Finite Group Databasebeta

← #30602592

Subgroups — #30602592

Order 15616.

1370 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519840C212yesmaximal
2C2×#16840635C212yesmaximal
3#21461480C211yesmaximal, characteristic
4#21460997C211yesmaximal, characteristic
5C2×C4×Q8×D61C211yesmaximal, characteristic
6C22×D61⋊C8C2212yes—
7C22×C4⋊#6923C2212yes—
8C2×#14519840—24no—
9C2×#14519840—24no—
10C2×#14519840C418yescomplemented
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4—21no—
13#16840635—24no—
14#16840628—21no—
15#16840628—21no—
16C22×Q8×D61C2211yescharacteristic
17C2×Q8×Dic61C411yescharacteristic, complemented
18C2×C4×Dic122—21no—
19C2×C4×Dic122C411yescharacteristic, complemented
20C2×#14519742—21no—
21C2×#14519742C411yescharacteristic, complemented
22C4×Q8×D61—24no—
23C4×Q8×D61—24no—
24C2×C4⋊C4×D61—21no—
25C2×C4⋊C4×D61C2211yescharacteristic
26C2×C42×D61—21no—
27C2×C42×D61C2211yescharacteristic
28C2×C4×Q8×C61C411yescharacteristic, complemented
29C23×#6923—22no—
30C2×D61⋊C8—22no—
31C2×D61⋊C8—22no—
32C2×D61⋊C8—22no—
33C2×D61⋊C8—22no—
34C2×D61⋊C8—22no—
35C2×D61⋊C8C2×C414yes—
36C22×#92420—22no—
37C2×C4⋊#6923—22no—
38C2×C4⋊#6923—22no—
39C2×C4⋊#6923—22no—
40C2×C4⋊#6923—22no—
41C2×C4⋊#6923—22no—
42C2×C4⋊#6923C2×C414yes—
43#14519840—216no—
44(C2×#6923)⋊C4—42no—
45(C2×#6923)⋊C4—42no—
46C22×Dic122—21no—
47C22×Dic122C2×C411yescharacteristic
48C2×Q8×D61—22no—
49C2×Q8×D61—22no—
50C2×Q8×D61—21no—
51C2×Q8×D61—21no—
52C2×Q8×D61C2×C414yes—
53C2×Q8×D61D414yes—
54C2×Q8×D61D412yes—
55C2×Q8×D61Q812yes—
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×C4×D61C2311yescharacteristic
61Q8×Dic61—22no—
62Q8×Dic61—22no—
63C4×Dic122—28no—
64C4×Dic122—22no—
65C4×Dic122—22no—
66#14519742—28no—
67#14519742—22no—
68#14519742—22no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2×C411yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C4⋊C4×D61—28no—
75C4⋊C4×D61—22no—
76C4⋊C4×D61D414yes—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61C2×C411yescharacteristic
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C42×D61—28no—
84C42×D61—22no—
85C42×D61—22no—
86C22×Q8×C61C2×C411yescharacteristic
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2×C411yescharacteristic
91C2×C42×C61—21no—
92C2×C42×C61C2×C411yescharacteristic
93C23×D61C2×D411yescharacteristic
94C22×#6923—24no—
95C22×#6923—42no—
96C22×#6923—42no—
97C22×#6923—42no—
98C22×#6923—42no—
99C22×#6923—42no—
100C22×#6923—42no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61C4212yes—
105C2×Q8×C61C22⋊C412yes—
106C2×Q8×C61C4⋊C412yes—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61C22⋊C411yescharacteristic
110C22×C4×C61C22⋊C411yescharacteristic
111C22×C4×C61C22×C411yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—21no—
114C22×Dic61C22⋊C411yescharacteristic
115C22×Dic61C22⋊C411yescharacteristic
116C22×Dic61C2×D411yescharacteristic
117Q8×D61—24no—
118Q8×D61—24no—
119Q8×D61—24no—
120Q8×D61—24no—
121Q8×D61—24no—
122Q8×D61—24no—
123Q8×D61C22⋊C418yes—
124Q8×D61C4⋊C418yes—
125C2×C4×D61—24no—
126C2×C4×D61—24no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—24no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—42no—
137C2×C4×D61—41no—
138C2×C4×D61—41no—
139C2×C4×D61SD1612yes—
140C2×C4×D61Q1612yes—
141C2×C4×D61C22×C412yes—
142C2×C4×D61C2×D411yescharacteristic
143C2×C4×D61C2×D412yes—
144C2×C4×D61C2×D412yes—
145C2×C4×D61C2×Q811yescharacteristic
146C2×C4×D61#4111yescharacteristic
147C2×C4×D61#4111yescharacteristic
148C2×C4×D61#4112yes—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—21no—
152C2×Dic122—24no—
153C2×Dic122—24no—
154C2×Dic122—22no—
155C2×Dic122—22no—
156C2×Dic122C4212yes—
157C2×Dic122C22⋊C412yes—
158C2×Dic122C4⋊C412yes—
159C2×#92420—24no—
160C2×#92420—42no—
161C2×#92420—42no—
162C4⋊#6923—24no—
163C4⋊#6923—24no—
164C4⋊#6923—42no—
165C4⋊#6923—42no—
166D61⋊C8—24no—
167D61⋊C8—24no—
168D61⋊C8—42no—
169D61⋊C8—42no—
170C4⋊C4×C61—24no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C22⋊C412yes—
173C42×C61—24no—
174C42×C61—21no—
175C42×C61—21no—
176C4⋊Dic61—24no—
177C4⋊Dic61—21no—
178C4⋊Dic61C22⋊C412yes—
179#11758196—24no—
180#11758196—24no—
181#11758196—24no—
182C4×Dic61—24no—
183C4×Dic61—24no—
184C4×Dic61—24no—
185C4×Dic61—21no—
186C4×Dic61—21no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189C23×C61C2×C22⋊C411yescharacteristic
190C22×D61C4×D412yes—
191C22×D61#12012yes—
192C22×D61#12111yescharacteristic
193C22×D61C22⋊Q811yescharacteristic
194C22×D61C2×SD1611yescharacteristic
195C22×D61C2×Q1611yescharacteristic
196C22×D61#13512yes—
197C22×D61AGL(1,ℤ/8)11yescharacteristic
198C22×D61#13712yes—
199C22×D61#13711yescharacteristic
200C2×#6923—44no—
201C2×#6923—44no—
202C2×#6923—44no—
203C2×#6923—82no—
204C2×#6923—82no—
205C2×#6923—82no—
206C2×#6923—82no—
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61#9514yescomplemented
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61#10312yescomplemented
223C2×C4×C61C2×C4211yescharacteristic
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C2×C4⋊C411yescharacteristic
227C2×C4×C61#11711yescharacteristic
228C2×C4×C61#11711yescharacteristic
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61#10312yescomplemented
241C2×Dic61C4×D412yes—
242C2×Dic61#12012yes—
243C2×Dic61#12111yescharacteristic
244C2×Dic61C22⋊Q811yescharacteristic
245C4×D61—28no—
246C4×D61—28no—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—42no—
250C4×D61—44no—
251C4×D61—42no—
252C4×D61—44no—
253C4×D61C2×C22⋊C412yes—
254C4×D61C2×C4⋊C412yes—
255C4×D61C4×D412yes—
256C4×D61C4×D412yes—
257C4×D61#12112yes—
258C4×D61C22⋊Q812yes—
259C4×D61#12312yes—
260C4×D61#12312yes—
261Dic122—22no—
262Dic122—22no—
263Dic122—22no—
264Dic122—28no—
265Dic122—28no—
266Dic122#9514yescomplemented
267#92420—44no—
268#10789—611nomaximal, Sylow
269C22×C61C4×C22⋊C411yescharacteristic
270C22×C61#37911yescharacteristic
271C22×C61#38011yescharacteristic
272C22×C61C2×#10311yescharacteristic
273C22×C61#41611yescharacteristic
274C22×C61#41711yescharacteristic
275C22×C61#41911yescharacteristic
276C2×D61#38512yes—
277C2×D61#38612yes—
278C2×D61C4×SD1612yes—
279C2×D61C4×Q1612yes—
280C2×D61#44012yes—
281C2×D61#44112yes—
282C2×D61#44812yes—
283C2×D61#44912yes—
284C2×D61#45112yes—
285C2×D61#45212yes—
286C2×D61#46011yescharacteristic
287C2×D61#46211yescharacteristic
288C2×D61#46412yes—
289C2×D61#47511yescharacteristic
290C2×D61#47711yescharacteristic
291C2×D61#47912yes—
292#6923—84no—
293C4×C61—24no—
294C4×C61—22no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61C2×#9511yescharacteristic
298C4×C61C4×C22⋊C411yescharacteristic
299C4×C61#38011yescharacteristic
300C4×C61#38111yescharacteristic
301Dic61—24no—
302Dic61—22no—
303Dic61—41no—
304Dic61—42no—
305Dic61#38512yes—
306Dic61#38612yes—
307C22×C4×Q8—611no—
308C22×#103—612no—
309C2×#340—612no—
310C2×#336—611no—
311C4⋊#95—611no—
312C2×C61C22⋊#9511yescharacteristic
313C2×C61C4×#10311yescharacteristic
314C2×C61#176311yescharacteristic
315C2×C61#178711yescharacteristic
316C2×C61#178811yescharacteristic
317C2×C61#180311yescharacteristic
318C2×C61#180511yescharacteristic
319D61#186312yes—
320D61#186512yes—
321D61#187812yes—
322D61#188012yes—
323C23×Q8—611no—
324C23×C8—612no—
325C2×C4×Q8—611no—
326C2×C4×Q8—611no—
327C2×C4×Q8—611no—
328C2×C4×Q8—1221no—
329C2×C4×Q8—1221no—
330C2×C4×Q8—1224no—
331C2×C4×Q8—1224no—
332C2×C4×Q8#692311yescharacteristic, complemented
333C22×C4⋊C4—611no—
334C22×C4⋊C4—612no—
335C22×C4⋊C4—1221no—
336C22×C42—611no—
337C22×C42—1221no—
338C2×#103—618no—
339C2×#103—1224no—
340C2×#103—1224no—
341C2×#95—1221no—
342C2×#95—1221no—
343#340—1224no—
344#336—1221no—
345#336—1221no—
346C61#1078911yescharacteristic, complemented, Sylow
347C22×Q8—614no—
348C22×Q8—612no—
349C22×Q8—612no—
350C22×Q8—614no—
351C22×Q8—611no—
352C22×Q8—1221no—
353C22×Q8—1222no—
354C22×Q8—1221no—
355C22×Q8—1222no—
356C22×Q8—1221no—
357C22×Q8C2×#692311yescharacteristic
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—1221no—
362C23×C4—1221no—
363C23×C4—1222no—
364C22×C8—614no—
365C22×C8—1222no—
366C22×C8—1222no—
367C22×C8—1222no—
368C22×C8—1222no—
369C22×C8—1222no—
370C22×C8—1222no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—1222no—
374C4×Q8—1222no—
375C4×Q8—1228no—
376C4×Q8—1222no—
377C4×Q8—1228no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—614no—
384C2×C4⋊C4—614no—
385C2×C4⋊C4—1222no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1222no—
392C2×C4⋊C4—1222no—
393C2×C4⋊C4—1222no—
394C2×C4⋊C4—1222no—
395C2×C4⋊C4—1222no—
396C2×C4⋊C4C2×#692311yescharacteristic
397C2×C42—21no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—1222no—
402C2×C42—1222no—
403C2×C42—1228no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42C2×#692311yescharacteristic
408#103—12216no—
409#95—2442no—
410#95—2442no—
411C24—611no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—618no—
416C2×Q8—618no—
417C2×Q8—612no—
418C2×Q8—612no—
419C2×Q8—612no—
420C2×Q8—1224no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×Q8—1224no—
424C2×Q8—1224no—
425C2×Q8—1224no—
426C2×Q8—1224no—
427C2×Q8—1224no—
428C2×Q8—1224no—
429C2×Q8—1224no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8C4×#692312yes—
434C2×Q8C4⋊#692312yes—
435C2×Q8C22⋊#692312yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—612no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—611no—
451C22×C4—1222no—
452C22×C4—1224no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1224no—
457C22×C4—1221no—
458C22×C4—1224no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1224no—
465C22×C4—2442no—
466C22×C4—2441no—
467C22×C4—2441no—
468C22×C4—2442no—
469C22×C4—2442no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4—2442no—
474C22×C4C22⋊#692311yescharacteristic
475C22×C4C22⋊#692311yescharacteristic
476C22×C4C22×#692311yescharacteristic
477C2×C8—1224no—
478C2×C8—1224no—
479C2×C8—1224no—
480C2×C8—2442no—
481C2×C8—2442no—
482C2×C8—2442no—
483C2×C8—2442no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—612no—
487C4⋊C4—1224no—
488C4⋊C4—1224no—
489C4⋊C4—1224no—
490C4⋊C4—1224no—
491C4⋊C4—1224no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—2442no—
495C4⋊C4—2442no—
496C4⋊C4C22⋊#692312yes—
497C42—24no—
498C42—21no—
499C42—21no—
500C42—1222no—
501C42—1222no—
502C42—1224no—
503C42—1224no—
504C42—1224no—
505C42—1221no—
506C42—1221no—
507C23—611no—
508C23—611no—
509C23—611no—
510C23—611no—
511C23—612no—
512C23—611no—
513C23—611no—
514C23—612no—
515C23—612no—
516C23—612no—
517C23C2×C22⋊#692311yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—22no—
521Q8—614no—
522Q8—1228no—
523Q8—1228no—
524Q8—1222no—
525Q8—1222no—
526Q8—1222no—
527Q8(C2×#6923)⋊C414yescomplemented
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—1228no—
555C2×C4—1228no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1221no—
560C2×C4—1222no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1222no—
564C2×C4—1221no—
565C2×C4—4882no—
566C2×C4—4882no—
567C2×C4—4882no—
568C2×C4—4882no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4—2444no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4#1451984012yescomplemented
579C2×C4C2×C4×#692311yescharacteristic
580C2×C4C2×C22⋊#692311yescharacteristic
581C2×C4C2×C22⋊#692311yescharacteristic
582C2×C4C2×C4⋊#692311yescharacteristic
583C2×C4#1451985611yescharacteristic
584C2×C4#1451985611yescharacteristic
585C8—2444no—
586C22—612no—
587C22—612no—
588C22—612no—
589C22—612no—
590C22—612no—
591C22—612no—
592C22—612no—
593C22—612no—
594C22—612no—
595C22—612no—
596C22—611no—
597C22—611no—
598C22—611no—
599C22—611no—
600C22—612no—
601C22—612no—
602C22C4×C22⋊#692311yescharacteristic
603C22#1684070311yescharacteristic
604C22#1684070411yescharacteristic
605C22C2×#1451984011yescharacteristic
606C22#1684075211yescharacteristic
607C22#1684075311yescharacteristic
608C22#1684075511yescharacteristic
609C4—24no—
610C4—22no—
611C4—41no—
612C4—42no—
613C4—612no—
614C4—612no—
615C4—1224no—
616C4—1222no—
617C4—4884no—
618C4—2441no—
619C4—2442no—
620C4C2×(C2×#6923)⋊C411yescharacteristic
621C4C4×C22⋊#692311yescharacteristic
622C4#1684070411yescharacteristic
623C4#1684070511yescharacteristic
624C2—612no—
625C2—612no—
626C2—612no—
627C2—612no—
628C2#2146099311yescharacteristic
629C2C4×#1451984011yescharacteristic
630C2#2146159111yescharacteristic
631C2#2146165811yescharacteristic
632C2#2146165911yescharacteristic
633C2#2146168811yescharacteristic
634C2#2146169011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#10789
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
10. C2×#14519840C4semidirect
17. C2×Q8×Dic61C4semidirect
19. C2×C4×Dic122C4semidirect
21. C2×#14519742C4semidirect
28. C2×C4×Q8×C61C4semidirect
210. Q8×C61#95semidirect
222. C2×C4×C61#103semidirect
240. C2×Dic61#103semidirect
266. Dic122#95semidirect
332. C2×C4×Q8#6923semidirect
346. C61#10789semidirect
527. Q8(C2×#6923)⋊C4semidirect
578. C2×C4#14519840semidirect

Maximal subgroups

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

subgroupindex
C22×#145198402
C2×#168406352
#214614802
#214609972
C2×C4×Q8×D612
#1078961