Finite Group Databasebeta

← #30603100

Subgroups — #30603100

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840742C211yesmaximal, characteristic
2C2×#16840736C211yesmaximal, characteristic
3C2×#16840728C211yesmaximal, characteristic, complemented
4C2×#16840645C211yesmaximal, characteristic, complemented
5C2×#16840645C211yesmaximal, characteristic, complemented
6#21461487C211yesmaximal, characteristic, complemented
7C2×#16839743C211yesmaximal, characteristic
8C22×#11758213C2211yescharacteristic
9C22×D61⋊C8C2211yescharacteristic
10C2×#14519856C2211yescharacteristic
11C2×#14519856C2211yescharacteristic
12#16840742—22no—
13#16840742—22no—
14C2×#14519835C414yes—
15#16840736—21no—
16#16840736—21no—
17#16840736—22no—
18C2×C22⋊#92420C414yescomplemented
19#16840728—22no—
20#16840728—22no—
21C2×#14519831—21no—
22C2×C8×#6923—21no—
23#16840645—22no—
24#16840645—22no—
25(C2×#6923)⋊C8—22no—
26C22×#11758224C2211yescharacteristic
27C2×M16×D61C2211yescharacteristic
28C2×M16×D61C2211yescharacteristic
29#16839743—22no—
30#16839743—22no—
31C2×#14519695C411yescharacteristic, complemented
32C2×#14519695C411yescharacteristic, complemented
33C2×#14519694C411yescharacteristic, complemented
34C2×#100×C61C411yescharacteristic, complemented
35C2×#11758213—21no—
36C2×#11758213—21no—
37C2×#11758213—21no—
38C2×#11758213—21no—
39C2×#11758213—21no—
40C2×#11758213—21no—
41C2×D61⋊C8—21no—
42C2×D61⋊C8—21no—
43C2×D61⋊C8—21no—
44C2×D61⋊C8—21no—
45C2×D61⋊C8D411yescharacteristic
46C2×D61⋊C8D411yescharacteristic
47C2×D61⋊C8D412yes—
48C22×#92420C2×C412yes—
49C22×#92420C2×C412yes—
50#14519856—21no—
51#14519856—21no—
52#14519856—21no—
53#14519856—21no—
54#14519856—21no—
55#14519856—21no—
56#14519856D412yes—
57#14519856D412yes—
58C2×C4⋊#6923—21no—
59C2×C4⋊#6923—21no—
60C2×C22⋊#6923C2×C412yes—
61C2×C22⋊#6923C2×C412yes—
62C2×C4×#6923—21no—
63C2×C4×#6923—21no—
64#14519835—28no—
65C22⋊#92420—24no—
66C22⋊#92420—24no—
67#14519831—24no—
68C8×#6923—24no—
69C2×#11758224—21no—
70C2×#11758224—21no—
71C2×#11758224—21no—
72C2×#11758224—21no—
73C2×#11758224—21no—
74C2×#11758224—21no—
75C2×#11758224—21no—
76C2×#11758224—21no—
77C2×#11758224—21no—
78C2×#11758224—21no—
79C2×#11758224—21no—
80C2×#11758224—21no—
81C22×Dic122C2×C411yescharacteristic
82C22×C22⋊D61C2×C412yes—
83C23×Dic61C2×C412yes—
84C22×C4×D61C2311yescharacteristic
85C2×#11758194C2×C411yescharacteristic
86C2×#11758194C2×C411yescharacteristic
87C2×#11758189—21no—
88C2×#11758189—21no—
89M16×D61—21no—
90M16×D61—21no—
91M16×D61—21no—
92M16×D61—21no—
93M16×D61—21no—
94M16×D61—21no—
95M16×D61D412yes—
96M16×D61D412yes—
97C2×C8×D61—21no—
98C2×C8×D61—21no—
99#14519695—22no—
100#14519695—22no—
101#14519694—22no—
102C22×D4×C61C2×C411yescharacteristic
103C2×M16×C61C2×C411yescharacteristic
104C2×M16×C61C2×C411yescharacteristic
105#100×C61—22no—
106C24×C61C4212yes—
107C23×D61C22×C411yescharacteristic
108C22×#6923—22no—
109C22×#6923—22no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C22×C4×C61C22×C411yescharacteristic
117C2×C22⋊D61—22no—
118C2×C22⋊D61—22no—
119C2×C22⋊D61—22no—
120C2×C22⋊D61—22no—
121C2×C22⋊D61—22no—
122C2×C22⋊D61—22no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61C4212yes—
130C22×Dic61C22×C411yescharacteristic
131#11758224—41no—
132#11758224—41no—
133#11758224—41no—
134#11758224—41no—
135#11758224—41no—
136#11758224—41no—
137#11758224—41no—
138#11758224—41no—
139#11758224—41no—
140#11758224—41no—
141#11758224—41no—
142#11758224—41no—
143#11758224—41no—
144#11758224—41no—
145#11758224—41no—
146#11758224—41no—
147C2×C4×D61C2×D411yescharacteristic
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61#4111yescharacteristic
154C2×C4×D61#4111yescharacteristic
155C2×C4×D61#4111yescharacteristic
156C2×C4×D61#4111yescharacteristic
157C2×C4×D61#4111yescharacteristic
158C2×C4×D61#4111yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C22⋊#6923—22no—
166C22⋊#6923—22no—
167C22⋊#6923—22no—
168C22⋊#6923C22⋊C414yes—
169C2×#92420—22no—
170C2×#92420—22no—
171C2×#92420—22no—
172C2×#92420—22no—
173C2×#92420—22no—
174C2×#92420C22⋊C412yes—
175C2×#92420C22⋊C412yes—
176C4⋊#6923—22no—
177C4⋊#6923—22no—
178C4⋊#6923—41no—
179C4⋊#6923—41no—
180C4×#6923—22no—
181C4×#6923—22no—
182C4×#6923—41no—
183C4×#6923—41no—
184#11758213—41no—
185#11758213—41no—
186#11758213—41no—
187#11758213—41no—
188D61⋊C8—21no—
189D61⋊C8—21no—
190D61⋊C8—22no—
191D61⋊C8—22no—
192D61⋊C8—21no—
193D61⋊C8—21no—
194M16×C61—21no—
195M16×C61—21no—
196M16×C61—21no—
197M16×C61C22⋊C412yes—
198C2×C8×C61—21no—
199C2×C8×C61—21no—
200#11758194—21no—
201#11758194—21no—
202#11758194—21no—
203#11758194C22⋊C412yes—
204C2×#92418—21no—
205C2×#92418—21no—
206#11758189—22no—
207#11758189—22no—
208#11758189—41no—
209#11758189—41no—
210C8×D61—22no—
211C8×D61—22no—
212C8×D61—41no—
213C8×D61—41no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—44no—
218C23×C61C2×C4211yescharacteristic
219C22×D61—42no—
220C22×D61C2×C22⋊C411yescharacteristic
221C22×D61#11711yescharacteristic
222C22×D61C4×D411yescharacteristic
223C22×D61C4×D411yescharacteristic
224C22×D61C4×D411yescharacteristic
225C22×D61C4×D411yescharacteristic
226C2×#6923—24no—
227C2×#6923—44no—
228C2×#6923—44no—
229C2×#6923—42no—
230D4×C61—41no—
231D4×C61—41no—
232D4×C61—41no—
233D4×C61—41no—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61#11711yescharacteristic
236C2×C4×C61C4×D411yescharacteristic
237C2×C4×C61C4×D411yescharacteristic
238C2×C4×C61C4×D411yescharacteristic
239C2×C4×C61C4×D411yescharacteristic
240C22⋊D61—42no—
241C22⋊D61—42no—
242C22⋊D61—42no—
243C22⋊D61—42no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61#11711yescharacteristic
253C2×Dic61C4×D411yescharacteristic
254C2×Dic61C4×D411yescharacteristic
255C2×Dic61C4×D411yescharacteristic
256C2×Dic61C4×D411yescharacteristic
257C4×D61#12011yescharacteristic
258C4×D61#12111yescharacteristic
259C4×D61#12111yescharacteristic
260C4×D61#12111yescharacteristic
261C4×D61#12111yescharacteristic
262C4×D61#12111yescharacteristic
263C4×D61#12111yescharacteristic
264C4×D61#12311yescharacteristic
265C4×D61#12311yescharacteristic
266C4×D61#12311yescharacteristic
267C4×D61#12411yescharacteristic
268C4×D61#12411yescharacteristic
269C4×D61#12411yescharacteristic
270C4×D61#12611yescharacteristic
271C4×D61#12611yescharacteristic
272C4×D61#12711yescharacteristic
273Dic122—41no—
274Dic122—41no—
275Dic122—41no—
276Dic122—41no—
277#92420—22no—
278#92420—22no—
279#92420—42no—
280C8×C61—22no—
281C8×C61—41no—
282#92418—22no—
283#92418—41no—
284#10961—611nomaximal, Sylow
285C22×C61—22no—
286C22×C61—42no—
287C22×C61—44no—
288C22×C61—82no—
289C22×C61—82no—
290C22×C61C4×C22⋊C411yescharacteristic
291C22×C61C4×C22⋊C411yescharacteristic
292C22×C61C4×C22⋊C411yescharacteristic
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—41no—
299C2×D61#38611yescharacteristic
300C2×D61#38811yescharacteristic
301C2×D61#38811yescharacteristic
302C2×D61#39011yescharacteristic
303#6923—44no—
304#6923—82no—
305C4×C61#38611yescharacteristic
306C4×C61#38811yescharacteristic
307C4×C61#38811yescharacteristic
308C4×C61#39011yescharacteristic
309Dic61—42no—
310Dic61#38611yescharacteristic
311Dic61#38811yescharacteristic
312Dic61#38811yescharacteristic
313Dic61#39011yescharacteristic
314C2×#413—611no—
315C2×#410—611no—
316C2×#408—611no—
317C2×#405—611no—
318C2×#344—611no—
319C2×#344—611no—
320#1729—611no—
321C2×C61—21no—
322C2×C61—21no—
323C2×C61—82no—
324C2×C61#143711yescharacteristic
325C2×C61#175712yes—
326D61—21no—
327D61—21no—
328D61—21no—
329D61—21no—
330C22×#41—611no—
331C22×M16—611no—
332C22×M16—611no—
333C22×M16—611no—
334C23×C8—611no—
335C2×#117—611no—
336C2×#117—611no—
337#413—1222no—
338#413—1222no—
339C2×#101—611no—
340C2×#101—611no—
341C2×#100—611no—
342C2×#100#692311yescharacteristic, complemented
343#410—1222no—
344#410—1222no—
345C2×#99—614no—
346#408—1222no—
347#408—1221no—
348#408—1221no—
349C2×C22⋊C8—614no—
350#405—1222no—
351#405—1222no—
352C2×#97—1221no—
353C2×C4×C8—1221no—
354#344—1222no—
355#344—1222no—
356#336—1222no—
357C61#1096111yescharacteristic, complemented, Sylow
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—1221no—
361C2×#41—1221no—
362C2×#41—1221no—
363C2×#41—1221no—
364C2×#41—1221no—
365C2×#41—1221no—
366C2×#41—1221no—
367C2×#41—1221no—
368C2×#41—1221no—
369C2×#41—1221no—
370C22×Q8—611no—
371C22×D4—612no—
372C22×D4C2×#692311yescharacteristic
373C23×C4—611no—
374C23×C4—612no—
375C2×M16—611no—
376C2×M16—612no—
377C2×M16—612no—
378C2×M16—611no—
379C2×M16—1221no—
380C2×M16—1221no—
381C2×M16—1221no—
382C2×M16—1221no—
383C2×M16—1221no—
384C2×M16—1221no—
385C2×M16—1221no—
386C2×M16—1221no—
387C2×M16—1221no—
388C2×M16—1221no—
389C2×M16—1221no—
390C2×M16—1221no—
391C2×M16—1221no—
392C2×M16—1221no—
393C2×M16C2×#692311yescharacteristic
394C2×M16C2×#692311yescharacteristic
395C22×C8—612no—
396C22×C8—612no—
397C22×C8—611no—
398C22×C8—611no—
399C22×C8—612no—
400C22×C8—1221no—
401C22×C8—1221no—
402C22×C8—1221no—
403C22×C8—1221no—
404C22×C8—1221no—
405C22×C8—1221no—
406#117—612no—
407#117—612no—
408#117—1221no—
409#117—1221no—
410#117—1221no—
411#117—1221no—
412#117—1221no—
413#117—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C22⋊C4—612no—
417C2×C22⋊C4—612no—
418C2×C42—1221no—
419C2×C42—1221no—
420#101—1222no—
421#101—1222no—
422#100—22no—
423#100—1222no—
424#99—1228no—
425C22⋊C8—1224no—
426C22⋊C8—1224no—
427#97—1224no—
428C4×C8—1224no—
429C24—611no—
430C24C4×#692312yes—
431#41—2441no—
432#41—2441no—
433#41—2441no—
434#41—2441no—
435#41—2441no—
436#41—2441no—
437#41—2441no—
438#41—2441no—
439#41—2441no—
440#41—2441no—
441#41—2441no—
442#41—2441no—
443#41—2441no—
444#41—2441no—
445#41—2441no—
446#41—2441no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—1221no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—1222no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1222no—
464C2×D4—1222no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—1222no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4C22×#692311yescharacteristic
488M16—21no—
489M16—21no—
490M16—21no—
491M16—612no—
492M16—1222no—
493M16—1222no—
494M16—1221no—
495M16—1221no—
496M16—1221no—
497M16—2441no—
498M16—2441no—
499M16—2441no—
500M16—2441no—
501M16—2441no—
502M16—2441no—
503M16C22⋊#692312yes—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—612no—
507C2×C8—612no—
508C2×C8—1222no—
509C2×C8—1222no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1222no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1222no—
516C2×C8—1222no—
517C2×C8—1222no—
518C2×C8—1221no—
519C2×C8—1222no—
520C2×C8—1222no—
521C2×C8—1222no—
522C2×C8—1221no—
523C2×C8—2441no—
524C2×C8—2441no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—2441no—
528C4⋊C4—2441no—
529C22⋊C4—614no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C42—1222no—
534C42—1222no—
535C42—2441no—
536C42—2441no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—44no—
541C23—611no—
542C23—611no—
543C23—611no—
544C23—611no—
545C23—611no—
546C23—611no—
547C23—2442no—
548C23C2×C4×#692311yescharacteristic
549Q8—2441no—
550Q8—2441no—
551Q8—2441no—
552Q8—2441no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—2442no—
558D4—2442no—
559D4—2442no—
560D4—2442no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—611no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—1224no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4—2444no—
593C2×C4—2444no—
594C2×C4C2×C22⋊#692311yescharacteristic
595C2×C4#1451985611yescharacteristic
596C2×C4D4×#692311yescharacteristic
597C2×C4D4×#692311yescharacteristic
598C2×C4D4×#692311yescharacteristic
599C2×C4D4×#692311yescharacteristic
600C8—22no—
601C8—41no—
602C8—1222no—
603C8—1222no—
604C8—1222no—
605C8—2442no—
606C8—2441no—
607C22—22no—
608C22—42no—
609C22—44no—
610C22—82no—
611C22—82no—
612C22—611no—
613C22—611no—
614C22—611no—
615C22—611no—
616C22—1222no—
617C22—1222no—
618C22—2441no—
619C22—2441no—
620C22—2441no—
621C22—2441no—
622C22C4×C22⋊#692311yescharacteristic
623C22C22⋊C4×#692311yescharacteristic
624C22C22⋊C4×#692311yescharacteristic
625C4—611no—
626C4—611no—
627C4—611no—
628C4—611no—
629C4—4882no—
630C4—2444no—
631C4—2442no—
632C4#1684071011yescharacteristic
633C4#1684071211yescharacteristic
634C4#1684071211yescharacteristic
635C4#1684071411yescharacteristic
636C2—21no—
637C2—21no—
638C2—82no—
639C2—1221no—
640C2—1221no—
641C2—1221no—
642C2—1221no—
643C2#2146100011yescharacteristic
644C2#2146157612yes—
subgroupG/H
centerC22C22⋊C4×#6923
commutatorC23×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×#100×C61C4
socleC22×C61C4×C22⋊C4
prime radicalC22×#11758224C22
cosocleC22×C4×D61C23
hypercenterC2×#100#6923
hypocenterC61#10961
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
3. C2×#16840728C2semidirect
4. C2×#16840645C2semidirect
5. C2×#16840645C2semidirect
6. #21461487C2semidirect
18. C2×C22⋊#92420C4semidirect
31. C2×#14519695C4semidirect
32. C2×#14519695C4semidirect
33. C2×#14519694C4semidirect
34. C2×#100×C61C4semidirect
342. C2×#100#6923semidirect
357. C61#10961semidirect

Maximal subgroups

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

subgroupindex
C2×#168407422
C2×#168407362
C2×#168407282
C2×#168406452
#214614872
C2×#168397432
#1096161