Finite Group Databasebeta

← #29743419

Subgroups — #29743419

Order 15616.

878 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840063C211yesmaximal, characteristic
2C2×#16839864C211yesmaximal, characteristic
3#21449680C211yesmaximal, characteristic, complemented
4C4⋊C4×D244C211yesmaximal, characteristic
5#21447361C211yesmaximal, characteristic, complemented
6#21447292C211yesmaximal, characteristic, complemented
7#21447213C211yesmaximal, characteristic, complemented
8#21447158C211yesmaximal, characteristic, complemented
9#21446557C211yesmaximal, characteristic, complemented
10#21446152C211yesmaximal, characteristic
11#21446152C211yesmaximal, characteristic
12#21446083C211yesmaximal, characteristic
13#21446076C211yesmaximal, characteristic, complemented
14#21446069C211yesmaximal, characteristic
15C61×#1917C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4⋊C4×D61—21no—
23#16840063—24no—
24#16839864—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27C2×#14519711C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519709C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36#16839703—21no—
37#16839693—21no—
38#16839668C2211yescharacteristic
39#16839668C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839657—21no—
45#16839644—21no—
46#16839636—21no—
47#16839636—21no—
48#16839632C2211yescharacteristic
49#16839619—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C2×#11758190—22no—
67C2×#11758190—22no—
68C2×#11758190Q814yes—
69#14519769—22no—
70#14519769—41no—
71C4×Dic122—22no—
72C4×Dic122—41no—
73C4×D244—22no—
74C4×D244—22no—
75C4×D244—22no—
76C4×D244—22no—
77C4×D244—41no—
78C4×D244—41no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C4⋊C4×D61—44no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99#14519711—22no—
100#14519711—22no—
101#14519711—22no—
102#14519709—22no—
103#14519709—41no—
104#14519702—22no—
105#14519702—22no—
106#14519702—41no—
107#14519699—22no—
108#14519699—22no—
109#14519699D414yes—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
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—
128#97×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×D244C2×D412yes—
150C2×D244C2×Q812yes—
151C2×D244#4112yes—
152C2×C4×D61—24no—
153C2×C4×D61—42no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—81no—
160C2×C4×D61—81no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C2×Q812yes—
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×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C2×D412yes—
178C2×C8×C61C2×Q812yes—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—41no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—21no—
187#11758198—24no—
188#11758198—42no—
189#11758198—42no—
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—21no—
200C4⋊Dic61—42no—
201C4⋊Dic61—81no—
202C4⋊Dic61C2×D412yes—
203C4⋊Dic61#4112yes—
204#11758196—42no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208#11758196—42no—
209#11758196—42no—
210#11758196—42no—
211C4×Dic61—42no—
212#11758190—28no—
213C23×C61C22×D411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
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×C61AGL(1,ℤ/8)12yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×Q811yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C612−1+411yescharacteristic
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
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—
266D244—22no—
267D244—22no—
268D244—22no—
269D244C22⋊Q814yes—
270C4×D61—48no—
271C4×D61—84no—
272C4×D61—84no—
273Dic122—22no—
274Dic122—22no—
275Dic122—41no—
276Dic122—41no—
277C8×C61—22no—
278C8×C61—22no—
279C8×C61—41no—
280C8×C61—41no—
281#26398—611nomaximal, Sylow
282C22×C61#52511yescharacteristic
283C22×C61D4211yescharacteristic
284C22×C61D4×Q811yescharacteristic
285C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C61#57511yescharacteristic
287C22×C61#57711yescharacteristic
288C22×C61#57811yescharacteristic
289C2×D61—24no—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C4×C61—22no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61C2×C22⋊Q811yescharacteristic
303C4×C61#53911yescharacteristic
304C4×C61#54711yescharacteristic
305C4×C61#55411yescharacteristic
306Dic61—42no—
307Dic61—42no—
308Dic61—42no—
309Dic61—81no—
310Dic61—81no—
311Dic61—81no—
312Dic61—81no—
313C2×#478—611no—
314C2×#440—611no—
315#2453—611no—
316D4×C4⋊C4—611no—
317#2096—611no—
318#2083—611no—
319#2068—611no—
320#2059—611no—
321#1943—611no—
322#1917D6111yescharacteristic, complemented
323#1877—611no—
324#1877—611no—
325#1864—611no—
326#1863—611no—
327#1862—611no—
328C2×C61#244911yescharacteristic
329C2×C61#311411yescharacteristic
330C2×C61#312011yescharacteristic
331C2×C61#326911yescharacteristic
332C2×C61#327211yescharacteristic
333C2×C61#338611yescharacteristic
334C2×C61#338811yescharacteristic
335D61—44no—
336C22×SD16—611no—
337C2×#125—611no—
338C2×C4×Q8—611no—
339C2×C4×D4—611no—
340C2×C4×D4—611no—
341C2×C4×D4—611no—
342C22×C4⋊C4—1221no—
343#478—1224no—
344#440—1224no—
345C2×#107—611no—
346C2×#107—611no—
347C2×#107—611no—
348C2×#106—611no—
349C2×#106—611no—
350C2×C4⋊C8C2×D6111yescharacteristic
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#102—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#97C2×D6111yescharacteristic
358#400—1221no—
359#398—1221no—
360#391—611no—
361#391—611no—
362#389—611no—
363#389—611no—
364#389—611no—
365#389—611no—
366#389—1221no—
367#387—1221no—
368#385—1221no—
369#385—1221no—
370#384—611no—
371#380—1221no—
372C4×C4⋊C4C2×D6111yescharacteristic
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#336C2×D6111yescharacteristic
378#336C2×D6111yescharacteristic
379#336C2×D6111yescharacteristic
380#336C2×D6111yescharacteristic
381C61#2639811yescharacteristic, complemented, Sylow
382C22×Q8—611no—
383C22×D4—611no—
384C23×C4—1221no—
385C23×C4—1221no—
386C23×C4—1221no—
387C2×SD16—614no—
388C2×SD16—1222no—
389C2×SD16—1222no—
390C22×C8C22×D6111yescharacteristic
391C22×C8C22×D6111yescharacteristic
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394#125—1222no—
395#125—2441no—
396C4×Q8—1222no—
397C4×Q8—2441no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—2441no—
403C4×D4—2441no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1221no—
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—2444no—
420C2×C4⋊C4D24411yescharacteristic
421C2×C4⋊C4D24411yescharacteristic
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C42—1221no—
426C2×C42C22×D6111yescharacteristic
427C2×C42C22×D6111yescharacteristic
428C2×C42C22×D6111yescharacteristic
429#107—1222no—
430#107—1222no—
431#107—1222no—
432#106—1222no—
433#106—2441no—
434C4⋊C8—22no—
435#103—1222no—
436#103—1222no—
437#103—2441no—
438#102—614no—
439#102—1222no—
440#102—1222no—
441#97—22no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95D24411yescharacteristic
451#95D24411yescharacteristic
452C24—1221no—
453C2×Q8—612no—
454C2×Q8—1221no—
455C2×Q8—1221no—
456C2×Q8—1221no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×D4—612no—
460C2×D4—612no—
461C2×D4—612no—
462C2×D4—1221no—
463C2×D4—1221no—
464C2×D4—1221no—
465C22×C4—1224no—
466C22×C4—1221no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—4881no—
474C22×C4—4881no—
475C22×C4—2442no—
476C22×C4—2442no—
477C22×C4—2442no—
478C22×C4—2442no—
479C22×C4—2442no—
480C22×C4—2442no—
481C22×C4C2×D24411yescharacteristic
482C22×C4C2×D24411yescharacteristic
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4C23×D6111yescharacteristic
488SD16—1228no—
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×C8—21no—
498C2×C8—21no—
499C2×C8D4×D6112yes—
500C2×C8Q8×D6112yes—
501C4⋊C4—22no—
502C4⋊C4—41no—
503C4⋊C4—612no—
504C4⋊C4—612no—
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—1221no—
514C4⋊C4—1221no—
515C4⋊C4—4881no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C22⋊C4—1224no—
525C22⋊C4—2442no—
526C22⋊C4—2442no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—2442no—
534C23—1222no—
535C23—1222no—
536C23—1222no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23C22×D24411yescharacteristic
542Q8—1222no—
543Q8—1222no—
544Q8—2441no—
545Q8—2441no—
546D4—614no—
547D4—1222no—
548D4—1222no—
549D4—1222no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
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—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—4884no—
571C2×C4—4884no—
572C2×C4—2448no—
573C2×C4—2441no—
574C2×C4—2441no—
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×C4#1451980512yes—
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C2×Q8×D6111yescharacteristic
594C2×C4#41×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#1451982311yescharacteristic
597C2×C4#1451982411yescharacteristic
598C8—22no—
599C8—22no—
600C8—41no—
601C8—41no—
602C22—1224no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22—2442no—
607C22—2442no—
608C22—2442no—
609C22#1684027411yescharacteristic
610C22D4×D24411yescharacteristic
611C22Q8×D24411yescharacteristic
612C22C2×#1451980511yescharacteristic
613C22#1684056211yescharacteristic
614C22#1684057411yescharacteristic
615C22#1684057911yescharacteristic
616C4—22no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—4881no—
623C4—4881no—
624C4—4881no—
625C4—4881no—
626C4—2442no—
627C4—2442no—
628C4—2442no—
629C4C22⋊Q8×D6111yescharacteristic
630C4#1684035311yescharacteristic
631C4#1684041011yescharacteristic
632C4#1684046811yescharacteristic
633C2—2444no—
634C2#2144963311yescharacteristic
635C2#2145595911yescharacteristic
636C2#2145603511yescharacteristic
637C2#2145737911yescharacteristic
638C2#2145742111yescharacteristic
639C2#2145874211yescharacteristic
640C2#2145876611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1917C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1917D61
hypocenterC61#26398
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
3. #21449680C2semidirect
5. #21447361C2semidirect
6. #21447292C2semidirect
7. #21447213C2semidirect
8. #21447158C2semidirect
9. #21446557C2semidirect
13. #21446076C2semidirect
15. C61×#1917C2semidirect
322. #1917D61semidirect
381. C61#26398semidirect

Maximal subgroups

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

subgroupindex
C2×#168400632
C2×#168398642
#214496802
C4⋊C4×D2442
#214473612
#214472922
#214472132
#214471582
#214465572
#214461522
#214460832
#214460762
#214460692
C61×#19172
#2639861