Finite Group Databasebeta

← #29738866

Subgroups — #29738866

Order 15616.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839803C211yesmaximal, characteristic
2#21449528C211yesmaximal, characteristic
3#21449517C211yesmaximal, characteristic
4#21447298C211yesmaximal, characteristic, complemented
5#21447293C211yesmaximal, characteristic, complemented
6#21447154C211yesmaximal, characteristic, complemented
7#21447149C211yesmaximal, characteristic, complemented
8#21446557C211yesmaximal, characteristic, complemented
9#21446553C211yesmaximal, characteristic, complemented
10#21446015C211yesmaximal, characteristic
11#21446010C211yesmaximal, characteristic
12#21445974C211yesmaximal, characteristic, complemented
13#21445849C211yesmaximal, characteristic
14#21445604C211yesmaximal, characteristic
15C2×C61×#501C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519753—21no—
18C2×#14519735C2211yescharacteristic
19C2×C4⋊C4×D61C2211yescharacteristic
20#16839803—24no—
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839697—21no—
30#16839695—21no—
31#16839684—21no—
32#16839681—21no—
33#16839661C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839658—21no—
38#16839643—21no—
39#16839633—21no—
40#16839631C2211yescharacteristic
41#16839631C2211yescharacteristic
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4×#11758198C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×#125×C61C2211yescharacteristic
55C61×#501—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C2×#106×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61C2311yescharacteristic
63C2×#11758189—22no—
64C2×#11758189—22no—
65C2×#11758189Q814yescomplemented
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519753—44no—
69#14519735—22no—
70#14519735—41no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C4⋊C4×D61—22no—
85C4⋊C4×D61—41no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198D411yescharacteristic
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519711—22no—
94#14519710—22no—
95#14519709—22no—
96#14519708—22no—
97#14519708—41no—
98#14519708—41no—
99#14519689Q814yescomplemented
100C8×Dic61—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×Q8×C61—21no—
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C4⋊Q8×C61—22no—
113C4⋊Q8×C61—41no—
114#125×C61—22no—
115#125×C61—41no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124#107×C61—22no—
125#107×C61D414yes—
126#106×C61—22no—
127#106×C61—22no—
128#97×C61—22no—
129C23×D61C2×D411yescharacteristic
130C2×Q8×C61—42no—
131C2×Q8×C61—42no—
132C2×Q8×C61—81no—
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×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61—81no—
154C2×C4×D61C2×Q812yes—
155C2×C4×D61C2×Q812yes—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C2×D412yes—
159C2×C8×C61C2×Q812yes—
160C2×C8×C61C2×Q812yes—
161C2×C8×C61#4112yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61C2×D412yes—
172C4⋊C4×C61C2×D412yes—
173C42×C61—21no—
174C42×C61—21no—
175#11758198—22no—
176#11758198—22no—
177#11758198—42no—
178#11758198—42no—
179#11758198—81no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—42no—
189C4⋊Dic61—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193#11758196—42no—
194#11758196—81no—
195#11758196—81no—
196#11758196—81no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418C2×Q812yes—
205C2×#92418C2×Q812yes—
206#11758189—28no—
207C23×C61C22×D411yescharacteristic
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61C4⋊Q812yes—
215Q8×C61—84no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×Q811yescharacteristic
236C2×C4×C61C22×Q811yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61H(5,𝔽2)11yescharacteristic
239C2×C4×C612−1+411yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61C4⋊Q812yes—
259C2×Dic61AGL(1,ℤ/8)12yes—
260C2×Dic61#13712yes—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—84no—
266C8×C61—22no—
267C8×C61C22⋊Q814yescomplemented
268#92418—22no—
269#92418—22no—
270#26078—611nomaximal, Sylow
271C22×C61C2×C4⋊Q811yescharacteristic
272C22×C61D4211yescharacteristic
273C22×C61#54711yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57711yescharacteristic
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—41no—
285C4×C61—42no—
286C4×C61—41no—
287C4×C61—42no—
288C4×C61—41no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61C2×C22⋊Q811yescharacteristic
292C4×C61#54411yescharacteristic
293C4×C61D4×Q811yescharacteristic
294C4×C61#55411yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—22no—
298Dic61—81no—
299Dic61—81no—
300Dic61—81no—
301Dic61—81no—
302C2×#501D6111yescharacteristic, complemented
303C2×#428—611no—
304#2441—611no—
305#2440—611no—
306#2084—611no—
307#2083—611no—
308#2059—611no—
309#2058—611no—
310#1943—611no—
311#1942—611no—
312#1854—611no—
313#1853—611no—
314#1844—611no—
315#1809—611no—
316#1753—611no—
317C2×C61#244011yescharacteristic
318C2×C61#316311yescharacteristic
319C2×C61#316511yescharacteristic
320C2×C61#327211yescharacteristic
321C2×C61#327811yescharacteristic
322C2×C61#334411yescharacteristic
323C2×C61#334511yescharacteristic
324D61—42no—
325C22×M16—611no—
326C2×C4⋊Q8C2×D6111yescharacteristic
327C2×#125C2×D6111yescharacteristic
328C2×C22⋊Q8—1221no—
329C2×#117—611no—
330C22×C4⋊C4—611no—
331#501—24no—
332#428—1224no—
333C2×#107—611no—
334C2×#107—611no—
335C2×#107C2×D6111yescharacteristic
336C2×#107C2×D6111yescharacteristic
337C2×#106—611no—
338C2×#106—611no—
339C2×#106—611no—
340C2×#106—611no—
341C2×#106C2×D6111yescharacteristic
342C2×#106C2×D6111yescharacteristic
343C2×C22⋊C8—611no—
344C2×#97C2×D6111yescharacteristic
345C2×C4×C8—611no—
346#399—1221no—
347#398—1221no—
348#396—1221no—
349#395—1221no—
350#389—611no—
351#389—611no—
352#389—611no—
353#389—611no—
354#389—1221no—
355#387—1221no—
356#385—1221no—
357#384—611no—
358#384—611no—
359#380—611no—
360#380—611no—
361C4×C22⋊C4—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#336—611no—
369#336—611no—
370C61#2607811yescharacteristic, complemented, Sylow
371C22×Q8—21no—
372C23×C4—611no—
373C23×C4—1221no—
374C2×M16—614no—
375C2×M16—1222no—
376C2×M16—1222no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8C22×D6111yescharacteristic
380C22×C8C22×D6111yescharacteristic
381C4⋊Q8—22no—
382C4⋊Q8—41no—
383#125—22no—
384#125—41no—
385C22⋊Q8—2444no—
386#117—1222no—
387#117—2441no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—2441no—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C4⋊C4C22×D6111yescharacteristic
408C2×C4⋊C4C22×D6111yescharacteristic
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—611no—
411C2×C22⋊C4—611no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C42—611no—
416C2×C42—1221no—
417C2×C42C22×D6111yescharacteristic
418#107—22no—
419#107—1222no—
420#107—1222no—
421#107C22⋊D6114yes—
422#106—22no—
423#106—22no—
424#106—1222no—
425#106—1222no—
426#106—2441no—
427#106—2441no—
428C22⋊C8—614no—
429#97—22no—
430C4×C8—1222no—
431#95—611no—
432#95—611no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438#95—1221no—
439C24—611no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—81no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—4881no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—1228no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—612no—
470C2×C8—612no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8Q8×D6112yes—
476C2×C8Q8×D6112yes—
477C2×C8#1175822612yes—
478C2×C8C2×C22⋊D6112yes—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—4881no—
497C4⋊C4—4881no—
498C4⋊C4—4881no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4—2442no—
503C4⋊C4—2442no—
504C4⋊C4—2442no—
505C4⋊C4C2×C22⋊D6112yes—
506C4⋊C4C2×C22⋊D6112yes—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—4881no—
510C22⋊C4—2442no—
511C22⋊C4—2442no—
512C42—21no—
513C42—21no—
514C42—1221no—
515C42—1221no—
516C42—2442no—
517C23—612no—
518C23—1221no—
519C23—1221no—
520C23—1221no—
521C23—1221no—
522C23—1221no—
523C23—1221no—
524C23C2×D4×D6111yescharacteristic
525Q8—84no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—4884no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4C22×C22⋊D6111yescharacteristic
571C2×C4C2×Q8×D6111yescharacteristic
572C2×C4C2×Q8×D6111yescharacteristic
573C2×C4C2×#1175822611yescharacteristic
574C2×C4#1451982211yescharacteristic
575C2×C4#1451982511yescharacteristic
576C8—22no—
577C8—1222no—
578C8—1222no—
579C8#1451975314yescomplemented
580C22—1222no—
581C22—1222no—
582C22—1222no—
583C22—2441no—
584C22—2441no—
585C22—2441no—
586C22—2441no—
587C22C4⋊Q8×D6111yescharacteristic
588C22D4×C22⋊D6111yescharacteristic
589C22#1684041511yescharacteristic
590C22AGL(1,ℤ/8)×D6111yescharacteristic
591C22#137×D6111yescharacteristic
592C22#1684057111yescharacteristic
593C22#1684057211yescharacteristic
594C4—41no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—1222no—
601C4—1222no—
602C4—1222no—
603C4—4881no—
604C4—4881no—
605C4—4881no—
606C4—4881no—
607C4C2×#1451975311yescharacteristic
608C4#1684038911yescharacteristic
609C4Q8×C22⋊D6111yescharacteristic
610C4#1684047311yescharacteristic
611C2—2442no—
612C2#2144951711yescharacteristic
613C2#2145634611yescharacteristic
614C2#2145636411yescharacteristic
615C2#2145741311yescharacteristic
616C2#2145747911yescharacteristic
617C2#2145827311yescharacteristic
618C2#2145828811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#501D61
hypocenterC61#26078
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
4. #21447298C2semidirect
5. #21447293C2semidirect
6. #21447154C2semidirect
7. #21447149C2semidirect
8. #21446557C2semidirect
9. #21446553C2semidirect
12. #21445974C2semidirect
15. C2×C61×#501C2semidirect
65. C2×#11758189Q8semidirect
99. #14519689Q8semidirect
267. C8×C61C22⋊Q8semidirect
302. C2×#501D61semidirect
370. C61#26078semidirect
579. C8#14519753semidirect

Maximal subgroups

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

subgroupindex
C2×#168398032
#214495282
#214495172
#214472982
#214472932
#214471542
#214471492
#214465572
#214465532
#214460152
#214460102
#214459742
#214458492
#214456042
C2×C61×#5012
#2607861