Finite Group Databasebeta

← #30622192

Subgroups — #30622192

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840801C212yesmaximal
2#21462348C211yesmaximal, characteristic, complemented
3D4×C4⋊#6923C211yesmaximal, characteristic
4#21461946C211yesmaximal, characteristic, complemented
5#21461930C211yesmaximal, characteristic, complemented
6#21461910C212yesmaximal, complemented
7#21461812C212yesmaximal
8#21461783C212yesmaximal, complemented
9#21461616C211yesmaximal, characteristic, complemented
10C4×#14519839C211yesmaximal, characteristic
11#461×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2212yes—
13C2×D4×#6923C2212yes—
14C22×C4⋊#6923—21no—
15#16840801—28no—
16C2×#14519846C2212yes—
17C2×#14519840C2212yes—
18C2×#14519840C2212yes—
19C2×#14519839C2211yescharacteristic
20C2×#14519839C2211yescharacteristic
21C2×#14519831C2212yes—
22C4×D61⋊C8C2211yescharacteristic
23(C2×#6923)⋊Q8C2212yes—
24#16840713—21no—
25#16840713C2212yes—
26#16840711—22no—
27#16840709—22no—
28#16840708C2211yescharacteristic
29#16840704—21no—
30C4⋊C4×#6923—21no—
31C4×C4⋊#6923C2211yescharacteristic
32C4×C4⋊#6923C2211yescharacteristic
33#16840636C2212yes—
34#16840635C2211yescharacteristic
35#16840635C2211yescharacteristic
36(C2×#6923)⋊C8C2212yes—
37C2×SD16×D61C2212yes—
38C4⋊Q8×D61C2211yescharacteristic
39C4×D4×D61C2211yescharacteristic
40#16839968C411yescharacteristic, complemented
41#16839967C411yescharacteristic, complemented
42#16839966C412yescomplemented
43#16839965C412yescomplemented
44#16839964C411yescharacteristic, complemented
45C4⋊C8×D61C2211yescharacteristic
46#103×D61C2212yes—
47C61×#461C411yescharacteristic, complemented
48C23×#6923—22no—
49C2×D61⋊C8C2311yescharacteristic
50C2×D61⋊C8C2311yescharacteristic
51Q8×#6923—24no—
52Q8×#6923—42no—
53D4×#6923—24no—
54D4×#6923—42no—
55C2×C4⋊#6923—22no—
56C2×C4⋊#6923—22no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—44no—
60C2×C4⋊#6923D411yescharacteristic
61C2×C4⋊#6923D411yescharacteristic
62C2×C4⋊#6923C2312yes—
63C2×C4⋊#6923C2311yescharacteristic
64C2×C4⋊#6923C2311yescharacteristic
65C2×C22⋊#6923—22no—
66C2×C4×#6923—22no—
67C2×C4×#6923—21no—
68C2×C4×#6923C2312yes—
69#14519846—24no—
70#14519840—24no—
71#14519840—42no—
72#14519839—22no—
73#14519839D414yes—
74#14519831—24no—
75C4×#92420—21no—
76C4×#92420—21no—
77(C2×#6923)⋊C4—22no—
78(C2×#6923)⋊C4—22no—
79(C2×#6923)⋊C4—22no—
80(C2×#6923)⋊C4—21no—
81(C2×#6923)⋊C4D412yes—
82C2×Q8×D61C2312yes—
83C2×D4×D61C2311yescharacteristic
84C22×C4×D61—21no—
85C2×#11758190C2×C412yes—
86C2×#11758200C2×C412yes—
87C2×#11758201C2×C412yes—
88SD16×D61—24no—
89SD16×D61—24no—
90C2×C8×D61C2312yes—
91#14519777C2×C411yescharacteristic, complemented
92#14519776—21no—
93#14519776—21no—
94#14519775C2×C412yescomplemented
95C4×D244C2×C411yescharacteristic
96D4×Dic61C2×C411yescharacteristic
97#14519739C2×C411yescharacteristic
98C4×C22⋊D61—21no—
99#14519737—21no—
100C4⋊C4×D61—21no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61C2311yescharacteristic
103C22⋊C4×D61—21no—
104C42×D61C2311yescharacteristic
105C4⋊#92418C2×C411yescharacteristic, complemented
106#14519706C2×C412yescomplemented
107#14519702C2×C412yescomplemented
108#14519701C2×C412yescomplemented
109#14519700C2×C412yescomplemented
110C2×SD16×C61C2×C412yes—
111C4⋊Q8×C61C2×C411yescharacteristic, complemented
112C4×D4×C61C2×C411yescharacteristic
113C4⋊C8×C61C2×C411yescharacteristic, complemented
114#103×C61C2×C412yescomplemented
115C23×D61—21no—
116C22×#6923—21no—
117C22×#6923—21no—
118C22×#6923—22no—
119C22×#6923—44no—
120C22×#6923—44no—
121C22×#6923—82no—
122C22×#6923C2×D412yes—
123C22×#6923C2×D412yes—
124C2×Q8×C61C22×C412yes—
125C2×D4×C61C22×C411yescharacteristic
126C22×C4×C61—21no—
127C2×C22⋊D61—21no—
128C22×Dic61—21no—
129Q8×D61—22no—
130Q8×D61—22no—
131Q8×D61—22no—
132Q8×D61—22no—
133D4×D61—21no—
134D4×D61—21no—
135D4×D61C2×D412yes—
136D4×D61#4112yes—
137C2×D244C22×C411yescharacteristic
138C2×C4×D61—21no—
139C2×C4×D61—22no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—42no—
143C2×C4×D61—42no—
144C2×C4×D61C2×D411yescharacteristic
145C2×C4×D61C2×D411yescharacteristic
146C2×C4×D61C2411yescharacteristic
147C2×Dic122—22no—
148C2×Dic122C22×C412yes—
149C22⋊#6923—44no—
150C2×#92420—21no—
151C2×#92420—21no—
152C4⋊#6923—22no—
153C4⋊#6923—22no—
154C4⋊#6923—22no—
155C4⋊#6923—21no—
156C4⋊#6923—21no—
157C4⋊#6923—22no—
158C4⋊#6923—21no—
159C4⋊#6923—44no—
160C4⋊#6923—44no—
161C4⋊#6923—42no—
162C4⋊#6923—81no—
163C4⋊#6923C2×D412yes—
164C4×#6923—22no—
165C4×#6923—22no—
166C4×#6923—44no—
167C4×#6923—42no—
168D61⋊C8—21no—
169D61⋊C8—21no—
170D61⋊C8—21no—
171D61⋊C8C2×D412yes—
172SD16×C61—24no—
173C2×C8×C61C22×C412yes—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C22×C411yescharacteristic
177C22⋊C4×C61—21no—
178C42×C61C22×C411yescharacteristic
179C22⋊Dic61—21no—
180#11758201—24no—
181#11758200—24no—
182#11758198—21no—
183#11758198—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61C22×C411yescharacteristic
187#11758196—22no—
188#11758196—22no—
189#11758196—21no—
190C4×Dic61C22×C411yescharacteristic
191C4×Dic61#4111yescharacteristic
192C4×Dic61#4111yescharacteristic
193C2×#92418C22×C412yes—
194#11758190—24no—
195C8×D61—22no—
196C8×D61—22no—
197C23×C61—21no—
198C22×D61—21no—
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—41no—
202C22×D61—41no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61C22×D411yescharacteristic
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—22no—
209C2×#6923—22no—
210C2×#6923—22no—
211C2×#6923—22no—
212C2×#6923—22no—
213C2×#6923—41no—
214C2×#6923—41no—
215C2×#6923—42no—
216C2×#6923—41no—
217C2×#6923—41no—
218C2×#6923—42no—
219C2×#6923—42no—
220C2×#6923—41no—
221C2×#6923—88no—
222Q8×C61—22no—
223Q8×C61—22no—
224D4×C61—21no—
225D4×C61C4×D412yes—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—42no—
229C2×C4×C61C4×D411yescharacteristic
230C2×C4×C61C4×D411yescharacteristic
231C2×C4×C61C23×C411yescharacteristic
232C22⋊D61—22no—
233C22⋊D61—22no—
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—42no—
237C2×Dic61C4×D411yescharacteristic
238C2×Dic61C4×D411yescharacteristic
239C2×Dic61C2×#4111yescharacteristic
240D244—21no—
241D244C4×D412yes—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—42no—
249C4×D61—42no—
250C4×D61AGL(1,ℤ/8)12yes—
251C4×D61#13712yes—
252C4×D61C22×D411yescharacteristic
253C4×D61C2×#4111yescharacteristic
254C4×D61H(5,𝔽2)11yescharacteristic
255C4×D612−1+411yescharacteristic
256Dic122—22no—
257Dic122—22no—
258Dic122—42no—
259Dic122—42no—
260#92420—22no—
261#92420—41no—
262C8×C61—22no—
263#92418—22no—
264#23002—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61C2×C4×D411yescharacteristic
269C2×D61—22no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61D4211yescharacteristic
277C2×D61#54811yescharacteristic
278C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
279C2×D61C2×#13711yescharacteristic
280C2×D61#57711yescharacteristic
281C2×D61#57811yescharacteristic
282#6923—42no—
283#6923—42no—
284#6923—42no—
285#6923—81no—
286#6923—82no—
287C4×C61—21no—
288C4×C61—42no—
289C4×C61—41no—
290C4×C61#44012yescomplemented
291C4×C61C2×C4×D411yescharacteristic
292C4×C61#52011yescharacteristic
293Dic61—21no—
294Dic61—42no—
295Dic61—41no—
296Dic61#44012yescomplemented
297Dic61#54711yescharacteristic
298Dic61#55311yescharacteristic
299C2×#461—611no—
300C2×#440—612no—
301#2365—611no—
302D4×C4⋊C4—611no—
303#1956—611no—
304#1947—611no—
305#1935—612no—
306#1877—612no—
307#1863—612no—
308#1770—611no—
309C4×#102—611no—
310C2×C61—42no—
311C2×C61#236111yescharacteristic
312C2×C61C2×#44011yescharacteristic
313C2×C61#294711yescharacteristic
314D61—42no—
315D61#327211yescharacteristic
316D61#327611yescharacteristic
317D61#336711yescharacteristic
318D61#337211yescharacteristic
319C22×SD16—612no—
320C2×C4⋊Q8—611no—
321C2×C4×Q8—612no—
322C2×C4×D4—611no—
323C2×C4×D4—612no—
324C22×C4⋊C4—1221no—
325#461—612no—
326#461—611no—
327#461—611no—
328#461—611no—
329#461—612no—
330#461#692311yescharacteristic, complemented
331#440—1228no—
332C2×#107—612no—
333C2×C4⋊C8—611no—
334C2×#103—612no—
335C2×#103—612no—
336C2×#103—612no—
337C2×#102—611no—
338C2×#102—611no—
339C2×#97—612no—
340C2×C4×C8—611no—
341#391—612no—
342#389—612no—
343#389—1221no—
344#387—1222no—
345#385—1222no—
346#384—611no—
347#380—1221no—
348C4×C4⋊C4—611no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—1221no—
351#340—611no—
352#340—612no—
353#340—611no—
354#336—612no—
355C61#2300211yescharacteristic, complemented, Sylow
356C22×Q8—612no—
357C22×D4—611no—
358C23×C4—1221no—
359C23×C4—1222no—
360C2×SD16—612no—
361C2×SD16—612no—
362C2×SD16—612no—
363C2×SD16—1224no—
364C2×SD16—1224no—
365C2×SD16C2×#692312yes—
366C22×C8—611no—
367C22×C8—612no—
368C22×C8—611no—
369C4⋊Q8—612no—
370C4⋊Q8—611no—
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8C2×#692311yescharacteristic, complemented
374C4×Q8—1224no—
375C4×Q8—2442no—
376C4×D4—611no—
377C4×D4—611no—
378C4×D4—611no—
379C4×D4—1221no—
380C4×D4—1221no—
381C4×D4—1224no—
382C4×D4—2442no—
383C4×D4C2×#692311yescharacteristic
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4—2444no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1222no—
399C2×C42—611no—
400C2×C42—612no—
401C2×C42—1222no—
402C2×C42—1221no—
403#107—1224no—
404C4⋊C8—612no—
405C4⋊C8—611no—
406C4⋊C8C2×#692311yescharacteristic, complemented
407#103—612no—
408#103—612no—
409#103—612no—
410#103—1224no—
411#103—2442no—
412#103C2×#692312yescomplemented
413#102—614no—
414#102—1222no—
415#97—1224no—
416C4×C8—1221no—
417C4×C8—1221no—
418#95—612no—
419#95—1221no—
420#95—1222no—
421#95—1222no—
422#95—1222no—
423C24—1221no—
424C2×Q8—612no—
425C2×Q8—1222no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1222no—
430C2×Q8C22×#692312yes—
431C2×D4—612no—
432C2×D4—612no—
433C2×D4—611no—
434C2×D4—1221no—
435C2×D4—1221no—
436C2×D4—1221no—
437C2×D4C22×#692311yescharacteristic
438C22×C4—21no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—1221no—
445C22×C4—1222no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1222no—
452C22×C4—4882no—
453C22×C4—2442no—
454C22×C4—2442no—
455C22×C4—2444no—
456C22×C4—2444no—
457SD16—24no—
458SD16—1224no—
459SD16—1224no—
460SD16—1224no—
461C2×C8—612no—
462C2×C8—612no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8C22×#692312yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—612no—
474C4⋊C4—611no—
475C4⋊C4—1221no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1221no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—2442no—
489C4⋊C4—2444no—
490C4⋊C4—2444no—
491C4⋊C4C22×#692311yescharacteristic
492C22⋊C4—21no—
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C22⋊C4—1221no—
496C22⋊C4—2444no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—1222no—
501C42—1222no—
502C42—2442no—
503C42—2444no—
504C42C22×#692311yescharacteristic
505C23—21no—
506C23—611no—
507C23—1222no—
508C23—1222no—
509C23—1221no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514Q8—22no—
515Q8—22no—
516Q8—1222no—
517Q8—1222no—
518Q8—2442no—
519Q8—2442no—
520D4—21no—
521D4—612no—
522D4—1222no—
523D4—1222no—
524D4—1221no—
525D4D4×#692312yes—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1221no—
548C2×C4—1222no—
549C2×C4—4888no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2441no—
557C2×C4—2442no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2442no—
565C2×C4D4×#692311yescharacteristic
566C2×C4D4×#692311yescharacteristic
567C2×C4C23×#692311yescharacteristic
568C8—22no—
569C8—1222no—
570C8—1222no—
571C8—2441no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—611no—
576C22—611no—
577C22—611no—
578C22—611no—
579C22—611no—
580C22—611no—
581C22—1222no—
582C22—2442no—
583C22—2442no—
584C22—2442no—
585C22—2442no—
586C22—2441no—
587C22—2441no—
588C22C2×D4×#692311yescharacteristic
589C4—21no—
590C4—41no—
591C4—42no—
592C4—612no—
593C4—611no—
594C4—611no—
595C4—1221no—
596C4—4882no—
597C4—4881no—
598C4—2442no—
599C4—2442no—
600C4—2442no—
601C4—2441no—
602C4—2442no—
603C4#1684080112yescomplemented
604C4C2×D4×#692311yescharacteristic
605C4#1684083111yescharacteristic
606C2—42no—
607C2—611no—
608C2—611no—
609C2—611no—
610C2—611no—
611C2—2442no—
612C2#2146234411yescharacteristic
613C2C2×#1684080111yescharacteristic
614C2#2146257711yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#461C4
socleC22×C61C2×C4×D4
prime radicalC2×D4×D61C23
cosocleC2×C4×D61C24
hypercenter#461#6923
hypocenterC61#23002
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. #21462348C2semidirect
4. #21461946C2semidirect
5. #21461930C2semidirect
6. #21461910C2semidirect
8. #21461783C2semidirect
9. #21461616C2semidirect
40. #16839968C4semidirect
41. #16839967C4semidirect
42. #16839966C4semidirect
43. #16839965C4semidirect
44. #16839964C4semidirect
47. C61×#461C4semidirect
91. #14519777C2×C4semidirect
94. #14519775C2×C4semidirect
105. C4⋊#92418C2×C4semidirect
106. #14519706C2×C4semidirect
107. #14519702C2×C4semidirect
108. #14519701C2×C4semidirect
109. #14519700C2×C4semidirect
111. C4⋊Q8×C61C2×C4semidirect
113. C4⋊C8×C61C2×C4semidirect
114. #103×C61C2×C4semidirect
290. C4×C61#440semidirect
296. Dic61#440semidirect
330. #461#6923semidirect
355. C61#23002semidirect
373. C4⋊Q8C2×#6923semidirect
406. C4⋊C8C2×#6923semidirect
412. #103C2×#6923semidirect
603. C4#16840801semidirect

Maximal subgroups

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

subgroupindex
C2×#168408012
#214623482
D4×C4⋊#69232
#214619462
#214619302
#214619102
#214618122
#214617832
#214616162
C4×#145198392
#461×D612
#2300261