteristic
323C4×C61#54611yescharacteristic
324C4×C61#55111yescharacteristic
325C4×C61#55111yescharacteristic
326Dic61—22no—
327Dic61—42no—
328Dic61—41no—
329Dic61—41no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333C2×#451—611no—
334C2×#433—611no—
335#2625—611no—
336#2310—611no—
337#2068—611no—
338#2066—611no—
339#2056—611no—
340#2056D6111yescharacteristic, complemented
341#1971—611no—
342#1969—611no—
343#1935—611no—
344#1933—611no—
345#1873—611no—
346#1854—611no—
347#1787—611no—
348C2×C61#241111yescharacteristic
349C2×C61#305911yescharacteristic
350C2×C61#306311yescharacteristic
351C2×C61#312111yescharacteristic
352C2×C61#312711yescharacteristic
353C2×C61#322311yescharacteristic
354C2×C61#322511yescharacteristic
355D61—42no—
356C23×Q8—611no—
357C22×Q16—611no—
358C22×Q16C2×D6111yescharacteristic
359C2×C22⋊Q8—611no—
360C2×#117—611no—
361C2×#117—611no—
362#451—1224no—
363#433—1224no—
364C2×#107—611no—
365C2×#106—611no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8C2×D6111yescharacteristic
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103C2×D6111yescharacteristic
375C2×#103C2×D6111yescharacteristic
376C2×C22⋊C8—611no—
377C2×C22⋊C8—611no—
378C2×#97—611no—
379C2×C4×C8—611no—
380#400—1221no—
381#398—1221no—
382#397—1221no—
383#397—1221no—
384#391—611no—
385#391—611no—
386#391—611no—
387#391—611no—
388#391—1221no—
389#391—1221no—
390#391C2×D6111yescharacteristic
391#391C2×D6111yescharacteristic
392#389—611no—
393#389—611no—
394#381—1221no—
395#381—1221no—
396#380—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340C2×D6111yescharacteristic
401C61#2451411yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8—1221no—
404C22×Q8—1222no—
405C22×Q8—1222no—
406C22×Q8—1222no—
407C22×Q8—1222no—
408C22×Q8—2441no—
409C22×Q8—2441no—
410C22×Q8C22×D6111yescharacteristic
411C22×Q8C22×D6111yescharacteristic
412C23×C4—611no—
413C23×C4—1221no—
414C2×Q16—22no—
415C2×Q16—22no—
416C2×Q16—41no—
417C2×Q16—1222no—
418C2×Q16—1222no—
419C2×Q16—2441no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C22⋊Q8—1222no—
425C22⋊Q8—1222no—
426#117—1222no—
427#117—1222no—
428#117—2441no—
429#117—2441no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—611no—
441C2×C22⋊C4—611no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C42—611no—
446C2×C42—1221no—
447C2×C42C22×D6111yescharacteristic
448#107—2441no—
449#106—2441no—
450C4⋊C8—22no—
451C4⋊C8—1222no—
452#103—22no—
453#103—41no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#103—2441no—
458#103—2441no—
459#103—2441no—
460C22⋊C8—614no—
461C22⋊C8—1222no—
462#97—1222no—
463C4×C8—1222no—
464#95—21no—
465#95—21no—
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—21no—
491C2×Q8—612no—
492C2×Q8—1224no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—2442no—
499C2×Q8—2442no—
500C2×Q8—2444no—
501C2×Q8—2444no—
502C2×Q8—2444no—
503C2×Q8—2444no—
504C2×Q8#1175822412yes—
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×C8D4×D6112yes—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—4881no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—4881no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C42—21no—
563C42—21no—
564C42—1221no—
565C42—1221no—
566C42—2442no—
567C23—612no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23—1221no—
573C23—1221no—
574C23C2×D4×D6111yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579Q8—41no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—1222no—
584Q8—1222no—
585Q8—2448no—
586Q8—2441no—
587Q8—2441no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
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—612no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—2448no—
621C2×C4—2441no—
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×C4C2×D4×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4#41×D6111yescharacteristic
635C2×C4C2×#1175822411yescharacteristic
636C2×C4#1451982211yescharacteristic
637C2×C4#1451982311yescharacteristic
638C8—22no—
639C8—41no—
640C8—1222no—
641C8—2441no—
642C22—1222no—
643C22—1222no—
644C22—1222no—
645C22—2441no—
646C22—2441no—
647C22—2441no—
648C22—2441no—
649C22#121×D6111yescharacteristic
650C22#1684027211yescharacteristic
651C22#1684032811yescharacteristic
652C22#1684057111yescharacteristic
653C22#1684057511yescharacteristic
654C22#1684057611yescharacteristic
655C22#1684057811yescharacteristic
656C4—41no—
657C4—41no—
658C4—42no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—1222no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4—2441no—
667C4—2441no—
668C4—2442no—
669C4#1684039911yescharacteristic
670C4#1684040511yescharacteristic
671C4#1684043511yescharacteristic
672C4#1684044011yescharacteristic
673C2—2442no—
674C2#2144919711yescharacteristic
675C2#2145546911yescharacteristic
676C2#2145550711yescharacteristic
677C2#2145604711yescharacteristic
678C2#2145611711yescharacteristic
679C2#2145689911yescharacteristic
680C2#2145693511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2056C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2056D61
hypocenterC61#24514
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2056

Complements

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

NH ≅ G/Ntype
5. #21447213C2semidirect
6. #21447200C2semidirect
7. #21447141C2semidirect
8. #21446687C2semidirect
9. #21446681C2semidirect
10. #21446511C2semidirect
11. #21446496C2semidirect
15. C61×#2056C2semidirect
340. #2056D61semidirect
401. C61#24514semidirect

Maximal subgroups

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

subgroupindex
C2×#168399162
C2×#168398222
#214516192
#214484022
#214472132
#214472002
#214471412
#214466872
#214466812
#214465112
#214464962
#214461282
#214460212
#214457522
C61×#20562
#2451461