Finite Group Databasebeta

← #29743044

Subgroups — #29743044

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×#16840048C211yesmaximal, characteristic, complemented
2C2×C4×#11758190C211yesmaximal, characteristic
3#21449669C211yesmaximal, characteristic
4C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
5#21447351C211yesmaximal, characteristic
6#21447297C211yesmaximal, characteristic, complemented
7#21447192C211yesmaximal, characteristic
8#21447153C211yesmaximal, characteristic, complemented
9#21446547C211yesmaximal, characteristic, complemented
10#21446173C211yesmaximal, characteristic
11#21446173C211yesmaximal, characteristic
12#21446090C211yesmaximal, characteristic, complemented
13#21446090C211yesmaximal, characteristic, complemented
14#21446083C211yesmaximal, characteristic
15C61×#1916C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519755—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23#16840048—24no—
24C4×#11758190—24no—
25C2×#14519709C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519709C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519709C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36#16839684—21no—
37#16839675—21no—
38#16839664C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839659—21no—
45#16839659—21no—
46#16839636—21no—
47#16839632C2211yescharacteristic
48#16839631—21no—
49C4⋊C4×Dic61—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190Q814yescomplemented
68#14519777—22no—
69#14519777—41no—
70#14519755—44no—
71C4×Dic122—22no—
72C4×Dic122—22no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—41no—
76C4×Dic122—41no—
77C4×D244—22no—
78C4×D244—41no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103#14519709—22no—
104#14519709—22no—
105#14519709—22no—
106#14519709—22no—
107#14519709—41no—
108#14519702—22no—
109#14519702—22no—
110#14519702D414yes—
111#14519699—22no—
112#14519699—22no—
113#14519699—41no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C4⋊C8×C61—22no—
128C4×C8×C61—22no—
129#95×C61D411yescharacteristic
130#95×C61D411yescharacteristic
131C23×D61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C2×Q812yes—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61—81no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—42no—
159C2×Dic122—42no—
160C2×Dic122—81no—
161C2×Dic122C2×D412yes—
162C2×Dic122C2×Q812yes—
163C2×Dic122#4112yes—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D412yes—
175C2×C8×C61C2×Q812yes—
176C4⋊C4×C61—41no—
177C4⋊C4×C61SD1614yescomplemented
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184#11758198—42no—
185#11758198—42no—
186#11758198—42no—
187#11758198—42no—
188#11758198—81no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—42no—
200C4⋊Dic61—42no—
201C4⋊Dic61—81no—
202C4⋊Dic61C2×D412yes—
203C4⋊Dic61#4112yes—
204#11758196—24no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208#11758196—42no—
209#11758196—81no—
210#11758196—81no—
211C4×Dic61—24no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214#11758190—28no—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61C2×SD1612yes—
239C2×C4×C61C2×SD1612yes—
240C2×C4×C61C2×SD1612yes—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×Q811yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61H(5,𝔽2)11yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268D244—22no—
269D244—22no—
270D244—41no—
271D244—41no—
272C4×D61—84no—
273Dic122—22no—
274Dic122—22no—
275Dic122—22no—
276Dic122—84no—
277Dic122C22⋊Q814yescomplemented
278C8×C61—22no—
279C8×C61—22no—
280C8×C61—41no—
281C8×C61—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—22no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61C2×C22⋊Q811yescharacteristic
304C4×C61#53911yescharacteristic
305C4×C61#54711yescharacteristic
306C4×C61#55411yescharacteristic
307Dic61—24no—
308Dic61—42no—
309Dic61—42no—
310Dic61—81no—
311Dic61—81no—
312Dic61—81no—
313Dic61—81no—
314C2×#475—611no—
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#1916D6111yescharacteristic, complemented
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⋊Q8—611no—
339C2×C22⋊Q8—1221no—
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×D4—611no—
344#475—1224no—
345C4×SD16—1224no—
346C2×#106—611no—
347C2×#106—611no—
348C2×#106—611no—
349C2×#106—611no—
350C2×#106—611no—
351C2×C4⋊C8C2×D6111yescharacteristic
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×C4×C8C2×D6111yescharacteristic
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—1221no—
373C4×C4⋊C4C2×D6111yescharacteristic
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#336C2×D6111yescharacteristic
379#336C2×D6111yescharacteristic
380#336C2×D6111yescharacteristic
381#336C2×D6111yescharacteristic
382C61#2637311yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8—1221no—
385C22×D4—611no—
386C23×C4—1221no—
387C2×SD16—614no—
388C2×SD16—1222no—
389C2×SD16—1222no—
390C22×C8C22×D6111yescharacteristic
391C22×C8C22×D6111yescharacteristic
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C4⋊Q8—1222no—
395C4⋊Q8—2441no—
396C22⋊Q8—2444no—
397C4×Q8—1222no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—2441no—
402C4×Q8—2441no—
403C4×D4—1222no—
404C4×D4—2441no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—1221no—
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⋊C4—1221no—
421C2×C4⋊C4D24411yescharacteristic
422C2×C4⋊C4D24411yescharacteristic
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42C22×D6111yescharacteristic
432C2×C42C22×D6111yescharacteristic
433C2×C42C22×D6111yescharacteristic
434#106—1222no—
435#106—1222no—
436#106—1222no—
437#106—1222no—
438#106—2441no—
439C4⋊C8—22no—
440#103—614no—
441#103—1222no—
442#103—1222no—
443#102—1222no—
444#102—1222no—
445#102—2441no—
446C4×C8—22no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95D24411yescharacteristic
452#95D24411yescharacteristic
453C24—1221no—
454C2×Q8—612no—
455C2×Q8—612no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—4881no—
461C2×Q8—2442no—
462C2×Q8—2442no—
463C2×D4—612no—
464C2×D4—1221no—
465C2×D4—1221no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—4881no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4C2×D24411yescharacteristic
480C22×C4C2×D24411yescharacteristic
481C22×C4C2×D24411yescharacteristic
482C22×C4C2×D24411yescharacteristic
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C23×D6111yescharacteristic
486SD16—1228no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8D4×D6112yes—
498C2×C8Q8×D6112yes—
499C4⋊C4—41no—
500C4⋊C4—612no—
501C4⋊C4—612no—
502C4⋊C4—1224no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—4881no—
514C4⋊C4—4881no—
515C4⋊C4—4881no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4#1175819014yescomplemented
523C22⋊C4—4881no—
524C22⋊C4—2442no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C22⋊C4—2442no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—1224no—
535C42—2442no—
536C42—2442no—
537C23—1222no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23C22×D24411yescharacteristic
545Q8—614no—
546Q8—1222no—
547Q8—1222no—
548Q8—1222no—
549Q8—4884no—
550D4—1222no—
551D4—1222no—
552D4—2441no—
553D4—2441no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—4884no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4C2×#1175819012yes—
592C2×C4C2×#1175819012yes—
593C2×C4C2×#1175819012yes—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4C2×Q8×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4#41×D6111yescharacteristic
598C2×C4#1451982311yescharacteristic
599C2×C4#1451982411yescharacteristic
600C8—22no—
601C8—22no—
602C8—41no—
603C8—41no—
604C22—4881no—
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22#1684027411yescharacteristic
612C22D4×D24411yescharacteristic
613C22Q8×D24411yescharacteristic
614C22C22×#1175819011yescharacteristic
615C22#1684056211yescharacteristic
616C22#1684057411yescharacteristic
617C22#1684057411yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—1224no—
625C4—4881no—
626C4—4881no—
627C4—4881no—
628C4—4881no—
629C4—2442no—
630C4—2442no—
631C4C22⋊Q8×D6111yescharacteristic
632C4#1684035311yescharacteristic
633C4#1684041011yescharacteristic
634C4#1684046811yescharacteristic
635C2—4882no—
636C2#2144963311yescharacteristic
637C2#2145585111yescharacteristic
638C2#2145609811yescharacteristic
639C2#2145740911yescharacteristic
640C2D4×#1175819011yescharacteristic
641C2Q8×#1175819011yescharacteristic
642C2#2145876611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1916C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1916D61
hypocenterC61#26373
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

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

NH ≅ G/Ntype
1. C2×#16840048C2semidirect
4. C4⋊C4×Dic122C2semidirect
6. #21447297C2semidirect
8. #21447153C2semidirect
9. #21446547C2semidirect
12. #21446090C2semidirect
13. #21446090C2semidirect
15. C61×#1916C2semidirect
67. C2×#11758190Q8semidirect
177. C4⋊C4×C61SD16semidirect
277. Dic122C22⋊Q8semidirect
323. #1916D61semidirect
382. C61#26373semidirect
522. C4⋊C4#11758190semidirect

Maximal subgroups

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

subgroupindex
C2×#168400482
C2×C4×#117581902
#214496692
C4⋊C4×Dic1222
#214473512
#214472972
#214471922
#214471532
#214465472
#214461732
#214460902
#214460832
C61×#19162
#2637361