Finite Group Databasebeta

← #29724351

Subgroups — #29724351

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#21450127C211yesmaximal, characteristic
2D4×C4⋊Dic61C211yesmaximal, characteristic
3#21447378C211yesmaximal, characteristic, complemented
4#21447292C211yesmaximal, characteristic, complemented
5#21447242C211yesmaximal, characteristic, complemented
6#21447242C211yesmaximal, characteristic, complemented
7#21446391C211yesmaximal, characteristic, complemented
8#21446391C211yesmaximal, characteristic, complemented
9#21446009C211yesmaximal, characteristic, complemented
10#21446009C211yesmaximal, characteristic, complemented
11#21445906C211yesmaximal, characteristic, complemented
12#21445906C211yesmaximal, characteristic, complemented
13#21445848C211yesmaximal, characteristic, complemented
14C2×#16839525C211yesmaximal, characteristic, complemented
15C61×#1869C211yesmaximal, characteristic
16C2×#14519758—21no—
17C2×D4×Dic61—21no—
18C22×C4⋊Dic61C2211yescharacteristic
19C22×C4⋊Dic61C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25#16839692—21no—
26#16839687—21no—
27#16839661—21no—
28#16839660C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839637—21no—
31#16839637—21no—
32#16839632C2211yescharacteristic
33#16839623—21no—
34#16839622C2211yescharacteristic
35#16839622C2211yescharacteristic
36#16839622C2211yescharacteristic
37#16839622C2211yescharacteristic
38#16839622C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40#16839525C2211yescharacteristic
41#16839525C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C23×C8×C61C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C23×Dic61—21no—
60C23×Dic61—21no—
61C23×Dic61—21no—
62#14519758—44no—
63D4×Dic61—44no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—41no—
74C2×C4⋊Dic61—41no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91#14519711—22no—
92#14519711—22no—
93#14519711—41no—
94#14519711—41no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×D4×C61D411yescharacteristic
104C23×C4×C61C2311yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C22×C8×C61—22no—
107C22×C8×C61—22no—
108C22×C8×C61—41no—
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C22⋊C4×C61D411yescharacteristic
117C2×C22⋊C4×C61D411yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C22⋊C8×C61—22no—
120C22⋊C8×C61—22no—
121C22⋊C8×C61—41no—
122C24×C61C2×D411yescharacteristic
123C24×C61C2×D411yescharacteristic
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61D814yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61#4112yes—
142C22×C4×C61#4112yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—42no—
153C22×Dic61—42no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61—42no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—81no—
160C22×Dic61—81no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—44no—
173C2×C8×C61#4112yes—
174C4⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C42×C61—21no—
178C42×C61—21no—
179C22⋊Dic61—42no—
180C22⋊Dic61—42no—
181C22⋊Dic61—42no—
182C22⋊Dic61—42no—
183C22⋊Dic61—42no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
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—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61—42no—
202C4⋊Dic61—42no—
203C4⋊Dic61—81no—
204C4⋊Dic61#4112yes—
205#11758196—42no—
206#11758196—42no—
207C4×Dic61—42no—
208C23×C61—21no—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61#12312yes—
219C23×C61C2×D812yes—
220C23×C61C2×D812yes—
221C23×C61#13712yes—
222C23×C61C22×D411yescharacteristic
223D4×C61—28no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61#12312yes—
243C2×C4×C61C2×D812yes—
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61H(5,𝔽2)11yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—41no—
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—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—84no—
275C2×Dic61—84no—
276C2×Dic61—84no—
277C8×C61—22no—
278C8×C61—41no—
279C8×C61—41no—
280C8×C61—41no—
281#24933—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61—22no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61#48014yes—
296C22×C61C2×#12311yescharacteristic
297C22×C61#54611yescharacteristic
298C22×C61#54711yescharacteristic
299C22×C61C22×D811yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57811yescharacteristic
303C4×C61—22no—
304C4×C61—22no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61#52911yescharacteristic
308C4×C61#53911yescharacteristic
309C4×C61#55111yescharacteristic
310C4×C61#55211yescharacteristic
311Dic61—42no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
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#1918—611no—
327#1869D6111yescharacteristic
328#1852—611no—
329#1852—611no—
330#1823—611no—
331#1823—611no—
332#1808—611no—
333C2×#340—611no—
334C2×C61—24no—
335C2×C61—42no—
336C2×C61#241911yescharacteristic
337C2×C61C2×#48011yescharacteristic
338C2×C61#309711yescharacteristic
339C2×C61#329411yescharacteristic
340C2×C61#331211yescharacteristic
341C2×C61#333411yescharacteristic
342C2×C61#334411yescharacteristic
343C23×C8C2×D6111yescharacteristic
344C2×#123—1221no—
345C2×C4×D4—1221no—
346C2×C4×D4C2×D6111yescharacteristic
347C22×C4⋊C4—611no—
348C22×C4⋊C4—611no—
349C22×C4⋊C4—611no—
350C2×#107—611no—
351C2×#107—611no—
352C2×#107—611no—
353C2×#107—611no—
354C2×C22⋊C8C2×D6111yescharacteristic
355C2×C22⋊C8C2×D6111yescharacteristic
356C2×C22⋊C8C2×D6111yescharacteristic
357#398—1221no—
358#397—1221no—
359#389—611no—
360#389—611no—
361#389—1221no—
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⋊C4—611no—
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#340—611no—
382#340—611no—
383#340—611no—
384#336C2×D6111yescharacteristic
385#336C2×D6111yescharacteristic
386C61#2493311yescharacteristic, complemented, Sylow
387C22×D4D24411yescharacteristic
388C23×C4—1221no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4C22×D6111yescharacteristic
392C23×C4C22×D6111yescharacteristic
393C22×C8—22no—
394C22×C8—22no—
395C22×C8—41no—
396C22×C8C22×D6111yescharacteristic
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399C22×C8C22×D6111yescharacteristic
400#123—2444no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—2444no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1222no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—2441no—
424C2×C4⋊C4—2441no—
425C2×C4⋊C4D24411yescharacteristic
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4D24411yescharacteristic
432C2×C22⋊C4D24411yescharacteristic
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435#107—1222no—
436#107—1222no—
437#107—2441no—
438#107—2441no—
439C22⋊C8—22no—
440C22⋊C8—22no—
441C22⋊C8—41no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450C24C2×D24411yescharacteristic
451C24C2×D24411yescharacteristic
452C2×D4—22no—
453C2×D4—22no—
454C2×D4D48814yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
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—4881no—
474C22×C4—4881no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4C2×D24411yescharacteristic
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×D24411yescharacteristic
486C22×C4#1175822412yes—
487C22×C4#1175822412yes—
488C22×C4C23×D6111yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—44no—
501C2×C8#1175822412yes—
502C4⋊C4—22no—
503C4⋊C4—612no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
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—2442no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C22⋊C4—2442no—
533C42—21no—
534C42—21no—
535C42—2442no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23#1451976112yes—
547C23C2×D48812yes—
548C23C2×D48812yes—
549C23#1451981012yes—
550C23C22×D24411yescharacteristic
551D4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—1222no—
571C2×C4—4884no—
572C2×C4—4884no—
573C2×C4—4884no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
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#1451976112yes—
598C2×C4C2×D48812yes—
599C2×C4#41×D6111yescharacteristic
600C2×C4C2×#1175822411yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4#1451982211yescharacteristic
604C2×C4#1451982411yescharacteristic
605C8—22no—
606C8—41no—
607C8—41no—
608C8—41no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22#1684007614yes—
623C22C2×#1451976111yescharacteristic
624C22#1684040711yescharacteristic
625C22#1684041611yescharacteristic
626C22C22×D48811yescharacteristic
627C22C2×#1451981011yescharacteristic
628C22#1684057411yescharacteristic
629C22#1684057911yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—4881no—
635C4—4881no—
636C4—4881no—
637C4—4881no—
638C4—4881no—
639C4—4881no—
640C4—4881no—
641C4—2442no—
642C4#1684029311yescharacteristic
643C4#1684035711yescharacteristic
644C4#1684044311yescharacteristic
645C4#1684044911yescharacteristic
646C2—24no—
647C2—42no—
648C2#2144929411yescharacteristic
649C2C2×#1684007611yescharacteristic
650C2#2145578911yescharacteristic
651C2#2145763311yescharacteristic
652C2#2145787211yescharacteristic
653C2#2145816711yescharacteristic
654C2#2145827611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1869C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#1869D61
hypocenterC61#24933
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
3. #21447378C2semidirect
4. #21447292C2semidirect
5. #21447242C2semidirect
6. #21447242C2semidirect
7. #21446391C2semidirect
8. #21446391C2semidirect
9. #21446009C2semidirect
10. #21446009C2semidirect
11. #21445906C2semidirect
12. #21445906C2semidirect
13. #21445848C2semidirect
14. C2×#16839525C2semidirect
386. C61#24933semidirect

Maximal subgroups

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

subgroupindex
#214501272
D4×C4⋊Dic612
#214473782
#214472922
#214472422
#214463912
#214460092
#214459062
#214458482
C2×#168395252
C61×#18692
#2493361