Finite Group Databasebeta

← #29738093

Subgroups — #29738093

Order 15616.

786 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450846C211yesmaximal, characteristic, complemented
2#21448764C211yesmaximal, characteristic
3#21447208C211yesmaximal, characteristic
4#21447206C211yesmaximal, characteristic
5#21446954C211yesmaximal, characteristic
6#21446952C211yesmaximal, characteristic
7#21446769C211yesmaximal, characteristic, complemented
8#21446576C211yesmaximal, characteristic, complemented
9#21446534C211yesmaximal, characteristic, complemented
10#21446390C211yesmaximal, characteristic, complemented
11#21446389C211yesmaximal, characteristic, complemented
12#21445945C211yesmaximal, characteristic, complemented
13#21445818C211yesmaximal, characteristic
14#21445633C211yesmaximal, characteristic
15C61×#1988C211yesmaximal, characteristic, complemented
16C2×#14519776C2211yescharacteristic
17C2×#14519775—21no—
18C2×#14519739C2211yescharacteristic
19C2×#14519734—21no—
20C2×#14519711C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839682—21no—
30#16839681—21no—
31#16839664C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839657—21no—
36#16839653—21no—
37#16839651—21no—
38#16839631C2211yescharacteristic
39#16839631C2211yescharacteristic
40#16839621—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4⋊Q8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519776—22no—
65#14519776—41no—
66#14519775—44no—
67#14519739—22no—
68#14519739Q814yescomplemented
69#14519734—44no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519711—22no—
92#14519710—22no—
93#14519706—22no—
94#14519706—41no—
95#14519699—22no—
96#14519699—41no—
97#14519697—22no—
98#14519697—41no—
99C8×Dic61—22no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×Q8×C61—21no—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C4⋊Q8×C61—22no—
110C4⋊Q8×C61—41no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118C4⋊C8×C61—41no—
119#95×C61—21no—
120C23×D61—21no—
121C2×Q8×C61—42no—
122C2×Q8×C61—42no—
123C2×Q8×C61—81no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244—21no—
142C2×D244—21no—
143C2×D244C2×Q812yes—
144C2×C4×D61—24no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122—81no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61C2×Q812yes—
164C4⋊C4×C61#4112yes—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—42no—
168#11758198—24no—
169#11758198—42no—
170#11758198—42no—
171#11758198—81no—
172#11758198—81no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61#4112yes—
179#11758196—22no—
180#11758196—42no—
181#11758196—42no—
182#11758196—42no—
183#11758196—42no—
184#11758196—42no—
185#11758196—81no—
186#11758196—81no—
187#11758196D814yes—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—42no—
191C4×Dic61—42no—
192C4×Dic61C2×Q812yes—
193C4×Dic61#4112yes—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C23×C61C22×D411yescharacteristic
201C22×D61—22no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208Q8×C61—84no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61C2×D812yes—
227C2×C4×C61C22×Q811yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61C22⋊Q812yes—
250C2×Dic61C22⋊Q812yes—
251C2×Dic61C2×D812yes—
252C2×Dic61C2×D812yes—
253C2×Dic61#13712yes—
254D244—22no—
255D244—22no—
256D244—41no—
257D244—41no—
258C4×D61—48no—
259Dic122—84no—
260C8×C61—41no—
261C8×C61—41no—
262#92418—41no—
263#92418—41no—
264#26020—611nomaximal, Sylow
265C22×C61C2×C22⋊Q811yescharacteristic
266C22×C61#54611yescharacteristic
267C22×C61#54811yescharacteristic
268C22×C61C22×D811yescharacteristic
269C22×C61C2×#13711yescharacteristic
270C22×C61#57711yescharacteristic
271C22×C61#57711yescharacteristic
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—81no—
276C2×D61—81no—
277C2×D61—81no—
278C2×D61—81no—
279C4×C61—22no—
280C4×C61—41no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—81no—
284C4×C61—81no—
285C4×C61#53311yescharacteristic
286C4×C61#54411yescharacteristic
287C4×C61#55511yescharacteristic
288C4×C61#55611yescharacteristic
289Dic61—22no—
290Dic61—22no—
291Dic61—41no—
292Dic61—42no—
293Dic61—81no—
294Dic61—81no—
295Dic61#47414yes—
296#2551—611no—
297#2354—611no—
298#2067—611no—
299#2067—611no—
300#2023—611no—
301#2023—611no—
302#1988—611no—
303#1988D6111yescharacteristic, complemented
304#1947—611no—
305#1939—611no—
306#1918—611no—
307#1918—611no—
308#1834—611no—
309#1802—611no—
310C4×#102—611no—
311C2×C61#243911yescharacteristic
312C2×C61C2×#47411yescharacteristic
313C2×C61#308411yescharacteristic
314C2×C61#329411yescharacteristic
315C2×C61#332911yescharacteristic
316C2×C61#335811yescharacteristic
317C2×C61#336911yescharacteristic
318D61—82no—
319C2×C4⋊Q8—611no—
320C2×C4⋊Q8—1221no—
321C2×C4⋊Q8C2×D6111yescharacteristic
322C2×C4×D4—611no—
323C2×#117—1221no—
324C2×#107—611no—
325C2×#107—611no—
326C2×C4⋊C8—611no—
327C2×C4⋊C8—611no—
328C2×C4⋊C8C2×D6111yescharacteristic
329C2×#102—611no—
330C2×#102—611no—
331C2×#102—611no—
332C2×#102—611no—
333C2×C4×C8—611no—
334#395—1221no—
335#395—1221no—
336#390—611no—
337#390—611no—
338#389—611no—
339#389—611no—
340#389—1221no—
341#388—1221no—
342#388—1221no—
343#384—611no—
344#384—611no—
345#380—1221no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4—611no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4C2×D6111yescharacteristic
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340C2×D6111yescharacteristic
357#340C2×D6111yescharacteristic
358#340C2×D6111yescharacteristic
359#340C2×D6111yescharacteristic
360#336—611no—
361#336—611no—
362C61#2602011yescharacteristic, complemented, Sylow
363C22×Q8—21no—
364C22×Q8—1221no—
365C22×D4—611no—
366C23×C4—1221no—
367C22×C8—611no—
368C22×C8—611no—
369C22×C8C22×D6111yescharacteristic
370C22×C8C22×D6111yescharacteristic
371C4⋊Q8—22no—
372C4⋊Q8—41no—
373C4⋊Q8—1222no—
374C4⋊Q8—2441no—
375C4⋊Q8—2444no—
376C4×D4—614no—
377C4×D4—1222no—
378#117—2444no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6111yescharacteristic
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—21no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407#107—1222no—
408#107—1222no—
409C4⋊C8—41no—
410C4⋊C8—1222no—
411C4⋊C8—2441no—
412#102—1222no—
413#102—1222no—
414#102—2441no—
415#102—2441no—
416C4×C8—1222no—
417#95—21no—
418#95—611no—
419#95—611no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424C24—1221no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—81no—
428C2×Q8—4881no—
429C2×Q8—2442no—
430C2×Q8—2442no—
431C2×D4—612no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—1224no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4C23×D6111yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—614no—
474C4⋊C4—612no—
475C4⋊C4—1222no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—4881no—
482C4⋊C4—4881no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C4⋊C4#1175822212yes—
489C4⋊C4Q8×D6112yes—
490C22⋊C4—1224no—
491C22⋊C4—4881no—
492C22⋊C4—4881no—
493C22⋊C4—2442no—
494C22⋊C4—2442no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—612no—
499C42—612no—
500C42—1221no—
501C42—1221no—
502C42—2442no—
503C42—2442no—
504C23—1222no—
505C23—2441no—
506C23—2441no—
507C23—2441no—
508C23—2441no—
509C23—2441no—
510C23—2441no—
511C23C2×D4×D6111yescharacteristic
512Q8—84no—
513Q8—4884no—
514D4—1222no—
515D4—1222no—
516D4—2441no—
517D4—2441no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1222no—
548C2×C4—2448no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4D8×D6112yes—
558C2×C4C2×Q8×D6111yescharacteristic
559C2×C4#41×D6111yescharacteristic
560C2×C4#41×D6111yescharacteristic
561C2×C4C2×#1175822211yescharacteristic
562C2×C4#1451982211yescharacteristic
563C2×C4#1451982511yescharacteristic
564C8—41no—
565C8—41no—
566C8—2441no—
567C8—2441no—
568C22—4881no—
569C22—4881no—
570C22—4881no—
571C22—4881no—
572C22—2442no—
573C22—2442no—
574C22—2442no—
575C22C22⋊Q8×D6111yescharacteristic
576C22#1684040211yescharacteristic
577C22#1684042211yescharacteristic
578C22C2×D8×D6111yescharacteristic
579C22#137×D6111yescharacteristic
580C22#1684056811yescharacteristic
581C22#1684057111yescharacteristic
582C4—22no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—614no—
589C4—1222no—
590C4—1222no—
591C4—4881no—
592C4—4881no—
593C4—2442no—
594C4—2441no—
595C4#1684031611yescharacteristic
596C4#1684038911yescharacteristic
597C4#1684047411yescharacteristic
598C4#1684048411yescharacteristic
599C2—4882no—
600C2#2144950711yescharacteristic
601C2#474×D6111yescharacteristic
602C2#2145567811yescharacteristic
603C2#2145762811yescharacteristic
604C2#2145810911yescharacteristic
605C2#2145845311yescharacteristic
606C2#2145856211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1988D61
hypocenterC61#26020
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. #21450846C2semidirect
7. #21446769C2semidirect
8. #21446576C2semidirect
9. #21446534C2semidirect
10. #21446390C2semidirect
11. #21446389C2semidirect
12. #21445945C2semidirect
15. C61×#1988C2semidirect
68. #14519739Q8semidirect
303. #1988D61semidirect
362. C61#26020semidirect

Maximal subgroups

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

subgroupindex
#214508462
#214487642
#214472082
#214472062
#214469542
#214469522
#214467692
#214465762
#214465342
#214463902
#214463892
#214459452
#214458182
#214456332
C61×#19882
#2602061