Finite Group Databasebeta

← #29697741

Subgroups — #29697741

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839804C211yesmaximal, characteristic
2#21448702C211yesmaximal, characteristic
3#21448694C211yesmaximal, characteristic
4#21446557C211yesmaximal, characteristic, complemented
5#21446552C211yesmaximal, characteristic, complemented
6#21446412C211yesmaximal, characteristic, complemented
7#21446405C211yesmaximal, characteristic, complemented
8#21446398C211yesmaximal, characteristic, complemented
9#21446391C211yesmaximal, characteristic, complemented
10#21445915C211yesmaximal, characteristic
11#21445906C211yesmaximal, characteristic
12#21445852C211yesmaximal, characteristic
13#21445716C211yesmaximal, characteristic, complemented
14C4×#14519709C211yesmaximal, characteristic, complemented
15C61×#1753C211yesmaximal, characteristic
16C2×#14519736C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18#16839804C418yescomplemented
19C2×#14519711C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25#16839660C2211yescharacteristic
26#16839660C2211yescharacteristic
27#16839660C2211yescharacteristic
28#16839660C2211yescharacteristic
29#16839650—21no—
30#16839650—21no—
31#16839645—21no—
32#16839645—21no—
33#16839626—21no—
34#16839622C2211yescharacteristic
35#16839622C2211yescharacteristic
36#16839620—21no—
37C4×C4⋊Dic61C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C22⋊C4×Dic61—21no—
42#16839607—21no—
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C4×C22⋊C4×C61C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C23×Dic61—21no—
61#14519736—22no—
62#14519736C2×C414yes—
63C2×C4⋊Dic61—21no—
64C2×C4⋊Dic61—22no—
65C2×C4⋊Dic61C2×C414yes—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86#14519711—22no—
87#14519711C2×C414yes—
88#14519711C2×C414yes—
89#14519709—22no—
90#14519709C2×C414yes—
91#14519709C2×C414yes—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102C23×C4×C61C2311yescharacteristic
103C2×M16×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61C2×C414yes—
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C22⋊C8×C61—22no—
115#97×C61—22no—
116C4×C8×C61—22no—
117#95×C61D411yescharacteristic
118#95×C61D411yescharacteristic
119C24×C61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61#4112yes—
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—24no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—81no—
142C22×Dic61—81no—
143M16×C61—28no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61C22×C412yes—
153C2×C8×C61C22×C412yes—
154C2×C8×C61#4112yes—
155C2×C8×C61#4112yes—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—41no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C22⋊Dic61—24no—
163C22⋊Dic61—42no—
164C22⋊Dic61—81no—
165C4⋊Dic61—24no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61C22×C412yes—
179C4⋊Dic61C22×C412yes—
180C4⋊Dic61C22×C412yes—
181C4⋊Dic61C22×C412yes—
182#11758196—81no—
183#11758196—81no—
184#11758196—81no—
185#11758196—81no—
186C4×Dic61—24no—
187C4×Dic61—42no—
188C4×Dic61—42no—
189C4×Dic61—81no—
190C4×Dic61—81no—
191C23×C61—21no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61C4×D412yes—
198C23×C61C22×D411yescharacteristic
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61C4×D412yes—
217C2×C4×C61#13712yes—
218C2×C4×C61C23×C411yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C612−1+411yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—84no—
249C8×C61—22no—
250C8×C61—41no—
251C8×C61—41no—
252C8×C61#11714yes—
253#22621—611nomaximal, Sylow
254C22×C61—22no—
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—41no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61C2×C4×D411yescharacteristic
262C22×C61#54711yescharacteristic
263C22×C61#54711yescharacteristic
264C22×C61C2×#13711yescharacteristic
265C22×C61#57511yescharacteristic
266C22×C61#57611yescharacteristic
267C22×C61#57711yescharacteristic
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61C2×#11711yescharacteristic
274C4×C61#51911yescharacteristic
275C4×C61#55211yescharacteristic
276C4×C61#55211yescharacteristic
277Dic61—42no—
278Dic61—42no—
279Dic61—42no—
280Dic61—42no—
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284Dic61—81no—
285C2×#428—611no—
286#2346—611no—
287#2345—611no—
288#1943—611no—
289#1942—611no—
290#1921—611no—
291#1920—611no—
292#1919—611no—
293#1918—611no—
294#1825—611no—
295#1823—611no—
296#1809—611no—
297#1776—611no—
298C4×#106—611no—
299#1753D6111yescharacteristic
300C2×C61—42no—
301C2×C61#234411yescharacteristic
302C2×C61C4×#13711yescharacteristic
303C2×C61#294611yescharacteristic
304C2×C61#334411yescharacteristic
305C2×C61#334711yescharacteristic
306C2×C61#335111yescharacteristic
307C2×C61#335711yescharacteristic
308C22×M16C2×D6111yescharacteristic
309C2×#117—611no—
310C22×C4⋊C4—611no—
311#428—618no—
312C2×#107—611no—
313C2×#107—611no—
314C2×#107—611no—
315C2×#106—611no—
316C2×#106—611no—
317C2×#106—611no—
318C2×C22⋊C8C2×D6111yescharacteristic
319C2×#97C2×D6111yescharacteristic
320C2×C4×C8C2×D6111yescharacteristic
321#389—611no—
322#389—611no—
323#389—611no—
324#389—611no—
325#388—1221no—
326#388—1221no—
327#387—1221no—
328#387—1221no—
329#382—1221no—
330#380—611no—
331#380—611no—
332#380—1221no—
333C4×C4⋊C4—611no—
334C4×C4⋊C4—611no—
335C4×C4⋊C4—611no—
336C4×C4⋊C4—611no—
337C4×C22⋊C4—1221no—
338C4×C22⋊C4C2×D6111yescharacteristic
339#376—1221no—
340#340—611no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#336C2×D6111yescharacteristic
351#336C2×D6111yescharacteristic
352C61#2262111yescharacteristic, complemented, Sylow
353C23×C4—1221no—
354C23×C4C22×D6111yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D6114yes—
358C22×C8C22×D6111yescharacteristic
359C22×C8C22×D6111yescharacteristic
360C22×C8C22×D6111yescharacteristic
361C22×C8C22×D6111yescharacteristic
362#117—614no—
363#117—1222no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4D24411yescharacteristic
383C2×C22⋊C4D24411yescharacteristic
384C2×C42—1221no—
385C2×C42—1221no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42C22×D6111yescharacteristic
390C2×C42C22×D6111yescharacteristic
391#107—614no—
392#107—614no—
393#107—1222no—
394#106—614no—
395#106—614no—
396#106—1222no—
397C22⋊C8—22no—
398#97—22no—
399C4×C8—22no—
400#95—1221no—
401#95—1221no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95D24411yescharacteristic
411#95D24411yescharacteristic
412C24C2×D24411yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—1224no—
418C22×C4—1221no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—4881no—
427C22×C4—4881no—
428C22×C4C2×C4×D6112yes—
429C22×C4C2×D24411yescharacteristic
430C22×C4C2×D24411yescharacteristic
431C22×C4C2×D24411yescharacteristic
432C22×C4C2×D24411yescharacteristic
433C22×C4C2×D24411yescharacteristic
434C22×C4#1175822212yes—
435C22×C4C23×D6111yescharacteristic
436M16—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8C2×C4×D6112yes—
446C2×C8C2×C4×D6112yes—
447C2×C8#1175822412yes—
448C2×C8#1175822412yes—
449C4⋊C4—612no—
450C4⋊C4—612no—
451C4⋊C4—612no—
452C4⋊C4—612no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1224no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—4881no—
467C4⋊C4—4881no—
468C4⋊C4—4881no—
469C4⋊C4—4881no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—1224no—
473C22⋊C4—4881no—
474C22⋊C4—2442no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—1224no—
480C42—4881no—
481C42—4881no—
482C42—2442no—
483C42—2442no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C4×D24412yes—
491C23C22×D24411yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—1222no—
510C2×C4—1222no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—4884no—
514C2×C4—2441no—
515C2×C4—2441no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4C4×D24412yes—
535C2×C4#1451981012yes—
536C2×C4C22×C4×D6111yescharacteristic
537C2×C4C2×#1175822411yescharacteristic
538C2×C4C2×#1175822411yescharacteristic
539C2×C4C2×#1175822211yescharacteristic
540C2×C4#1451982411yescharacteristic
541C2×C4#1451982411yescharacteristic
542C8—22no—
543C8—41no—
544C8—41no—
545C8#1451973314yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×C4×D24411yescharacteristic
554C22#1684041611yescharacteristic
555C22#1684041611yescharacteristic
556C22C2×#1451981011yescharacteristic
557C22#1684056211yescharacteristic
558C22#1684056711yescharacteristic
559C22#1684057411yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—4881no—
566C4—4881no—
567C4—4881no—
568C4—4881no—
569C4—2442no—
570C4—2442no—
571C4—2442no—
572C4—2442no—
573C4C2×#1451973311yescharacteristic
574C4#1684023911yescharacteristic
575C4#1684045211yescharacteristic
576C4#1684045211yescharacteristic
577C2—42no—
578C2#2144868611yescharacteristic
579C2C4×#1451981011yescharacteristic
580C2#2145465311yescharacteristic
581C2#2145827611yescharacteristic
582C2#2145832111yescharacteristic
583C2#2145838111yescharacteristic
584C2#2145844611yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1753C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1753D61
hypocenterC61#22621
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #21446557C2semidirect
5. #21446552C2semidirect
6. #21446412C2semidirect
7. #21446405C2semidirect
8. #21446398C2semidirect
9. #21446391C2semidirect
13. #21445716C2semidirect
14. C4×#14519709C2semidirect
18. #16839804C4semidirect
352. C61#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#168398042
#214487022
#214486942
#214465572
#214465522
#214464122
#214464052
#214463982
#214463912
#214459152
#214459062
#214458522
#214457162
C4×#145197092
C61×#17532
#2262161