Finite Group Databasebeta

← #29690333

Subgroups — #29690333

Order 15616.

1010 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
1C2×#16839725C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic
3#407×D61C211yesmaximal, characteristic
4#21446026C211yesmaximal, characteristic, complemented
5#21446025C212yesmaximal, complemented
6#21446024C211yesmaximal, characteristic, complemented
7#21446023C211yesmaximal, characteristic, complemented
8#21446022C212yesmaximal, complemented
9#21445860C212yesmaximal, complemented
10#21445858C212yesmaximal, complemented
11C61×#1855C211yesmaximal, characteristic, complemented
12C23×C4×D61C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C22×#11758189C2212yes—
15C2×M16×D61C2211yescharacteristic
16C2×M16×D61C2211yescharacteristic
17C2×#14519713C2212yescomplemented
18C2×#14519712C2212yescomplemented
19C2×#14519712C2212yescomplemented
20#16839726C2211yescharacteristic
21#16839725—21no—
22#16839725—21no—
23#16839725C2212yes—
24#16839725C2211yescharacteristic
25#16839725C2211yescharacteristic
26#16839725C2212yes—
27#16839724—21no—
28#16839724—21no—
29#16839724C412yes—
30#16839724C412yes—
31#16839724C412yes—
32#16839724C412yes—
33C2×C22⋊#92418C2211yescharacteristic
34C2×#14519689—21no—
35C2×#14519689—21no—
36C2×#14519689C2212yes—
37C22⋊C8×D61—21no—
38C22⋊C8×D61—21no—
39#16839527C2212yescomplemented
40#16839526C2212yescomplemented
41#16839526C2212yescomplemented
42C22×M16×C61C2211yescharacteristic
43C2×#108×C61C2212yescomplemented
44C61×#407C2211yescharacteristic
45C2×C22⋊C8×C61C2211yescharacteristic
46C61×#341C2212yescomplemented
47C24×D61Q811yescharacteristic
48C23×Dic61Q811yescharacteristic
49C22×C4×D61—42no—
50C22×C4×D61—42no—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61D411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61Q811yescharacteristic
60C22×C4×D61Q811yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C2×#11758194—21no—
64C2×#11758194—42no—
65C2×#11758194C2312yes—
66C2×#11758194C2311yescharacteristic
67C2×#11758194C2311yescharacteristic
68C2×#11758189—21no—
69C2×#11758189—21no—
70C2×#11758189—42no—
71C2×#11758189—42no—
72C2×#11758189C2×C414yes—
73C2×#11758189C2×C414yes—
74M16×D61—42no—
75M16×D61—42no—
76C22×#92418—21no—
77C22×#92418C2312yes—
78C2×C8×D61—21no—
79C2×C8×D61—21no—
80#14519713—42no—
81#14519712—42no—
82#14519712—42no—
83C22⋊#92418—21no—
84C22⋊#92418—21no—
85C22⋊#92418C2×C412yes—
86C22⋊#92418C2×C412yes—
87#14519689—21no—
88#14519689—21no—
89#14519689—21no—
90#14519689—21no—
91#14519689—22no—
92#14519689—22no—
93#14519689—22no—
94#14519689—22no—
95#14519689C2×C414yes—
96#14519689C2×C414yes—
97C23×C4×C61C2311yescharacteristic
98C2×M16×C61—21no—
99C2×M16×C61—42no—
100C2×M16×C61C2312yes—
101C2×M16×C61C2311yescharacteristic
102C2×M16×C61C2311yescharacteristic
103C22×C8×C61—21no—
104C22×C8×C61C2312yes—
105#108×C61—42no—
106C22⋊C8×C61—21no—
107C22⋊C8×C61—21no—
108C22⋊C8×C61C2×C412yes—
109C22⋊C8×C61C2×C412yes—
110C24×C61C2×Q811yescharacteristic
111C23×D61—42no—
112C23×D61—42no—
113C23×D61—42no—
114C23×D61C4⋊C411yescharacteristic
115C23×D61C4⋊C411yescharacteristic
116C23×D61C4⋊C411yescharacteristic
117C23×D61C4⋊C411yescharacteristic
118C23×D61C2×Q811yescharacteristic
119C23×D61C2×Q811yescharacteristic
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×Q811yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—42no—
129C22×Dic61C4⋊C411yescharacteristic
130C22×Dic61C4⋊C411yescharacteristic
131C22×Dic61C4⋊C411yescharacteristic
132C22×Dic61C4⋊C411yescharacteristic
133C22×Dic61C2×Q811yescharacteristic
134C22×Dic61C2×Q811yescharacteristic
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—42no—
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×C4×D61—42no—
143C2×C4×D61—41no—
144C2×C4×D61—41no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61—41no—
148C2×C4×D61—41no—
149C2×C4×D61—41no—
150C2×C4×D61—41no—
151C2×C4×D61—42no—
152C2×C4×D61—42no—
153C2×C4×D61C4⋊C411yescharacteristic
154C2×C4×D61C4⋊C411yescharacteristic
155C2×C4×D61C4⋊C411yescharacteristic
156C2×C4×D61C4⋊C411yescharacteristic
157C2×C4×D61C4⋊C411yescharacteristic
158C2×C4×D61C4⋊C411yescharacteristic
159C2×C4×D61C4⋊C411yescharacteristic
160C2×C4×D61C4⋊C411yescharacteristic
161C2×C4×D61C22×C411yescharacteristic
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C22×C411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×C4×D61C2×D411yescharacteristic
169M16×C61—22no—
170M16×C61—22no—
171M16×C61—22no—
172M16×C61—22no—
173M16×C61—42no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—22no—
179C2×C8×C61—42no—
180C2×C8×C61C22×C414yes—
181#11758194—22no—
182#11758194—22no—
183#11758194—22no—
184#11758194—22no—
185#11758194—42no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—42no—
192C2×#92418C22×C414yes—
193#11758189—44no—
194#11758189—44no—
195#11758189—42no—
196#11758189—42no—
197C8×D61—42no—
198C8×D61—42no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—41no—
202C23×C61C2×C4⋊C411yescharacteristic
203C23×C61C2×C4⋊C411yescharacteristic
204C23×C61C22×Q811yescharacteristic
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—416no—
224C22×D61C2×C4⋊C411yescharacteristic
225C22×D61C2×C4⋊C411yescharacteristic
226C22×D61C2×C4⋊C411yescharacteristic
227C22×D61C2×C4⋊C411yescharacteristic
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61C2×C4⋊C411yescharacteristic
236C2×C4×C61C2×C4⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61H(5,𝔽2)12yes—
240C2×C4×C61H(5,𝔽2)12yes—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61C2×C4⋊C411yescharacteristic
248C2×Dic61C2×C4⋊C411yescharacteristic
249C2×Dic61C2×C4⋊C411yescharacteristic
250C2×Dic61C2×C4⋊C411yescharacteristic
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—21no—
256C4×D61—21no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—21no—
260C4×D61—21no—
261C4×D61—21no—
262C4×D61—21no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61—41no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C8×C61—24no—
272C8×C61—41no—
273C8×C61—41no—
274C8×C61—42no—
275#92418—24no—
276#92418—41no—
277#92418—41no—
278#92418—42no—
279#21949—611nomaximal, Sylow
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—44no—
288C22×C61#43012yes—
289C22×C61C22×C4⋊C411yescharacteristic
290C22×C61#54312yes—
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—44no—
303C2×D61—44no—
304C2×D61—44no—
305C2×D61—44no—
306C2×D61—44no—
307C2×D61—44no—
308C2×D61#43012yes—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61#51812yes—
312C4×C61#53411yescharacteristic
313C4×C61#53411yescharacteristic
314Dic61—21no—
315Dic61—21no—
316Dic61—21no—
317Dic61—21no—
318C2×#407—611no—
319C2×#407—611no—
320C2×#407—611no—
321#1855—612no—
322#1855—611no—
323#1855—611no—
324#1855—612no—
325#1855—611no—
326#1855D6111yescharacteristic, complemented
327#1811—612no—
328#1811—612no—
329C2×C61—22no—
330C2×C61—22no—
331C2×C61—41no—
332C2×C61#231511yescharacteristic
333C2×C61C2×#43011yescharacteristic
334C2×C61C2×#43011yescharacteristic
335D61—22no—
336D61—41no—
337D61—41no—
338D61—41no—
339C24×C4—611no—
340C22×M16—612no—
341C22×M16—611no—
342C22×M16—611no—
343C22×M16—611no—
344C22×M16C2×D6111yescharacteristic
345C2×#108—612no—
346C2×#108—612no—
347C2×#108—612no—
348C2×#108C2×D6112yescomplemented
349#407—612no—
350#407—612no—
351#407—612no—
352#407—612no—
353#407—611no—
354#407—611no—
355#407—611no—
356#407—612no—
357#407—612no—
358#407—1221no—
359#407—1221no—
360#407—1221no—
361#407—1221no—
362#407C2×D6111yescharacteristic
363C2×C22⋊C8—611no—
364C2×C22⋊C8—612no—
365C2×C22⋊C8—1221no—
366C2×C22⋊C8—1221no—
367C2×C22⋊C8—1221no—
368C2×C22⋊C8—1221no—
369C2×C22⋊C8C2×D6111yescharacteristic
370#341—612no—
371#341—612no—
372#341—612no—
373#341C2×D6112yescomplemented
374C61#2194911yescharacteristic, complemented, Sylow
375C25—611no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—611no—
387C23×C4—611no—
388C23×C4—611no—
389C23×C4—2442no—
390C23×C4—2442no—
391C23×C4C22×D6111yescharacteristic
392C2×M16—21no—
393C2×M16—42no—
394C2×M16—614no—
395C2×M16—614no—
396C2×M16—611no—
397C2×M16—612no—
398C2×M16—611no—
399C2×M16—1221no—
400C2×M16—1221no—
401C2×M16—1221no—
402C2×M16—2442no—
403C2×M16—2442no—
404C2×M16—2442no—
405C2×M16—2442no—
406C2×M16—2442no—
407C2×M16C22×D6111yescharacteristic
408C2×M16C22×D6111yescharacteristic
409C2×M16C22×D6112yes—
410C22×C8—21no—
411C22×C8—612no—
412C22×C8—1221no—
413C22×C8—1221no—
414C22×C8—1221no—
415C22×C8C22×D6112yes—
416#108—42no—
417#108—2442no—
418#108—2442no—
419#108—2442no—
420C22⋊C8—21no—
421C22⋊C8—21no—
422C22⋊C8—612no—
423C22⋊C8—612no—
424C22⋊C8—614no—
425C22⋊C8—614no—
426C22⋊C8—1221no—
427C22⋊C8—1221no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1222no—
431C22⋊C8—1222no—
432C22⋊C8—1222no—
433C22⋊C8—1222no—
434C22⋊C8—1221no—
435C22⋊C8—1221no—
436C22⋊C8C4×D6112yes—
437C22⋊C8C4×D6112yes—
438C24—611no—
439C24—611no—
440C24—611no—
441C24—611no—
442C24—611no—
443C24—611no—
444C24—2442no—
445C24—2442no—
446C24—2442no—
447C24Q8×D6111yescharacteristic
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2441no—
481C22×C4—2441no—
482C22×C4—2441no—
483C22×C4—2441no—
484C22×C4—2441no—
485C22×C4—2441no—
486C22×C4—2441no—
487C22×C4—2441no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4C2×C4×D6111yescharacteristic
492C22×C4C2×C4×D6111yescharacteristic
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4Q8×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497M16—22no—
498M16—22no—
499M16—22no—
500M16—22no—
501M16—42no—
502M16—1222no—
503M16—1222no—
504M16—1222no—
505M16—1222no—
506M16—2444no—
507M16—2444no—
508M16—2442no—
509M16—2442no—
510M16—2442no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—42no—
517C2×C8—614no—
518C2×C8—1222no—
519C2×C8—1222no—
520C2×C8—1221no—
521C2×C8—1222no—
522C2×C8—1221no—
523C2×C8—2442no—
524C2×C8—2442no—
525C2×C8—2442no—
526C2×C8C2×C4×D6114yes—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—611no—
531C23—611no—
532C23—611no—
533C23—611no—
534C23—1222no—
535C23—1222no—
536C23—1222no—
537C23—1222no—
538C23—1222no—
539C23—1222no—
540C23—1222no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—24416no—
553C23C4⋊C4×D6111yescharacteristic
554C23C4⋊C4×D6111yescharacteristic
555C23C2×Q8×D6111yescharacteristic
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
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×C4—2441no—
592C2×C4—2441no—
593C2×C4C4⋊C4×D6111yescharacteristic
594C2×C4C4⋊C4×D6111yescharacteristic
595C2×C4C22×C4×D6111yescharacteristic
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4#1451982212yes—
598C2×C4#1451982312yes—
599C8—24no—
600C8—42no—
601C8—41no—
602C8—41no—
603C8—1224no—
604C8—2441no—
605C8—2441no—
606C8—2442no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—44no—
615C22—612no—
616C22—1221no—
617C22—1221no—
618C22—1221no—
619C22—1221no—
620C22—1221no—
621C22—1221no—
622C22—1221no—
623C22—1221no—
624C22—1221no—
625C22—1221no—
626C22—1221no—
627C22—2444no—
628C22—2444no—
629C22—2444no—
630C22—2444no—
631C22—2444no—
632C22—2444no—
633C22#1683980912yes—
634C22C2×C4⋊C4×D6111yescharacteristic
635C22#1684038712yes—
636C4—21no—
637C4—21no—
638C4—1221no—
639C4—1221no—
640C4—1221no—
641C4—1221no—
642C4#1684023412yes—
643C4#1684032211yescharacteristic
644C4#1684032311yescharacteristic
645C2—22no—
646C2—22no—
647C2—41no—
648C2—1222no—
649C2—2441no—
650C2—2441no—
651C2—2441no—
652C2#2144843011yescharacteristic
653C2#430×D6111yescharacteristic
654C2C2×#1683980911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1855C2
socleC22×C61C22×C4⋊C4
prime radicalC24×D61Q8
cosocleC22×C4×C61C24
hypercenter#1855D61
hypocenterC61#21949
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1855

Complements

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

NH ≅ G/Ntype
4. #21446026C2semidirect
5. #21446025C2semidirect
6. #21446024C2semidirect
7. #21446023C2semidirect
8. #21446022C2semidirect
9. #21445860C2semidirect
10. #21445858C2semidirect
11. C61×#1855C2semidirect
17. C2×#14519713C22semidirect
18. C2×#14519712C22semidirect
19. C2×#14519712C22semidirect
39. #16839527C22semidirect
40. #16839526C22semidirect
41. #16839526C22semidirect
43. C2×#108×C61C22semidirect
46. C61×#341C22semidirect
326. #1855D61semidirect
348. C2×#108C2×D61semidirect
373. #341C2×D61semidirect
374. C61#21949semidirect

Maximal subgroups

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

subgroupindex
C2×#168397252
C2×#168397242
#407×D612
#214460262
#214460252
#214460242
#214460232
#214460222
#214458602
#214458582
C61×#18552
#2194961