Finite Group Databasebeta

← #30603021

Subgroups — #30603021

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519842C211yesmaximal, characteristic
2C2×#16840737C211yesmaximal, characteristic
3C2×#16840727C211yesmaximal, characteristic, complemented
4C2×#16840638C211yesmaximal, characteristic, complemented
5C2×#16840632C211yesmaximal, characteristic, complemented
6C2×(C2×#6923)⋊C8C211yesmaximal, characteristic, complemented
7C2×#16839757C211yesmaximal, characteristic
8C22×#11758213C2211yescharacteristic
9C22×#11758213C2211yescharacteristic
10C22×D61⋊C8C2211yescharacteristic
11C22×C4×#6923C2211yescharacteristic
12C2×#14519842—22no—
13C2×#14519842—22no—
14C2×#14519842C414yescomplemented
15#16840737—22no—
16#16840737C414yescomplemented
17C2×#14519834—21no—
18C2×#14519834—21no—
19#16840727—22no—
20#16840727—22no—
21C2×#14519832—21no—
22C2×C4×#92420—21no—
23#16840638—22no—
24#16840632—22no—
25(C2×#6923)⋊C8—22no—
26C22×#11758226C2211yescharacteristic
27C22×C8×D61C2211yescharacteristic
28C2×#14519735C2211yescharacteristic
29#16839757—22no—
30#16839757—22no—
31C2×#14519701C411yescharacteristic, complemented
32C2×#14519699C411yescharacteristic, complemented
33C2×#14519697C411yescharacteristic, complemented
34C2×#103×C61C411yescharacteristic, complemented
35C23×#6923—21no—
36C2×#11758213—21no—
37C2×#11758213—21no—
38C2×#11758213—21no—
39C2×#11758213—21no—
40C2×#11758213—21no—
41C2×#11758213—21no—
42C2×#11758213—21no—
43C2×#11758213—21no—
44C2×#11758213C2×C412yes—
45C2×#11758213C2×C412yes—
46C2×#11758213D412yes—
47C2×#11758213D412yes—
48C2×D61⋊C8—21no—
49C2×D61⋊C8—21no—
50C2×D61⋊C8—21no—
51C2×D61⋊C8—21no—
52C2×D61⋊C8C2×C412yes—
53C2×D61⋊C8D412yes—
54C22×#92420—21no—
55C22×#92420—21no—
56C22×#92420D411yescharacteristic
57C22×#92420D411yescharacteristic
58C2×C4×#6923—21no—
59C2×C4×#6923—21no—
60C2×C4×#6923—21no—
61C2×C4×#6923—21no—
62C2×C4×#6923—21no—
63C2×C4×#6923C2×C412yes—
64#14519842—28no—
65#14519834—24no—
66#14519834—24no—
67#14519832—24no—
68C4×#92420—24no—
69C2×#11758226—21no—
70C2×#11758226—21no—
71C2×#11758226—21no—
72C2×#11758226—21no—
73C2×#11758226—21no—
74C2×#11758226—21no—
75C2×#11758226—21no—
76C2×#11758226—21no—
77C2×#11758226—21no—
78C2×#11758226—21no—
79C2×#11758226C2×C412yes—
80C2×#11758226C2×C412yes—
81C22×D244—21no—
82C22×D244C2×C411yescharacteristic
83C22×C4×D61—21no—
84C22×C4×D61C2311yescharacteristic
85C22×#92418C2×C411yescharacteristic
86C2×C8×D61—21no—
87C2×C8×D61—21no—
88C2×C8×D61—21no—
89C2×C8×D61—21no—
90C2×C8×D61—21no—
91C2×C8×D61—21no—
92#14519735—21no—
93#14519735—21no—
94#14519735D412yes—
95#14519735D412yes—
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99#14519701—22no—
100#14519699—22no—
101#14519697—22no—
102C22×Q8×C61C2×C411yescharacteristic
103C22×C8×C61C2×C411yescharacteristic
104C2×C4⋊C4×C61C2×C411yescharacteristic
105#103×C61—22no—
106C23×D61—21no—
107C23×D61#4111yescharacteristic
108C22×#6923—22no—
109C22×#6923—22no—
110C22×#6923—22no—
111C22×#6923—41no—
112C22×#6923—41no—
113C22×#6923—41no—
114C22×#6923—41no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61C4212yes—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C22×C411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61C2×D411yescharacteristic
126#11758226—22no—
127#11758226—22no—
128#11758226—22no—
129#11758226—22no—
130#11758226—22no—
131#11758226—22no—
132#11758226—22no—
133#11758226—22no—
134#11758226—41no—
135#11758226—41no—
136#11758226—41no—
137#11758226—41no—
138#11758226—41no—
139#11758226—41no—
140#11758226—41no—
141#11758226—41no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—22no—
147C2×D244—22no—
148C2×D244—21no—
149C2×D244—41no—
150C2×D244—41no—
151C2×D244—41no—
152C2×D244—41no—
153C2×D244C4212yes—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—41no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×C4×D61—41no—
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C22×C411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61#4111yescharacteristic
168C2×C4×D61#4111yescharacteristic
169C2×C4×D61#4111yescharacteristic
170C2×C4×D61#4111yescharacteristic
171C2×C4×D61#4111yescharacteristic
172C2×#92420—22no—
173C2×#92420—21no—
174C2×#92420—22no—
175C2×#92420—21no—
176C2×#92420—22no—
177C2×#92420—21no—
178C2×#92420—21no—
179C2×#92420—22no—
180C2×#92420—41no—
181C2×#92420—41no—
182C2×#92420C22⋊C412yes—
183C2×#92420C22⋊C412yes—
184C4×#6923—22no—
185C4×#6923—22no—
186C4×#6923—41no—
187C4×#6923—41no—
188#11758213—22no—
189#11758213—22no—
190#11758213—22no—
191#11758213—22no—
192#11758213—22no—
193#11758213—41no—
194#11758213—41no—
195#11758213C22⋊C414yes—
196D61⋊C8—22no—
197D61⋊C8—22no—
198D61⋊C8—22no—
199D61⋊C8—22no—
200C2×C8×C61—21no—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61C22⋊C412yes—
205#11758198—22no—
206#11758198—22no—
207C4⋊Dic61—21no—
208C4⋊Dic61C22⋊C412yes—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C8×D61—41no—
215C8×D61—41no—
216C8×D61—41no—
217C8×D61—41no—
218C23×C61C4×D411yescharacteristic
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—22no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61#11711yescharacteristic
231C22×D61C4×D411yescharacteristic
232C22×D61#12111yescharacteristic
233C22×D61#12311yescharacteristic
234C22×D61#12411yescharacteristic
235C22×D61#12611yescharacteristic
236C2×#6923—24no—
237C2×#6923—42no—
238C2×#6923—42no—
239C2×#6923—42no—
240C2×#6923—42no—
241C2×#6923—42no—
242C2×#6923—42no—
243Q8×C61—22no—
244Q8×C61—22no—
245Q8×C61—41no—
246Q8×C61—41no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61C2×C4211yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61#11711yescharacteristic
256C2×C4×C61C4×D411yescharacteristic
257C2×C4×C61C4×D411yescharacteristic
258C2×C4×C61C4×D411yescharacteristic
259C2×Dic61—22no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C4×D411yescharacteristic
264C2×Dic61#12011yescharacteristic
265C2×Dic61#12111yescharacteristic
266C2×Dic61#12111yescharacteristic
267C2×Dic61#12311yescharacteristic
268D244—22no—
269D244—22no—
270D244—42no—
271D244—42no—
272D244—42no—
273D244—42no—
274D244—41no—
275D244—41no—
276C4×D61—42no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—42no—
280C4×D61C2×C22⋊C411yescharacteristic
281C4×D61#11711yescharacteristic
282C4×D61C4×D411yescharacteristic
283C4×D61C4×D411yescharacteristic
284C4×D61C4×D411yescharacteristic
285C4×D61C4×D411yescharacteristic
286C4×D61C4×D411yescharacteristic
287C4×D61C4×D411yescharacteristic
288C4×D61#12111yescharacteristic
289C4×D61#12111yescharacteristic
290C4×D61#12111yescharacteristic
291C4×D61#12311yescharacteristic
292C4×D61#12411yescharacteristic
293C4×D61#12411yescharacteristic
294C4×D61#12611yescharacteristic
295C4×D61#12711yescharacteristic
296#92420—22no—
297#92420—22no—
298#92420—24no—
299#92420—42no—
300C8×C61—41no—
301#92418—41no—
302#10934—611nomaximal, Sylow
303C22×C61C4×C22⋊C411yescharacteristic
304C22×C61#38611yescharacteristic
305C22×C61#38811yescharacteristic
306C22×C61#44311yescharacteristic
307C22×C61#44311yescharacteristic
308C22×C61#44411yescharacteristic
309C22×C61#44411yescharacteristic
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—21no—
317C2×D61—21no—
318C2×D61—21no—
319C2×D61—21no—
320C2×D61—21no—
321C2×D61—21no—
322C2×D61—42no—
323C2×D61—42no—
324C2×D61—42no—
325C2×D61—81no—
326C2×D61—81no—
327C2×D61—81no—
328C2×D61—81no—
329C2×D61#38811yescharacteristic
330C2×D61#38811yescharacteristic
331C2×D61#38811yescharacteristic
332C2×D61#39011yescharacteristic
333#6923—44no—
334C4×C61—42no—
335C4×C61—42no—
336C4×C61C4×C22⋊C411yescharacteristic
337C4×C61C4×C22⋊C411yescharacteristic
338C4×C61#38811yescharacteristic
339C4×C61#39011yescharacteristic
340Dic61—42no—
341Dic61#38611yescharacteristic
342Dic61#38611yescharacteristic
343Dic61#38811yescharacteristic
344Dic61#39011yescharacteristic
345C22×#104—611no—
346C2×#416—611no—
347C2×#408—611no—
348C2×#405—611no—
349C2×#341—611no—
350C2×#339—611no—
351C2×#336—611no—
352C2×C61#143711yescharacteristic
353C2×C61#176411yescharacteristic
354C2×C61#176511yescharacteristic
355C2×C61#187611yescharacteristic
356C2×C61#187611yescharacteristic
357C2×C61#192911yescharacteristic
358C2×C61#192911yescharacteristic
359D61—21no—
360D61—21no—
361D61—21no—
362D61—21no—
363D61—82no—
364C22×#41—611no—
365C22×M16—611no—
366C22×M16—611no—
367C23×C8—611no—
368C23×C8—611no—
369C2×#117—611no—
370C22×C42—611no—
371C2×#104—614no—
372C2×#104—1222no—
373C2×#104—1222no—
374#416—1222no—
375#416—1222no—
376C2×#103—611no—
377C2×#103#692311yescharacteristic, complemented
378C2×#102—611no—
379C2×#102—611no—
380#408—614no—
381#408—1222no—
382C2×C22⋊C8—1221no—
383C2×C22⋊C8—1221no—
384#405—1222no—
385#405—1222no—
386C2×#97—1221no—
387C2×C4×C8—1221no—
388#341—1222no—
389#339—1222no—
390#336—1222no—
391C61#1093411yescharacteristic, complemented, Sylow
392C2×#41—612no—
393C2×#41—612no—
394C2×#41—1221no—
395C2×#41—1221no—
396C2×#41—1221no—
397C2×#41—1221no—
398C2×#41—1221no—
399C2×#41—1221no—
400C2×#41—1221no—
401C2×#41—1221no—
402C2×#41—1221no—
403C2×#41—1221no—
404C22×Q8C2×#692311yescharacteristic
405C22×D4—611no—
406C22×D4—1221no—
407C23×C4—611no—
408C23×C4—1221no—
409C23×C4—1221no—
410C2×M16—612no—
411C2×M16—612no—
412C2×M16—612no—
413C2×M16—612no—
414C2×M16—1221no—
415C2×M16—1221no—
416C2×M16—1221no—
417C2×M16—1221no—
418C2×M16—1221no—
419C2×M16—1221no—
420C2×M16—1221no—
421C2×M16—1221no—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8—612no—
425C22×C8—612no—
426C22×C8—611no—
427C22×C8—1221no—
428C22×C8—1221no—
429C22×C8—1221no—
430C22×C8—1221no—
431C22×C8—1221no—
432C22×C8—1221no—
433C22×C8—1221no—
434C22×C8—1221no—
435C22×C8—1221no—
436C22×C8—1221no—
437C22×C8—1221no—
438C22×C8—1221no—
439C22×C8C2×#692311yescharacteristic
440#117—612no—
441#117—612no—
442#117—1221no—
443#117—1221no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4C2×#692311yescharacteristic
446C2×C22⋊C4—1221no—
447C2×C42—612no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42—1221no—
454#104—1228no—
455#103—22no—
456#103—1222no—
457#102—1222no—
458#102—1222no—
459C22⋊C8—1224no—
460C22⋊C8—1224no—
461#97—1224no—
462C4×C8—1224no—
463C24—611no—
464C24—1221no—
465#41—1222no—
466#41—1222no—
467#41—1222no—
468#41—1222no—
469#41—1222no—
470#41—1222no—
471#41—1222no—
472#41—1222no—
473#41—2441no—
474#41—2441no—
475#41—2441no—
476#41—2441no—
477#41—2441no—
478#41—2441no—
479#41—2441no—
480#41—2441no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8C4×#692312yes—
487C2×D4—612no—
488C2×D4—1222no—
489C2×D4—1222no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—2441no—
496C2×D4—2441no—
497C2×D4—2441no—
498C2×D4—2441no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—1222no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1221no—
520C22×C4—2441no—
521C22×C4—2441no—
522C22×C4—2441no—
523C22×C4—2441no—
524C22×C4—2441no—
525C22×C4—2441no—
526C22×C4—2441no—
527C22×C4—2441no—
528C22×C4C22×#692311yescharacteristic
529M16—614no—
530M16—1222no—
531M16—1222no—
532M16—1222no—
533M16—1222no—
534M16—1222no—
535M16—2441no—
536M16—2441no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—612no—
541C2×C8—612no—
542C2×C8—1221no—
543C2×C8—1222no—
544C2×C8—1221no—
545C2×C8—1222no—
546C2×C8—1222no—
547C2×C8—1222no—
548C2×C8—1222no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1222no—
552C2×C8—1222no—
553C2×C8—1222no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8—1221no—
557C2×C8—2441no—
558C2×C8—2441no—
559C2×C8—2441no—
560C2×C8—2441no—
561C2×C8—2441no—
562C2×C8—2441no—
563C4⋊C4—21no—
564C4⋊C4—612no—
565C4⋊C4—1221no—
566C4⋊C4C22⋊#692312yes—
567C22⋊C4—1222no—
568C22⋊C4—1222no—
569C42—1222no—
570C42—1222no—
571C42—1222no—
572C42—1222no—
573C42—2441no—
574C42—2441no—
575C23—611no—
576C23—611no—
577C23—611no—
578C23—611no—
579C23—611no—
580C23—611no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23—1222no—
586C23—2441no—
587C23—2441no—
588C23—2441no—
589C23—2441no—
590C23—2441no—
591C23—2441no—
592C23D4×#692311yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—41no—
596Q8—41no—
597D4—1222no—
598D4—1222no—
599D4—2442no—
600D4—2442no—
601D4—2442no—
602D4—2442no—
603D4—2441no—
604D4—2441no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—611no—
614C2×C4—611no—
615C2×C4—611no—
616C2×C4—611no—
617C2×C4—611no—
618C2×C4—611no—
619C2×C4—611no—
620C2×C4—611no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—611no—
624C2×C4—611no—
625C2×C4—611no—
626C2×C4—611no—
627C2×C4—611no—
628C2×C4—611no—
629C2×C4—611no—
630C2×C4—611no—
631C2×C4—611no—
632C2×C4—611no—
633C2×C4—1224no—
634C2×C4—1222no—
635C2×C4—2442no—
636C2×C4—2442no—
637C2×C4—2442no—
638C2×C4—2442no—
639C2×C4—2442no—
640C2×C4—2442no—
641C2×C4—2442no—
642C2×C4—2442no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4C2×C4×#692311yescharacteristic
648C2×C4C2×C22⋊#692311yescharacteristic
649C2×C4#1451985611yescharacteristic
650C2×C4D4×#692311yescharacteristic
651C2×C4D4×#692311yescharacteristic
652C2×C4D4×#692311yescharacteristic
653C8—41no—
654C8—1222no—
655C8—1222no—
656C8—1224no—
657C8—2442no—
658C8—2441no—
659C22—611no—
660C22—611no—
661C22—611no—
662C22—611no—
663C22—1221no—
664C22—1221no—
665C22—1221no—
666C22—1221no—
667C22—1221no—
668C22—1221no—
669C22—1221no—
670C22—1221no—
671C22—1221no—
672C22—1221no—
673C22—1221no—
674C22—1221no—
675C22—4881no—
676C22—4881no—
677C22—4881no—
678C22—4881no—
679C22—2442no—
680C22—2442no—
681C22—2442no—
682C22C22⋊C4×#692311yescharacteristic
683C22#1684071011yescharacteristic
684C22#1684071211yescharacteristic
685C22#1684080511yescharacteristic
686C22#1684080611yescharacteristic
687C22#1684080711yescharacteristic
688C22#1684080811yescharacteristic
689C4—42no—
690C4—42no—
691C4—611no—
692C4—611no—
693C4—611no—
694C4—611no—
695C4—2444no—
696C4—2442no—
697C4C4×C22⋊#692311yescharacteristic
698C4C22⋊C4×#692311yescharacteristic
699C4#1684071211yescharacteristic
700C4#1684071411yescharacteristic
701C2—1221no—
702C2—1221no—
703C2—1221no—
704C2—1221no—
705C2—4882no—
706C2#2146100011yescharacteristic
707C2#2146159611yescharacteristic
708C2#2146160011yescharacteristic
709C2#2146180811yescharacteristic
710C2#2146180911yescharacteristic
711C2#2146190211yescharacteristic
712C2#2146190311yescharacteristic
subgroupG/H
centerC23D4×#6923
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×#103×C61C4
socleC23×C61C4×D4
prime radicalC22×#11758226C22
cosocleC22×C4×D61C23
hypercenterC2×#103#6923
hypocenterC61#10934
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#16840727C2semidirect
4. C2×#16840638C2semidirect
5. C2×#16840632C2semidirect
6. C2×(C2×#6923)⋊C8C2semidirect
14. C2×#14519842C4semidirect
16. #16840737C4semidirect
31. C2×#14519701C4semidirect
32. C2×#14519699C4semidirect
33. C2×#14519697C4semidirect
34. C2×#103×C61C4semidirect
377. C2×#103#6923semidirect
391. C61#10934semidirect

Maximal subgroups

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

subgroupindex
C22×#145198422
C2×#168407372
C2×#168407272
C2×#168406382
C2×#168406322
C2×(C2×#6923)⋊C82
C2×#168397572
#1093461