Finite Group Databasebeta

← #30603294

Subgroups — #30603294

Order 15616.

1230 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519846C211yesmaximal, characteristic, complemented
2C2×#16840704C212yesmaximal
3C2×#16840636C212yesmaximal, complemented
4C2×#16840636C211yesmaximal, characteristic, complemented
5C2×C22⋊C8×D61C211yesmaximal, characteristic
6C22×C4⋊#6923C2212yes—
7C22×C4⋊#6923C2212yes—
8C22×C22⋊#6923—22no—
9C2×#14519846—22no—
10C2×#14519846—24no—
11#16840704—24no—
12#16840704—24no—
13C2×(C2×#6923)⋊C4—22no—
14#16840636—22no—
15#16840636—24no—
16C23×C4×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×C22⋊#92418C411yescharacteristic, complemented
20C2×#14519689C411yescharacteristic, complemented
21C2×#14519689C411yescharacteristic, complemented
22C22⋊C8×D61—22no—
23C22⋊C8×D61—22no—
24C2×C22⋊C8×C61C411yescharacteristic, complemented
25C23×#6923—22no—
26C23×#6923—22no—
27C2×C4⋊#6923—22no—
28C2×C4⋊#6923—22no—
29C2×C4⋊#6923—22no—
30C2×C4⋊#6923—22no—
31C2×C4⋊#6923—22no—
32C2×C4⋊#6923—22no—
33C2×C4⋊#6923—22no—
34C2×C4⋊#6923—22no—
35C2×C4⋊#6923—22no—
36C2×C4⋊#6923—22no—
37C2×C4⋊#6923D414yes—
38C2×C4⋊#6923D414yes—
39C2×C22⋊#6923—28no—
40C2×C22⋊#6923—44no—
41C2×C22⋊#6923—44no—
42#14519846—28no—
43(C2×#6923)⋊C4—28no—
44C24×D61D411yescharacteristic
45C23×Dic61Q811yescharacteristic
46C22×C4×D61—22no—
47C22×C4×D61—22no—
48C22×C4×D61—21no—
49C22×C4×D61—21no—
50C22×C4×D61—21no—
51C22×C4×D61—21no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61Q811yescharacteristic
57C22×C4×D61C2311yescharacteristic
58C22×#92418C2×C411yescharacteristic
59C22×#92418C2×C411yescharacteristic
60C2×C8×D61—22no—
61C2×C8×D61—22no—
62C2×C8×D61—21no—
63C2×C8×D61—21no—
64C2×C8×D61—22no—
65C2×C8×D61—22no—
66C2×C8×D61D412yes—
67C2×C8×D61D412yes—
68C22⋊#92418—22no—
69#14519689—22no—
70#14519689—22no—
71C23×C4×C61C2×C411yescharacteristic
72C22×C8×C61C2×C411yescharacteristic
73C22×C8×C61C2×C411yescharacteristic
74C22⋊C8×C61—22no—
75C24×C61C4⋊C411yescharacteristic
76C23×D61—22no—
77C23×D61—21no—
78C23×D61—21no—
79C23×D61—22no—
80C23×D61—21no—
81C23×D61—21no—
82C23×D61—22no—
83C23×D61—22no—
84C23×D61C4⋊C411yescharacteristic
85C23×D61D812yes—
86C23×D61Q1612yes—
87C23×D61C2×D411yescharacteristic
88C22×#6923—24no—
89C22×#6923—24no—
90C22×#6923—42no—
91C22×#6923—42no—
92C22×#6923—42no—
93C22×#6923—42no—
94C22×#6923—42no—
95C22×#6923—42no—
96C22×#6923—42no—
97C22×#6923—42no—
98C22×#6923—42no—
99C22×#6923—42no—
100C22×#6923—42no—
101C22×#6923—42no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61C4⋊C411yescharacteristic
107C22×C4×C61C22×C411yescharacteristic
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61—22no—
112C22×Dic61C4⋊C411yescharacteristic
113C22×Dic61C2×Q811yescharacteristic
114C2×C4×D61—22no—
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—22no—
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—22no—
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—22no—
126C2×C4×D61—41no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—42no—
131C2×C4×D61—41no—
132C2×C4×D61—42no—
133C2×C4×D61—41no—
134C2×C4×D61—42no—
135C2×C4×D61—41no—
136C2×C4×D61C2×D412yes—
137C2×C4×D61C2×D411yescharacteristic
138C2×C4×D61C2×D411yescharacteristic
139C2×C4×D61C2×D412yes—
140C2×C4×D61#4111yescharacteristic
141C2×C4×D61#4112yes—
142C2×C4×D61#4112yes—
143C2×C4×D61#4111yescharacteristic
144C22⋊#6923—416no—
145C4⋊#6923—24no—
146C4⋊#6923—24no—
147C4⋊#6923—24no—
148C4⋊#6923—24no—
149C4⋊#6923—42no—
150C4⋊#6923—42no—
151C4⋊#6923—42no—
152C4⋊#6923—42no—
153C2×C8×C61—22no—
154C2×C8×C61—21no—
155C2×C8×C61—22no—
156C2×C8×C61C22⋊C412yescomplemented
157C2×#92418—22no—
158C2×#92418—21no—
159C2×#92418—22no—
160C2×#92418C22⋊C412yescomplemented
161C8×D61—22no—
162C8×D61—24no—
163C8×D61—22no—
164C8×D61—42no—
165C8×D61—41no—
166C8×D61—41no—
167C23×C61—21no—
168C23×C61—22no—
169C23×C61—21no—
170C23×C61—22no—
171C23×C61#10712yescomplemented
172C23×C61C2×C4⋊C411yescharacteristic
173C22×D61—21no—
174C22×D61—22no—
175C22×D61—21no—
176C22×D61—22no—
177C22×D61—24no—
178C22×D61—24no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—24no—
184C22×D61—24no—
185C22×D61—41no—
186C22×D61—42no—
187C22×D61—41no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—42no—
194C22×D61—42no—
195C22×D61#10712yescomplemented
196C22×D61#12012yes—
197C22×D61#12111yescharacteristic
198C22×D61#12111yescharacteristic
199C22×D61#12312yes—
200C22×D61C2×D811yescharacteristic
201C22×D61C2×Q1611yescharacteristic
202C22×D61#13511yescharacteristic
203C22×D61#13511yescharacteristic
204C22×D61AGL(1,ℤ/8)12yes—
205C22×D61#13712yes—
206C2×#6923—44no—
207C2×#6923—44no—
208C2×#6923—44no—
209C2×#6923—44no—
210C2×#6923—44no—
211C2×#6923—44no—
212C2×#6923—82no—
213C2×#6923—82no—
214C2×#6923—82no—
215C2×#6923—82no—
216C2×#6923—82no—
217C2×#6923—82no—
218C2×#6923—82no—
219C2×#6923—82no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—41no—
223C2×C4×C61—42no—
224C2×C4×C61—41no—
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61#11711yescharacteristic
227C2×C4×C61C4×D412yes—
228C2×C4×C61C4×D412yes—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic61—41no—
232C2×Dic61—42no—
233C2×Dic61—41no—
234C2×Dic61C22⋊Q812yes—
235C2×Dic61C22⋊Q811yescharacteristic
236C2×Dic61C22⋊Q811yescharacteristic
237C2×Dic61C22⋊Q812yes—
238C4×D61—41no—
239C4×D61—42no—
240C4×D61—41no—
241C4×D61—41no—
242C4×D61—42no—
243C4×D61—41no—
244C4×D61—41no—
245C4×D61—42no—
246C4×D61—41no—
247C4×D61#12011yescharacteristic
248C4×D61#12112yes—
249C4×D61#12112yes—
250C4×D61#12112yes—
251C4×D61#12311yescharacteristic
252C4×D61#12312yes—
253C4×D61#12412yes—
254C4×D61#12411yescharacteristic
255C4×D61#12612yes—
256C4×D61#12711yescharacteristic
257C8×C61—22no—
258C8×C61—41no—
259#92418—22no—
260#92418—41no—
261#11096—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61—24no—
264C22×C61—41no—
265C22×C61—42no—
266C22×C61—41no—
267C22×C61#38011yescharacteristic
268C22×C61#38512yes—
269C22×C61C2×#10711yescharacteristic
270C22×C61#42711yescharacteristic
271C22×C61#42812yes—
272C2×D61—22no—
273C2×D61—24no—
274C2×D61—41no—
275C2×D61—42no—
276C2×D61—41no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—44no—
282C2×D61—44no—
283C2×D61#39211yescharacteristic
284C2×D61#39412yes—
285C2×D61#39711yescharacteristic
286C2×D61#44712yes—
287C2×D61#44912yes—
288C2×D61#45112yes—
289C2×D61#45212yes—
290C2×D61#46611yescharacteristic
291C2×D61#46611yescharacteristic
292C2×D61#46711yescharacteristic
293C2×D61#46711yescharacteristic
294C2×D61#46812yes—
295C2×D61#46911yescharacteristic
296C2×D61#47011yescharacteristic
297C2×D61#48012yes—
298C2×D61#48212yes—
299C2×D61#48412yes—
300C2×D61#48512yes—
301#6923—84no—
302#6923—84no—
303C4×C61—41no—
304C4×C61#38611yescharacteristic
305C4×C61#38812yes—
306C4×C61#39011yescharacteristic
307Dic61—41no—
308Dic61#39311yescharacteristic
309Dic61#39612yes—
310Dic61#39911yescharacteristic
311C22×#107—611no—
312C22×C22⋊C8—611no—
313C2×#380—612no—
314C2×#340—612no—
315C2×#340—611no—
316C2×C61—42no—
317C2×C61#143911yescharacteristic
318C2×C61#180811yescharacteristic
319C2×C61#180911yescharacteristic
320C2×C61#185212yes—
321C2×C61#185412yes—
322D61—42no—
323D61#199911yescharacteristic
324D61#200111yescharacteristic
325D61#203312yes—
326D61#203612yes—
327D61#207511yescharacteristic
328D61#207611yescharacteristic
329C24×C4—611no—
330C23×C8—611no—
331C23×C8—611no—
332C22×C4⋊C4—612no—
333C22×C4⋊C4—612no—
334C22×C22⋊C4—1222no—
335C2×#107—1224no—
336C2×#107—1222no—
337C2×C22⋊C8—611no—
338C2×C22⋊C8—611no—
339C2×C22⋊C8—611no—
340C2×C22⋊C8—1222no—
341C2×C22⋊C8—1222no—
342C2×C22⋊C8#692311yescharacteristic, complemented
343#380—1224no—
344#380—1224no—
345C2×#95—1222no—
346#340—1224no—
347#340—1222no—
348C61#1109611yescharacteristic, complemented, Sylow
349C25—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1221no—
356C23×C4—1221no—
357C23×C4—1222no—
358C23×C4—1222no—
359C23×C4—1222no—
360C23×C4—1222no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4—1222no—
364C23×C4—1222no—
365C23×C4C2×#692311yescharacteristic
366C22×C8—611no—
367C22×C8—612no—
368C22×C8—612no—
369C22×C8—611no—
370C22×C8—1222no—
371C22×C8—1222no—
372C22×C8—1222no—
373C22×C8—1221no—
374C22×C8—1221no—
375C22×C8—1222no—
376C22×C8C2×#692311yescharacteristic
377C22×C8C2×#692311yescharacteristic
378C2×C4⋊C4—614no—
379C2×C4⋊C4—614no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1222no—
387C2×C4⋊C4—1222no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C22⋊C4—1228no—
391C2×C22⋊C4—2444no—
392C2×C22⋊C4—2444no—
393#107—1228no—
394C22⋊C8—22no—
395C22⋊C8—1222no—
396C22⋊C8—1222no—
397C22⋊C8—1222no—
398#95—1228no—
399C24—611no—
400C24—612no—
401C24—612no—
402C24—611no—
403C24—1222no—
404C24—1222no—
405C24—1221no—
406C24—1221no—
407C24—1222no—
408C24—1221no—
409C24—1221no—
410C24—1222no—
411C24C4⋊#692311yescharacteristic
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—611no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1222no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—2441no—
445C22×C4—2442no—
446C22×C4—2441no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4—2441no—
453C22×C4—2441no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4—2442no—
466C22×C4C4⋊#692311yescharacteristic
467C22×C4C22×#692311yescharacteristic
468C2×C8—22no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—612no—
472C2×C8—1222no—
473C2×C8—1224no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1221no—
477C2×C8—1222no—
478C2×C8—2442no—
479C2×C8—2441no—
480C2×C8—2441no—
481C2×C8C22⋊#692312yescomplemented
482C4⋊C4—1224no—
483C4⋊C4—1224no—
484C4⋊C4—1224no—
485C4⋊C4—1224no—
486C4⋊C4—2442no—
487C4⋊C4—2442no—
488C4⋊C4—2442no—
489C4⋊C4—2442no—
490C22⋊C4—24416no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—22no—
495C23—611no—
496C23—611no—
497C23—612no—
498C23—612no—
499C23—611no—
500C23—611no—
501C23—611no—
502C23—611no—
503C23—612no—
504C23—612no—
505C23—612no—
506C23—1224no—
507C23—1224no—
508C23—1224no—
509C23—1224no—
510C23—1222no—
511C23—1222no—
512C23—1222no—
513C23—1222no—
514C23—1221no—
515C23—1222no—
516C23—1221no—
517C23—1222no—
518C23—2441no—
519C23—2442no—
520C23—2441no—
521C23—2442no—
522C23—2442no—
523C23—2442no—
524C23—2442no—
525C23—2441no—
526C23—2442no—
527C23—2441no—
528C23#1451984612yescomplemented
529C23C2×C4⋊#692311yescharacteristic
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—611no—
540C2×C4—612no—
541C2×C4—611no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—4882no—
552C2×C4—4882no—
553C2×C4—4882no—
554C2×C4—4882no—
555C2×C4—4882no—
556C2×C4—4882no—
557C2×C4—4882no—
558C2×C4—4882no—
559C2×C4—2441no—
560C2×C4—2442no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2442no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2442no—
567C2×C4—2441no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2444no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4—2444no—
574C2×C4—2441no—
575C2×C4—2442no—
576C2×C4—2441no—
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4#1451985611yescharacteristic
579C2×C4D4×#692312yes—
580C2×C4D4×#692312yes—
581C8—22no—
582C8—41no—
583C8—1222no—
584C8—2441no—
585C22—24no—
586C22—22no—
587C22—41no—
588C22—42no—
589C22—41no—
590C22—612no—
591C22—612no—
592C22—612no—
593C22—612no—
594C22—611no—
595C22—612no—
596C22—611no—
597C22—612no—
598C22—612no—
599C22—612no—
600C22—612no—
601C22—611no—
602C22—611no—
603C22—611no—
604C22—611no—
605C22—611no—
606C22—612no—
607C22—611no—
608C22—1224no—
609C22—1222no—
610C22—2442no—
611C22—2444no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22—2444no—
616C22—2441no—
617C22—2442no—
618C22—2441no—
619C22#1684070411yescharacteristic
620C22#1684070912yes—
621C22C2×#1451984611yescharacteristic
622C22#1684077211yescharacteristic
623C22#1684077312yes—
624C4—41no—
625C4—611no—
626C4—612no—
627C4—611no—
628C4—4884no—
629C4—4884no—
630C4—2441no—
631C4#1684071011yescharacteristic
632C4#1684071212yes—
633C4#1684071411yescharacteristic
634C2—42no—
635C2—611no—
636C2—612no—
637C2—611no—
638C2—611no—
639C2—611no—
640C2—612no—
641C2—2442no—
642C2#2146100211yescharacteristic
643C2#2146169611yescharacteristic
644C2#2146169711yescharacteristic
645C2#2146176812yes—
646C2#2146177012yes—
subgroupG/H
centerC23C2×C4⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊C8×C61C4
socleC23×C61C2×C4⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC2×C22⋊C8#6923
hypocenterC61#11096
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×#14519846C2semidirect
3. C2×#16840636C2semidirect
4. C2×#16840636C2semidirect
19. C2×C22⋊#92418C4semidirect
20. C2×#14519689C4semidirect
21. C2×#14519689C4semidirect
24. C2×C22⋊C8×C61C4semidirect
156. C2×C8×C61C22⋊C4semidirect
160. C2×#92418C22⋊C4semidirect
171. C23×C61#107semidirect
195. C22×D61#107semidirect
342. C2×C22⋊C8#6923semidirect
348. C61#11096semidirect
481. C2×C8C22⋊#6923semidirect
528. C23#14519846semidirect

Maximal subgroups

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

subgroupindex
C22×#145198462
C2×#168407042
C2×#168406362
C2×C22⋊C8×D612
#1109661