Finite Group Databasebeta

← #30546249

Subgroups — #30546249

Order 15616.

1930 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
1C4×#14519825C211yesmaximal, characteristic
2#21454945C213yesmaximal, complemented
3#21454943C213yesmaximal, complemented
4#21454933C213yesmaximal, complemented
5#21454929C211yesmaximal, characteristic, complemented
6#21454908C211yesmaximal, characteristic, complemented
7#21454907C211yesmaximal, characteristic, complemented
8#21454901C213yesmaximal, complemented
9#21454834C213yesmaximal, complemented
10#21454832C213yesmaximal, complemented
11#21454781C213yesmaximal, complemented
12C8×#11758226C211yesmaximal, characteristic, complemented
13Q8×#11758189C211yesmaximal, characteristic, complemented
14#446×D61C213yesmaximal, complemented
15Q8×M16×C61C211yesmaximal, characteristic, complemented
16C2×#14519825C411yescharacteristic
17#16840244C413yes—
18#16840244C411yescharacteristic
19#16840243C413yes—
20#16840243C413yes—
21#16840240C411yescharacteristic
22#16840238C411yescharacteristic
23#16840238C413yes—
24C4×#11758226C2211yescharacteristic
25C4×#11758226C2213yes—
26C4×#11758222C2213yes—
27C4×#11758222C2213yes—
28C4×Q8×D61C2213yes—
29C4×Q8×D61C2211yescharacteristic
30#16839893C2213yes—
31#16839893C2213yes—
32#16839892C2213yes—
33#16839892C2213yes—
34#16839891C2216yes—
35#16839890C2213yes—
36#16839890C2213yes—
37#16839889C2213yes—
38#16839889C2213yes—
39Q8×#92418C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41#16839838C2213yes—
42#16839838C2211yescharacteristic
43#16839835C2213yes—
44#16839835C2213yes—
45#16839832C2216yes—
46#16839831C2216yes—
47#16839831C2213yes—
48#16839831C2213yes—
49C8×D244C2213yes—
50C8×D244C2213yes—
51#16839827C2213yes—
52#16839827C2213yes—
53#16839819C2213yes—
54#16839818C2216yes—
55#16839818C2213yes—
56#16839817C2216yes—
57#16839815C2213yes—
58#16839814C2213yes—
59#16839813C2213yes—
60#16839813C2213yes—
61#16839784C2213yes—
62C4⋊C8×D61C2213yes—
63C4⋊C8×D61C2216yes—
64C4×#11758189C2213yes—
65#97×D61C2216yes—
66C4×C8×D61C2213yes—
67C2×C4×Q8×C61C2211yescharacteristic
68C61×#446C2213yes—
69C61×#446C2213yes—
70C8×Q8×C61C2211yescharacteristic
71C8×Q8×C61C2211yescharacteristic
72C61×#423C2213yes—
73C4×M16×C61C2213yes—
74#14519825—28no—
75C2×#11758226C2×C413yes—
76C2×#11758226C2×C411yescharacteristic
77C2×#11758222C2×C413yes—
78C2×#11758222C2×C413yes—
79C2×Q8×D61C2×C411yescharacteristic
80C2×Q8×D61C2×C413yes—
81C2×#11758189C2311yescharacteristic
82C2×#11758189C2313yes—
83C2×C8×D61C2313yes—
84C2×C8×D61C2313yes—
85C2×C8×D61C2316yes—
86Q8×Dic61C2×C413yes—
87Q8×Dic61C2×C411yescharacteristic
88Q8×Dic61C2×C413yes—
89Q8×Dic61C2×C411yescharacteristic
90C4×Dic122C2×C413yes—
91C4×Dic122C2×C413yes—
92C4×Dic122C2313yes—
93C4×Dic122C2313yes—
94#14519742C2×C416yes—
95#14519742C2×C413yes—
96#14519742C2×C413yes—
97C4×D244C2×C413yes—
98C4×D244C2×C413yes—
99C4×D244C2313yes—
100C4×D244C2313yes—
101#14519739C2×C416yes—
102#14519739C2×C413yes—
103#14519739C2×C413yes—
104C4×C22⋊D61C2×C413yes—
105C4×C22⋊D61C2×C413yes—
106C4×C22⋊D61C2×C416yes—
107C4×C22⋊D61C2313yes—
108C4×C22⋊D61C2311yescharacteristic
109#14519736C2×C413yes—
110#14519736C2×C413yes—
111#14519735C2×C413yes—
112#14519735C2×C416yes—
113#14519735C2×C413yes—
114#14519734C2×C413yes—
115#14519734C2×C416yes—
116#14519734C2×C413yes—
117C4⋊C4×D61C2×C416yes—
118C4⋊C4×D61C2×C413yes—
119C4⋊C4×D61C2×C413yes—
120C42×D61C2313yes—
121C42×D61C2316yes—
122C42×D61C2313yes—
123C4⋊#92418C2313yes—
124C4⋊#92418C2313yes—
125C4⋊#92418C2313yes—
126C4⋊#92418C2313yes—
127#14519706C2313yes—
128#14519706C2313yes—
129#14519706C2316yes—
130#14519706C2311yescharacteristic
131#14519706C2313yes—
132C22⋊#92418C2311yescharacteristic
133C22⋊#92418C2313yes—
134#14519689C2311yescharacteristic
135#14519689C2313yes—
136#14519689C2313yes—
137#14519689C2313yes—
138#14519689C2316yes—
139#14519688C2316yes—
140#14519687C2313yes—
141#14519687C2316yes—
142#14519687C2313yes—
143C4×#92418C2313yes—
144C4×#92418C2313yes—
145C8×Dic61C2311yescharacteristic
146C8×Dic61C2313yes—
147C22×Q8×C61C2×C411yescharacteristic
148C2×M16×C61C2311yescharacteristic
149C2×M16×C61C2313yes—
150C4×Q8×C61C2×C413yes—
151C4×Q8×C61C2×C411yescharacteristic
152C4×Q8×C61C2311yescharacteristic
153C4×Q8×C61C2313yes—
154C2×C4⋊C4×C61C2×C413yes—
155C2×C42×C61C2313yes—
156C4⋊C8×C61C2313yes—
157C4⋊C8×C61C2316yes—
158C4⋊C8×C61C2313yes—
159#97×C61C2313yes—
160#97×C61C2313yes—
161C4×C8×C61C2313yes—
162C4×C8×C61C2313yes—
163C2×Q8×C61—24no—
164C2×Q8×C61C22×C413yes—
165C2×Q8×C61C22×C411yescharacteristic
166C22×C4×C61C22×C413yes—
167C22×C4×C61C22×C413yes—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61C22×C413yes—
170C2×C22⋊D61C22×C411yescharacteristic
171#11758226—212no—
172#11758226M1618yescomplemented
173Q8×D61—24no—
174Q8×D61—212no—
175#11758222—212no—
176#11758222—212no—
177C2×D244C22×C413yes—
178C2×D244C22×C413yes—
179C2×C4×D61C22×C416yes—
180C2×C4×D61C22×C413yes—
181C2×C4×D61C22×C413yes—
182C2×C4×D61C22×C413yes—
183C2×C4×D61C22×C416yes—
184C2×C4×D61C22×C413yes—
185C2×C4×D61C2413yes—
186C2×C4×D61C2411yescharacteristic
187C2×Dic122C22×C413yes—
188C2×Dic122C22×C413yes—
189M16×C61—22no—
190M16×C61—43no—
191C2×C8×C61C2411yescharacteristic
192C2×C8×C61C2413yes—
193C2×C8×C61C2413yes—
194C2×C8×C61C2411yescharacteristic
195C4⋊C4×C61C22×C416yes—
196C4⋊C4×C61C22×C413yes—
197C4⋊C4×C61C22×C413yes—
198C42×C61C22×C413yes—
199C42×C61C22×C413yes—
200C42×C61C2413yes—
201C42×C61C2413yes—
202C22⋊Dic61C22×C413yes—
203C22⋊Dic61C22×C411yescharacteristic
204#11758198C22×C411yescharacteristic
205#11758198C22×C413yes—
206#11758198C22×C413yes—
207#11758198C22×C416yes—
208#11758198C22×C413yes—
209C4⋊Dic61C22×C413yes—
210C4⋊Dic61C22×C413yes—
211C4⋊Dic61C22×C413yes—
212C4⋊Dic61C22×C413yes—
213#11758196C22×C416yes—
214#11758196C22×C413yes—
215#11758196C22×C413yes—
216#11758196C22×C413yes—
217#11758196C22×C411yescharacteristic
218C4×Dic61C22×C416yes—
219C4×Dic61C22×C413yes—
220C4×Dic61C22×C413yes—
221C4×Dic61C22×C416yes—
222C4×Dic61C22×C413yes—
223C4×Dic61C22×C413yes—
224C4×Dic61C2411yescharacteristic
225C4×Dic61C2413yes—
226C2×#92418C2411yescharacteristic
227C2×#92418C2413yes—
228C2×#92418C2411yescharacteristic
229C2×#92418C2413yes—
230#11758189—22no—
231#11758189—43no—
232C8×D61—26no—
233C8×D61—43no—
234C8×D61—46no—
235C23×C61C23×C411yescharacteristic
236C22×D61C23×C411yescharacteristic
237C22×D61C23×C413yes—
238Q8×C61—26no—
239Q8×C61C2×M1614yes—
240C2×C4×C61—26no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—43no—
244C2×C4×C61C23×C413yes—
245C2×C4×C61C23×C413yes—
246C2×C4×C61C23×C413yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C23×C413yes—
249C2×C4×C61C2511yescharacteristic
250C22⋊D61—26no—
251C22⋊D61—22no—
252C2×Dic61C23×C413yes—
253C2×Dic61C23×C411yescharacteristic
254C2×Dic61C23×C413yes—
255C2×Dic61C23×C411yescharacteristic
256C2×Dic61C23×C413yes—
257C2×Dic61C23×C411yescharacteristic
258D244—26no—
259D244C2×M16112yes—
260C4×D61—212no—
261C4×D61—26no—
262C4×D61—23no—
263C4×D61—21no—
264C4×D61—23no—
265C4×D61—21no—
266C4×D61—43no—
267C4×D61—46no—
268C4×D61—43no—
269C4×D61C2×M16112yes—
270Dic122—26no—
271Dic122—26no—
272C8×C61—21no—
273C8×C61—23no—
274C8×C61—23no—
275C8×C612−1+412yescomplemented
276#92418—23no—
277#92418—21no—
278#92418—21no—
279#92418—23no—
280#60343—611nomaximal, Sylow
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61C24×C411yescharacteristic
285C2×D61—23no—
286C2×D61—23no—
287C2×D61—21no—
288C2×D61—21no—
289C2×D61—21no—
290C2×D61C22×M1616yes—
291C4×C61—21no—
292C4×C61—23no—
293C4×C61—23no—
294C4×C61—23no—
295C4×C61C22×M1616yes—
296C4×C61C2×2−1+411yescharacteristic
297C4×C61#58511yescharacteristic
298Dic61—23no—
299Dic61—21no—
300Dic61—23no—
301Dic61—21no—
302Dic61—23no—
303Dic61C22×M1612yes—
304C4×2−1+4—611no—
305#2990—613no—
306#2990—613no—
307#2989—613no—
308#2988—611no—
309#2984—611no—
310#2984—611no—
311#2983—613no—
312#2970—613no—
313#2970—613no—
314#2965—613no—
315C8×#41—611no—
316Q8×M16—611no—
317Q8×M16D6111yescharacteristic, complemented
318C2×#446—613no—
319C2×C61—41no—
320C2×C61C4×2−1+411yescharacteristic
321C2×C61C23×M1611yescharacteristic
322C2×C61#357311yescharacteristic
323D61—26no—
324D61—41no—
325C2×2−1+4—611no—
326#520—611no—
327#520—613no—
328#520—613no—
329#520—613no—
330#519—611no—
331#519—613no—
332#519—611no—
333C4×#41—611no—
334C4×#41—613no—
335C4×#41—613no—
336C4×#41—613no—
337C2×C4×Q8—613no—
338C2×C4×Q8—611no—
339C2×C4×Q8C2×D6111yescharacteristic
340#446—613no—
341#446—613no—
342#446—616no—
343#446—613no—
344#446—613no—
345#446—613no—
346#446—613no—
347#446—613no—
348#446—613no—
349#446C2×D6113yes—
350#446C2×D6113yes—
351C8×Q8—611no—
352C8×Q8—611no—
353C8×Q8C2×D6111yescharacteristic
354C8×Q8C2×D6111yescharacteristic
355#436—611no—
356#436—613no—
357#435—613no—
358#435—613no—
359#435—616no—
360#435—616no—
361#435—613no—
362#435—613no—
363C8×D4—613no—
364C8×D4—613no—
365C8×D4—613no—
366C8×D4—613no—
367#432—613no—
368#432—616no—
369#432—613no—
370#432—616no—
371#431—613no—
372#431—613no—
373#431—613no—
374#431—613no—
375#423—613no—
376#423C2×D6113yes—
377C2×C4⋊C8—613no—
378C2×C4⋊C8—616no—
379C4×M16—613no—
380C4×M16C2×D6113yes—
381C2×#97—616no—
382C2×C4×C8—613no—
383C61#6034311yescharacteristic, complemented, Sylow
3842−1+4—1228no—
385C2×#41—611no—
386C2×#41—613no—
387C2×#41—613no—
388C2×#41—613no—
389C22×Q8—613no—
390C22×Q8—611no—
391C22×Q8C4×D6111yescharacteristic
392C2×M16—611no—
393C2×M16—613no—
394C2×M16C22×D6111yescharacteristic
395C2×M16C22×D6113yes—
396C22×C8—613no—
397C22×C8—613no—
398C22×C8—616no—
399C4×Q8—613no—
400C4×Q8—611no—
401C4×Q8—613no—
402C4×Q8—611no—
403C4×Q8—616no—
404C4×Q8—613no—
405C4×Q8—613no—
406C4×Q8—613no—
407C4×Q8—613no—
408C4×Q8—613no—
409C4×Q8—613no—
410C4×Q8C4×D6113yes—
411C4×Q8C4×D6111yescharacteristic
412C4×Q8C22×D6111yescharacteristic
413C4×Q8C22×D6113yes—
414C4×D4—613no—
415C4×D4—613no—
416C4×D4—616no—
417C4×D4—613no—
418C4×D4—611no—
419C4×D4—613no—
420C4×D4—613no—
421C4×D4—616no—
422C4×D4—613no—
423C4×D4—613no—
424C4×D4—613no—
425C4×D4—613no—
426#117—613no—
427#117—613no—
428#117—613no—
429#117—613no—
430#117—616no—
431#117—613no—
432#117—613no—
433#117—616no—
434C2×C4⋊C4—613no—
435C2×C4⋊C4—616no—
436C2×C4⋊C4—613no—
437C2×C4⋊C4C4×D6113yes—
438C2×C42—613no—
439C2×C42—616no—
440C2×C42—613no—
441C2×C42C22×D6113yes—
442C4⋊C8—613no—
443C4⋊C8—613no—
444C4⋊C8—616no—
445C4⋊C8—611no—
446C4⋊C8—613no—
447C4⋊C8—613no—
448C4⋊C8—613no—
449C4⋊C8—613no—
450C4⋊C8—613no—
451C4⋊C8C22×D6113yes—
452C4⋊C8C22×D6116yes—
453C4⋊C8C22×D6113yes—
454C22⋊C8—611no—
455C22⋊C8—613no—
456C22⋊C8—611no—
457C22⋊C8—613no—
458C22⋊C8—616no—
459C22⋊C8—613no—
460C22⋊C8—613no—
461#97—613no—
462#97—616no—
463#97—613no—
464#97—616no—
465#97C22×D6113yes—
466#97C22×D6113yes—
467C4×C8—611no—
468C4×C8—613no—
469C4×C8—613no—
470C4×C8—613no—
471C4×C8C22×D6113yes—
472C4×C8C22×D6113yes—
473#41—618no—
474#41—12212no—
475#41—12212no—
476#41—12212no—
477C2×Q8—24no—
478C2×Q8—613no—
479C2×Q8—613no—
480C2×Q8—12212no—
481C2×Q8—1224no—
482C2×Q8C2×C4×D6113yes—
483C2×Q8C2×C4×D6111yescharacteristic
484C2×D4—611no—
485C2×D4—613no—
486C2×D4—613no—
487C2×D4—613no—
488C22×C4—613no—
489C22×C4—616no—
490C22×C4—613no—
491C22×C4—613no—
492C22×C4—611no—
493C22×C4—613no—
494C22×C4—616no—
495C22×C4—613no—
496C22×C4C2×C4×D6113yes—
497C22×C4C2×C4×D6113yes—
498C22×C4C23×D6111yescharacteristic
499M16—22no—
500M16—43no—
501M16—1222no—
502M16—2443no—
503C2×C8—611no—
504C2×C8—611no—
505C2×C8—613no—
506C2×C8—613no—
507C2×C8—1226no—
508C2×C8—2446no—
509C2×C8—2443no—
510C2×C8C23×D6113yes—
511C2×C8C23×D6111yescharacteristic
512C2×C8C23×D6113yes—
513C2×C8C23×D6111yescharacteristic
514C4⋊C4—613no—
515C4⋊C4—613no—
516C4⋊C4—616no—
517C4⋊C4—613no—
518C4⋊C4—613no—
519C4⋊C4—611no—
520C4⋊C4—613no—
521C4⋊C4—613no—
522C4⋊C4—613no—
523C4⋊C4C2×C4×D6116yes—
524C4⋊C4C2×C4×D6113yes—
525C4⋊C4C2×C4×D6113yes—
526C22⋊C4—613no—
527C22⋊C4—611no—
528C22⋊C4—616no—
529C22⋊C4—613no—
530C22⋊C4—613no—
531C22⋊C4—613no—
532C22⋊C4—611no—
533C42—613no—
534C42—616no—
535C42—613no—
536C42—613no—
537C42—616no—
538C42—613no—
539C42—611no—
540C42—613no—
541C42C2×C4×D6113yes—
542C42C2×C4×D6113yes—
543C42C23×D6113yes—
544C42C23×D6113yes—
545C23—613no—
546C23—611no—
547C23C22×C4×D6111yescharacteristic
548Q8—26no—
549Q8—1226no—
550Q8—1226no—
551Q8M16×D6114yes—
552D4—6112no—
553D4—1226no—
554D4—1222no—
555D4—1226no—
556C2×C4—26no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—43no—
560C2×C4—611no—
561C2×C4—6112no—
562C2×C4—613no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—613no—
566C2×C4—613no—
567C2×C4—12212no—
568C2×C4—1226no—
569C2×C4—1223no—
570C2×C4—1221no—
571C2×C4—1223no—
572C2×C4—1221no—
573C2×C4—2443no—
574C2×C4—2446no—
575C2×C4—2443no—
576C2×C4C22×C4×D6113yes—
577C2×C4C22×C4×D6113yes—
578C2×C4C22×C4×D6113yes—
579C2×C4C22×C4×D6111yescharacteristic
580C2×C4C22×C4×D6113yes—
581C2×C4C24×D6111yescharacteristic
582C8—21no—
583C8—23no—
584C8—23no—
585C8—1223no—
586C8—1221no—
587C8—1221no—
588C8—1223no—
589C8#1451982512yescomplemented
590C22—21no—
591C22—21no—
592C22—21no—
593C22—616no—
594C22—1223no—
595C22—1223no—
596C22—1221no—
597C22—1221no—
598C22—1221no—
599C22C23×C4×D6111yescharacteristic
600C4—23no—
601C4—23no—
602C4—23no—
603C4—21no—
604C4—612no—
605C4—1223no—
606C4—1221no—
607C4—1223no—
608C4—1223no—
609C4—1221no—
610C4C2×M16×D6116yes—
611C4C2×#1451982511yescharacteristic
612C4#1684059811yescharacteristic
613C2—41no—
614C2—1226no—
615C2—2441no—
616C2C4×#1451982511yescharacteristic
617C2C22×M16×D6111yescharacteristic
618C2#2146053211yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D61
commutator, socleC22×C61C24×C4
FittingQ8×M16×C61C2
prime radicalC2×#14519825C4
cosocleC2×C4×C61C25
hypercenterQ8×M16D61
hypocenterC61#60343
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #21454945C2semidirect
3. #21454943C2semidirect
4. #21454933C2semidirect
5. #21454929C2semidirect
6. #21454908C2semidirect
7. #21454907C2semidirect
8. #21454901C2semidirect
9. #21454834C2semidirect
10. #21454832C2semidirect
11. #21454781C2semidirect
12. C8×#11758226C2semidirect
13. Q8×#11758189C2semidirect
14. #446×D61C2semidirect
15. Q8×M16×C61C2semidirect
172. #11758226M16semidirect
275. C8×C612−1+4semidirect
317. Q8×M16D61semidirect
383. C61#60343semidirect
589. C8#14519825semidirect

Maximal subgroups

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

subgroupindex
C4×#145198252
#214549452
#214549432
#214549332
#214549292
#214549082
#214549072
#214549012
#214548342
#214548322
#214547812
C8×#117582262
Q8×#117581892
#446×D612
Q8×M16×C612
#6034361