Finite Group Databasebeta

← #29587174

Subgroups — #29587174

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839778C211yesmaximal, characteristic
2#21454336C211yesmaximal, characteristic
3#21448307C211yesmaximal, characteristic
4#21448301C211yesmaximal, characteristic, complemented
5C2×#16839518C211yesmaximal, characteristic, complemented
6#21445512C211yesmaximal, characteristic, complemented
7C61×#1733C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C22×#11758194C2211yescharacteristic
10C2×#14519736C2211yescharacteristic
11C22×C4×Dic61C2211yescharacteristic
12#16839778—22no—
13#16839778C414yescomplemented
14C2×#14519703—22no—
15#16839770—21no—
16#16839766—21no—
17#16839761—22no—
18#16839756—22no—
19#16839667—21no—
20#16839663—21no—
21#16839658—22no—
22#16839639—22no—
23#16839634—22no—
24C4×#11758196—22no—
25C4×#11758198—22no—
26#16839518—22no—
27#16839516—22no—
28C22×M16×C61C2211yescharacteristic
29C2×#117×C61C2211yescharacteristic
30C2×#117×C61C2211yescharacteristic
31C61×#339—22no—
32C2×#11758222—21no—
33C2×#11758222—21no—
34C2×#11758222—21no—
35C2×#11758222—21no—
36C2×#11758222—21no—
37C2×#11758222—21no—
38C2×#11758222—21no—
39C2×#11758222—21no—
40C2×#11758222—21no—
41C2×#11758222—21no—
42C2×#11758222C2×C412yes—
43C2×#11758222Q812yes—
44C22×Dic122D411yescharacteristic
45C22×D244D411yescharacteristic
46C22×C22⋊D61—21no—
47C23×Dic61D412yes—
48C22×C4×D61—21no—
49C2×#11758194—21no—
50C2×#11758194—21no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53C2×#11758194—21no—
54C2×#11758194—21no—
55C22×#92418D412yes—
56#14519736—21no—
57#14519736—21no—
58#14519736C2×C412yes—
59#14519736D412yes—
60C2×C4⋊Dic61—21no—
61C2×#11758196—24no—
62C2×C22⋊Dic61—21no—
63C2×#11758198—24no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—21no—
69C2×C4×Dic61—21no—
70C2×C4×Dic61D412yes—
71#14519703—28no—
72#14519700—24no—
73#14519697—24no—
74#14519684—24no—
75#14519684—24no—
76C23×C4×C61C2311yescharacteristic
77C2×M16×C61—21no—
78C2×M16×C61—21no—
79C2×M16×C61—21no—
80C2×M16×C61—21no—
81C2×M16×C61—21no—
82C2×M16×C61C2×C412yes—
83C22×C8×C61—21no—
84#117×C61—21no—
85#117×C61—21no—
86#117×C61—21no—
87#117×C61—21no—
88#117×C61—21no—
89#117×C61—21no—
90#117×C61—21no—
91#117×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D412yes—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C42×C61—21no—
97C2×C42×C61D412yes—
98C24×C61C2×D411yescharacteristic
99C23×D61—21no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61C22×C411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×Q811yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×C4×C61#4111yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C2×C22⋊D61—22no—
117C2×C22⋊D61—22no—
118C2×C22⋊D61—41no—
119C2×C22⋊D61—41no—
120C2×C22⋊D61—41no—
121C2×C22⋊D61—41no—
122C22×Dic61—21no—
123C22×Dic61—22no—
124C22×Dic61—22no—
125C22×Dic61—22no—
126C22×Dic61—22no—
127C22×Dic61—44no—
128#11758222—22no—
129#11758222—22no—
130#11758222—22no—
131#11758222—22no—
132#11758222—22no—
133#11758222—22no—
134#11758222—22no—
135#11758222—22no—
136#11758222—41no—
137#11758222—41no—
138#11758222—41no—
139#11758222—41no—
140#11758222—41no—
141#11758222—41no—
142#11758222—41no—
143#11758222—41no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244C4⋊C412yescomplemented
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—41no—
153C2×C4×D61—41no—
154C2×C4×D61—41no—
155C2×C4×D61—41no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C4⋊C412yescomplemented
162M16×C61—22no—
163M16×C61—22no—
164M16×C61—41no—
165M16×C61—41no—
166C2×C8×C61—22no—
167C2×C8×C61—41no—
168C2×C8×C61—41no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61—41no—
173C22⋊C4×C61—41no—
174C22⋊C4×C61—41no—
175C22⋊C4×C61—41no—
176C22⋊C4×C61—41no—
177C42×C61—22no—
178C42×C61—22no—
179C42×C61—41no—
180C42×C61—41no—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183#11758198—48no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186#11758196—48no—
187C4×Dic61—22no—
188C4×Dic61—41no—
189C4×Dic61—41no—
190C4×Dic61—41no—
191C4×Dic61—41no—
192C4×Dic61C22⋊C414yescomplemented
193#11758194—41no—
194#11758194—41no—
195#11758194—41no—
196#11758194—41no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C23×C61—42no—
201C23×C61C4×D411yescharacteristic
202C23×C61#12011yescharacteristic
203C23×C61#12011yescharacteristic
204C23×C61#12111yescharacteristic
205C23×C61#12111yescharacteristic
206C23×C61C22⋊Q811yescharacteristic
207C23×C61#12411yescharacteristic
208C22×D61—22no—
209C22×D61—44no—
210C22×D61—41no—
211C22×D61—41no—
212C2×C4×C61—44no—
213C2×C4×C61—44no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61C2×C22⋊C411yescharacteristic
217C2×C4×C61C2×C4⋊C411yescharacteristic
218C2×C4×C61#11711yescharacteristic
219C2×C4×C61C4×D411yescharacteristic
220C2×C4×C61C4×D411yescharacteristic
221C2×C4×C61C4×D411yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61#12111yescharacteristic
226C2×C4×C61#12111yescharacteristic
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61C22⋊Q811yescharacteristic
230C2×C4×C61C22⋊Q811yescharacteristic
231C2×C4×C61C22⋊Q811yescharacteristic
232C2×C4×C61C22⋊Q811yescharacteristic
233C2×C4×C61C22⋊Q811yescharacteristic
234C2×C4×C61#12311yescharacteristic
235C2×C4×C61#12311yescharacteristic
236C2×C4×C61#12311yescharacteristic
237C2×C4×C61#12311yescharacteristic
238C2×C4×C61#12311yescharacteristic
239C2×C4×C61#12411yescharacteristic
240C2×C4×C61#12411yescharacteristic
241C2×C4×C61#12611yescharacteristic
242C2×C4×C61#12611yescharacteristic
243C2×C4×C61#12711yescharacteristic
244C22⋊D61—42no—
245C22⋊D61—42no—
246C22⋊D61—42no—
247C22⋊D61—42no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—22no—
252C2×Dic61—24no—
253C2×Dic61—44no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260D244—22no—
261D244—22no—
262D244—41no—
263D244—41no—
264C4×D61—42no—
265C4×D61—42no—
266C4×D61—42no—
267C4×D61—42no—
268Dic122—22no—
269Dic122—22no—
270Dic122—41no—
271Dic122—41no—
272C8×C61—42no—
273#92418—42no—
274#11083—611nomaximal, Sylow
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61#38511yescharacteristic
283C22×C61#38611yescharacteristic
284C22×C61#38811yescharacteristic
285C22×C61#39211yescharacteristic
286C22×C61#39311yescharacteristic
287C22×C61#39411yescharacteristic
288C22×C61#39611yescharacteristic
289C2×D61—44no—
290C2×D61—42no—
291C2×D61—82no—
292C2×D61—82no—
293C4×C61—82no—
294C4×C61—82no—
295C4×C61#38011yescharacteristic
296C4×C61#38511yescharacteristic
297C4×C61#38811yescharacteristic
298C4×C61#39011yescharacteristic
299C4×C61#39411yescharacteristic
300C4×C61#39611yescharacteristic
301C4×C61#39711yescharacteristic
302C4×C61#39911yescharacteristic
303Dic61—44no—
304Dic61—42no—
305Dic61—44no—
306C2×#421—611no—
307#2896—611no—
308#2297—611no—
309#2296—611no—
310#1733D6111yescharacteristic, complemented
311C2×H(3,ℤ/4)—611no—
312#1728—611no—
313C2×C61—21no—
314C2×C61—21no—
315C2×C61—21no—
316C2×C61—21no—
317C2×C61—21no—
318C2×C61—21no—
319C2×C61#143911yescharacteristic
320C2×C61#201012yes—
321D61—82no—
322C22×#41—611no—
323C22×M16—611no—
324C22×M16C2×D6111yescharacteristic
325C2×#117—611no—
326C2×#117C2×D6111yescharacteristic
327C2×#117C2×D6111yescharacteristic
328C22×C42—611no—
329#421—614no—
330#421—1222no—
331C2×#104—1222no—
332#419—1221no—
333#418—1221no—
334#417—1222no—
335#416—1222no—
336#391—1221no—
337#390—1221no—
338#389—1222no—
339#386—1222no—
340#385—1222no—
341C4×C4⋊C4—1222no—
342C4×C22⋊C4—1222no—
343#339—22no—
344H(3,ℤ/4)—1222no—
345#336—1222no—
346C61#1108311yescharacteristic, complemented, Sylow
347C2×#41—612no—
348C2×#41—612no—
349C2×#41—1221no—
350C2×#41—1221no—
351C2×#41—1221no—
352C2×#41—1221no—
353C2×#41—1221no—
354C2×#41—1221no—
355C2×#41—1221no—
356C2×#41—1221no—
357C2×#41—1221no—
358C2×#41—1221no—
359C22×Q8—611no—
360C22×D4—611no—
361C22×D4—1221no—
362C23×C4—612no—
363C23×C4—1221no—
364C23×C4C22×D6111yescharacteristic
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—1221no—
371C2×M16—1221no—
372C2×M16—1221no—
373C2×M16—1221no—
374C2×M16—1221no—
375C2×M16—1221no—
376C2×M16C4×D6112yes—
377C22×C8—21no—
378C22×C8—612no—
379#117—21no—
380#117—21no—
381#117—21no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—21no—
386#117—21no—
387#117—612no—
388#117—612no—
389#117—1221no—
390#117—1221no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1224no—
394C2×C4⋊C4D24412yes—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1224no—
399C2×C42—21no—
400C2×C42—612no—
401C2×C42—1221no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42C22⋊D6112yes—
408#104—1228no—
409#103—1224no—
410#102—1224no—
411#95—1224no—
412#95—1224no—
413C24—1221no—
414C24D4×D6111yescharacteristic
415#41—1222no—
416#41—1222no—
417#41—1222no—
418#41—1222no—
419#41—1222no—
420#41—1222no—
421#41—1222no—
422#41—1222no—
423#41—2441no—
424#41—2441no—
425#41—2441no—
426#41—2441no—
427#41—2441no—
428#41—2441no—
429#41—2441no—
430#41—2441no—
431C2×Q8—612no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8—1221no—
437C2×D4—612no—
438C2×D4—1222no—
439C2×D4—1222no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—2441no—
446C2×D4—2441no—
447C2×D4—2441no—
448C2×D4—2441no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—1221no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—2444no—
459C22×C4—2441no—
460C22×C4—2441no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4C2×C4×D6111yescharacteristic
464C22×C4C2×D24411yescharacteristic
465C22×C4#1175822211yescharacteristic
466C22×C4#1175822211yescharacteristic
467C22×C4#1175822211yescharacteristic
468C22×C4#1175822211yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471C22×C4D4×D6111yescharacteristic
472C22×C4D4×D6111yescharacteristic
473C22×C4#1175822411yescharacteristic
474C22×C4Q8×D6111yescharacteristic
475C22×C4#1175822611yescharacteristic
476C22×C4C2×C22⋊D6111yescharacteristic
477M16—22no—
478M16—22no—
479M16—41no—
480M16—41no—
481M16—2441no—
482M16—2441no—
483M16—2441no—
484M16—2441no—
485C2×C8—22no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—1222no—
489C2×C8—1222no—
490C2×C8—1222no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—2448no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C22⋊C4—2448no—
505C42—22no—
506C42—22no—
507C42—41no—
508C42—41no—
509C42—614no—
510C42—1222no—
511C42—2441no—
512C42—2441no—
513C42—2441no—
514C42—2441no—
515C23—42no—
516C23—1222no—
517C23—2444no—
518C23—2441no—
519C23—2441no—
520C23#1451973911yescharacteristic
521C23#1451974511yescharacteristic
522C23#1451974711yescharacteristic
523C23#1451975011yescharacteristic
524C23#1451975011yescharacteristic
525C23#1451975511yescharacteristic
526C23#1451976311yescharacteristic
527Q8—1222no—
528Q8—1222no—
529Q8—2441no—
530Q8—2441no—
531D4—1222no—
532D4—1222no—
533D4—2442no—
534D4—2442no—
535D4—2442no—
536D4—2442no—
537D4—2441no—
538D4—2441no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—1222no—
547C2×C4—1224no—
548C2×C4—2444no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4C22⋊C4×D6111yescharacteristic
560C2×C4C4⋊C4×D6111yescharacteristic
561C2×C4#1451973411yescharacteristic
562C2×C4#1451973711yescharacteristic
563C2×C4C4×C22⋊D6111yescharacteristic
564C2×C4C4×D24411yescharacteristic
565C2×C4#1451974711yescharacteristic
566C2×C4#1451974911yescharacteristic
567C2×C4#1451975011yescharacteristic
568C2×C4#1451975011yescharacteristic
569C2×C4#1451975011yescharacteristic
570C2×C4#1451975111yescharacteristic
571C2×C4#1451975211yescharacteristic
572C2×C4#1451975311yescharacteristic
573C2×C4#1451975511yescharacteristic
574C2×C4#1451975511yescharacteristic
575C2×C4#1451975511yescharacteristic
576C2×C4#1451975711yescharacteristic
577C2×C4#1451975911yescharacteristic
578C2×C4#1451976011yescharacteristic
579C2×C4#1451976211yescharacteristic
580C2×C4#1451976211yescharacteristic
581C2×C4#1451976211yescharacteristic
582C2×C4#1451976311yescharacteristic
583C2×C4#1451976511yescharacteristic
584C2×C4#1451977111yescharacteristic
585C2×C4#1451977211yescharacteristic
586C2×C4#1451977311yescharacteristic
587C8—42no—
588C8—2442no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—4882no—
597C22—4882no—
598C22—2444no—
599C22—2442no—
600C22#1683963311yescharacteristic
601C22#1683964011yescharacteristic
602C22#1683965111yescharacteristic
603C22#1683967111yescharacteristic
604C22#1683967511yescharacteristic
605C22#1683967711yescharacteristic
606C22#1683968611yescharacteristic
607C4—82no—
608C4—82no—
609C4—2444no—
610C4—2442no—
611C4—2444no—
612C4#1683962111yescharacteristic
613C4#1683963611yescharacteristic
614C4#1683965311yescharacteristic
615C4#1683966211yescharacteristic
616C4#1683968011yescharacteristic
617C4#1683968411yescharacteristic
618C4#1683969111yescharacteristic
619C4#1683969711yescharacteristic
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—21no—
626C2—4882no—
627C2#2144406311yescharacteristic
628C2#2144688512yes—
subgroupG/H
centerC2×C4#14519773
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1733C2
socleC22×C61#392
prime radicalC22×#11758222C22
cosocleC23×C4×C61C23
hypercenter#1733D61
hypocenterC61#11083
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1733

Complements

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

NH ≅ G/Ntype
4. #21448301C2semidirect
5. C2×#16839518C2semidirect
6. #21445512C2semidirect
7. C61×#1733C2semidirect
13. #16839778C4semidirect
149. C2×D244C4⋊C4semidirect
161. C2×Dic122C4⋊C4semidirect
192. C4×Dic61C22⋊C4semidirect
310. #1733D61semidirect
346. C61#11083semidirect

Maximal subgroups

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

subgroupindex
C2×#168397782
#214543362
#214483072
#214483012
C2×#168395182
#214455122
C61×#17332
#1108361