Finite Group Databasebeta

← #29743035

Subgroups — #29743035

Order 15616.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839847C211yesmaximal, characteristic
2#21449665C211yesmaximal, characteristic
3Q8×#11758196C211yesmaximal, characteristic, complemented
4#21447349C211yesmaximal, characteristic
5#21447298C211yesmaximal, characteristic, complemented
6#21447192C211yesmaximal, characteristic
7#21447149C211yesmaximal, characteristic, complemented
8#21446545C211yesmaximal, characteristic, complemented
9#21446373C211yesmaximal, characteristic, complemented
10#21446174C211yesmaximal, characteristic
11#21446172C211yesmaximal, characteristic
12#21446089C211yesmaximal, characteristic, complemented
13#21446085C211yesmaximal, characteristic, complemented
14#21446077C211yesmaximal, characteristic
15C2×C61×#475C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519753—21no—
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21#16839847—24no—
22C2×#14519709C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30#16839684—21no—
31#16839673—21no—
32#16839664C2211yescharacteristic
33#16839663C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839660—21no—
36#16839660C2211yescharacteristic
37#16839658—21no—
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839633—21no—
41#16839631—21no—
42#16839631C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#475—24no—
56C2×#106×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C2×#11758201Q814yescomplemented
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519753—44no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—41no—
73#14519739—22no—
74#14519739—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196D411yescharacteristic
87C2×#11758196D411yescharacteristic
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97#14519709—22no—
98#14519708—22no—
99#14519708—41no—
100#14519701—22no—
101#14519699—22no—
102#14519699—22no—
103#14519697—41no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×Q8×C61—21no—
112C22×Q8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C4⋊Q8×C61—22no—
116C4⋊Q8×C61—41no—
117C4×Q8×C61—22no—
118C4×Q8×C61—41no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C42×C61—21no—
125C2×C42×C61C2311yescharacteristic
126#106×C61—22no—
127#106×C61—22no—
128C4⋊C8×C61—22no—
129#103×C61—22no—
130#103×C61D414yes—
131C23×D61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—42no—
136C2×Q8×C61—42no—
137C2×Q8×C61—81no—
138C2×Q8×C61C2×D412yes—
139C2×Q8×C61C2×Q812yes—
140C2×Q8×C61#4112yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244C2×Q812yes—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—81no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C2×D412yes—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—42no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61C2×D412yes—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—42no—
181#11758201—28no—
182#11758198—42no—
183#11758198—42no—
184#11758198—42no—
185#11758198—42no—
186#11758198—81no—
187C4⋊Dic61—24no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—42no—
194C4⋊Dic61—42no—
195C4⋊Dic61#4112yes—
196#11758196—41no—
197#11758196—42no—
198#11758196—42no—
199#11758196—81no—
200#11758196—81no—
201#11758196—81no—
202#11758196SD1614yescomplemented
203C4×Dic61—24no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—21no—
208C4×Dic61—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C2×Q812yes—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223Q8×C61—22no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—84no—
227Q8×C61C22⋊Q814yescomplemented
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C2×SD1612yes—
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×C4×C612−1+411yescharacteristic
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61C2×SD1612yes—
272C2×Dic61C2×SD1612yes—
273D244—22no—
274D244—22no—
275D244—41no—
276D244—41no—
277C4×D61—84no—
278C8×C61—22no—
279C8×C61—41no—
280#92418—22no—
281#92418—41no—
282#26373—611nomaximal, Sylow
283C22×C61#52511yescharacteristic
284C22×C61D4211yescharacteristic
285C22×C61D4×Q811yescharacteristic
286C22×C61C22×SD1611yescharacteristic
287C22×C61#57511yescharacteristic
288C22×C61#57711yescharacteristic
289C22×C61#57711yescharacteristic
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C4×C61—24no—
298C4×C61—22no—
299C4×C61—41no—
300C4×C61—42no—
301C4×C61—81no—
302C4×C61—81no—
303C4×C61C2×C22⋊Q811yescharacteristic
304C4×C61#53911yescharacteristic
305C4×C61#54711yescharacteristic
306C4×C61#55411yescharacteristic
307Dic61—22no—
308Dic61—22no—
309Dic61—41no—
310Dic61—42no—
311Dic61—41no—
312Dic61—81no—
313Dic61—81no—
314C2×#475D6111yescharacteristic, complemented
315C2×C4×SD16—611no—
316#2452—611no—
317Q8×C4⋊C4—611no—
318#2094—611no—
319#2084—611no—
320#2065—611no—
321#2058—611no—
322#1941—611no—
323#1916—611no—
324#1880—611no—
325#1880—611no—
326#1865—611no—
327#1865—611no—
328#1864—611no—
329C2×C61#244911yescharacteristic
330C2×C61#310311yescharacteristic
331C2×C61#312511yescharacteristic
332C2×C61#327111yescharacteristic
333C2×C61D4×SD1611yescharacteristic
334C2×C61Q8×SD1611yescharacteristic
335C2×C61#338811yescharacteristic
336D61—82no—
337C22×SD16—611no—
338C2×C4⋊Q8C2×D6111yescharacteristic
339C2×C22⋊Q8—1221no—
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×Q8C2×D6111yescharacteristic
343C2×C4×D4—611no—
344#475—24no—
345C4×SD16—1224no—
346C2×#106—611no—
347C2×#106—611no—
348C2×#106—611no—
349C2×#106C2×D6111yescharacteristic
350C2×#106C2×D6111yescharacteristic
351C2×C4⋊C8C2×D6111yescharacteristic
352C2×#103—611no—
353C2×#103C2×D6111yescharacteristic
354C2×#103C2×D6111yescharacteristic
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×C4×C8—611no—
359#396—1221no—
360#393—1221no—
361#390—611no—
362#390—611no—
363#389—611no—
364#389—611no—
365#389—611no—
366#389—611no—
367#389—1221no—
368#389—1221no—
369#385—1221no—
370#384—611no—
371#384—1221no—
372C4×C4⋊C4—611no—
373C4×C4⋊C4—1221no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#336—611no—
379#336—611no—
380#336—611no—
381#336—611no—
382C61#2637311yescharacteristic, complemented, Sylow
383C22×Q8—21no—
384C22×Q8C22×D6111yescharacteristic
385C22×D4—611no—
386C23×C4—1221no—
387C2×SD16—614no—
388C2×SD16—1222no—
389C2×SD16—1222no—
390C22×C8—611no—
391C22×C8—611no—
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—22no—
395C4⋊Q8—41no—
396C22⋊Q8—2444no—
397C4×Q8—22no—
398C4×Q8—41no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—2441no—
403C4×D4—1222no—
404C4×D4—2441no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4C22×D6111yescharacteristic
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—21no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434#106—22no—
435#106—22no—
436#106—1222no—
437#106—1222no—
438#106—2441no—
439C4⋊C8—22no—
440#103—22no—
441#103—1222no—
442#103C22⋊D6114yes—
443#102—1222no—
444#102—1222no—
445#102—2441no—
446C4×C8—1222no—
447#95—611no—
448#95—611no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453C24—1221no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—81no—
460C2×Q8Q8×D6112yes—
461C2×Q8#1175822612yes—
462C2×Q8C2×C22⋊D6112yes—
463C2×D4—612no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—4881no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4D4×D6111yescharacteristic
484C22×C4D4×D6111yescharacteristic
485C22×C4C23×D6111yescharacteristic
486SD16—1228no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—612no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8C2×C22⋊D6112yes—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—614no—
507C4⋊C4—612no—
508C4⋊C4—1224no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—4881no—
515C4⋊C4—4881no—
516C4⋊C4—4881no—
517C4⋊C4—2441no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4C2×C22⋊D6112yes—
523C22⋊C4—4881no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C22⋊C4—2442no—
528C42—21no—
529C42—21no—
530C42—42no—
531C42—1224no—
532C42—1221no—
533C42—1221no—
534C42—1221no—
535C42—1221no—
536C42—2442no—
537C23—1222no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23C2×D4×D6111yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—22no—
548Q8—84no—
549Q8#1451975314yescomplemented
550D4—1222no—
551D4—1222no—
552D4—2441no—
553D4—2441no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1222no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—4884no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4SD16×D6112yes—
594C2×C4C22×C22⋊D6111yescharacteristic
595C2×C4C2×Q8×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822611yescharacteristic
598C2×C4#1451982211yescharacteristic
599C2×C4#1451982411yescharacteristic
600C8—22no—
601C8—41no—
602C8—1222no—
603C8—2441no—
604C22—4881no—
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22#1684027511yescharacteristic
612C22D4×C22⋊D6111yescharacteristic
613C22#1684042511yescharacteristic
614C22C2×SD16×D6111yescharacteristic
615C22#1684056111yescharacteristic
616C22#1684056811yescharacteristic
617C22#1684057111yescharacteristic
618C4—22no—
619C4—24no—
620C4—42no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—1222no—
625C4—1222no—
626C4—4881no—
627C4—4881no—
628C4—2441no—
629C4—2442no—
630C4—2441no—
631C4C2×#1451975311yescharacteristic
632C4#1684036111yescharacteristic
633C4#1684041111yescharacteristic
634C4#1684046611yescharacteristic
635C2—4882no—
636C2#2144962411yescharacteristic
637C2#2145585211yescharacteristic
638C2#2145610311yescharacteristic
639C2#2145739811yescharacteristic
640C2SD16×C22⋊D6111yescharacteristic
641C2#2145867811yescharacteristic
642C2#2145877111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#475C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#475D61
hypocenterC61#26373
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. Q8×#11758196C2semidirect
5. #21447298C2semidirect
7. #21447149C2semidirect
8. #21446545C2semidirect
9. #21446373C2semidirect
12. #21446089C2semidirect
13. #21446085C2semidirect
15. C2×C61×#475C2semidirect
65. C2×#11758201Q8semidirect
202. #11758196SD16semidirect
227. Q8×C61C22⋊Q8semidirect
314. C2×#475D61semidirect
382. C61#26373semidirect
549. Q8#14519753semidirect

Maximal subgroups

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

subgroupindex
C2×#168398472
#214496652
Q8×#117581962
#214473492
#214472982
#214471922
#214471492
#214465452
#214463732
#214461742
#214461722
#214460892
#214460852
#214460772
C2×C61×#4752
#2637361