Finite Group Databasebeta

← #29743410

Subgroups — #29743410

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839859C211yesmaximal, characteristic
2#21449676C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic
4#21447359C211yesmaximal, characteristic, complemented
5#21447293C211yesmaximal, characteristic, complemented
6#21447213C211yesmaximal, characteristic, complemented
7#21447154C211yesmaximal, characteristic, complemented
8#21446555C211yesmaximal, characteristic, complemented
9#21446380C211yesmaximal, characteristic, complemented
10#21446153C211yesmaximal, characteristic
11#21446151C211yesmaximal, characteristic
12#21446078C211yesmaximal, characteristic
13#21446070C211yesmaximal, characteristic, complemented
14#21446068C211yesmaximal, characteristic
15C2×C61×#478C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C22×#11758196—21no—
21#16839859—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519698C2211yescharacteristic
29C2×#14519687C2211yescharacteristic
30#16839700—21no—
31#16839695—21no—
32#16839668C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839658—21no—
37#16839658C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839643—21no—
40#16839635—21no—
41#16839634—21no—
42#16839631C2211yescharacteristic
43#16839620—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×C4×D4×C61C2211yescharacteristic
55C61×#478—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#102×C61C2211yescharacteristic
60C2×#102×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C23×Dic61—21no—
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200Q814yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519742—22no—
70#14519742—41no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73D4×Dic61—22no—
74D4×Dic61—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—44no—
86C2×#11758196D411yescharacteristic
87C2×#11758196D411yescharacteristic
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519710—22no—
95#14519708—41no—
96#14519702—22no—
97#14519702—22no—
98#14519700—41no—
99#14519698—22no—
100#14519687—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×D4×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115#125×C61—22no—
116#125×C61—41no—
117C4×D4×C61—22no—
118C4×D4×C61—41no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#107×C61—22no—
127#106×C61—22no—
128C4⋊C8×C61—22no—
129#102×C61—22no—
130#102×C61D414yes—
131C24×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61C2×D412yes—
136C2×D4×C61C2×Q812yes—
137C2×D4×C61#4112yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—42no—
142C22×C4×C61—42no—
143C22×C4×C61—81no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—24no—
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—81no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C2×Q812yes—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61C2×D412yes—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61—42no—
180C4⋊C4×C61C2×D412yes—
181C22⋊C4×C61—42no—
182C42×C61—21no—
183C42×C61—21no—
184C22⋊Dic61—24no—
185C22⋊Dic61—42no—
186#11758200—28no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—42no—
193C4⋊Dic61#4112yes—
194#11758196—22no—
195#11758196—41no—
196#11758196—42no—
197#11758196—42no—
198#11758196—42no—
199#11758196—42no—
200#11758196—42no—
201#11758196—81no—
202C4×Dic61—21no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—42no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×Q812yes—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61—41no—
220C23×C61C22×D411yescharacteristic
221D4×C61—22no—
222D4×C61—22no—
223D4×C61—22no—
224D4×C61C22⋊Q814yes—
225C2×C4×C61—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
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×C61—84no—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×Q811yescharacteristic
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—22no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—22no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—48no—
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—84no—
272C2×Dic61AGL(1,ℤ/8)12yes—
273Dic122—22no—
274Dic122—22no—
275Dic122—41no—
276Dic122—41no—
277C8×C61—22no—
278C8×C61—41no—
279#92418—22no—
280#92418—41no—
281#26398—611nomaximal, Sylow
282C22×C61—24no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61#52511yescharacteristic
290C22×C61D4211yescharacteristic
291C22×C61D4×Q811yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61#57511yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57811yescharacteristic
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61C2×C22⋊Q811yescharacteristic
302C4×C61#53911yescharacteristic
303C4×C61#54711yescharacteristic
304C4×C61#55411yescharacteristic
305Dic61—22no—
306Dic61—42no—
307Dic61—41no—
308Dic61—42no—
309Dic61—41no—
310Dic61—41no—
311Dic61—81no—
312Dic61—81no—
313C2×#478D6111yescharacteristic
314C2×#440—611no—
315#2453—611no—
316D4×C4⋊C4—611no—
317#2096—611no—
318#2083—611no—
319#2068—611no—
320#2059—611no—
321#1943—611no—
322#1917—611no—
323#1877—611no—
324#1877—611no—
325#1864—611no—
326#1863—611no—
327#1862—611no—
328C2×C61—44no—
329C2×C61#244911yescharacteristic
330C2×C61#311411yescharacteristic
331C2×C61#312011yescharacteristic
332C2×C61#326911yescharacteristic
333C2×C61#327211yescharacteristic
334C2×C61#338611yescharacteristic
335C2×C61#338811yescharacteristic
336C22×SD16—611no—
337C2×#125C2×D6111yescharacteristic
338C2×C4×Q8—611no—
339C2×C4×D4—611no—
340C2×C4×D4—611no—
341C2×C4×D4C2×D6111yescharacteristic
342C22×C4⋊C4—1221no—
343#478—24no—
344#440—1224no—
345C2×#107—611no—
346C2×#107—611no—
347C2×#107C2×D6111yescharacteristic
348C2×#106—611no—
349C2×#106C2×D6111yescharacteristic
350C2×C4⋊C8C2×D6111yescharacteristic
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#102—611no—
355C2×#102C2×D6111yescharacteristic
356C2×#102C2×D6111yescharacteristic
357C2×#97—611no—
358#400—1221no—
359#398—1221no—
360#391—611no—
361#391—611no—
362#389—611no—
363#389—611no—
364#389—611no—
365#389—611no—
366#389—1221no—
367#387—1221no—
368#385—1221no—
369#385—1221no—
370#384—611no—
371#380—1221no—
372C4×C4⋊C4—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#336—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381C61#2639811yescharacteristic, complemented, Sylow
382C22×Q8—611no—
383C22×D4C22×D6111yescharacteristic
384C23×C4—21no—
385C23×C4—1221no—
386C23×C4—1221no—
387C2×SD16—614no—
388C2×SD16—1222no—
389C2×SD16—1222no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394#125—22no—
395#125—41no—
396C4×Q8—1222no—
397C4×Q8—2441no—
398C4×D4—22no—
399C4×D4—41no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—1222no—
403C4×D4—2441no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—2444no—
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42C22×D6111yescharacteristic
429#107—22no—
430#107—1222no—
431#107—1222no—
432#106—22no—
433#106—2441no—
434C4⋊C8—22no—
435#103—1222no—
436#103—1222no—
437#103—2441no—
438#102—22no—
439#102—1222no—
440#102C22⋊D6114yes—
441#97—1222no—
442#95—611no—
443#95—611no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452C24—21no—
453C2×Q8—612no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4Q8×D6112yes—
463C2×D4#1175822612yes—
464C2×D4C2×C22⋊D6112yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—81no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—1224no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—4881no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4D4×D6111yescharacteristic
486C22×C4D4×D6111yescharacteristic
487C22×C4C23×D6111yescharacteristic
488SD16—1228no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—612no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8C2×C22⋊D6112yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—612no—
509C4⋊C4—1222no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—4881no—
516C4⋊C4—2441no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4C2×C22⋊D6112yes—
524C22⋊C4—42no—
525C22⋊C4—1224no—
526C22⋊C4—2442no—
527C42—21no—
528C42—21no—
529C42—1221no—
530C42—1221no—
531C42—1221no—
532C42—1221no—
533C42—2442no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D6111yescharacteristic
542Q8—1222no—
543Q8—1222no—
544Q8—2441no—
545Q8—2441no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4#1451975314yes—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—612no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1222no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1222no—
580C2×C4—4884no—
581C2×C4—2448no—
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×C4C22×C22⋊D6111yescharacteristic
593C2×C4C2×Q8×D6111yescharacteristic
594C2×C4#41×D6111yescharacteristic
595C2×C4C2×#1175822611yescharacteristic
596C2×C4#1451982211yescharacteristic
597C2×C4#1451982411yescharacteristic
598C8—22no—
599C8—41no—
600C8—1222no—
601C8—2441no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#1684027511yescharacteristic
610C22D4×C22⋊D6111yescharacteristic
611C22#1684042511yescharacteristic
612C22AGL(1,ℤ/8)×D6111yescharacteristic
613C22#1684055911yescharacteristic
614C22#1684057211yescharacteristic
615C22#1684057611yescharacteristic
616C4—41no—
617C4—42no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—1222no—
622C4—4881no—
623C4—4881no—
624C4—2441no—
625C4—2442no—
626C4—2442no—
627C4—2441no—
628C4—2441no—
629C4C2×#1451975311yescharacteristic
630C4#1684036111yescharacteristic
631C4#1684041111yescharacteristic
632C4#1684046611yescharacteristic
633C2—44no—
634C2#2144962411yescharacteristic
635C2#2145596211yescharacteristic
636C2#2145603811yescharacteristic
637C2#2145737411yescharacteristic
638C2#2145741311yescharacteristic
639C2#2145873511yescharacteristic
640C2#2145876811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#478C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#478D61
hypocenterC61#26398
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
2. #21449676C2semidirect
4. #21447359C2semidirect
5. #21447293C2semidirect
6. #21447213C2semidirect
7. #21447154C2semidirect
8. #21446555C2semidirect
9. #21446380C2semidirect
13. #21446070C2semidirect
381. C61#26398semidirect

Maximal subgroups

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

subgroupindex
C2×#168398592
#214496762
D4×#117581962
#214473592
#214472932
#214472132
#214471542
#214465552
#214463802
#214461532
#214461512
#214460782
#214460702
#214460682
C2×C61×#4782
#2639861