Finite Group Databasebeta

← #29697762

Subgroups — #29697762

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×#16839801C211yesmaximal, characteristic
2#21448708C211yesmaximal, characteristic
3#21448684C211yesmaximal, characteristic
4#21446557C211yesmaximal, characteristic, complemented
5#21446550C211yesmaximal, characteristic, complemented
6#21446411C211yesmaximal, characteristic, complemented
7#21446404C211yesmaximal, characteristic, complemented
8#21446395C211yesmaximal, characteristic, complemented
9#21446388C211yesmaximal, characteristic, complemented
10#21445912C211yesmaximal, characteristic
11#21445908C211yesmaximal, characteristic
12#21445853C211yesmaximal, characteristic
13#106×Dic61C211yesmaximal, characteristic, complemented
14#21445604C211yesmaximal, characteristic
15C61×#1776C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C2×#14519735C2211yescharacteristic
18C2×#14519734—21no—
19C2×C4⋊C4×D61C2211yescharacteristic
20#16839801C418yescomplemented
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839660C2211yescharacteristic
28#16839660C2211yescharacteristic
29#16839658C2211yescharacteristic
30#16839658C2211yescharacteristic
31#16839653—21no—
32#16839651—21no—
33#16839646—21no—
34#16839644—21no—
35#16839629—21no—
36#16839624—21no—
37#16839619C2211yescharacteristic
38#16839619C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196—21no—
42C4×#11758198C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61C2311yescharacteristic
62C2×#11758189—22no—
63C2×#11758189—22no—
64C2×#11758189C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519735—22no—
68#14519735C2×C414yes—
69#14519734—44no—
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×#11758196—21no—
79C4⋊C4×D61—22no—
80C4⋊C4×D61C2×C414yes—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198D411yescharacteristic
85C2×#11758198D411yescharacteristic
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519711C2×C414yes—
91#14519710C2×C414yes—
92#14519709—22no—
93#14519708C2×C414yes—
94#14519689—22no—
95C8×Dic61—22no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C42×C61—21no—
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#107×C61—22no—
116#106×C61C2×C414yes—
117#97×C61—22no—
118#95×C61—21no—
119#95×C61—21no—
120C23×D61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—24no—
140C2×C4×D61—42no—
141C2×C4×D61—42no—
142C2×C4×D61C22×C412yes—
143C2×C4×D61#4112yes—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61C22×C412yes—
148C2×C8×C61#4112yes—
149C2×C8×C61#4112yes—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61C22×C412yes—
157C4⋊C4×C61C22×C412yes—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61—42no—
161C42×C61—81no—
162#11758198—24no—
163#11758198—22no—
164#11758198—41no—
165#11758198—42no—
166#11758198—81no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61C22×C412yes—
174C4⋊Dic61C22×C412yes—
175#11758196—24no—
176#11758196—42no—
177#11758196—42no—
178#11758196—42no—
179#11758196—42no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—24no—
183C4×Dic61—42no—
184C4×Dic61—81no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418C22×C412yes—
191#11758189—28no—
192C23×C61C22×D411yescharacteristic
193C22×D61—21no—
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61C4×D412yes—
200C2×C4×C61—22no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
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—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
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×Dic61C4×D412yes—
244C2×Dic61AGL(1,ℤ/8)12yes—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—48no—
250C8×C61—41no—
251C8×C61#11714yescomplemented
252#92418—22no—
253#92418—41no—
254#22622—611nomaximal, Sylow
255C22×C61C2×C4×D411yescharacteristic
256C22×C61#54711yescharacteristic
257C22×C61#54711yescharacteristic
258C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C61#57511yescharacteristic
260C22×C61#57711yescharacteristic
261C22×C61#57811yescharacteristic
262C2×D61—22no—
263C2×D61—22no—
264C2×D61—22no—
265C2×D61—41no—
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61C2×#11711yescharacteristic
276C4×C61#51911yescharacteristic
277C4×C61#55211yescharacteristic
278C4×C61#55211yescharacteristic
279Dic61—22no—
280Dic61—22no—
281Dic61—41no—
282Dic61—42no—
283Dic61—42no—
284Dic61—81no—
285Dic61—81no—
286C2×#428—611no—
287#2347—611no—
288#2344—611no—
289#1943—611no—
290#1942—611no—
291#1921—611no—
292#1920—611no—
293#1919—611no—
294#1918—611no—
295#1825—611no—
296#1824—611no—
297#1809—611no—
298#1776D6111yescharacteristic, complemented
299C4×#106—611no—
300#1753—611no—
301C2×C61#234411yescharacteristic
302C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C61#294711yescharacteristic
304C2×C61#334511yescharacteristic
305C2×C61#334911yescharacteristic
306C2×C61#335111yescharacteristic
307C2×C61#335311yescharacteristic
308D61—42no—
309C22×M16—611no—
310C2×#117—611no—
311C2×#117—1221no—
312C22×C4⋊C4—611no—
313#428—618no—
314C2×#107—611no—
315C2×#107—611no—
316C2×#107C2×D6111yescharacteristic
317C2×#106—611no—
318C2×#106—611no—
319C2×#106C2×D6111yescharacteristic
320C2×C22⋊C8—611no—
321C2×#97C2×D6111yescharacteristic
322C2×C4×C8—611no—
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#383—1221no—
332#381—1221no—
333#380—611no—
334#380—611no—
335C4×C4⋊C4—611no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4—1221no—
338C4×C4⋊C4C2×D6111yescharacteristic
339C4×C4⋊C4C2×D6111yescharacteristic
340C4×C22⋊C4—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#340C2×D6111yescharacteristic
350#340C2×D6111yescharacteristic
351#336—611no—
352#336—611no—
353C61#2262211yescharacteristic, complemented, Sylow
354C23×C4—611no—
355C23×C4—1221no—
356C2×M16—614no—
357C2×M16—1222no—
358C2×M16—1222no—
359C22×C8—611no—
360C22×C8—611no—
361C22×C8C22×D6111yescharacteristic
362C22×C8C22×D6111yescharacteristic
363#117—614no—
364#117—1222no—
365#117—2444no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—614no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—611no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—22no—
394#107—614no—
395#107—614no—
396#106—614no—
397#106—1222no—
398#106C4×D6114yes—
399C22⋊C8—1222no—
400#97—22no—
401C4×C8—1222no—
402#95—21no—
403#95—21no—
404#95—611no—
405#95—611no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414C24—611no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—611no—
424C22×C4—1224no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—2442no—
434C22×C4—2442no—
435C22×C4D4×D6111yescharacteristic
436C22×C4D4×D6111yescharacteristic
437C22×C4C23×D6111yescharacteristic
438M16—1228no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—612no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—1221no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8C2×C4×D6112yes—
449C2×C8#1175822212yes—
450C2×C8#1175822212yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—612no—
458C4⋊C4—612no—
459C4⋊C4—1224no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—2442no—
467C4⋊C4—2442no—
468C4⋊C4—2442no—
469C4⋊C4—2442no—
470C4⋊C4C2×C4×D6112yes—
471C4⋊C4C2×C4×D6112yes—
472C22⋊C4—1222no—
473C22⋊C4—1224no—
474C22⋊C4—4881no—
475C22⋊C4—2441no—
476C22⋊C4—2442no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—81no—
481C42—1224no—
482C42—1221no—
483C42—1221no—
484C42—4881no—
485C42—2442no—
486C23—612no—
487C23—1221no—
488C23—1221no—
489C23—1221no—
490C23—1221no—
491C23—1221no—
492C23—1221no—
493C23C2×D4×D6111yescharacteristic
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—1221no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
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×C4—2441no—
537C2×C4—2441no—
538C2×C4C22×C4×D6111yescharacteristic
539C2×C4#41×D6111yescharacteristic
540C2×C4C2×#1175822211yescharacteristic
541C2×C4C2×#1175822211yescharacteristic
542C2×C4#1451982411yescharacteristic
543C2×C4#1451982511yescharacteristic
544C8—41no—
545C8—1222no—
546C8—2441no—
547C8#1451973414yescomplemented
548C22—1222no—
549C22—1222no—
550C22—1222no—
551C22—2441no—
552C22—2441no—
553C22—2441no—
554C22—2441no—
555C22C4×D4×D6111yescharacteristic
556C22#1684041311yescharacteristic
557C22#1684041711yescharacteristic
558C22AGL(1,ℤ/8)×D6111yescharacteristic
559C22#1684056111yescharacteristic
560C22#1684057211yescharacteristic
561C22#1684057811yescharacteristic
562C4—42no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—1222no—
569C4—1222no—
570C4—4881no—
571C4—4881no—
572C4—2442no—
573C4—2442no—
574C4—2441no—
575C4C2×#1451973411yescharacteristic
576C4#1684023811yescharacteristic
577C4#1684045111yescharacteristic
578C4#1684045311yescharacteristic
579C2—2442no—
580C2#2144868411yescharacteristic
581C2#2145462411yescharacteristic
582C2#2145465911yescharacteristic
583C2#2145829411yescharacteristic
584C2#2145835411yescharacteristic
585C2#2145838011yescharacteristic
586C2#2145839911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1776C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1776D61
hypocenterC61#22622
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #21446557C2semidirect
5. #21446550C2semidirect
6. #21446411C2semidirect
7. #21446404C2semidirect
8. #21446395C2semidirect
9. #21446388C2semidirect
13. #106×Dic61C2semidirect
15. C61×#1776C2semidirect
20. #16839801C4semidirect
251. C8×C61#117semidirect
298. #1776D61semidirect
353. C61#22622semidirect
547. C8#14519734semidirect

Maximal subgroups

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

subgroupindex
C2×#168398012
#214487082
#214486842
#214465572
#214465502
#214464112
#214464042
#214463952
#214463882
#214459122
#214459082
#214458532
#106×Dic612
#214456042
C61×#17762
#2262261