Finite Group Databasebeta

← #29702044

Subgroups — #29702044

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840041C211yesmaximal, characteristic
2#21448906C211yesmaximal, characteristic, complemented
3#21448728C211yesmaximal, characteristic
4#21446790C211yesmaximal, characteristic, complemented
5#21446590C211yesmaximal, characteristic, complemented
6#21446554C211yesmaximal, characteristic, complemented
7#21446553C211yesmaximal, characteristic, complemented
8#21446414C211yesmaximal, characteristic, complemented
9#21446081C211yesmaximal, characteristic
10#21446069C211yesmaximal, characteristic
11#21445692C211yesmaximal, characteristic, complemented
12#21445647C211yesmaximal, characteristic
13#21445643C211yesmaximal, characteristic
14#21445633C211yesmaximal, characteristic
15C61×#1918C211yesmaximal, characteristic, complemented
16C2×#14519776C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20#16840041C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839668—21no—
33#16839666—21no—
34#16839661C2211yescharacteristic
35#16839661C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839659—21no—
38#16839636—21no—
39#16839633—21no—
40#16839627C2211yescharacteristic
41#16839619—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196—21no—
45C4×#11758196C2211yescharacteristic
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C61×#389C2211yescharacteristic
55C4×C4⋊C4×C61C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519776—22no—
68#14519776C2×C414yescomplemented
69C4×D244—22no—
70C4×D244—41no—
71#14519739—22no—
72#14519739—22no—
73#14519739—41no—
74#14519739C2×C414yes—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196D411yescharacteristic
83C2×#11758196D411yescharacteristic
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2311yescharacteristic
91#14519711—41no—
92#14519710C2×C414yescomplemented
93#14519708—41no—
94#14519706C2×C414yescomplemented
95#14519699—22no—
96#14519699C2×C414yes—
97#14519697C2×C414yes—
98#14519697Q814yescomplemented
99#14519688—41no—
100#14519687—22no—
101C8×Dic61—41no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684D411yescharacteristic
110#14519684D411yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#107×C61C2×C414yescomplemented
121#95×C61—21no—
122#95×C61—21no—
123C23×D61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61C2×D411yescharacteristic
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—21no—
141C2×D244C22×C412yes—
142C2×D244C2×Q812yes—
143C2×D244#4112yes—
144C2×C4×D61—24no—
145C2×C4×D61—42no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—81no—
150C2×C4×D61—81no—
151C2×C4×D61—81no—
152C2×C4×D61—81no—
153C2×Dic122—24no—
154C2×Dic122—81no—
155C2×Dic122—81no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C22×C412yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—81no—
168C4⋊C4×C61C22×C412yes—
169C4⋊C4×C61C22×C412yes—
170C4⋊C4×C61C2×Q812yes—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—81no—
174#11758198—24no—
175#11758198—42no—
176#11758198—42no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—81no—
183C4⋊Dic61—81no—
184C4⋊Dic61C22×C412yes—
185#11758196—24no—
186#11758196—22no—
187#11758196—41no—
188#11758196—81no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—42no—
192C4×Dic61—81no—
193C4×Dic61C22×C412yes—
194C4×Dic61#4112yes—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418C22×C412yes—
200C2×#92418C2×Q812yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—22no—
203C22×D61—22no—
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61C23×C411yescharacteristic
228C2×C4×C61C22×Q811yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—22no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61C4×D412yes—
252C2×Dic61C4×D412yes—
253C2×Dic61#13712yes—
254D244—22no—
255D244—22no—
256D244—22no—
257D244C4×Q814yes—
258C4×D61—84no—
259C4×D61—84no—
260C4×D61—84no—
261Dic122—84no—
262C8×C61—22no—
263C8×C61—41no—
264#92418—22no—
265#92418—22no—
266#22979—611nomaximal, Sylow
267C22×C61C2×C4×D411yescharacteristic
268C22×C61#54611yescharacteristic
269C22×C61D4×Q811yescharacteristic
270C22×C61C2×#13711yescharacteristic
271C22×C61#57511yescharacteristic
272C22×C61#57611yescharacteristic
273C22×C61#57611yescharacteristic
274C2×D61—24no—
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C4×C61—42no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61C2×C4×Q811yescharacteristic
288C4×C61#52011yescharacteristic
289C4×C61#55211yescharacteristic
290C4×C61#55411yescharacteristic
291Dic61—22no—
292Dic61—22no—
293Dic61—22no—
294Dic61—42no—
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298C2×#474—611no—
299#2369—611no—
300#2349—611no—
301#1993—611no—
302#1949—611no—
303#1943—611no—
304#1942—611no—
305#1921—611no—
306#1918D6111yescharacteristic, complemented
307#1864—611no—
308#1862—611no—
309#1771—611no—
310#1762—611no—
311#1762—611no—
312C4×#102—611no—
313C2×C61#236011yescharacteristic
314C2×C61C4×#13711yescharacteristic
315C2×C61#294811yescharacteristic
316C2×C61#331411yescharacteristic
317C2×C61#331911yescharacteristic
318C2×C61#338511yescharacteristic
319C2×C61#338911yescharacteristic
320D61—44no—
321C2×C4⋊Q8—611no—
322C2×C4×D4—611no—
323C2×C4×D4—611no—
324C2×C4×D4—611no—
325#474—618no—
326C2×#107—611no—
327C2×#107—611no—
328C2×#107C2×D6111yescharacteristic
329C2×#106—611no—
330C2×C4⋊C8—611no—
331C2×#102—611no—
332C2×#102—611no—
333C2×#102—611no—
334C2×#102—611no—
335C2×#97—611no—
336C2×#97—611no—
337C2×C4×C8—611no—
338#391—1221no—
339#391—1221no—
340#389—611no—
341#389—611no—
342#389—611no—
343#389—1221no—
344#389C2×D6111yescharacteristic
345#385—1221no—
346#385—1221no—
347#382—611no—
348#380—1221no—
349C4×C4⋊C4—611no—
350C4×C4⋊C4—611no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—1221no—
353C4×C4⋊C4C2×D6111yescharacteristic
354C2×#95—1221no—
355#340—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340C2×D6111yescharacteristic
361#340C2×D6111yescharacteristic
362#340C2×D6111yescharacteristic
363#336—611no—
364#336C2×D6111yescharacteristic
365C61#2297911yescharacteristic, complemented, Sylow
366C22×Q8—1221no—
367C22×D4—611no—
368C23×C4—1221no—
369C23×C4—1221no—
370C23×C4—1221no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4⋊Q8—614no—
376C4⋊Q8—1222no—
377C4×D4—614no—
378C4×D4—1222no—
379C4×D4—1222no—
380C4×D4—1222no—
381C4×D4—2441no—
382C4×D4—2441no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C42—21no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42C22×D6111yescharacteristic
406#107—614no—
407#107—2441no—
408#107C4×D6114yescomplemented
409#106—2441no—
410C4⋊C8—614no—
411#102—614no—
412#102—614no—
413#102—614no—
414#102—1222no—
415#97—1222no—
416#97—2441no—
417C4×C8—2441no—
418#95—21no—
419#95—21no—
420#95—611no—
421#95—611no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429C24—1221no—
430C2×Q8—1224no—
431C2×Q8—4881no—
432C2×Q8—4881no—
433C2×D4—612no—
434C2×D4—612no—
435C2×D4—612no—
436C2×D4—1221no—
437C2×D4—1221no—
438C2×D4—1221no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—1224no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—4881no—
452C22×C4—4881no—
453C22×C4—4881no—
454C22×C4—4881no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4—2442no—
458C22×C4—2442no—
459C22×C4D4×D6111yescharacteristic
460C22×C4D4×D6111yescharacteristic
461C22×C4C23×D6111yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8C2×C4×D6112yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—81no—
480C4⋊C4—612no—
481C4⋊C4—1222no—
482C4⋊C4—1224no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—4881no—
489C4⋊C4—4881no—
490C4⋊C4—4881no—
491C4⋊C4—2441no—
492C4⋊C4C2×C4×D6112yes—
493C4⋊C4C2×C4×D6112yes—
494C4⋊C4Q8×D6112yes—
495C22⋊C4—1224no—
496C22⋊C4—2442no—
497C22⋊C4—2442no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42—612no—
502C42—612no—
503C42—1221no—
504C42—1221no—
505C42—4881no—
506C42—2442no—
507C23—1222no—
508C23—1222no—
509C23—1222no—
510C23—2441no—
511C23—2441no—
512C23—2441no—
513C23—2441no—
514C23C2×D4×D6111yescharacteristic
515Q8—4884no—
516D4—614no—
517D4—1222no—
518D4—1222no—
519D4—1222no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1222no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—4884no—
552C2×C4—4884no—
553C2×C4—4884no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4C22×C4×D6111yescharacteristic
563C2×C4C2×Q8×D6111yescharacteristic
564C2×C4#41×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4#1451982311yescharacteristic
567C2×C4#1451982411yescharacteristic
568C8—22no—
569C8—41no—
570C8—1222no—
571C8—1222no—
572C22—1224no—
573C22—2442no—
574C22—2442no—
575C22—2442no—
576C22—2442no—
577C22—2442no—
578C22—2442no—
579C22C4×D4×D6111yescharacteristic
580C22#1684040411yescharacteristic
581C22#1684042511yescharacteristic
582C22#137×D6111yescharacteristic
583C22#1684056111yescharacteristic
584C22#1684056311yescharacteristic
585C22#1684056411yescharacteristic
586C4—41no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—41no—
591C4—81no—
592C4—1222no—
593C4—1222no—
594C4—1222no—
595C4—4881no—
596C4—4881no—
597C4—2441no—
598C4—2442no—
599C4C4×Q8×D6111yescharacteristic
600C4#1684024211yescharacteristic
601C4#1684044811yescharacteristic
602C4#1684046811yescharacteristic
603C2—2444no—
604C2#2144882711yescharacteristic
605C2#2145463511yescharacteristic
606C2#2145466611yescharacteristic
607C2#2145790311yescharacteristic
608C2#2145797811yescharacteristic
609C2#2145873211yescharacteristic
610C2#2145877211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1918C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1918D61
hypocenterC61#22979
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #21448906C2semidirect
4. #21446790C2semidirect
5. #21446590C2semidirect
6. #21446554C2semidirect
7. #21446553C2semidirect
8. #21446414C2semidirect
11. #21445692C2semidirect
15. C61×#1918C2semidirect
20. #16840041C4semidirect
68. #14519776C2×C4semidirect
92. #14519710C2×C4semidirect
94. #14519706C2×C4semidirect
98. #14519697Q8semidirect
120. #107×C61C2×C4semidirect
306. #1918D61semidirect
365. C61#22979semidirect
408. #107C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400412
#214489062
#214487282
#214467902
#214465902
#214465542
#214465532
#214464142
#214460812
#214460692
#214456922
#214456472
#214456432
#214456332
C61×#19182
#2297961