Finite Group Databasebeta

← #29697786

Subgroups — #29697786

Order 15616.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839804C211yesmaximal, characteristic
2#21448702C211yesmaximal, characteristic
3#21448686C211yesmaximal, characteristic
4#21446562C211yesmaximal, characteristic, complemented
5#21446547C211yesmaximal, characteristic, complemented
6#21446412C211yesmaximal, characteristic, complemented
7#21446405C211yesmaximal, characteristic, complemented
8#21446398C211yesmaximal, characteristic, complemented
9#21446391C211yesmaximal, characteristic, complemented
10#21445915C211yesmaximal, characteristic
11#21445910C211yesmaximal, characteristic
12#21445852C211yesmaximal, characteristic
13#21445716C211yesmaximal, characteristic, complemented
14C4×#14519711C211yesmaximal, characteristic, complemented
15C61×#1753C211yesmaximal, characteristic
16C2×#14519736C2211yescharacteristic
17C2×#14519733—21no—
18C22×C4⋊Dic61C2211yescharacteristic
19#16839804C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519711C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26#16839660C2211yescharacteristic
27#16839660C2211yescharacteristic
28#16839660C2211yescharacteristic
29#16839660C2211yescharacteristic
30#16839650—21no—
31#16839650—21no—
32#16839647—21no—
33#16839647—21no—
34#16839626—21no—
35#16839623—21no—
36#16839622C2211yescharacteristic
37#16839622C2211yescharacteristic
38C4×C4⋊Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
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—
63#14519733—44no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61C2×C414yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87#14519711—22no—
88#14519711C2×C414yes—
89#14519711C2×C414yes—
90#14519709—22no—
91#14519709C2×C414yes—
92#14519709C2×C414yes—
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—
102#14519684—21no—
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—22no—
105C2×M16×C61—22no—
106C2×M16×C61C2×C414yes—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115C22⋊C8×C61—22no—
116#97×C61—22no—
117C4×C8×C61—22no—
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C22×C412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61#4112yes—
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—24no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144M16×C61—28no—
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×C61—21no—
153C2×C8×C61C22×C412yes—
154C2×C8×C61C22×C412yes—
155C2×C8×C61#4112yes—
156C2×C8×C61#4112yes—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—41no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C22⋊Dic61—24no—
164C22⋊Dic61—42no—
165C22⋊Dic61—81no—
166C4⋊Dic61—24no—
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⋊Dic61—21no—
179C4⋊Dic61C22×C412yes—
180C4⋊Dic61C22×C412yes—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C22×C412yes—
183#11758196—42no—
184#11758196—42no—
185#11758196—81no—
186#11758196—81no—
187C4×Dic61—24no—
188C4×Dic61—42no—
189C4×Dic61—42no—
190C4×Dic61—42no—
191C4×Dic61—42no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61C4×D412yes—
199C23×C61C22×D411yescharacteristic
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
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×C61—21no—
217C2×C4×C61C4×D412yes—
218C2×C4×C61AGL(1,ℤ/8)12yes—
219C2×C4×C61C23×C411yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×C4×C612−1+411yescharacteristic
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—48no—
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—41no—
249C2×Dic61—41no—
250C8×C61—22no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61#11714yescomplemented
254#22624—611nomaximal, Sylow
255C22×C61—22no—
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—41no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61C2×C4×D411yescharacteristic
263C22×C61#54711yescharacteristic
264C22×C61#54711yescharacteristic
265C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C61#57511yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57811yescharacteristic
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61C2×#11711yescharacteristic
275C4×C61#51911yescharacteristic
276C4×C61#55211yescharacteristic
277C4×C61#55211yescharacteristic
278Dic61—42no—
279Dic61—42no—
280Dic61—42no—
281Dic61—42no—
282Dic61—81no—
283Dic61—81no—
284Dic61—81no—
285Dic61—81no—
286C2×#428—611no—
287#2346—611no—
288#2344—611no—
289#1944—611no—
290#1941—611no—
291#1921—611no—
292#1920—611no—
293#1919—611no—
294#1918—611no—
295#1825—611no—
296#1824—611no—
297#1809—611no—
298#1776—611no—
299C4×#107—611no—
300#1753D6111yescharacteristic
301C2×C61—42no—
302C2×C61#234411yescharacteristic
303C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C61#294711yescharacteristic
305C2×C61#334311yescharacteristic
306C2×C61#334811yescharacteristic
307C2×C61#335211yescharacteristic
308C2×C61#335411yescharacteristic
309C22×M16C2×D6111yescharacteristic
310C2×#117—611no—
311C2×#117—1221no—
312C22×C4⋊C4—611no—
313#428—618no—
314C2×#107—611no—
315C2×#107—611no—
316C2×#107—611no—
317C2×#106—611no—
318C2×#106—611no—
319C2×#106—611no—
320C2×C22⋊C8C2×D6111yescharacteristic
321C2×#97C2×D6111yescharacteristic
322C2×C4×C8C2×D6111yescharacteristic
323#389—611no—
324#389—611no—
325#389—611no—
326#389—611no—
327#388—1221no—
328#388—1221no—
329#387—1221no—
330#387—1221no—
331#382—1221no—
332#381—1221no—
333#380—611no—
334#380—611no—
335C4×C4⋊C4—611no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4—611no—
338C4×C4⋊C4—611no—
339C4×C22⋊C4C2×D6111yescharacteristic
340#376—1221no—
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#340—611no—
351#336C2×D6111yescharacteristic
352#336C2×D6111yescharacteristic
353C61#2262411yescharacteristic, complemented, Sylow
354C23×C4—1221no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C4×D6114yes—
359C22×C8C22×D6111yescharacteristic
360C22×C8C22×D6111yescharacteristic
361C22×C8C22×D6111yescharacteristic
362C22×C8C22×D6111yescharacteristic
363#117—614no—
364#117—1222no—
365#117—2444no—
366C2×C4⋊C4—614no—
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—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4D24411yescharacteristic
385C2×C22⋊C4D24411yescharacteristic
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42C22×D6111yescharacteristic
392C2×C42C22×D6111yescharacteristic
393#107—614no—
394#107—614no—
395#107—1222no—
396#106—614no—
397#106—614no—
398#106—1222no—
399C22⋊C8—22no—
400#97—22no—
401C4×C8—22no—
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#95—1221no—
411#95—1221no—
412#95D24411yescharacteristic
413#95D24411yescharacteristic
414C24C2×D24411yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—1221no—
420C22×C4—1221no—
421C22×C4—1221no—
422C22×C4—1221no—
423C22×C4—1224no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—2442no—
429C22×C4—2442no—
430C22×C4C2×C4×D6112yes—
431C22×C4C2×D24411yescharacteristic
432C22×C4C2×D24411yescharacteristic
433C22×C4C2×D24411yescharacteristic
434C22×C4C2×D24411yescharacteristic
435C22×C4C2×D24411yescharacteristic
436C22×C4#1175822212yes—
437C22×C4C23×D6111yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8C2×C4×D6112yes—
448C2×C8C2×C4×D6112yes—
449C2×C8#1175822412yes—
450C2×C8#1175822412yes—
451C4⋊C4—612no—
452C4⋊C4—612no—
453C4⋊C4—612no—
454C4⋊C4—612no—
455C4⋊C4—1221no—
456C4⋊C4—1221no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1224no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—4881no—
469C4⋊C4—4881no—
470C4⋊C4—2442no—
471C4⋊C4—2442no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—1224no—
475C22⋊C4—4881no—
476C22⋊C4—2442no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—1224no—
482C42—2442no—
483C42—2442no—
484C42—2442no—
485C42—2442no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×D24412yes—
493C23C22×D24411yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—1222no—
512C2×C4—1222no—
513C2×C4—1222no—
514C2×C4—1222no—
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—2448no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4C4×D24412yes—
537C2×C4#1451980512yes—
538C2×C4C22×C4×D6111yescharacteristic
539C2×C4C2×#1175822411yescharacteristic
540C2×C4C2×#1175822411yescharacteristic
541C2×C4C2×#1175822211yescharacteristic
542C2×C4#1451982411yescharacteristic
543C2×C4#1451982411yescharacteristic
544C8—22no—
545C8—41no—
546C8—41no—
547C8#1451973314yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×C4×D24411yescharacteristic
556C22#1684041611yescharacteristic
557C22#1684041611yescharacteristic
558C22C2×#1451980511yescharacteristic
559C22#1684056211yescharacteristic
560C22#1684057411yescharacteristic
561C22#1684057911yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—4881no—
568C4—4881no—
569C4—4881no—
570C4—4881no—
571C4—2442no—
572C4—2442no—
573C4—2442no—
574C4—2442no—
575C4C2×#1451973311yescharacteristic
576C4#1684023911yescharacteristic
577C4#1684045211yescharacteristic
578C4#1684045211yescharacteristic
579C2—42no—
580C2#2144868611yescharacteristic
581C2C4×#1451980511yescharacteristic
582C2#2145466411yescharacteristic
583C2#2145826111yescharacteristic
584C2#2145833611yescharacteristic
585C2#2145839311yescharacteristic
586C2#2145841511yescharacteristic
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#22624
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. #21446562C2semidirect
5. #21446547C2semidirect
6. #21446412C2semidirect
7. #21446405C2semidirect
8. #21446398C2semidirect
9. #21446391C2semidirect
13. #21445716C2semidirect
14. C4×#14519711C2semidirect
19. #16839804C4semidirect
253. C8×C61#117semidirect
353. C61#22624semidirect
547. C8#14519733semidirect

Maximal subgroups

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

subgroupindex
C2×#168398042
#214487022
#214486862
#214465622
#214465472
#214464122
#214464052
#214463982
#214463912
#214459152
#214459102
#214458522
#214457162
C4×#145197112
C61×#17532
#2262461