Finite Group Databasebeta

← #29689874

Subgroups — #29689874

Order 15616.

1082 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840365C211yesmaximal, characteristic
2#21454454C211yesmaximal, characteristic, complemented
3#21454382C211yesmaximal, characteristic, complemented
4#21446810C212yesmaximal, complemented
5#21446657C212yesmaximal, complemented
6#21446283C211yesmaximal, characteristic, complemented
7#21446282C211yesmaximal, characteristic, complemented
8#21446280C211yesmaximal, characteristic, complemented
9#21445810C212yesmaximal, complemented
10#21445809C212yesmaximal, complemented
11C61×#1900C211yesmaximal, characteristic, complemented
12C22×#11758194C2211yescharacteristic
13#16840365—22no—
14#16840365—22no—
15#16840365—22no—
16#16840365—22no—
17C2×#14519773C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519761C412yes—
20C2×#14519759C412yes—
21C2×#14519751C2211yescharacteristic
22C2×#14519749C412yes—
23C2×#14519748C412yes—
24C2×D4×Dic61C2211yescharacteristic
25C2×#14519739C2211yescharacteristic
26C2×#14519736C2211yescharacteristic
27#16839820—22no—
28#16839820—22no—
29#16839790—22no—
30#16839788—22no—
31#16839785C2212yes—
32#16839785C2212yes—
33#16839785C2212yes—
34C2×#14519694C2212yes—
35C2×#14519693C2212yes—
36C2×#14519693C2212yes—
37#16839726C2211yescharacteristic
38M16×Dic61C2212yes—
39#16839531C2212yes—
40#16839530C2212yes—
41#16839530C2212yes—
42C22×M16×C61C2211yescharacteristic
43C2×#121×C61C2211yescharacteristic
44C2×#100×C61C2212yes—
45C61×#407C2211yescharacteristic
46C61×#343C2212yes—
47C22×D244C2311yescharacteristic
48C22×C22⋊D61C2×C412yes—
49C22×C22⋊D61C2×C412yes—
50C22×C22⋊D61C2×C412yes—
51C23×Dic61C2×C412yes—
52C22×C4×D61C2×C412yes—
53C2×#11758194—21no—
54C2×#11758194—42no—
55C2×#11758194C2312yes—
56C2×#11758194C2312yes—
57C22×#92418—21no—
58#14519773—22no—
59#14519773—22no—
60#14519768—22no—
61#14519768—22no—
62#14519761—24no—
63#14519761—24no—
64#14519759—24no—
65#14519759—24no—
66#14519751—21no—
67#14519751—21no—
68#14519751—21no—
69#14519751—21no—
70#14519749—24no—
71#14519749—24no—
72#14519748—24no—
73#14519748—24no—
74D4×Dic61—22no—
75D4×Dic61—22no—
76#14519739—22no—
77#14519739—22no—
78#14519736D412yes—
79#14519736D412yes—
80#14519736D412yes—
81#14519736D412yes—
82C2×C4⋊Dic61C2×C412yes—
83C2×C4⋊Dic61C2×C411yescharacteristic
84C2×C4⋊Dic61C2×C411yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196C2×C412yes—
87C2×C22⋊Dic61C2×C412yes—
88C2×C22⋊Dic61C2×C411yescharacteristic
89C2×C22⋊Dic61C2×C411yescharacteristic
90C2×#11758198C2×C412yes—
91C2×#11758198C2×C412yes—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519706—24no—
95#14519706—24no—
96#14519706—24no—
97#14519694—42no—
98#14519693—42no—
99#14519693—42no—
100C22⋊#92418—21no—
101C22⋊#92418—21no—
102#14519687—22no—
103C8×Dic61—22no—
104C22×D4×C61C2×C412yes—
105C22×D4×C61C2311yescharacteristic
106C23×C4×C61C2311yescharacteristic
107C2×M16×C61—21no—
108C2×M16×C61—42no—
109C2×M16×C61C2312yes—
110C2×M16×C61C2312yes—
111C22×C8×C61—21no—
112#121×C61—21no—
113#121×C61—21no—
114#121×C61—21no—
115#121×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61C2×C412yes—
118#100×C61—42no—
119C22⋊C8×C61—21no—
120C22⋊C8×C61—21no—
121C24×C61C22×C412yes—
122C24×C61C22×C411yescharacteristic
123C23×D61C22×C412yes—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—82no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C412yes—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—22no—
141C2×C22⋊D61—22no—
142C2×C22⋊D61—24no—
143C2×C22⋊D61—24no—
144C2×C22⋊D61—82no—
145C2×C22⋊D61—82no—
146C2×C22⋊D61—82no—
147C22×Dic61—24no—
148C22×Dic61—82no—
149C22×Dic61C22×C412yes—
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—22no—
157C2×D244—22no—
158C2×C4×D61—24no—
159C2×C4×D61—82no—
160M16×C61—22no—
161M16×C61—22no—
162M16×C61—22no—
163M16×C61—22no—
164M16×C61—42no—
165C2×C8×C61—22no—
166C2×C8×C61—22no—
167C2×C8×C61—42no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊Dic61—24no—
173C22⋊Dic61C22⋊C412yes—
174C22⋊Dic61C22⋊C412yes—
175C22⋊Dic61C22⋊C412yes—
176C22⋊Dic61C22⋊C412yes—
177#11758198—22no—
178#11758198—22no—
179#11758198—24no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—24no—
183C4⋊Dic61C22⋊C412yes—
184C4⋊Dic61C22⋊C412yes—
185C4⋊Dic61C22⋊C412yes—
186C4⋊Dic61C22⋊C412yes—
187#11758196—24no—
188C4×Dic61C2×D412yes—
189C4×Dic61C2×D412yes—
190C4×Dic61C2×D412yes—
191C4×Dic61C2×D412yes—
192#11758194—22no—
193#11758194—22no—
194#11758194—22no—
195#11758194—22no—
196#11758194—42no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—42no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—24no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—42no—
206C23×C61—42no—
207C23×C61C2×C22⋊C411yescharacteristic
208C23×C61C2×C22⋊C411yescharacteristic
209C23×C61C23×C411yescharacteristic
210C22×D61—22no—
211C22×D61—24no—
212C22×D61—42no—
213C22×D61—42no—
214D4×C61—41no—
215D4×C61—41no—
216D4×C61—42no—
217D4×C61—82no—
218D4×C61—82no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—42no—
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)12yes—
233C2×C4×C612−1+412yes—
234C22⋊D61—84no—
235C22⋊D61—84no—
236C22⋊D61—82no—
237C22⋊D61—82no—
238C2×Dic61—22no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—84no—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×C22⋊C412yes—
249D244—41no—
250D244—41no—
251D244—42no—
252C4×D61—84no—
253C8×C61—42no—
254C8×C61—42no—
255#92418—42no—
256#92418—42no—
257#21882—611nomaximal, Sylow
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—82no—
270C22×C61—84no—
271C22×C61C22×C22⋊C411yescharacteristic
272C22×C61#52012yes—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—84no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—42no—
280C4×C61#53511yescharacteristic
281C4×C61#53611yescharacteristic
282C4×C61#53712yes—
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—22no—
287Dic61—42no—
288C2×#540—611no—
289#2921—611no—
290#2905—611no—
291#1998—612no—
292#1965—612no—
293#1900—611no—
294#1900—611no—
295#1900—611no—
296#1900D6111yescharacteristic, complemented
297#1800—612no—
298#1800—612no—
299C2×C61—22no—
300C2×C61—41no—
301C2×C61—41no—
302C2×C61—82no—
303C2×C61#231111yescharacteristic
304C2×C61#288811yescharacteristic
305C2×C61#288811yescharacteristic
306D61—82no—
307C22×M16—611no—
308C22×M16C2×D6111yescharacteristic
309#540—1222no—
310#540—1222no—
311#540—1222no—
312#540—1222no—
313C2×#127—611no—
314C2×#125—611no—
315C2×#123—612no—
316C2×#123—612no—
317C2×#121—612no—
318C2×#121—612no—
319C2×#121—611no—
320C2×#121C2×D6111yescharacteristic
321C2×C4×D4—611no—
322C2×C4×D4—611no—
323C2×#117—611no—
324#433—1222no—
325#433—1222no—
326#424—1222no—
327#424—1222no—
328#423—612no—
329#423—612no—
330#423—612no—
331C2×#100—612no—
332C2×#100—612no—
333C2×#100—612no—
334C2×#100C2×D6112yes—
335#407—611no—
336#407C2×D6111yescharacteristic
337C4×M16—612no—
338#343—612no—
339#343—612no—
340#343—612no—
341#343C2×D6112yes—
342C61#2188211yescharacteristic, complemented, Sylow
343C22×D4—612no—
344C22×D4—612no—
345C22×D4—612no—
346C22×D4—611no—
347C22×D4C4×D6112yes—
348C22×D4C22×D6111yescharacteristic
349C23×C4—612no—
350C23×C4—612no—
351C23×C4C22×D6111yescharacteristic
352C2×M16—21no—
353C2×M16—42no—
354C2×M16—612no—
355C2×M16—612no—
356C2×M16—1221no—
357C2×M16—2442no—
358C2×M16C22×D6112yes—
359C2×M16C22×D6112yes—
360C22×C8—21no—
361C22×C8—1221no—
362#127—1222no—
363#127—1222no—
364#125—1222no—
365#125—1222no—
366#123—1224no—
367#123—1224no—
368#123—1224no—
369#123—1224no—
370#121—21no—
371#121—21no—
372#121—21no—
373#121—21no—
374#121—1224no—
375#121—1224no—
376#121—1224no—
377#121—1224no—
378#121—1221no—
379#121—1221no—
380#121—1221no—
381#121—1221no—
382C4×D4—1222no—
383C4×D4—1222no—
384C4×D4—1222no—
385C4×D4—1222no—
386#117—612no—
387#117—612no—
388#117—612no—
389#117—612no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C22⋊C4—611no—
397C2×C22⋊C4—611no—
398C2×C22⋊C4—612no—
399C2×C22⋊C4—612no—
400C2×C22⋊C4—612no—
401C2×C22⋊C4C4×D6112yes—
402C2×C42—611no—
403C2×C42—611no—
404C4⋊C8—1224no—
405C4⋊C8—1224no—
406C4⋊C8—1224no—
407#100—42no—
408#100—2442no—
409#100—2442no—
410#100—2442no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415#97—1222no—
416C4×C8—1222no—
417C24—612no—
418C24C2×C4×D6112yes—
419C24C2×C4×D6111yescharacteristic
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—82no—
427C2×D4—1224no—
428C2×D4—1224no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1222no—
434C2×D4—1222no—
435C2×D4—4882no—
436C2×D4—4882no—
437C2×D4—4882no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—612no—
445C22×C4—1224no—
446C22×C4—1224no—
447C22×C4—4882no—
448C22×C4—4882no—
449C22×C4C2×C4×D6111yescharacteristic
450C22×C4C2×C4×D6112yes—
451C22×C4D4×D6111yescharacteristic
452C22×C4D4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455C22×C4C23×D6111yescharacteristic
456M16—22no—
457M16—22no—
458M16—22no—
459M16—22no—
460M16—42no—
461M16—1222no—
462M16—1222no—
463M16—1222no—
464M16—1222no—
465M16—2442no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—42no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—2442no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—612no—
475C4⋊C4—612no—
476C4⋊C4—612no—
477C4⋊C4—612no—
478C4⋊C4—1224no—
479C4⋊C4—1224no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—612no—
485C22⋊C4—612no—
486C22⋊C4—612no—
487C22⋊C4—612no—
488C22⋊C4—1224no—
489C22⋊C4—1224no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C42—612no—
493C42—612no—
494C42—612no—
495C42—612no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—1224no—
504C23—1222no—
505C23—2442no—
506C23—2442no—
507C23C22⋊C4×D6111yescharacteristic
508C23C22⋊C4×D6111yescharacteristic
509C23C22×C4×D6111yescharacteristic
510D4—41no—
511D4—41no—
512D4—42no—
513D4—82no—
514D4—82no—
515D4—4884no—
516D4—4884no—
517D4—4882no—
518D4—4882no—
519D4—2441no—
520D4—2441no—
521D4—2442no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—42no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—1222no—
536C2×C4—4884no—
537C2×C4—4884no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2442no—
543C2×C4C22⋊C4×D6111yescharacteristic
544C2×C4C22⋊C4×D6111yescharacteristic
545C2×C4C2×D4×D6111yescharacteristic
546C2×C4C2×D4×D6111yescharacteristic
547C2×C4#1451982212yes—
548C2×C4#1451982412yes—
549C8—42no—
550C8—42no—
551C8—2442no—
552C8—2442no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—82no—
565C22—84no—
566C22—4884no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22C2×C22⋊C4×D6111yescharacteristic
571C22#1684024112yes—
572C4—21no—
573C4—21no—
574C4—42no—
575C4—1222no—
576C4—1222no—
577C4—1222no—
578C4—1222no—
579C4—2442no—
580C4#1684032711yescharacteristic
581C4#1684033411yescharacteristic
582C4#1684033912yes—
583C2—22no—
584C2—41no—
585C2—41no—
586C2—82no—
587C2—4882no—
588C2#2144840611yescharacteristic
589C2#2145429511yescharacteristic
590C2#2145429611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1900C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840365C2
cosocleC22×C4×C61C24
hypercenter#1900D61
hypocenterC61#21882
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1900

Complements

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

NH ≅ G/Ntype
2. #21454454C2semidirect
3. #21454382C2semidirect
4. #21446810C2semidirect
5. #21446657C2semidirect
6. #21446283C2semidirect
7. #21446282C2semidirect
8. #21446280C2semidirect
9. #21445810C2semidirect
10. #21445809C2semidirect
11. C61×#1900C2semidirect
296. #1900D61semidirect
342. C61#21882semidirect

Maximal subgroups

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

subgroupindex
C2×#168403652
#214544542
#214543822
#214468102
#214466572
#214462832
#214462822
#214462802
#214458102
#214458092
C61×#19002
#2188261