Finite Group Databasebeta

← #29706880

Subgroups — #29706880

Order 15616.

2014 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519808C211yesmaximal, characteristic, complemented
2C2×#16840340C211yesmaximal, characteristic
3C2×#16839756C211yesmaximal, characteristic, complemented
4C22×#14519701C211yesmaximal, characteristic, complemented
5C2×#16839729C211yesmaximal, characteristic, complemented
6#21446827C214yesmaximal, complemented
7#21446820C214yesmaximal, complemented
8#21445540C211yesmaximal, characteristic, complemented
9C2×C61×#536C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519808—24no—
12C2×#14519808—24no—
13C2×#14519808—24no—
14C22×#11758202C2212yes—
15C22×#11758201C2212yes—
16C22×#11758194C2211yescharacteristic
17C23×#92418C2211yescharacteristic
18#16840340—24no—
19#16840340—24no—
20C2×#14519756C2212yes—
21C2×#14519753—21no—
22C2×#14519751C2212yes—
23C2×#14519749—22no—
24C2×#14519748—21no—
25C22×C4⋊Dic61C2211yescharacteristic
26#16839756—24no—
27C2×#14519701—24no—
28C2×#14519701C2214yes—
29C2×#14519700C2212yes—
30C2×#14519697C2212yes—
31#16839729—24no—
32C2×C22⋊#92418C2214yes—
33#16839524C2214yes—
34C22×#41×C61C2211yescharacteristic
35C23×Q8×C61C2211yescharacteristic
36C61×#536—24no—
37C61×#536—24no—
38C2×C4⋊Q8×C61—21no—
39C2×#124×C61—21no—
40C2×#123×C61—22no—
41C2×C22⋊Q8×C61C2214yes—
42C2×#117×C61C2211yescharacteristic
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222D414yes—
49C22×Dic122C2311yescharacteristic
50C22×D244C2311yescharacteristic
51C22×C22⋊D61—21no—
52C22×C22⋊D61—22no—
53C23×Dic61—21no—
54C22×C4×D61—21no—
55#14519808—232no—
56C2×#11758202—24no—
57C2×#11758202—24no—
58C2×#11758202—24no—
59C2×#11758201—24no—
60C2×#11758201—24no—
61C2×#11758201—24no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194D414yescomplemented
65C22×#92418—22no—
66C22×#92418—22no—
67C22×#92418—22no—
68C22×#92418C2312yes—
69C22×#92418C2312yes—
70#14519756—24no—
71#14519756—24no—
72#14519753—28no—
73#14519751—24no—
74#14519751—24no—
75#14519749—216no—
76#14519748—28no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—22no—
79C2×C4⋊Dic61D414yes—
80C2×C4⋊Dic61C2312yes—
81C2×#11758196—22no—
82C2×C22⋊Dic61—22no—
83C2×#11758198—22no—
84#14519701—28no—
85#14519700—24no—
86#14519697—24no—
87C22⋊#92418—28no—
88C2×#41×C61—22no—
89C2×#41×C61—22no—
90C2×#41×C61—22no—
91C2×#41×C61—22no—
92C2×#41×C61—22no—
93C2×#41×C61—22no—
94C22×Q8×C61—21no—
95C22×Q8×C61—22no—
96C22×Q8×C61—22no—
97C22×Q8×C61—22no—
98C22×Q8×C61—22no—
99C22×Q8×C61—22no—
100C22×Q8×C61D411yescharacteristic
101C22×Q8×C61D414yescomplemented
102C22×Q8×C61C2312yes—
103C22×D4×C61D411yescharacteristic
104C22×D4×C61D412yes—
105C23×C4×C61—21no—
106C23×C4×C61C2312yes—
107C23×C4×C61C2311yescharacteristic
108C4⋊Q8×C61—28no—
109#124×C61—28no—
110#123×C61—216no—
111C22⋊Q8×C61—28no—
112C22⋊Q8×C61—28no—
113#117×C61—22no—
114#117×C61D414yescomplemented
115C2×C4⋊C4×C61—24no—
116C2×C4⋊C4×C61C2312yes—
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—24no—
119C2×C42×C61—21no—
120C24×C61C2×D412yes—
121C24×C61C2×D411yescharacteristic
122C23×D61—21no—
123#41×C61—44no—
124#41×C61—44no—
125#41×C61—44no—
126#41×C61—44no—
127C2×Q8×C61—24no—
128C2×Q8×C61—22no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61—22no—
132C2×Q8×C61—28no—
133C2×Q8×C61—28no—
134C2×Q8×C61—22no—
135C2×Q8×C61—22no—
136C2×Q8×C61—22no—
137C2×Q8×C61—22no—
138C2×Q8×C61—42no—
139C2×Q8×C61—42no—
140C2×Q8×C61—44no—
141C2×Q8×C61—44no—
142C2×Q8×C61C2×D414yes—
143C2×D4×C61—24no—
144C2×D4×C61—24no—
145C2×D4×C61—24no—
146C2×D4×C61—22no—
147C2×D4×C61—22no—
148C2×D4×C61—22no—
149C22×C4×C61—24no—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61—22no—
153C22×C4×C61—22no—
154C22×C4×C61—22no—
155C22×C4×C61—22no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—42no—
159C22×C4×C61—42no—
160C22×C4×C61C2×D412yes—
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D412yes—
163C22×C4×C61C2×D412yes—
164C22×C4×C61C2×D412yes—
165C22×C4×C61C2×D412yes—
166C22×C4×C61C2×D412yes—
167C22×C4×C61#4112yes—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61—24no—
170C2×C22⋊D61—28no—
171C2×C22⋊D61—42no—
172C2×C22⋊D61—42no—
173C2×C22⋊D61—84no—
174C22×Dic61—21no—
175C22×Dic61—22no—
176C22×Dic61—24no—
177C22×Dic61—82no—
178#11758222—28no—
179#11758222—28no—
180#11758222—44no—
181#11758222—44no—
182C2×D244—21no—
183C2×D244—21no—
184C2×D244—21no—
185C2×D244—21no—
186C2×D244—21no—
187C2×D244C2×D412yes—
188C2×C4×D61—24no—
189C2×C4×D61—42no—
190C2×C4×D61—42no—
191C2×Dic122—21no—
192C2×Dic122—21no—
193C2×Dic122—21no—
194C2×Dic122—21no—
195C2×Dic122—21no—
196C2×Dic122C2×D412yes—
197C4⋊C4×C61—216no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61C2×D414yes—
200C22⋊C4×C61—216no—
201C22⋊C4×C61—24no—
202C42×C61—24no—
203C22⋊Dic61—28no—
204#11758202—216no—
205#11758201—216no—
206#11758198—28no—
207C4⋊Dic61—24no—
208C4⋊Dic61—22no—
209C4⋊Dic61C2×D414yes—
210#11758196—28no—
211#11758194—28no—
212C2×#92418—22no—
213C2×#92418—22no—
214C2×#92418—22no—
215C2×#92418—22no—
216C2×#92418—22no—
217C2×#92418—44no—
218C2×#92418C2×D414yes—
219C23×C61—22no—
220C23×C61—22no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—24no—
225C23×C61—24no—
226C23×C61#12012yes—
227C23×C61#12012yes—
228C23×C61#12112yes—
229C23×C61C22×D411yescharacteristic
230C22×D61—22no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—42no—
234C22×D61—42no—
235Q8×C61—24no—
236Q8×C61—24no—
237Q8×C61—42no—
238Q8×C61—42no—
239Q8×C61—48no—
240Q8×C61—44no—
241D4×C61—48no—
242D4×C61—44no—
243C2×C4×C61—22no—
244C2×C4×C61—24no—
245C2×C4×C61—24no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—42no—
252C2×C4×C61—48no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61#12014yes—
258C2×C4×C61#12012yes—
259C2×C4×C61#12012yes—
260C2×C4×C61#12114yes—
261C2×C4×C61#12114yes—
262C2×C4×C61#12112yes—
263C2×C4×C61#12714yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C22⋊D61—48no—
271C22⋊D61—88no—
272C2×Dic61—22no—
273C2×Dic61—24no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—44no—
277C2×Dic61—84no—
278D244—22no—
279D244—22no—
280D244—41no—
281D244—41no—
282C4×D61—48no—
283Dic122—22no—
284Dic122—22no—
285Dic122—41no—
286Dic122—41no—
287#92418—24no—
288#92418—42no—
289#23394—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—44no—
298C22×C61—44no—
299C22×C61#39214yes—
300C22×C61C2×#12011yescharacteristic
301C22×C61C2×#12011yescharacteristic
302C22×C61C2×#12111yescharacteristic
303C22×C61#57712yes—
304C22×C61#57812yes—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—82no—
309C2×D61—82no—
310C4×C61—42no—
311C4×C61—44no—
312C4×C61—44no—
313C4×C61#39214yes—
314C4×C61C2×#12011yescharacteristic
315C4×C61C2×#12111yescharacteristic
316C4×C61C2×#12111yescharacteristic
317C4×C61C2×#12711yescharacteristic
318Dic61—42no—
319Dic61—44no—
320C22×#137—611no—
321C2×#537—611no—
322C2×#536D6111yescharacteristic, complemented
323C2×#416—611no—
324C22×#103—611no—
325C2×#408—611no—
326#2001—614no—
327#2000—614no—
328#1736—611no—
329C2×C61—24no—
330C2×C61—82no—
331C2×C61C2×#39211yescharacteristic
332C2×C61#300911yescharacteristic
333C2×C61#301111yescharacteristic
334C2×C61#301312yes—
335C2×C61#306312yes—
336D61—82no—
337C22×#41—611no—
338C22×#41C2×D6111yescharacteristic
339C23×Q8C2×D6111yescharacteristic
340C2×#137—1224no—
341C2×#137—1224no—
342C2×#137—1224no—
343C22×Q16—612no—
344C22×SD16—612no—
345C22×M16—611no—
346C23×C8—611no—
347#537—1224no—
348#537—1224no—
349#536—24no—
350#536—24no—
351C2×C4⋊Q8—21no—
352C2×#124—21no—
353C2×#123—22no—
354C2×C22⋊Q8—612no—
355C2×C22⋊Q8—1221no—
356C2×C22⋊Q8C2×D6114yes—
357C2×#121—612no—
358C2×#121—1221no—
359C2×#121—1222no—
360C2×#117C2×D6111yescharacteristic
361C22×C4⋊C4—611no—
362#416—1224no—
363C2×#103—614no—
364C2×#103—612no—
365C2×#103—1224no—
366C2×#102—612no—
367#408—1224no—
368C2×C22⋊C8—614no—
369#340—614no—
370C61#2339411yescharacteristic, complemented, Sylow
371C2×#41—22no—
372C2×#41—22no—
373C2×#41—22no—
374C2×#41—22no—
375C2×#41—22no—
376C2×#41—22no—
377C2×#41—614no—
378C2×#41—1222no—
379C2×#41—1222no—
380C2×#41—1222no—
381C2×#41—1222no—
382C2×#41—1222no—
383C22×Q8—21no—
384C22×Q8—22no—
385C22×Q8—22no—
386C22×Q8—22no—
387C22×Q8—22no—
388C22×Q8—22no—
389C22×Q8—611no—
390C22×Q8C22⋊D6111yescharacteristic
391C22×Q8C22⋊D6114yescomplemented
392C22×Q8C22×D6112yes—
393C22×D4—611no—
394C22×D4—1222no—
395C22×D4—1221no—
396C22×D4C22⋊D6112yes—
397C22×D4C22⋊D6111yescharacteristic
398C23×C4—21no—
399C23×C4—1221no—
400C23×C4—1221no—
401C23×C4C22×D6112yes—
402C23×C4C22×D6111yescharacteristic
403#137—12232no—
404C2×Q16—1224no—
405C2×Q16—1224no—
406C2×Q16—1224no—
407C2×SD16—1224no—
408C2×SD16—1224no—
409C2×SD16—1224no—
410C2×M16—614no—
411C2×M16—1222no—
412C2×M16—1222no—
413C22×C8—612no—
414C22×C8—612no—
415C22×C8—1222no—
416C22×C8—1222no—
417C22×C8—1222no—
418C4⋊Q8—28no—
419#124—28no—
420#123—216no—
421C22⋊Q8—28no—
422C22⋊Q8—28no—
423C22⋊Q8—1224no—
424C22⋊Q8—1224no—
425C22⋊Q8—1228no—
426#121—1224no—
427#121—1224no—
428#121—1228no—
429#121—12216no—
430#117—22no—
431#117C22⋊D6114yescomplemented
432C2×C4⋊C4—24no—
433C2×C4⋊C4—614no—
434C2×C4⋊C4—612no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1222no—
438C2×C4⋊C4C22×D6112yes—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—24no—
441C2×C22⋊C4—1222no—
442C2×C22⋊C4—1222no—
443C2×C42—21no—
444#103—1228no—
445#103—1224no—
446#102—1224no—
447C22⋊C8—1228no—
448C24—1221no—
449C24C2×C22⋊D6112yes—
450C24C2×C22⋊D6111yescharacteristic
451#41—44no—
452#41—44no—
453#41—44no—
454#41—44no—
455#41—1228no—
456#41—1228no—
457#41—2444no—
458#41—2444no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—24no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—28no—
469C2×Q8—28no—
470C2×Q8—42no—
471C2×Q8—44no—
472C2×Q8—42no—
473C2×Q8—44no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8C2×C22⋊D6114yes—
481C2×D4—24no—
482C2×D4—24no—
483C2×D4—24no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—612no—
488C2×D4—1228no—
489C2×D4—1224no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—4884no—
496C2×D4—2442no—
497C2×D4—2442no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—1224no—
510C22×C4—1222no—
511C22×C4—1221no—
512C22×C4—1224no—
513C22×C4—4882no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4D4×D6112yes—
517C22×C4D4×D6112yes—
518C22×C4D4×D6112yes—
519C22×C4#1175822412yes—
520C22×C4C2×C22⋊D6112yes—
521C22×C4C2×C22⋊D6112yes—
522C22×C4C2×C22⋊D6112yes—
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C23×D6111yescharacteristic
525Q16—12216no—
526SD16—12216no—
527M16—1228no—
528C2×C8—614no—
529C2×C8—1222no—
530C2×C8—1222no—
531C2×C8—1222no—
532C2×C8—1222no—
533C2×C8—1222no—
534C2×C8—2444no—
535C4⋊C4—216no—
536C4⋊C4—22no—
537C4⋊C4—614no—
538C4⋊C4—1228no—
539C4⋊C4—1222no—
540C4⋊C4—1224no—
541C4⋊C4C2×C22⋊D6114yes—
542C22⋊C4—216no—
543C22⋊C4—24no—
544C22⋊C4—1228no—
545C22⋊C4—1228no—
546C42—24no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—1222no—
555C23—2441no—
556C23—2441no—
557C23—2442no—
558C23—2442no—
559C23#1451974512yes—
560C23#1451974612yes—
561C23#1451974912yes—
562C23C22×C22⋊D6111yescharacteristic
563Q8—24no—
564Q8—24no—
565Q8—42no—
566Q8—42no—
567Q8—48no—
568Q8—44no—
569Q8—1222no—
570Q8—1222no—
571Q8—2441no—
572Q8—2441no—
573D4—48no—
574D4—44no—
575D4—1222no—
576D4—1222no—
577D4—4888no—
578D4—2448no—
579D4—2441no—
580D4—2441no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—1224no—
596C2×C4—1222no—
597C2×C4—4884no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2444no—
601C2×C4—2448no—
602C2×C4#1451974514yes—
603C2×C4#1451974512yes—
604C2×C4#1451974612yes—
605C2×C4#1451974814yes—
606C2×C4#1451974914yes—
607C2×C4#1451974912yes—
608C2×C4#1451977314yes—
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C22×C22⋊D6111yescharacteristic
613C2×C4C22×C22⋊D6111yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C8—1224no—
616C8—2442no—
617C22—22no—
618C22—24no—
619C22—24no—
620C22—22no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—42no—
626C22—4882no—
627C22—4882no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22#1683967014yes—
632C22C2×#1451974511yescharacteristic
633C22C2×#1451974611yescharacteristic
634C22C2×#1451974911yescharacteristic
635C22#1684057012yes—
636C22#1684057712yes—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—2442no—
641C4—2444no—
642C4#1683967014yes—
643C4C2×#1451974511yescharacteristic
644C4C2×#1451974811yescharacteristic
645C4C2×#1451974911yescharacteristic
646C4C2×#1451977311yescharacteristic
647C2—24no—
648C2—82no—
649C2—4882no—
650C2C2×#1683967011yescharacteristic
651C2#2145509411yescharacteristic
652C2#2145511211yescharacteristic
653C2#2145513412yes—
654C2#2145550612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#536C2
socleC23×C61C22×D4
prime radicalC2×#16840340C2
hypercenterC2×#536D61
hypocenterC61#23394
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C22×#14519808C2semidirect
3. C2×#16839756C2semidirect
4. C22×#14519701C2semidirect
5. C2×#16839729C2semidirect
6. #21446827C2semidirect
7. #21446820C2semidirect
8. #21445540C2semidirect
9. C2×C61×#536C2semidirect
64. C2×#11758194D4semidirect
101. C22×Q8×C61D4semidirect
114. #117×C61D4semidirect
322. C2×#536D61semidirect
370. C61#23394semidirect
391. C22×Q8C22⋊D61semidirect
431. #117C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168403402
C2×#168397562
C22×#145197012
C2×#168397292
#214468272
#214468202
#214455402
C2×C61×#5362
#2339461