Finite Group Databasebeta

← #29697749

Subgroups — #29697749

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#21448706C211yesmaximal, characteristic
3#21448698C211yesmaximal, characteristic
4#21446555C211yesmaximal, characteristic, complemented
5#21446550C211yesmaximal, characteristic, complemented
6#21446410C211yesmaximal, characteristic, complemented
7#21446403C211yesmaximal, characteristic, complemented
8#21446396C211yesmaximal, characteristic, complemented
9#21446389C211yesmaximal, characteristic, complemented
10#21445911C211yesmaximal, characteristic
11#21445903C211yesmaximal, characteristic
12#21445718C211yesmaximal, characteristic, complemented
13#106×Dic61C211yesmaximal, characteristic, complemented
14#21445600C211yesmaximal, characteristic
15C61×#1809C211yesmaximal, characteristic
16C2×#14519736C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18#16839804C418yescomplemented
19C2×#14519711C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24C2×#14519687C2211yescharacteristic
25C2×C8×Dic61C2211yescharacteristic
26#16839658C2211yescharacteristic
27#16839658C2211yescharacteristic
28#16839658C2211yescharacteristic
29#16839658C2211yescharacteristic
30#16839656—21no—
31#16839656—21no—
32#16839647—21no—
33#16839647—21no—
34#16839626—21no—
35#16839620—21no—
36C4⋊C4×Dic61C2211yescharacteristic
37C4⋊C4×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C22⋊C4×Dic61—21no—
41C4×C22⋊Dic61C2211yescharacteristic
42#16839607—21no—
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C61×#380C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519736—22no—
64#14519736C2×C414yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61C2×C414yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61D411yescharacteristic
78C2×C22⋊Dic61D411yescharacteristic
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711C2×C414yes—
87#14519711C2×C414yes—
88#14519709C2×C414yes—
89#14519709C2×C414yes—
90C22⋊#92418—22no—
91#14519687—22no—
92C8×Dic61—22no—
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#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—22no—
105C2×M16×C61—22no—
106C2×M16×C61C2×C414yes—
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114C2×C22⋊C4×C61—21no—
115#107×C61—22no—
116#106×C61—22no—
117#95×C61—21no—
118#95×C61—21no—
119C24×C61C2×D411yescharacteristic
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61C22×C412yes—
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—81no—
138C22×Dic61—81no—
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143M16×C61—28no—
144C2×C8×C61—21no—
145C2×C8×C61—21no—
146C2×C8×C61C22×C412yes—
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×C61—21no—
157C4⋊C4×C61—21no—
158C22⋊C4×C61—42no—
159C22⋊C4×C61—81no—
160C22⋊Dic61—24no—
161C22⋊Dic61—22no—
162C22⋊Dic61—41no—
163C4⋊Dic61—24no—
164C4⋊Dic61—21no—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61C22×C412yes—
169C4⋊Dic61C22×C412yes—
170C4⋊Dic61C22×C412yes—
171C4⋊Dic61C22×C412yes—
172#11758196—81no—
173#11758196—81no—
174#11758196—81no—
175#11758196—81no—
176C4×Dic61—24no—
177C4×Dic61—21no—
178C4×Dic61—21no—
179C4×Dic61—21no—
180C4×Dic61—21no—
181C4×Dic61—42no—
182C4×Dic61—42no—
183C4×Dic61—81no—
184C4×Dic61—81no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
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—22no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
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×C61C4×D412yes—
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—22no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
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—84no—
248C2×Dic61#13712yes—
249C8×C61—22no—
250C8×C61#11714yes—
251#92418—41no—
252#92418—41no—
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—81no—
270C4×C61—81no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61C2×#11711yescharacteristic
274C4×C61#51911yescharacteristic
275C4×C61#55211yescharacteristic
276C4×C61#55211yescharacteristic
277Dic61—22no—
278Dic61—41no—
279Dic61—41no—
280Dic61—41no—
281Dic61—42no—
282Dic61—42no—
283Dic61—42no—
284Dic61—42no—
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#1809D6111yescharacteristic
297#1776—611no—
298C4×#106—611no—
299#1753—611no—
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×#107C2×D6111yescharacteristic
315C2×#106—611no—
316C2×#106—611no—
317C2×#106C2×D6111yescharacteristic
318C2×C22⋊C8—611no—
319C2×#97—611no—
320C2×C4×C8—611no—
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—1221no—
331#380C2×D6111yescharacteristic
332#380C2×D6111yescharacteristic
333C4×C4⋊C4—611no—
334C4×C4⋊C4—611no—
335C4×C4⋊C4—611no—
336C4×C4⋊C4—611no—
337C4×C22⋊C4—611no—
338C4×C22⋊C4—1221no—
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#340C2×D6111yescharacteristic
349#340C2×D6111yescharacteristic
350#336—611no—
351#336—611no—
352C61#2262111yescharacteristic, complemented, Sylow
353C23×C4—1221no—
354C23×C4C22×D6111yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D6114yes—
358C22×C8—611no—
359C22×C8—611no—
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—1221no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4—1221no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4C22×D6111yescharacteristic
376C2×C4⋊C4C22×D6111yescharacteristic
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—1221no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42—1221no—
390C2×C42—1221no—
391#107—22no—
392#107—614no—
393#107—614no—
394#106—22no—
395#106—614no—
396#106—614no—
397C22⋊C8—1222no—
398#97—1222no—
399C4×C8—1222no—
400#95—21no—
401#95—21no—
402#95—611no—
403#95—611no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412C24D4×D6111yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—1224no—
426C22×C4—1221no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—4881no—
431C22×C4—4881no—
432C22×C4C2×C4×D6112yes—
433C22×C4D4×D6111yescharacteristic
434C22×C4#1175822612yes—
435C22×C4C23×D6111yescharacteristic
436M16—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—1221no—
440C2×C8—1221no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8C2×C4×D6112yes—
446C2×C8C2×C4×D6112yes—
447C2×C8#1175822412yes—
448C2×C8#1175822412yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
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—612no—
460C4⋊C4—612no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1224no—
466C4⋊C4—4881no—
467C4⋊C4—4881no—
468C4⋊C4—4881no—
469C4⋊C4—4881no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C22⋊C4—1222no—
473C22⋊C4—1224no—
474C22⋊C4—2441no—
475C42—1221no—
476C42—1221no—
477C42—1221no—
478C42—1221no—
479C42—1224no—
480C42—4881no—
481C42—4881no—
482C42—2442no—
483C42—2442no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23#1451973912yes—
491C23C2×D4×D6111yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
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—612no—
511C2×C4—1221no—
512C2×C4—1221no—
513C2×C4—1222no—
514C2×C4—1222no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—4884no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4#1451973912yes—
536C2×C4C22×C4×D6111yescharacteristic
537C2×C4C2×#1175822411yescharacteristic
538C2×C4C2×#1175822411yescharacteristic
539C2×C4C2×#1175822611yescharacteristic
540C2×C4#1451982411yescharacteristic
541C2×C4#1451982411yescharacteristic
542C8—22no—
543C8—2441no—
544C8—2441no—
545C8#1451973314yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×#1451973911yescharacteristic
554C22#1684041211yescharacteristic
555C22#1684041211yescharacteristic
556C22#137×D6111yescharacteristic
557C22#1684055811yescharacteristic
558C22#1684056611yescharacteristic
559C22#1684057111yescharacteristic
560C4—22no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—1222no—
566C4—2441no—
567C4—2442no—
568C4—2442no—
569C4—2442no—
570C4—2442no—
571C4—2441no—
572C4—2441no—
573C4C2×#1451973311yescharacteristic
574C4#1684023911yescharacteristic
575C4#1684045411yescharacteristic
576C4#1684045411yescharacteristic
577C2—42no—
578C2#2144869011yescharacteristic
579C2#2145463711yescharacteristic
580C2#2145464911yescharacteristic
581C2#2145826911yescharacteristic
582C2#2145831411yescharacteristic
583C2#2145837411yescharacteristic
584C2#2145844011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1809C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1809D61
hypocenterC61#22621
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #21446555C2semidirect
5. #21446550C2semidirect
6. #21446410C2semidirect
7. #21446403C2semidirect
8. #21446396C2semidirect
9. #21446389C2semidirect
12. #21445718C2semidirect
13. #106×Dic61C2semidirect
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
#214487062
#214486982
#214465552
#214465502
#214464102
#214464032
#214463962
#214463892
#214459112
#214459032
#214457182
#106×Dic612
#214456002
C61×#18092
#2262161