Finite Group Databasebeta

← #29707611

Subgroups — #29707611

Order 15616.

1450 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454408C211yesmaximal, characteristic, complemented
2C22×#14519704C211yesmaximal, characteristic, complemented
3#21454338C211yesmaximal, characteristic, complemented
4C2×#16839746C211yesmaximal, characteristic, complemented
5#21448310C211yesmaximal, characteristic, complemented
6#21447033C214yesmaximal, complemented
7#21447026C214yesmaximal, complemented
8C2×#16839521C211yesmaximal, characteristic, complemented
9C61×#3449C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11C22×#11758194C2211yescharacteristic
12C2×#14519736C2211yescharacteristic
13C22×C4×Dic61C2211yescharacteristic
14#16839805C2214yescomplemented
15#16839802—22no—
16#16839798—22no—
17C2×#14519704—24no—
18C2×#14519704C2214yes—
19#16839771—21no—
20#16839767—21no—
21#16839763C2214yes—
22#16839758—22no—
23#16839746—24no—
24C2×#14519696C2212yescomplemented
25C2×#14519694C2212yescomplemented
26#16839669C2212yescomplemented
27#16839665C2212yescomplemented
28#16839642—22no—
29C4×C22⋊Dic61—22no—
30#16839521C2214yescomplemented
31C22×#41×C61C2211yescharacteristic
32C22×#41×C61C2211yescharacteristic
33C61×#537—22no—
34C61×#536—21no—
35C61×#536C2212yes—
36C61×#535—21no—
37C61×#535C2212yes—
38C61×#534—22no—
39C61×#532—22no—
40C61×#525—22no—
41C61×#525—22no—
42C2×#117×C61C2211yescharacteristic
43C23×Dic61—21no—
44C2×#11758194—21no—
45C2×#11758194—22no—
46C2×#11758194—21no—
47C2×#11758194—22no—
48C2×#11758194—22no—
49C2×#11758194D414yescomplemented
50C2×#11758194C2312yes—
51C2×#11758194C2312yes—
52C22×#92418—21no—
53C22×#92418—21no—
54#14519736—21no—
55#14519736—21no—
56#14519736—21no—
57#14519736—21no—
58C2×C4⋊Dic61C2312yes—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—24no—
61C2×C4×Dic61—22no—
62C2×C4×Dic61—22no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2312yes—
66#14519710—24no—
67#14519708—24no—
68#14519704—28no—
69#14519701—24no—
70#14519698—24no—
71#14519696—24no—
72#14519694—24no—
73#14519684—24no—
74C2×#41×C61—21no—
75C2×#41×C61—21no—
76C2×#41×C61—21no—
77C2×#41×C61—21no—
78C2×#41×C61—22no—
79C2×#41×C61—21no—
80C2×#41×C61—21no—
81C2×#41×C61—22no—
82C2×#41×C61—22no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—22no—
86C2×#41×C61—22no—
87C2×#41×C61—22no—
88C2×#41×C61D412yes—
89C2×#41×C61C2312yes—
90C22×Q8×C61D411yescharacteristic
91C22×Q8×C61C2311yescharacteristic
92C22×D4×C61—21no—
93C22×D4×C61D412yes—
94C22×D4×C61D411yescharacteristic
95C22×D4×C61C2311yescharacteristic
96C23×C4×C61—21no—
97C23×C4×C61D412yes—
98C23×C4×C61C2311yescharacteristic
99C4⋊Q8×C61—22no—
100#127×C61—22no—
101#124×C61—22no—
102#124×C61—22no—
103#123×C61—24no—
104#123×C61—24no—
105C22⋊Q8×C61—24no—
106C22⋊Q8×C61—24no—
107#121×C61—28no—
108#121×C61—24no—
109#121×C61—24no—
110#120×C61—24no—
111#120×C61—24no—
112C4×D4×C61—28no—
113#117×C61—21no—
114#117×C61—21no—
115#117×C61D412yes—
116#117×C61C2312yes—
117C2×C4⋊C4×C61D412yes—
118C2×C22⋊C4×C61—21no—
119C2×C42×C61—21no—
120C24×C61—21no—
121C24×C61C2×D412yes—
122C24×C61C2×D411yescharacteristic
123#41×C61—24no—
124#41×C61—24no—
125#41×C61—22no—
126#41×C61—22no—
127#41×C61—22no—
128#41×C61—22no—
129#41×C61—42no—
130#41×C61—42no—
131#41×C61—42no—
132#41×C61—42no—
133#41×C61—42no—
134#41×C61—42no—
135#41×C61—42no—
136#41×C61—42no—
137#41×C61—42no—
138#41×C61—42no—
139#41×C61—42no—
140#41×C61—42no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—22no—
149C2×Q8×C61—22no—
150C2×Q8×C61C2×D412yes—
151C2×D4×C61—22no—
152C2×D4×C61—22no—
153C2×D4×C61—22no—
154C2×D4×C61—24no—
155C2×D4×C61—24no—
156C2×D4×C61—22no—
157C2×D4×C61—28no—
158C2×D4×C61—21no—
159C2×D4×C61—22no—
160C2×D4×C61—22no—
161C2×D4×C61—21no—
162C2×D4×C61—21no—
163C2×D4×C61—21no—
164C2×D4×C61—21no—
165C2×D4×C61—21no—
166C2×D4×C61—21no—
167C2×D4×C61—42no—
168C2×D4×C61—42no—
169C2×D4×C61C2×D412yes—
170C22×C4×C61—21no—
171C22×C4×C61—22no—
172C22×C4×C61—22no—
173C22×C4×C61—22no—
174C22×C4×C61—22no—
175C22×C4×C61—24no—
176C22×C4×C61—24no—
177C22×C4×C61—22no—
178C22×C4×C61—42no—
179C22×C4×C61—42no—
180C22×C4×C61C2×D412yes—
181C22×C4×C61C2×D411yescharacteristic
182C22×C4×C61C2×D411yescharacteristic
183C22×C4×C61C2×D411yescharacteristic
184C22×C4×C61C2×D411yescharacteristic
185C22×C4×C61C2×D411yescharacteristic
186C22×C4×C61C2×D412yes—
187C22×C4×C61#4112yes—
188C22×C4×C61#4112yes—
189C22×C4×C61#4112yes—
190C22×C4×C61C2411yescharacteristic
191C22×Dic61—22no—
192C22×Dic61—22no—
193C22×Dic61—44no—
194C22×Dic61—42no—
195C4⋊C4×C61—28no—
196C4⋊C4×C61—22no—
197C4⋊C4×C61C2×D414yes—
198C22⋊C4×C61—28no—
199C22⋊C4×C61—28no—
200C22⋊C4×C61—22no—
201C22⋊C4×C61—22no—
202C42×C61—22no—
203C42×C61—22no—
204C22⋊Dic61—48no—
205C22⋊Dic61—41no—
206C22⋊Dic61—41no—
207C4⋊Dic61—22no—
208C4⋊Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—41no—
212C4×Dic61—41no—
213C4×Dic61—42no—
214#11758194—24no—
215#11758194—22no—
216#11758194—22no—
217#11758194—22no—
218#11758194—42no—
219#11758194C2×D414yes—
220C2×#92418—24no—
221C2×#92418—22no—
222C2×#92418—22no—
223C2×#92418—42no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—22no—
228C23×C61—42no—
229C23×C61—44no—
230C23×C61—42no—
231C23×C61—82no—
232C23×C61#12011yescharacteristic
233C23×C61#12011yescharacteristic
234C23×C61#12112yes—
235C23×C61#12312yes—
236C23×C61C22×D411yescharacteristic
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—41no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—41no—
243Q8×C61—42no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—42no—
247D4×C61—42no—
248D4×C61—44no—
249D4×C61—44no—
250D4×C61—44no—
251D4×C61—41no—
252D4×C61—41no—
253D4×C61—41no—
254D4×C61—41no—
255D4×C61—42no—
256D4×C61—88no—
257C2×C4×C61—24no—
258C2×C4×C61—22no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—22no—
262C2×C4×C61—42no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—44no—
266C2×C4×C61—44no—
267C2×C4×C61—44no—
268C2×C4×C61—82no—
269C2×C4×C61#12011yescharacteristic
270C2×C4×C61#12011yescharacteristic
271C2×C4×C61#12112yes—
272C2×C4×C61#12112yes—
273C2×C4×C61#12112yes—
274C2×C4×C61#12112yes—
275C2×C4×C61#12112yes—
276C2×C4×C61#12312yes—
277C2×C4×C61#12312yes—
278C2×C4×C61#12312yes—
279C2×C4×C61#12412yes—
280C2×C4×C61#12412yes—
281C2×C4×C61C22×D411yescharacteristic
282C2×C4×C61C22×D411yescharacteristic
283C2×C4×C61C22×D411yescharacteristic
284C2×C4×C61C2×#4111yescharacteristic
285C2×C4×C61C2×#4111yescharacteristic
286C2×C4×C61C2×#4111yescharacteristic
287C2×Dic61—24no—
288C2×Dic61—42no—
289C2×Dic61—48no—
290C2×Dic61—42no—
291C2×Dic61—44no—
292C2×Dic61—42no—
293#92418—24no—
294#92418—42no—
295#23492—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—44no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—44no—
305C22×C61—44no—
306C22×C61—82no—
307C22×C61—84no—
308C22×C61#39412yes—
309C22×C61#39412yes—
310C22×C61C2×#12011yescharacteristic
311C22×C61C2×#12111yescharacteristic
312C22×C61C2×#12311yescharacteristic
313C4×C61—42no—
314C4×C61—42no—
315C4×C61—44no—
316C4×C61#39412yes—
317C4×C61#39412yes—
318C4×C61C2×#12111yescharacteristic
319C4×C61C2×#12111yescharacteristic
320C4×C61C2×#12311yescharacteristic
321C4×C61C2×#12411yescharacteristic
322Dic61—44no—
323Dic61—82no—
324#3449D6111yescharacteristic
325#2911—611no—
326C22×#104—611no—
327#2896—611no—
328C2×#413—611no—
329#2297—611no—
330#2038—614no—
331#2037—614no—
332C2×#339—611no—
333C2×C61—21no—
334C2×C61—21no—
335C2×C61—22no—
336C2×C61—22no—
337C2×C61—82no—
338C2×C61—82no—
339C2×C61C2×#39411yescharacteristic
340C2×C61#301812yes—
341C22×#41C2×D6111yescharacteristic
342C22×#41C2×D6111yescharacteristic
343C22×M16—611no—
344C22×M16—611no—
345#537—22no—
346#536—21no—
347#536C2×D6112yes—
348#535—21no—
349#535C2×D6112yes—
350#534—22no—
351#532—22no—
352#525—22no—
353#525—22no—
354C2×#117—611no—
355C2×#117C2×D6111yescharacteristic
356C22×C42—611no—
357#428—614no—
358#428—1222no—
359#427—1222no—
360C2×#104—614no—
361C2×#104—1224no—
362#419—1221no—
363#418—1221no—
364#417—614no—
365#416—1222no—
366#413—1224no—
367C2×#101—612no—
368C2×#100—612no—
369#391—612no—
370#390—612no—
371#386—1222no—
372C4×C22⋊C4—1222no—
373#339—614no—
374C61#2349211yescharacteristic, complemented, Sylow
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—22no—
378C2×#41—22no—
379C2×#41—21no—
380C2×#41—21no—
381C2×#41—21no—
382C2×#41—21no—
383C2×#41—21no—
384C2×#41—21no—
385C2×#41—22no—
386C2×#41—22no—
387C2×#41—22no—
388C2×#41—22no—
389C2×#41C22⋊D6112yes—
390C2×#41C22×D6112yes—
391C22×Q8C22⋊D6111yescharacteristic
392C22×Q8C22×D6111yescharacteristic
393C22×D4—21no—
394C22×D4C22⋊D6112yes—
395C22×D4C22⋊D6111yescharacteristic
396C22×D4C22×D6111yescharacteristic
397C23×C4—21no—
398C23×C4—1221no—
399C23×C4C22⋊D6112yes—
400C23×C4C22×D6111yescharacteristic
401C2×M16—614no—
402C2×M16—612no—
403C2×M16—612no—
404C2×M16—1222no—
405C2×M16—1221no—
406C2×M16—1221no—
407C2×M16—1222no—
408C2×M16—1222no—
409C22×C8—1221no—
410C22×C8—1221no—
411C4⋊Q8—22no—
412#127—22no—
413#124—22no—
414#124—22no—
415#123—24no—
416#123—24no—
417C22⋊Q8—24no—
418C22⋊Q8—24no—
419#121—24no—
420#121—28no—
421#121—24no—
422#120—24no—
423#120—24no—
424C4×D4—28no—
425#117—21no—
426#117—21no—
427#117—1221no—
428#117—1221no—
429#117—1221no—
430#117—1221no—
431#117C22⋊D6112yes—
432#117C22×D6112yes—
433C2×C4⋊C4—612no—
434C2×C4⋊C4C22⋊D6112yes—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1224no—
438C2×C42—21no—
439C2×C42—612no—
440C2×C42—1222no—
441C2×C42—1222no—
442C2×C42—1221no—
443C2×C42—1221no—
444#107—1224no—
445#106—1224no—
446#104—1228no—
447#103—1224no—
448#102—1224no—
449#101—1224no—
450#100—1224no—
451#95—1224no—
452C24—21no—
453C24C2×C22⋊D6112yes—
454C24C2×C22⋊D6111yescharacteristic
455#41—22no—
456#41—22no—
457#41—22no—
458#41—22no—
459#41—24no—
460#41—24no—
461#41—42no—
462#41—42no—
463#41—42no—
464#41—42no—
465#41—42no—
466#41—42no—
467#41—42no—
468#41—42no—
469#41—42no—
470#41—42no—
471#41—42no—
472#41—42no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8C2×C22⋊D6112yes—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—28no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—22no—
498C2×D4—22no—
499C2×D4—42no—
500C2×D4—42no—
501C2×D4C2×C22⋊D6112yes—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—24no—
509C22×C4—24no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—2444no—
515C22×C4—2442no—
516C22×C4D4×D6112yes—
517C22×C4#1175822412yes—
518C22×C4#1175822412yes—
519C22×C4#1175822412yes—
520C22×C4C2×C22⋊D6112yes—
521C22×C4C2×C22⋊D6111yescharacteristic
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C2×C22⋊D6111yescharacteristic
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C23×D6111yescharacteristic
527M16—614no—
528M16—1222no—
529M16—1224no—
530M16—1222no—
531M16—1222no—
532M16—2442no—
533C2×C8—1222no—
534C2×C8—1224no—
535C2×C8—1222no—
536C2×C8—2442no—
537C4⋊C4—28no—
538C4⋊C4—22no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4C2×C22⋊D6114yes—
542C22⋊C4—28no—
543C22⋊C4—28no—
544C22⋊C4—22no—
545C22⋊C4—22no—
546C22⋊C4—2448no—
547C22⋊C4—2441no—
548C22⋊C4—2441no—
549C42—22no—
550C42—22no—
551C42—1222no—
552C42—1222no—
553C42—2441no—
554C42—2441no—
555C42—2442no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—42no—
561C23—44no—
562C23—42no—
563C23—82no—
564C23#1451974611yescharacteristic
565C23#1451974611yescharacteristic
566C23#1451974912yes—
567C23#1451975812yes—
568C23C22×C22⋊D6111yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574Q8—41no—
575Q8—42no—
576D4—22no—
577D4—22no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—44no—
583D4—41no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—42no—
588D4—88no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—82no—
601C2×C4—1224no—
602C2×C4—2444no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4—2448no—
606C2×C4—2442no—
607C2×C4#1451974611yescharacteristic
608C2×C4#1451974611yescharacteristic
609C2×C4#1451974812yes—
610C2×C4#1451974812yes—
611C2×C4#1451974912yes—
612C2×C4#1451974912yes—
613C2×C4#1451974912yes—
614C2×C4#1451975812yes—
615C2×C4#1451975812yes—
616C2×C4#1451975812yes—
617C2×C4#1451976412yes—
618C2×C4#1451976412yes—
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C22×C22⋊D6111yescharacteristic
621C2×C4C22×C22⋊D6111yescharacteristic
622C2×C4C2×#1175822411yescharacteristic
623C2×C4C2×#1175822411yescharacteristic
624C2×C4C2×#1175822411yescharacteristic
625C8—1224no—
626C8—2442no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—44no—
637C22—82no—
638C22—84no—
639C22#1683967612yes—
640C22#1683967612yes—
641C22C2×#1451974611yescharacteristic
642C22C2×#1451974911yescharacteristic
643C22C2×#1451975811yescharacteristic
644C4—42no—
645C4—42no—
646C4—44no—
647C4—4882no—
648C4—2444no—
649C4#1683967612yes—
650C4#1683967612yes—
651C4C2×#1451974811yescharacteristic
652C4C2×#1451974911yescharacteristic
653C4C2×#1451975811yescharacteristic
654C4C2×#1451976411yescharacteristic
655C2—21no—
656C2—21no—
657C2—22no—
658C2—22no—
659C2—82no—
660C2—82no—
661C2C2×#1683967611yescharacteristic
662C2#2145516812yes—
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3449C2
socleC22×C61C2×#120
hypercenter#3449D61
hypocenterC61#23492
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. #21454408C2semidirect
2. C22×#14519704C2semidirect
3. #21454338C2semidirect
4. C2×#16839746C2semidirect
5. #21448310C2semidirect
6. #21447033C2semidirect
7. #21447026C2semidirect
8. C2×#16839521C2semidirect
14. #16839805C22semidirect
24. C2×#14519696C22semidirect
25. C2×#14519694C22semidirect
26. #16839669C22semidirect
27. #16839665C22semidirect
30. #16839521C22semidirect
49. C2×#11758194D4semidirect
374. C61#23492semidirect

Maximal subgroups

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

subgroupindex
#214544082
C22×#145197042
#214543382
C2×#168397462
#214483102
#214470332
#214470262
C2×#168395212
C61×#34492
#2349261