Finite Group Databasebeta

← #30603293

Subgroups — #30603293

Order 15616.

1230 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519845C211yesmaximal, 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×#14519845—22no—
10C2×#14519845—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#14519845—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—22no—
81C23×D61—22no—
82C23×D61—22no—
83C23×D61C4⋊C411yescharacteristic
84C23×D61SD1614yes—
85C23×D61C2×D411yescharacteristic
86C22×#6923—24no—
87C22×#6923—24no—
88C22×#6923—42no—
89C22×#6923—42no—
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×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61C4⋊C411yescharacteristic
105C22×C4×C61C22×C411yescharacteristic
106C22×Dic61—21no—
107C22×Dic61—21no—
108C22×Dic61—22no—
109C22×Dic61—22no—
110C22×Dic61C4⋊C411yescharacteristic
111C22×Dic61C2×Q811yescharacteristic
112C2×C4×D61—22no—
113C2×C4×D61—21no—
114C2×C4×D61—21no—
115C2×C4×D61—22no—
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—22no—
124C2×C4×D61—41no—
125C2×C4×D61—42no—
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×C4×D61—42no—
129C2×C4×D61—41no—
130C2×C4×D61—42no—
131C2×C4×D61—41no—
132C2×C4×D61—42no—
133C2×C4×D61—41no—
134C2×C4×D61C2×D412yes—
135C2×C4×D61C2×D411yescharacteristic
136C2×C4×D61C2×D411yescharacteristic
137C2×C4×D61C2×D412yes—
138C2×C4×D61#4111yescharacteristic
139C2×C4×D61#4112yes—
140C2×C4×D61#4112yes—
141C2×C4×D61#4111yescharacteristic
142C22⋊#6923—416no—
143C4⋊#6923—24no—
144C4⋊#6923—24no—
145C4⋊#6923—24no—
146C4⋊#6923—24no—
147C4⋊#6923—42no—
148C4⋊#6923—42no—
149C4⋊#6923—42no—
150C4⋊#6923—42no—
151C2×C8×C61—22no—
152C2×C8×C61—21no—
153C2×C8×C61—22no—
154C2×C8×C61C22⋊C412yescomplemented
155C2×#92418—22no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418C22⋊C412yescomplemented
159C8×D61—22no—
160C8×D61—24no—
161C8×D61—22no—
162C8×D61—42no—
163C8×D61—41no—
164C8×D61—41no—
165C23×C61—21no—
166C23×C61—22no—
167C23×C61—21no—
168C23×C61—22no—
169C23×C61#10612yescomplemented
170C23×C61C2×C4⋊C411yescharacteristic
171C22×D61—21no—
172C22×D61—22no—
173C22×D61—21no—
174C22×D61—22no—
175C22×D61—24no—
176C22×D61—24no—
177C22×D61—24no—
178C22×D61—24no—
179C22×D61—24no—
180C22×D61—24no—
181C22×D61—41no—
182C22×D61—42no—
183C22×D61—41no—
184C22×D61—42no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—41no—
188C22×D61—41no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61#10612yescomplemented
192C22×D61#12012yes—
193C22×D61#12111yescharacteristic
194C22×D61#12111yescharacteristic
195C22×D61#12312yes—
196C22×D61C2×SD1612yes—
197C22×D61#13512yes—
198C22×D61AGL(1,ℤ/8)12yes—
199C22×D61#13712yes—
200C2×#6923—44no—
201C2×#6923—44no—
202C2×#6923—44no—
203C2×#6923—44no—
204C2×#6923—44no—
205C2×#6923—44no—
206C2×#6923—82no—
207C2×#6923—82no—
208C2×#6923—82no—
209C2×#6923—82no—
210C2×#6923—82no—
211C2×#6923—82no—
212C2×#6923—82no—
213C2×#6923—82no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—41no—
217C2×C4×C61—42no—
218C2×C4×C61—41no—
219C2×C4×C61C2×C22⋊C411yescharacteristic
220C2×C4×C61#11711yescharacteristic
221C2×C4×C61C4×D412yes—
222C2×C4×C61C4×D412yes—
223C2×Dic61—21no—
224C2×Dic61—22no—
225C2×Dic61—41no—
226C2×Dic61—42no—
227C2×Dic61—41no—
228C2×Dic61C22⋊Q812yes—
229C2×Dic61C22⋊Q811yescharacteristic
230C2×Dic61C22⋊Q811yescharacteristic
231C2×Dic61C22⋊Q812yes—
232C4×D61—41no—
233C4×D61—42no—
234C4×D61—41no—
235C4×D61—41no—
236C4×D61—42no—
237C4×D61—41no—
238C4×D61—41no—
239C4×D61—42no—
240C4×D61—41no—
241C4×D61#12011yescharacteristic
242C4×D61#12112yes—
243C4×D61#12112yes—
244C4×D61#12112yes—
245C4×D61#12311yescharacteristic
246C4×D61#12312yes—
247C4×D61#12412yes—
248C4×D61#12411yescharacteristic
249C4×D61#12612yes—
250C4×D61#12711yescharacteristic
251C8×C61—22no—
252C8×C61—41no—
253#92418—22no—
254#92418—41no—
255#11095—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—24no—
258C22×C61—41no—
259C22×C61—42no—
260C22×C61—41no—
261C22×C61#38011yescharacteristic
262C22×C61#38512yes—
263C22×C61C2×#10611yescharacteristic
264C22×C61#42711yescharacteristic
265C22×C61#42812yes—
266C2×D61—22no—
267C2×D61—24no—
268C2×D61—41no—
269C2×D61—42no—
270C2×D61—41no—
271C2×D61—44no—
272C2×D61—44no—
273C2×D61—44no—
274C2×D61—44no—
275C2×D61#39211yescharacteristic
276C2×D61#39412yes—
277C2×D61#39711yescharacteristic
278C2×D61#44812yes—
279C2×D61#44912yes—
280C2×D61#45012yes—
281C2×D61#45212yes—
282C2×D61#46512yes—
283C2×D61#46512yes—
284C2×D61#46812yes—
285C2×D61#46911yescharacteristic
286C2×D61#47011yescharacteristic
287C2×D61#48112yes—
288C2×D61#48212yes—
289C2×D61#48312yes—
290C2×D61#48512yes—
291#6923—84no—
292#6923—84no—
293C4×C61—41no—
294C4×C61#38611yescharacteristic
295C4×C61#38812yes—
296C4×C61#39011yescharacteristic
297Dic61—41no—
298Dic61#39311yescharacteristic
299Dic61#39612yes—
300Dic61#39911yescharacteristic
301C22×#106—611no—
302C22×C22⋊C8—611no—
303C2×#380—612no—
304C2×#340—612no—
305C2×#340—611no—
306C2×C61—42no—
307C2×C61#143911yescharacteristic
308C2×C61#180711yescharacteristic
309C2×C61#180911yescharacteristic
310C2×C61#185312yes—
311C2×C61#185412yes—
312D61—42no—
313D61#200012yes—
314D61#203412yes—
315D61#203512yes—
316D61#207712yes—
317C24×C4—611no—
318C23×C8—611no—
319C23×C8—611no—
320C22×C4⋊C4—612no—
321C22×C4⋊C4—612no—
322C22×C22⋊C4—1222no—
323C2×#106—1224no—
324C2×#106—1222no—
325C2×C22⋊C8—611no—
326C2×C22⋊C8—611no—
327C2×C22⋊C8—611no—
328C2×C22⋊C8—1222no—
329C2×C22⋊C8—1222no—
330C2×C22⋊C8#692311yescharacteristic, complemented
331#380—1224no—
332#380—1224no—
333C2×#95—1222no—
334#340—1224no—
335#340—1222no—
336C61#1109511yescharacteristic, complemented, Sylow
337C25—611no—
338C23×C4—611no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C23×C4—1221no—
344C23×C4—1221no—
345C23×C4—1222no—
346C23×C4—1222no—
347C23×C4—1222no—
348C23×C4—1222no—
349C23×C4—1221no—
350C23×C4—1221no—
351C23×C4—1222no—
352C23×C4—1222no—
353C23×C4C2×#692311yescharacteristic
354C22×C8—611no—
355C22×C8—612no—
356C22×C8—612no—
357C22×C8—611no—
358C22×C8—1222no—
359C22×C8—1222no—
360C22×C8—1222no—
361C22×C8—1221no—
362C22×C8—1221no—
363C22×C8—1222no—
364C22×C8C2×#692311yescharacteristic
365C22×C8C2×#692311yescharacteristic
366C2×C4⋊C4—614no—
367C2×C4⋊C4—614no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1222no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C22⋊C4—1228no—
379C2×C22⋊C4—2444no—
380C2×C22⋊C4—2444no—
381#106—1228no—
382C22⋊C8—22no—
383C22⋊C8—1222no—
384C22⋊C8—1222no—
385C22⋊C8—1222no—
386#95—1228no—
387C24—611no—
388C24—614no—
389C24—611no—
390C24—1222no—
391C24—1222no—
392C24—1221no—
393C24—1221no—
394C24—1222no—
395C24—1222no—
396C24—1222no—
397C24C4⋊#692311yescharacteristic
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—611no—
412C22×C4—1221no—
413C22×C4—1221no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—1222no—
417C22×C4—1221no—
418C22×C4—1221no—
419C22×C4—1222no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1222no—
423C22×C4—1222no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1224no—
429C22×C4—1224no—
430C22×C4—2441no—
431C22×C4—2442no—
432C22×C4—2441no—
433C22×C4—2442no—
434C22×C4—2442no—
435C22×C4—2442no—
436C22×C4—2442no—
437C22×C4—2442no—
438C22×C4—2441no—
439C22×C4—2441no—
440C22×C4—2442no—
441C22×C4—2442no—
442C22×C4—2442no—
443C22×C4—2442no—
444C22×C4—2442no—
445C22×C4—2442no—
446C22×C4—2442no—
447C22×C4—2442no—
448C22×C4—2442no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4—2442no—
452C22×C4C4⋊#692311yescharacteristic
453C22×C4C22×#692311yescharacteristic
454C2×C8—22no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—612no—
458C2×C8—1222no—
459C2×C8—1224no—
460C2×C8—1222no—
461C2×C8—1222no—
462C2×C8—1221no—
463C2×C8—1222no—
464C2×C8—2442no—
465C2×C8—2441no—
466C2×C8—2441no—
467C2×C8C22⋊#692312yescomplemented
468C4⋊C4—1224no—
469C4⋊C4—1224no—
470C4⋊C4—1224no—
471C4⋊C4—1224no—
472C4⋊C4—2442no—
473C4⋊C4—2442no—
474C4⋊C4—2442no—
475C4⋊C4—2442no—
476C22⋊C4—24416no—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—22no—
481C23—611no—
482C23—611no—
483C23—612no—
484C23—612no—
485C23—612no—
486C23—612no—
487C23—612no—
488C23—612no—
489C23—612no—
490C23—1224no—
491C23—1224no—
492C23—1224no—
493C23—1224no—
494C23—1224no—
495C23—1224no—
496C23—1221no—
497C23—1222no—
498C23—1221no—
499C23—1222no—
500C23—2441no—
501C23—2442no—
502C23—2441no—
503C23—2442no—
504C23—2442no—
505C23—2442no—
506C23—2442no—
507C23—2441no—
508C23—2442no—
509C23—2441no—
510C23#1451984512yescomplemented
511C23C2×C4⋊#692311yescharacteristic
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—611no—
518C2×C4—611no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—611no—
522C2×C4—612no—
523C2×C4—611no—
524C2×C4—612no—
525C2×C4—612no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—612no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—1221no—
532C2×C4—1222no—
533C2×C4—4882no—
534C2×C4—4882no—
535C2×C4—4882no—
536C2×C4—4882no—
537C2×C4—4882no—
538C2×C4—4882no—
539C2×C4—4882no—
540C2×C4—4882no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2442no—
549C2×C4—2441no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4—2444no—
553C2×C4—2444no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2441no—
557C2×C4—2442no—
558C2×C4—2441no—
559C2×C4C2×C22⋊#692311yescharacteristic
560C2×C4#1451985611yescharacteristic
561C2×C4D4×#692312yes—
562C2×C4D4×#692312yes—
563C8—22no—
564C8—41no—
565C8—1222no—
566C8—2441no—
567C22—24no—
568C22—22no—
569C22—41no—
570C22—42no—
571C22—41no—
572C22—612no—
573C22—612no—
574C22—612no—
575C22—612no—
576C22—611no—
577C22—612no—
578C22—611no—
579C22—612no—
580C22—612no—
581C22—612no—
582C22—612no—
583C22—612no—
584C22—612no—
585C22—611no—
586C22—612no—
587C22—611no—
588C22—1224no—
589C22—1222no—
590C22—2444no—
591C22—2444no—
592C22—2444no—
593C22—2444no—
594C22—2441no—
595C22—2442no—
596C22—2441no—
597C22#1684070411yescharacteristic
598C22#1684070912yes—
599C22C2×#1451984511yescharacteristic
600C22#1684077211yescharacteristic
601C22#1684077312yes—
602C4—41no—
603C4—611no—
604C4—612no—
605C4—611no—
606C4—4884no—
607C4—4884no—
608C4—2441no—
609C4#1684071011yescharacteristic
610C4#1684071212yes—
611C4#1684071411yescharacteristic
612C2—42no—
613C2—612no—
614C2—612no—
615C2—612no—
616C2—612no—
617C2—2442no—
618C2#2146100211yescharacteristic
619C2#2146169511yescharacteristic
620C2#2146169711yescharacteristic
621C2#2146176912yes—
622C2#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#11095
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×#14519845C2semidirect
3. C2×#16840636C2semidirect
4. C2×#16840636C2semidirect
19. C2×C22⋊#92418C4semidirect
20. C2×#14519689C4semidirect
21. C2×#14519689C4semidirect
24. C2×C22⋊C8×C61C4semidirect
154. C2×C8×C61C22⋊C4semidirect
158. C2×#92418C22⋊C4semidirect
169. C23×C61#106semidirect
191. C22×D61#106semidirect
330. C2×C22⋊C8#6923semidirect
336. C61#11095semidirect
467. C2×C8C22⋊#6923semidirect
510. C23#14519845semidirect

Maximal subgroups

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

subgroupindex
C22×#145198452
C2×#168407042
C2×#168406362
C2×C22⋊C8×D612
#1109561