Finite Group Databasebeta

← #29708704

Subgroups — #29708704

Order 15616.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839802C211yesmaximal, characteristic, complemented
2C2×#16839786C211yesmaximal, characteristic, complemented
3C2×#16839761C211yesmaximal, characteristic
4C2×#16839761C211yesmaximal, characteristic
5#21448309C211yesmaximal, characteristic
6#21447211C212yesmaximal, complemented
7#21447204C212yesmaximal, complemented
8#21447197C212yesmaximal, complemented
9#21447190C212yesmaximal, complemented
10#21445538C211yesmaximal, characteristic, complemented
11C2×C61×#389C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14C22×#11758194C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16#16839802—24no—
17C2×#14519710C2212yes—
18C2×#14519708C2212yes—
19#16839786—24no—
20C2×C4⋊#92418C2212yes—
21C2×C4⋊#92418C2212yes—
22#16839761—24no—
23#16839761—24no—
24C2×#14519700C2212yes—
25C2×#14519700C2212yes—
26C2×#14519697C2212yes—
27C2×#14519697C2212yes—
28#16839668C2211yescharacteristic
29#16839668C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839664C2211yescharacteristic
32#16839641—21no—
33#16839641—21no—
34C4×#11758198—21no—
35C4×#11758198—21no—
36#16839524C2212yes—
37#16839524C2212yes—
38C22×C4⋊C4×C61—21no—
39C22×C4⋊C4×C61C2211yescharacteristic
40C22×C4⋊C4×C61C2211yescharacteristic
41C22×C42×C61C2211yescharacteristic
42C61×#389—21no—
43C61×#389—21no—
44C61×#389C2212yes—
45C61×#389C2212yes—
46C2×#95×C61—21no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222—22no—
52C2×#11758222Q814yes—
53C22×Dic122C2311yescharacteristic
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C22×C4×D61—21no—
57C2×#11758194—22no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194D414yes—
63C22×#92418C2312yes—
64C22×#92418C2312yes—
65#14519736—22no—
66#14519736—22no—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C22⋊Dic61—21no—
70C2×#11758198—21no—
71C2×#11758198—21no—
72C2×#11758198—21no—
73C2×#11758198—21no—
74C2×C4×Dic61—21no—
75#14519710—24no—
76#14519708—24no—
77C4⋊#92418—24no—
78C4⋊#92418—24no—
79#14519700—24no—
80#14519700—24no—
81#14519697—24no—
82#14519697—24no—
83#14519684—21no—
84#14519684—21no—
85#14519684—21no—
86#14519684—21no—
87C23×C4×C61—21no—
88C23×C4×C61—21no—
89C23×C4×C61D411yescharacteristic
90C23×C4×C61D411yescharacteristic
91C23×C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—22no—
94C2×C4⋊C4×C61—22no—
95C2×C4⋊C4×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61—44no—
100C2×C4⋊C4×C61D414yes—
101C2×C4⋊C4×C61C2312yes—
102C2×C4⋊C4×C61C2312yes—
103C2×C42×C61—22no—
104C2×C42×C61—22no—
105C2×C42×C61D411yescharacteristic
106C2×C42×C61D411yescharacteristic
107C2×C42×C61C2311yescharacteristic
108C2×C42×C61C2311yescharacteristic
109#95×C61—22no—
110#95×C61—22no—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—24no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—41no—
121C22×C4×C61—41no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×Q812yes—
135C22×C4×C61#4112yes—
136C22×C4×C61#4112yes—
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C2×C22⋊D61—24no—
140C2×C22⋊D61—42no—
141C2×C22⋊D61—42no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145#11758222—28no—
146#11758222—28no—
147#11758222—44no—
148#11758222—44no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C2×Q812yes—
155C2×C4×D61—24no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122C2×Q812yes—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—44no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61—44no—
171C4⋊C4×C61C2×D414yes—
172C42×C61—22no—
173C42×C61—22no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C22⋊Dic61—42no—
179#11758198—42no—
180#11758198—42no—
181#11758198—42no—
182#11758198—42no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4×Dic61—42no—
188#11758194—28no—
189#11758194—44no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418C2×D414yes—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61#12112yes—
201C23×C61#12112yes—
202C23×C61C22⋊Q812yes—
203C23×C61C22×D411yescharacteristic
204C22×D61—22no—
205C22×D61—41no—
206C22×D61—41no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—24no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—44no—
223C2×C4×C61—44no—
224C2×C4×C61—48no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—84no—
231C2×C4×C61#12114yes—
232C2×C4×C61#12112yes—
233C2×C4×C61#12112yes—
234C2×C4×C61C22⋊Q814yes—
235C2×C4×C61C22⋊Q814yes—
236C2×C4×C61C22⋊Q812yes—
237C2×C4×C61#12614yes—
238C2×C4×C61AGL(1,ℤ/8)12yes—
239C2×C4×C61#13712yes—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×Q811yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C22⋊D61—48no—
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—
256D244—22no—
257D244—22no—
258D244—41no—
259D244—41no—
260C4×D61—48no—
261Dic122—22no—
262Dic122—22no—
263Dic122—41no—
264Dic122—41no—
265#92418—24no—
266#92418—42no—
267#23617—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—22no—
273C22×C61—22no—
274C22×C61#39614yes—
275C22×C61C2×#12111yescharacteristic
276C22×C61C2×#12111yescharacteristic
277C22×C61C2×C22⋊Q811yescharacteristic
278C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
279C22×C61C2×#13711yescharacteristic
280C22×C61#57511yescharacteristic
281C22×C61#57511yescharacteristic
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61—81no—
288C2×D61—81no—
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—44no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—82no—
295C4×C61#39614yes—
296C4×C61C2×#12111yescharacteristic
297C4×C61C2×C22⋊Q811yescharacteristic
298C4×C61C2×C22⋊Q811yescharacteristic
299C4×C61C2×#12611yescharacteristic
300Dic61—42no—
301Dic61—81no—
302Dic61—81no—
303C2×#428—611no—
304C2×#423—611no—
305C2×#417—611no—
306C2×#417—611no—
307#2297—611no—
308C2×#389D6111yescharacteristic, complemented
309#2068—612no—
310#2067—612no—
311#2066—612no—
312#2065—612no—
313#1736—611no—
314C2×C61—24no—
315C2×C61C2×#39611yescharacteristic
316C2×C61#303611yescharacteristic
317C2×C61#303711yescharacteristic
318C2×C61#305511yescharacteristic
319C2×C61#305611yescharacteristic
320C2×C61#307911yescharacteristic
321C2×C61#308011yescharacteristic
322D61—82no—
323C22×#41—611no—
324C22×M16—611no—
325C22×M16—611no—
326C2×#117—611no—
327C22×C4⋊C4—21no—
328C22×C4⋊C4C2×D6111yescharacteristic
329C22×C4⋊C4C2×D6111yescharacteristic
330C22×C42C2×D6111yescharacteristic
331#428—1224no—
332C2×#107—612no—
333C2×#106—612no—
334#423—1224no—
335C2×C4⋊C8—612no—
336C2×C4⋊C8—612no—
337#417—1224no—
338#417—1224no—
339C2×#103—612no—
340C2×#103—612no—
341C2×#102—612no—
342C2×#102—612no—
343#391—611no—
344#391—611no—
345#390—611no—
346#390—611no—
347#389—21no—
348#389—21no—
349#389C2×D6112yes—
350#389C2×D6112yes—
351#386—1221no—
352#386—1221no—
353C4×C22⋊C4—1221no—
354C4×C22⋊C4—1221no—
355C2×#95—21no—
356#340—612no—
357#340—612no—
358C61#2361711yescharacteristic, complemented, Sylow
359C2×#41—614no—
360C2×#41—1222no—
361C2×#41—1222no—
362C2×#41—1222no—
363C2×#41—1222no—
364C2×#41—1222no—
365C22×Q8—611no—
366C22×D4—611no—
367C22×D4—1221no—
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—1221no—
371C23×C4C22⋊D6111yescharacteristic
372C23×C4C22⋊D6111yescharacteristic
373C23×C4C22×D6111yescharacteristic
374C2×M16—614no—
375C2×M16—1222no—
376C2×M16—1222no—
377C2×M16—1222no—
378C2×M16—1222no—
379C2×M16—1222no—
380C22×C8—612no—
381C22×C8—612no—
382#117—1222no—
383#117—1222no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—44no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4D24414yes—
395C2×C4⋊C4C22×D6112yes—
396C2×C4⋊C4C22×D6112yes—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—1221no—
405C2×C42C22⋊D6111yescharacteristic
406C2×C42C22⋊D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408C2×C42C22×D6111yescharacteristic
409#107—1224no—
410#106—1224no—
411C4⋊C8—1224no—
412C4⋊C8—1224no—
413#103—1224no—
414#103—1224no—
415#102—1224no—
416#102—1224no—
417#95—22no—
418#95—22no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422#95—1221no—
423C24—1221no—
424C24C2×C22⋊D6111yescharacteristic
425#41—1228no—
426#41—1228no—
427#41—2444no—
428#41—2444no—
429C2×Q8—612no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×D4—612no—
436C2×D4—1224no—
437C2×D4—1221no—
438C2×D4—1221no—
439C2×D4—1221no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—2442no—
443C2×D4—2442no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—1221no—
459C22×C4—1224no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4C2×D24412yes—
465C22×C4#1175822212yes—
466C22×C4D4×D6112yes—
467C22×C4#1175822412yes—
468C22×C4Q8×D6112yes—
469C22×C4#1175822612yes—
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C23×D6111yescharacteristic
476M16—1228no—
477M16—2444no—
478C2×C8—614no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—1222no—
482C2×C8—1222no—
483C2×C8—1222no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4C2×D24414yes—
496C22⋊C4—2442no—
497C22⋊C4—2442no—
498C22⋊C4—2442no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—22no—
506C42—22no—
507C42—2442no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—1222no—
513C23—2441no—
514C23—2441no—
515C23—2441no—
516C23—2441no—
517C23—2441no—
518C23—2441no—
519C23#1451974912yes—
520C23#1451975112yes—
521C23#1451975312yes—
522C23C22×C22⋊D6111yescharacteristic
523Q8—1222no—
524Q8—1222no—
525Q8—2441no—
526Q8—2441no—
527D4—1222no—
528D4—1222no—
529D4—2448no—
530D4—2441no—
531D4—2441no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—1222no—
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—2448no—
562C2×C4#1451974912yes—
563C2×C4#1451975112yes—
564C2×C4#1451975214yes—
565C2×C4#1451975312yes—
566C2×C4#1451975514yes—
567C2×C4#1451975714yes—
568C2×C4#1451977214yes—
569C2×C4#1451980312yes—
570C2×C4#1451980812yes—
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C22×D24411yescharacteristic
573C2×C4C2×Q8×D6111yescharacteristic
574C2×C4C2×#1175822411yescharacteristic
575C2×C4C2×#1175822211yescharacteristic
576C2×C4C2×#1175822611yescharacteristic
577C8—1224no—
578C8—2442no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—4881no—
586C22—4881no—
587C22—4881no—
588C22—4881no—
589C22—2442no—
590C22—2442no—
591C22—2442no—
592C22#1683968414yes—
593C22C2×#1451974911yescharacteristic
594C22C2×#1451975111yescharacteristic
595C22C2×#1451975311yescharacteristic
596C22C2×#1451980311yescharacteristic
597C22C2×#1451980811yescharacteristic
598C22#1684056011yescharacteristic
599C22#1684056011yescharacteristic
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—44no—
605C4—82no—
606C4—4881no—
607C4—4881no—
608C4—2442no—
609C4#1683968414yes—
610C4C2×#1451975211yescharacteristic
611C4C2×#1451975511yescharacteristic
612C4C2×#1451975711yescharacteristic
613C4C2×#1451977211yescharacteristic
614C2—24no—
615C2—4882no—
616C2C2×#1683968411yescharacteristic
617C2#2145528311yescharacteristic
618C2#2145529111yescharacteristic
619C2#2145543411yescharacteristic
620C2#2145544511yescharacteristic
621C2#2145562811yescharacteristic
622C2#2145563711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#389C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC2×#389D61
hypocenterC61#23617
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
1. C2×#16839802C2semidirect
2. C2×#16839786C2semidirect
6. #21447211C2semidirect
7. #21447204C2semidirect
8. #21447197C2semidirect
9. #21447190C2semidirect
10. #21445538C2semidirect
11. C2×C61×#389C2semidirect
308. C2×#389D61semidirect
358. C61#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#168398022
C2×#168397862
C2×#168397612
#214483092
#214472112
#214472042
#214471972
#214471902
#214455382
C2×C61×#3892
#2361761