Finite Group Databasebeta

← #29650000

Subgroups — #29650000

Order 15616.

700 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455875C211yesmaximal, characteristic
2C8×#11758224C211yesmaximal, characteristic, complemented
3#21454526C211yesmaximal, characteristic, complemented
4#21444700C211yesmaximal, characteristic, complemented
5#21444696C211yesmaximal, characteristic, complemented
6#21444684C211yesmaximal, characteristic, complemented
7#21444680C211yesmaximal, characteristic, complemented
8C8×#11758190C211yesmaximal, characteristic, complemented
9C8×#11758200C211yesmaximal, characteristic, complemented
10C8×#11758199C211yesmaximal, characteristic, complemented
11#21444609C211yesmaximal, characteristic, complemented
12#21444255C211yesmaximal, characteristic, complemented
13#21444234C211yesmaximal, characteristic, complemented
14C8×#11758189C211yesmaximal, characteristic, complemented
15C8×D8×C61C211yesmaximal, characteristic, complemented
16C2×#14519783—21no—
17#16840309C2211yescharacteristic
18C4×#11758224C2211yescharacteristic
19#16840089C411yescharacteristic
20#16840085C411yescharacteristic
21#16840056C411yescharacteristic
22#16840050C411yescharacteristic
23#16839961C411yescharacteristic
24#16839955C411yescharacteristic
25#16839908C411yescharacteristic
26#16839904C411yescharacteristic
27C8×Dic122C2211yescharacteristic
28#16839886C2211yescharacteristic
29C4×#11758190C2211yescharacteristic
30C4×#11758200C2211yescharacteristic
31C4×#11758199C2211yescharacteristic
32#16839840C2211yescharacteristic
33#16839839C2211yescharacteristic
34#16839837—21no—
35C8×D244C2211yescharacteristic
36D4×#92418C2211yescharacteristic
37C8×C22⋊D61—21no—
38C8×C22⋊D61—21no—
39#16839825—21no—
40#16839814C2211yescharacteristic
41#16839813C2211yescharacteristic
42#16839812—21no—
43#16839784C2211yescharacteristic
44#16839727—21no—
45C4×#11758189C2211yescharacteristic
46C2×C8×Dic61—21no—
47C4×C8×D61C2211yescharacteristic
48#16839514C2211yescharacteristic
49#16839513C2211yescharacteristic
50#16839512C2211yescharacteristic
51#16839494C2211yescharacteristic
52#16839492C2211yescharacteristic
53#16839491C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55D4⋊#92418C2211yescharacteristic
56#16839489C2211yescharacteristic
57#16839484C2211yescharacteristic
58#16839483C2211yescharacteristic
59C8×#92418C2211yescharacteristic
60C4×D8×C61C2211yescharacteristic
61C8×D4×C61C2211yescharacteristic
62C8×D4×C61C2211yescharacteristic
63C61×#335C2211yescharacteristic
64C61×D4⋊C8C2211yescharacteristic
65C61×D4⋊C8C2211yescharacteristic
66C82×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758224C2×C411yescharacteristic
69C2×#11758190C2×C411yescharacteristic
70C2×#11758200C2×C411yescharacteristic
71C2×#11758199C2×C411yescharacteristic
72#14519783—44no—
73C2×#11758194—21no—
74C2×#11758189—21no—
75C2×#11758189—21no—
76C2×#11758189—21no—
77C22×#92418—21no—
78C2×C8×D61—21no—
79C2×C8×D61D411yescharacteristic
80C2×C8×D61D411yescharacteristic
81#14519776C2×C411yescharacteristic
82#14519773C2×C411yescharacteristic
83#14519763—21no—
84#14519752C2×C411yescharacteristic
85#14519750—21no—
86#14519748C2×C411yescharacteristic
87C4×Dic122C2311yescharacteristic
88#14519742C2×C411yescharacteristic
89C4×D244C2311yescharacteristic
90D4×Dic61C2×C411yescharacteristic
91C4×C22⋊D61—21no—
92C4×C22⋊D61—21no—
93#14519737—21no—
94#14519735C2×C411yescharacteristic
95#14519733—21no—
96C2×C4×Dic61—21no—
97C42×D61C2311yescharacteristic
98#14519710C2×C411yescharacteristic
99#14519709C2×C411yescharacteristic
100#14519708C2×C411yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519706—21no—
104#14519706—21no—
105#14519706—21no—
106#14519706C2×C411yescharacteristic
107#14519706C2×C411yescharacteristic
108#14519702C2×C411yescharacteristic
109#14519700C2×C411yescharacteristic
110#14519699C2×C411yescharacteristic
111#14519698C2×C411yescharacteristic
112#14519698C2×C411yescharacteristic
113#14519697C2×C411yescharacteristic
114C22⋊#92418—21no—
115C22⋊#92418—21no—
116#14519689—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689C2×C411yescharacteristic
120#14519689C2×C411yescharacteristic
121#14519687—21no—
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C4×#92418C2311yescharacteristic
125C8×Dic61—21no—
126C8×Dic61—21no—
127C8×Dic61—21no—
128C8×Dic61D411yescharacteristic
129C8×Dic61D411yescharacteristic
130C2×D8×C61C2×C411yescharacteristic
131C22×C8×C61—21no—
132C22×C8×C61—21no—
133C4×D4×C61C2311yescharacteristic
134C4×D4×C61C2311yescharacteristic
135#107×C61C2×C411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#102×C61C2×C411yescharacteristic
139#102×C61C2×C411yescharacteristic
140C22⋊C8×C61—21no—
141C22⋊C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144C4×C8×C61C2311yescharacteristic
145C2×D4×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C2×C22⋊D61—21no—
150C2×C22⋊D61—21no—
151C2×C22⋊D61—21no—
152C2×C22⋊D61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155#11758224—22no—
156#11758224—22no—
157#11758222—42no—
158#11758222—42no—
159C2×D244—21no—
160C2×D244C22×C411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61#4111yescharacteristic
165C2×Dic122—21no—
166C2×Dic122C22×C411yescharacteristic
167D8×C61—22no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—42no—
172C2×C8×C61—42no—
173C2×C8×C61C22×C411yescharacteristic
174C2×C8×C61C22×C411yescharacteristic
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61C22×C411yescharacteristic
178C4⋊C4×C61C22×C411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—21no—
183C22⋊Dic61—21no—
184#11758200—22no—
185#11758199—22no—
186#11758198—21no—
187#11758198—21no—
188#11758198—21no—
189C4⋊Dic61C22×C411yescharacteristic
190C4⋊Dic61C22×C411yescharacteristic
191#11758196—21no—
192#11758196—21no—
193#11758196—21no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61C22×C411yescharacteristic
198C4×Dic61C2×D411yescharacteristic
199C4×Dic61#4111yescharacteristic
200#11758194—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—42no—
205C2×#92418C22×C411yescharacteristic
206C2×#92418C22×C411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209#11758190—22no—
210#11758189—24no—
211#11758189—42no—
212#11758189—42no—
213C8×D61—22no—
214C8×D61—41no—
215C8×D61—42no—
216C23×C61—21no—
217C23×C61—21no—
218C22×D61—21no—
219C22×D61C4×D411yescharacteristic
220D4×C61—21no—
221D4×C61—21no—
222D4×C61—21no—
223D4×C61#13112yescomplemented
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61C23×C411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C22⋊D61—24no—
234C22⋊D61—42no—
235C22⋊D61—42no—
236C22⋊D61—42no—
237C2×Dic61—24no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—42no—
242C2×Dic61C4×D411yescharacteristic
243C2×Dic61C4×D411yescharacteristic
244C2×Dic61C4×D411yescharacteristic
245D244—21no—
246D244—21no—
247D244—42no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—22no—
253C4×D61—41no—
254C4×D61—41no—
255Dic122—21no—
256Dic122—42no—
257Dic122#13112yescomplemented
258C8×C61—22no—
259C8×C61—21no—
260C8×C61—21no—
261C8×C61—21no—
262C8×C61—41no—
263C8×C61—41no—
264C8×C61AGL(1,ℤ/8)12yes—
265#92418—22no—
266#92418—21no—
267#92418—21no—
268#92418—21no—
269#92418—21no—
270#92418—41no—
271#92418—41no—
272#17912—611nomaximal, Sylow
273C22×C61—22no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61C2×C4×D411yescharacteristic
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—41no—
284C2×D61—41no—
285C2×D61—41no—
286C4×C61—21no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61C2×#13111yescharacteristic
290C4×C61C2×#13111yescharacteristic
291C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C61#57511yescharacteristic
293Dic61—22no—
294Dic61—21no—
295Dic61—21no—
296Dic61—41no—
297Dic61—41no—
298Dic61#44312yes—
299#3106—611no—
300C8×#41—611no—
301#2931—611no—
302#1588—611no—
303#1588—611no—
304#1586—611no—
305#1586—611no—
306C8×SD16—611no—
307C8×SD16—611no—
308C8×D8—611no—
309C8×D8D6111yescharacteristic, complemented
310#1574—611no—
311#1489—611no—
312#1484—611no—
313C8×M16—611no—
314C2×C61—42no—
315C2×C61—81no—
316C2×C61#292911yescharacteristic
317C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
318C2×C61C2×#44311yescharacteristic
319D61—41no—
320D61—81no—
321C2×#131—1221no—
322#532—611no—
323C4×#41—611no—
324#482—611no—
325#482—611no—
326#477—611no—
327#476—611no—
328#460—611no—
329#459—611no—
330#449—611no—
331#449—611no—
332C8×Q8—611no—
333C8×Q8—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336C4×D8—611no—
337C4×D8C2×D6111yescharacteristic
338#436—611no—
339#436—611no—
340#436—1221no—
341C8×D4—611no—
342C8×D4—611no—
343C8×D4—1221no—
344C8×D4—1221no—
345C8×D4—1221no—
346C8×D4C2×D6111yescharacteristic
347C8×D4C2×D6111yescharacteristic
348#431—611no—
349#431—611no—
350#431—1221no—
351#423—611no—
352#408—1221no—
353C4×M16—611no—
354C2×C4×C8—611no—
355C2×C4×C8—1221no—
356#335—611no—
357#335C2×D6111yescharacteristic
358#334—611no—
359#334—611no—
360Q8⋊C8—611no—
361Q8⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8C2×D6111yescharacteristic
367D4⋊C8C2×D6111yescharacteristic
368#322—611no—
369#322—611no—
370C82—611no—
371C82C2×D6111yescharacteristic
372C61#1791211yescharacteristic, complemented, Sylow
373C2×#41—611no—
374C2×#41—1221no—
375C2×SD16—611no—
376C2×SD16—611no—
377C2×D8—611no—
378C2×D8C4×D6111yescharacteristic
379#131—2444no—
380C2×M16—1221no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—1221no—
384C22×C8—21no—
385C22×C8—21no—
386C22×C8—611no—
387C22×C8—611no—
388C22×C8—1221no—
389C22×C8—1221no—
390C4⋊Q8—611no—
391#127—611no—
392#124—1221no—
393#121—611no—
394#121—611no—
395#121—1221no—
396C4×Q8—611no—
397C4×Q8—611no—
398C4×D4—611no—
399C4×D4—611no—
400C4×D4—1221no—
401C4×D4—1221no—
402C4×D4—1221no—
403C4×D4C22×D6111yescharacteristic
404C4×D4C22×D6111yescharacteristic
405#117—611no—
406#117—1221no—
407C2×C42—611no—
408C2×C42—1221no—
409#107—611no—
410#107C4×D6111yescharacteristic
411#106—611no—
412#106—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—611no—
416C4⋊C8—611no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421C4⋊C8C22×D6111yescharacteristic
422#103—611no—
423#103—611no—
424#102—611no—
425#102—611no—
426#102—611no—
427#102—611no—
428#102C4×D6111yescharacteristic
429#102C4×D6111yescharacteristic
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—611no—
433C22⋊C8—611no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8—1221no—
437C22⋊C8—1221no—
438C22⋊C8—1221no—
439#97—1221no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—611no—
444C4×C8—611no—
445C4×C8—1221no—
446C4×C8—1221no—
447C4×C8—1221no—
448C4×C8C22×D6111yescharacteristic
449C4×C8C22×D6111yescharacteristic
450C4×C8C22×D6111yescharacteristic
451#41—1222no—
452#41—1222no—
453#41—2442no—
454#41—2442no—
455C2×Q8—611no—
456C2×Q8—1221no—
457C2×D4—611no—
458C2×D4—1221no—
459C2×D4—1221no—
460C2×D4—1221no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4C2×C4×D6111yescharacteristic
464C2×D4C2×C4×D6111yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473SD16—1222no—
474SD16—1222no—
475D8—22no—
476D8—1222no—
477M16—1224no—
478M16—2442no—
479M16—2442no—
480M16—2442no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—611no—
487C2×C8—611no—
488C2×C8—611no—
489C2×C8—611no—
490C2×C8—1221no—
491C2×C8—1222no—
492C2×C8—1221no—
493C2×C8—1221no—
494C2×C8—2441no—
495C2×C8—2442no—
496C2×C8—2442no—
497C2×C8C2×C4×D6111yescharacteristic
498C2×C8C2×C4×D6111yescharacteristic
499C2×C8D4×D6111yescharacteristic
500C2×C8D4×D6111yescharacteristic
501C4⋊C4—611no—
502C4⋊C4—611no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4C2×C4×D6111yescharacteristic
507C4⋊C4C2×C4×D6111yescharacteristic
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—1221no—
511C22⋊C4—1221no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1221no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—1221no—
519C42—1221no—
520C42—1221no—
521C42C23×D6111yescharacteristic
522C23—21no—
523C23—21no—
524C23—611no—
525C23—1221no—
526Q8—612no—
527Q8—1221no—
528Q8—2442no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—1224no—
533D4—1221no—
534D4—1221no—
535D4—2442no—
536D4—2442no—
537D4—2442no—
538D4—2442no—
539D4#1451978312yescomplemented
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—1222no—
550C2×C4—1224no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—2442no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4C22×C4×D6111yescharacteristic
562C2×C4C2×D4×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C8—22no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—41no—
569C8—41no—
570C8—1222no—
571C8—1221no—
572C8—1221no—
573C8—1221no—
574C8—1221no—
575C8—2441no—
576C8—2441no—
577C8#1451980112yes—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22—2441no—
588C22—2441no—
589C22—2441no—
590C22C4×D4×D6111yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—612no—
595C4—1221no—
596C4—1221no—
597C4—1222no—
598C4—2441no—
599C4—2441no—
600C4#131×D6111yescharacteristic
601C4C2×#1451978311yescharacteristic
602C4C2×#1451980111yescharacteristic
603C4#1684055911yescharacteristic
604C2—42no—
605C2—81no—
606C2—4881no—
607C2—2441no—
608C2#2145450811yescharacteristic
609C2C4×#1451980111yescharacteristic
610C2#443×D6111yescharacteristic
subgroupG/H
centerC2×C8D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC8×D8×C61C2
socleC22×C61C2×C4×D4
prime radical#21455875C2
cosocleC42×C61C24
hypercenterC8×D8D61
hypocenterC61#17912
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C8 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. C8×#11758224C2semidirect
3. #21454526C2semidirect
4. #21444700C2semidirect
5. #21444696C2semidirect
6. #21444684C2semidirect
7. #21444680C2semidirect
8. C8×#11758190C2semidirect
9. C8×#11758200C2semidirect
10. C8×#11758199C2semidirect
11. #21444609C2semidirect
12. #21444255C2semidirect
13. #21444234C2semidirect
14. C8×#11758189C2semidirect
15. C8×D8×C61C2semidirect
223. D4×C61#131semidirect
257. Dic122#131semidirect
309. C8×D8D61semidirect
372. C61#17912semidirect
539. D4#14519783semidirect

Maximal subgroups

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

subgroupindex
#214558752
C8×#117582242
#214545262
#214447002
#214446962
#214446842
#214446802
C8×#117581902
C8×#117582002
C8×#117581992
#214446092
#214442552
#214442342
C8×#117581892
C8×D8×C612
#1791261