Finite Group Databasebeta

← #29724358

Subgroups — #29724358

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450125C211yesmaximal, characteristic
2C4⋊C4×D244C211yesmaximal, characteristic
3#21447378C211yesmaximal, characteristic, complemented
4#21447290C211yesmaximal, characteristic, complemented
5#21447246C211yesmaximal, characteristic, complemented
6#21447246C211yesmaximal, characteristic, complemented
7#21446390C211yesmaximal, characteristic, complemented
8#21446110C211yesmaximal, characteristic
9#21446005C211yesmaximal, characteristic, complemented
10#21446005C211yesmaximal, characteristic, complemented
11#21445904C211yesmaximal, characteristic, complemented
12#21445904C211yesmaximal, characteristic, complemented
13#21445847C211yesmaximal, characteristic, complemented
14#340×D61C211yesmaximal, characteristic, complemented
15C61×#1918C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C2×#14519762—21no—
18C2×C4×D244—21no—
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29#16839695—21no—
30#16839691—21no—
31#16839660—21no—
32#16839660C2211yescharacteristic
33#16839636—21no—
34#16839636—21no—
35#16839632C2211yescharacteristic
36#16839624—21no—
37#16839619C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839619C2211yescharacteristic
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×D244D411yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C2×C8×D61—22no—
68C2×C8×D61—22no—
69C2×C8×D61—41no—
70#14519762—44no—
71C4×D244—22no—
72C4×D244—22no—
73C4×D244—44no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—41no—
90C4⋊C4×D61—41no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198D411yescharacteristic
97C2×#11758198D411yescharacteristic
98#14519710—22no—
99#14519710—41no—
100#14519710—41no—
101#14519689—22no—
102#14519689—22no—
103#14519689—41no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#107×C61—22no—
120#95×C61—21no—
121#95×C61—21no—
122C23×D61C2×D411yescharacteristic
123C23×D61C2×D411yescharacteristic
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×Dic61—21no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C2×D244—22no—
138C2×D244—22no—
139C2×D244D814yes—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—42no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—81no—
158C2×C4×D61—81no—
159C2×C4×D61#4112yes—
160C2×C4×D61#4112yes—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—42no—
175C4⋊C4×C61#4112yes—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—42no—
179#11758198—22no—
180#11758198—22no—
181#11758198—42no—
182#11758198—42no—
183#11758198—42no—
184#11758198—42no—
185#11758198—42no—
186C4⋊Dic61—22no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61—81no—
197#11758196—42no—
198#11758196—42no—
199#11758196—42no—
200#11758196—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418#4112yes—
207C8×D61—44no—
208C23×C61C22×D411yescharacteristic
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61#12312yes—
220C22×D61C2×D812yes—
221C22×D61C2×D812yes—
222C22×D61#13712yes—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61C2×D812yes—
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61#12312yes—
265D244—28no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—84no—
275C4×D61—84no—
276C4×D61—84no—
277C8×C61—41no—
278C8×C61—41no—
279#92418—22no—
280#92418—41no—
281#24933—611nomaximal, Sylow
282C22×C61C2×#12311yescharacteristic
283C22×C61#54611yescharacteristic
284C22×C61#54711yescharacteristic
285C22×C61C22×D811yescharacteristic
286C22×C61C2×#13711yescharacteristic
287C22×C61#57711yescharacteristic
288C22×C61#57811yescharacteristic
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—41no—
301C2×D61—41no—
302C2×D61#48014yes—
303C4×C61—22no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61—81no—
309C4×C61#52911yescharacteristic
310C4×C61#53911yescharacteristic
311C4×C61#55111yescharacteristic
312C4×C61#55211yescharacteristic
313Dic61—22no—
314Dic61—41no—
315Dic61—81no—
316Dic61—81no—
317Dic61—81no—
318Dic61—81no—
319#2486—611no—
320D4×C4⋊C4—611no—
321#2099—611no—
322#2083—611no—
323#2075—611no—
324#2075—611no—
325#1918—611no—
326#1918D6111yescharacteristic, complemented
327#1869—611no—
328#1852—611no—
329#1852—611no—
330#1823—611no—
331#1823—611no—
332#1808—611no—
333C2×#340—611no—
334C2×C61#241911yescharacteristic
335C2×C61C2×#48011yescharacteristic
336C2×C61#309711yescharacteristic
337C2×C61#329411yescharacteristic
338C2×C61#331211yescharacteristic
339C2×C61#333411yescharacteristic
340C2×C61#334411yescharacteristic
341D61—24no—
342D61—42no—
343C23×C8—611no—
344C2×#123—1221no—
345C2×C4×D4—611no—
346C2×C4×D4—1221no—
347C22×C4⋊C4—611no—
348C22×C4⋊C4—611no—
349C22×C4⋊C4—611no—
350C2×#107—611no—
351C2×#107—611no—
352C2×#107—611no—
353C2×#107C2×D6111yescharacteristic
354C2×C22⋊C8—611no—
355C2×C22⋊C8—611no—
356C2×C22⋊C8—611no—
357#398—1221no—
358#397—1221no—
359#389—611no—
360#389—1221no—
361#389C2×D6111yescharacteristic
362#385—1221no—
363#385—1221no—
364#384—611no—
365#381—1221no—
366#380—611no—
367#380—611no—
368#380—611no—
369#380—611no—
370#380—611no—
371C4×C4⋊C4C2×D6111yescharacteristic
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340C2×D6111yescharacteristic
382#340C2×D6111yescharacteristic
383#340C2×D6111yescharacteristic
384#336—611no—
385#336C2×D6111yescharacteristic
386C61#2493311yescharacteristic, complemented, Sylow
387C22×D4—611no—
388C23×C4—611no—
389C23×C4—611no—
390C23×C4—1221no—
391C23×C4—1221no—
392C23×C4—1221no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8—1222no—
396C22×C8—1222no—
397C22×C8—2441no—
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400#123—2444no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—2444no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1222no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—2441no—
421C2×C4⋊C4—2441no—
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C22⋊C4—611no—
427C2×C22⋊C4—611no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—21no—
434C2×C42C22×D6111yescharacteristic
435#107—22no—
436#107—1222no—
437#107—2441no—
438#107—2441no—
439C22⋊C8—1222no—
440C22⋊C8—1222no—
441C22⋊C8—2441no—
442#95—21no—
443#95—21no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450C24—611no—
451C24—611no—
452C2×D4—614no—
453C2×D4—1222no—
454C2×D4—1222no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—611no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—611no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—4881no—
478C22×C4—4881no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4—2442no—
486C22×C4D4×D6111yescharacteristic
487C22×C4D4×D6111yescharacteristic
488C22×C4C23×D6111yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—612no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—2444no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—42no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—4881no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C4⋊C4—2442no—
525C4⋊C4#1175822212yes—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C42—21no—
534C42—21no—
535C42—42no—
536C23—612no—
537C23—612no—
538C23—612no—
539C23—612no—
540C23—1221no—
541C23—1221no—
542C23—1221no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23C2×D4×D6111yescharacteristic
551D4—1228no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—612no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—4884no—
580C2×C4—4884no—
581C2×C4—4884no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4D8×D6112yes—
599C2×C4#41×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4#41×D6111yescharacteristic
602C2×C4C2×#1175822211yescharacteristic
603C2×C4#1451982311yescharacteristic
604C2×C4#1451982511yescharacteristic
605C8—41no—
606C8—41no—
607C8—1222no—
608C8—2441no—
609C22—614no—
610C22—1222no—
611C22—1222no—
612C22—1222no—
613C22—1222no—
614C22—1222no—
615C22—1222no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—2441no—
620C22—2441no—
621C22—2441no—
622C22—2441no—
623C22#123×D6111yescharacteristic
624C22#1684040811yescharacteristic
625C22#1684041411yescharacteristic
626C22C2×D8×D6111yescharacteristic
627C22#137×D6111yescharacteristic
628C22#1684056911yescharacteristic
629C22#1684057811yescharacteristic
630C4—22no—
631C4—41no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4—1222no—
637C4—4881no—
638C4—4881no—
639C4—4881no—
640C4—4881no—
641C4—2441no—
642C4#1684029011yescharacteristic
643C4#1684036011yescharacteristic
644C4#1684043711yescharacteristic
645C4#1684045311yescharacteristic
646C2—1224no—
647C2—2442no—
648C2#2144930111yescharacteristic
649C2#480×D6111yescharacteristic
650C2#2145578511yescharacteristic
651C2#2145763411yescharacteristic
652C2#2145787511yescharacteristic
653C2#2145816511yescharacteristic
654C2#2145828111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1918C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenter#1918D61
hypocenterC61#24933
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
3. #21447378C2semidirect
4. #21447290C2semidirect
5. #21447246C2semidirect
6. #21447246C2semidirect
7. #21446390C2semidirect
9. #21446005C2semidirect
10. #21446005C2semidirect
11. #21445904C2semidirect
12. #21445904C2semidirect
13. #21445847C2semidirect
14. #340×D61C2semidirect
15. C61×#1918C2semidirect
326. #1918D61semidirect
386. C61#24933semidirect

Maximal subgroups

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

subgroupindex
#214501252
C4⋊C4×D2442
#214473782
#214472902
#214472462
#214463902
#214461102
#214460052
#214459042
#214458472
#340×D612
C61×#19182
#2493361