Finite Group Databasebeta

← #29731802

Subgroups — #29731802

Order 15616.

812 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840170C211yesmaximal, characteristic, complemented
2C2×#16840064C211yesmaximal, characteristic
3#21450770C211yesmaximal, characteristic
4#21448917C211yesmaximal, characteristic, complemented
5#21447297C211yesmaximal, characteristic, complemented
6#21447293C211yesmaximal, characteristic, complemented
7#21447203C211yesmaximal, characteristic
8#21447193C211yesmaximal, characteristic
9#21446765C211yesmaximal, characteristic, complemented
10#21446581C211yesmaximal, characteristic, complemented
11#21446577C211yesmaximal, characteristic, complemented
12#21446541C211yesmaximal, characteristic, complemented
13#21446448C211yesmaximal, characteristic
14#21445646C211yesmaximal, characteristic
15C61×#1925C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519754—21no—
19C2×#14519754—21no—
20C2×D4×Dic61C2211yescharacteristic
21#16840170—24no—
22#16840064—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34#16839698—21no—
35#16839698—21no—
36#16839665C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839659—21no—
40#16839646—21no—
41#16839645—21no—
42#16839632C2211yescharacteristic
43#16839631C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196C2211yescharacteristic
47C4×C4⋊Dic61—21no—
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#390C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519776—22no—
66#14519776—22no—
67#14519768—22no—
68#14519768D414yes—
69#14519754—44no—
70#14519754—44no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
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×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519710—22no—
96#14519710—22no—
97#14519709—22no—
98#14519709—22no—
99#14519708—22no—
100C4⋊#92418—22no—
101#14519706—22no—
102#14519706—22no—
103#14519698—22no—
104#14519698—22no—
105#14519687—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C22×D4×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C22⋊C4×C61—21no—
119C2×C22⋊C4×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122#107×C61—22no—
123#102×C61—22no—
124#102×C61Q814yescomplemented
125#95×C61—21no—
126C24×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61C2×Q812yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
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—42no—
144C22×Dic61—42no—
145C22×Dic61—42no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×Dic122—42no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61C2×Q812yes—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61C2×D412yes—
164C4⋊C4×C61C2×Q812yes—
165C22⋊C4×C61—42no—
166C22⋊C4×C61—42no—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—42no—
170C22⋊Dic61—42no—
171C22⋊Dic61—42no—
172C22⋊Dic61—42no—
173C4⋊Dic61—21no—
174C4⋊Dic61—24no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—42no—
182C4⋊Dic61—42no—
183C4⋊Dic61—42no—
184C4⋊Dic61C2×D412yes—
185#11758196—22no—
186#11758196—24no—
187#11758196—22no—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191#11758196—81no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—81no—
195C4×Dic61—81no—
196C4×Dic61C2×D412yes—
197C4×Dic61#4112yes—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418#4112yes—
204C23×C61—22no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C22×D411yescharacteristic
212D4×C61—22no—
213D4×C61—22no—
214D4×C61—41no—
215D4×C61—41no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61AGL(1,ℤ/8)12yes—
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×Q811yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
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—84no—
259C2×Dic61#12112yes—
260C2×Dic61#12112yes—
261C2×Dic61AGL(1,ℤ/8)12yes—
262Dic122—84no—
263C8×C61—22no—
264C8×C61—41no—
265#92418—22no—
266#92418—41no—
267#25465—611nomaximal, Sylow
268C22×C61—42no—
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—81no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61C2×#12111yescharacteristic
276C22×C61#54711yescharacteristic
277C22×C61D4×Q811yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57811yescharacteristic
282C4×C61—22no—
283C4×C61—41no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61—81no—
287C4×C61#53311yescharacteristic
288C4×C61#54111yescharacteristic
289C4×C61#54811yescharacteristic
290C4×C61D4×Q811yescharacteristic
291Dic61—22no—
292Dic61—22no—
293Dic61—22no—
294Dic61—41no—
295Dic61—42no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299C2×#501—611no—
300C2×#478—611no—
301#2542—611no—
302#2370—611no—
303#2084—611no—
304#2083—611no—
305#2067—611no—
306#2065—611no—
307#1987—611no—
308#1948—611no—
309#1947—611no—
310#1940—611no—
311#1926—611no—
312#1925D6111yescharacteristic
313#1762—611no—
314C2×C61—82no—
315C2×C61#242911yescharacteristic
316C2×C61#303811yescharacteristic
317C2×C61#304611yescharacteristic
318C2×C61#334311yescharacteristic
319C2×C61#334511yescharacteristic
320C2×C61#338411yescharacteristic
321C2×C61#338611yescharacteristic
322C2×C4⋊Q8—611no—
323C2×#125—611no—
324C2×C22⋊Q8—1221no—
325C2×C22⋊Q8—1221no—
326C2×C4×D4—611no—
327#501—1224no—
328#478—1224no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#107C2×D6111yescharacteristic
332C2×#106—611no—
333C2×#106—611no—
334C2×#106—611no—
335C2×C4⋊C8—611no—
336C2×C4⋊C8—611no—
337C2×C4⋊C8—611no—
338C2×#102—611no—
339C2×#102—611no—
340C2×#102C2×D6111yescharacteristic
341C2×#102C2×D6111yescharacteristic
342C2×#97—611no—
343#399—1221no—
344#399—1221no—
345#390—611no—
346#390C2×D6111yescharacteristic
347#389—611no—
348#389—611no—
349#389—1221no—
350#387—1221no—
351#387—1221no—
352#384—611no—
353#384—611no—
354C4×C4⋊C4—611no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—1221no—
358C4×C4⋊C4C2×D6111yescharacteristic
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340C2×D6111yescharacteristic
365#336—611no—
366#336C2×D6111yescharacteristic
367C61#2546511yescharacteristic, complemented, Sylow
368C22×Q8—1221no—
369C22×D4C22×D6111yescharacteristic
370C23×C4—1221no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375C4⋊Q8—1222no—
376C4⋊Q8—1222no—
377#125—614no—
378#125—1222no—
379C22⋊Q8—2444no—
380C22⋊Q8—2444no—
381C4×D4—1222no—
382C4×D4—1222no—
383C2×C4⋊C4—611no—
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⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—21no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—1221no—
410C2×C42—1221no—
411C2×C42C22×D6111yescharacteristic
412#107—22no—
413#107—1222no—
414#107—1222no—
415#106—1222no—
416#106—1222no—
417#106—1222no—
418C4⋊C8—1222no—
419C4⋊C8—1222no—
420C4⋊C8—1222no—
421#102—22no—
422#102—1222no—
423#102—1222no—
424#102Dic12214yescomplemented
425#97—1222no—
426#95—21no—
427#95—611no—
428#95—611no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433C24—21no—
434C2×Q8—2442no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×Dic12212yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—2442no—
455C22×C4—2442no—
456C22×C4—2442no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—612no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8C2×Dic12212yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—612no—
477C4⋊C4—1222no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4—1221no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—4881no—
489C4⋊C4—2442no—
490C4⋊C4—2442no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4C2×Dic12212yes—
496C4⋊C4D4×D6112yes—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C22⋊C4—2442no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—612no—
506C42—612no—
507C42—1221no—
508C42—1221no—
509C42—4881no—
510C42—4881no—
511C23—22no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23C2×D4×D6111yescharacteristic
519Q8—4884no—
520D4—22no—
521D4—22no—
522D4—41no—
523D4—41no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
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—2441no—
553C2×C4—2441no—
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×C4—2441no—
563C2×C4#1451980112yes—
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C22×Dic12211yescharacteristic
566C2×C4#41×D6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C2×C4#1451982411yescharacteristic
569C2×C4#1451982411yescharacteristic
570C8—22no—
571C8—41no—
572C8—1222no—
573C8—2441no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22#121×D6111yescharacteristic
582C22#1684041011yescharacteristic
583C22D4×Dic12211yescharacteristic
584C22AGL(1,ℤ/8)×D6111yescharacteristic
585C22C2×#1451980111yescharacteristic
586C22#1684056911yescharacteristic
587C22#1684057811yescharacteristic
588C4—22no—
589C4—42no—
590C4—41no—
591C4—42no—
592C4—81no—
593C4—1222no—
594C4—1222no—
595C4—1222no—
596C4—4881no—
597C4—4881no—
598C4—4881no—
599C4—2442no—
600C4—2441no—
601C4#1684031511yescharacteristic
602C4#1684036911yescharacteristic
603C4#1684042011yescharacteristic
604C4D4×Dic12211yescharacteristic
605C2—82no—
606C2#2144941511yescharacteristic
607C2#2145529411yescharacteristic
608C2#2145537211yescharacteristic
609C2#2145826311yescharacteristic
610C2#2145829211yescharacteristic
611C2#2145871611yescharacteristic
612C2#2145874111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1925C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#1925D61
hypocenterC61#25465
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1925

Complements

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

NH ≅ G/Ntype
1. C2×#16840170C2semidirect
4. #21448917C2semidirect
5. #21447297C2semidirect
6. #21447293C2semidirect
9. #21446765C2semidirect
10. #21446581C2semidirect
11. #21446577C2semidirect
12. #21446541C2semidirect
124. #102×C61Q8semidirect
367. C61#25465semidirect
424. #102Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168401702
C2×#168400642
#214507702
#214489172
#214472972
#214472932
#214472032
#214471932
#214467652
#214465812
#214465772
#214465412
#214464482
#214456462
C61×#19252
#2546561