Finite Group Databasebeta

← #29571690

Subgroups — #29571690

Order 15616.

844 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×#16840460C211yesmaximal, characteristic
2#21445909C211yesmaximal, characteristic, complemented
3#21445804C211yesmaximal, characteristic, complemented
4#21444388C211yesmaximal, characteristic, complemented
5#21444386C211yesmaximal, characteristic, complemented
6#21444135C211yesmaximal, characteristic, complemented
7C61×#1824C211yesmaximal, characteristic, complemented
8#16840460—22no—
9#16840460—22no—
10#16840460—22no—
11#16840460—22no—
12C2×#14519772—21no—
13C2×#14519768C411yescharacteristic, complemented
14C2×#14519762—21no—
15C2×#14519752C411yescharacteristic, complemented
16C2×#14519751—21no—
17C2×#14519751C2211yescharacteristic
18C2×#14519739C411yescharacteristic, complemented
19C2×C4×C22⋊D61—21no—
20C2×C4×C22⋊D61C2211yescharacteristic
21C2×#14519735C411yescharacteristic, complemented
22C2×#14519691—21no—
23C2×C22⋊#92418C2211yescharacteristic
24#16839622C2211yescharacteristic
25#16839528—21no—
26#16839524—22no—
27#16839497—22no—
28#16839495—22no—
29#16839487—21no—
30#16839485—21no—
31C22×C4⋊C4×C61C2211yescharacteristic
32C2×C22⋊C8×C61C2211yescharacteristic
33C61×#380C2211yescharacteristic
34C61×#340—22no—
35C22×D244—21no—
36C22×D244C2×C411yescharacteristic
37C22×C22⋊D61—21no—
38C22×C22⋊D61D411yescharacteristic
39C22×C4×D61—21no—
40C22×C4×D61C2×C411yescharacteristic
41C22×#92418—21no—
42#14519772—24no—
43#14519772—24no—
44#14519768—22no—
45#14519768—22no—
46#14519762—24no—
47#14519762—24no—
48#14519752—22no—
49#14519752—22no—
50#14519751—24no—
51#14519751—24no—
52#14519751—21no—
53#14519751—21no—
54#14519751D412yes—
55#14519751D412yes—
56#14519739—22no—
57#14519739—22no—
58C4×C22⋊D61—24no—
59C4×C22⋊D61—24no—
60C4×C22⋊D61—21no—
61C4×C22⋊D61—21no—
62C4×C22⋊D61—21no—
63C4×C22⋊D61—21no—
64#14519735—22no—
65#14519735—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2×C411yescharacteristic
69C2×#11758196—21no—
70C2×#11758196D411yescharacteristic
71C2×C22⋊Dic61—21no—
72C2×C22⋊Dic61D411yescharacteristic
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198D411yescharacteristic
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2×C411yescharacteristic
80#14519691—42no—
81C22⋊#92418—21no—
82C22⋊#92418—21no—
83#14519684—21no—
84C23×C4×C61—21no—
85C23×C4×C61C2311yescharacteristic
86C22×C8×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2×C411yescharacteristic
92C2×C4⋊C4×C61C2×C411yescharacteristic
93C2×C4⋊C4×C61D412yes—
94C2×C4⋊C4×C61D412yes—
95C2×C22⋊C4×C61—21no—
96C22⋊C8×C61—21no—
97C22⋊C8×C61—21no—
98#95×C61—21no—
99C24×C61C2×D411yescharacteristic
100C23×D61—21no—
101C23×D61C22⋊C411yescharacteristic
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—42no—
107C22×C4×C61—41no—
108C22×C4×C61—82no—
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C2×C22⋊D61—22no—
116C2×C22⋊D61—22no—
117C2×C22⋊D61—21no—
118C2×C22⋊D61—21no—
119C2×C22⋊D61—21no—
120C2×C22⋊D61—21no—
121C2×C22⋊D61—21no—
122C2×C22⋊D61—21no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C22×Dic61—21no—
128C22×Dic61—41no—
129C22×Dic61C22⋊C411yescharacteristic
130C22×Dic61C22⋊C411yescharacteristic
131C22×Dic61C22⋊C411yescharacteristic
132C2×D244—21no—
133C2×D244—24no—
134C2×D244—42no—
135C2×D244—82no—
136C2×D244C22⋊C412yes—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—24no—
140C2×C4×D61—42no—
141C2×C4×D61—82no—
142C2×C8×C61—42no—
143C2×C8×C61—41no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—42no—
149C4⋊C4×C61C22⋊C412yes—
150C4⋊C4×C61C22⋊C412yes—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C22⋊Dic61—21no—
154C22⋊Dic61—21no—
155C22⋊Dic61—41no—
156C22⋊Dic61—41no—
157#11758198—24no—
158#11758198—24no—
159#11758198—24no—
160#11758198—22no—
161#11758198—22no—
162#11758198—21no—
163#11758198—21no—
164C4⋊Dic61—24no—
165C4⋊Dic61—21no—
166C4⋊Dic61—42no—
167C4⋊Dic61C22⋊C412yes—
168#11758196—24no—
169#11758196—21no—
170#11758196—21no—
171C4×Dic61—24no—
172C4×Dic61—21no—
173C4×Dic61—21no—
174C2×#92418—42no—
175C2×#92418—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61C2×C22⋊C411yescharacteristic
179C23×C61#12011yescharacteristic
180C23×C61#12011yescharacteristic
181C23×C61#13511yescharacteristic
182C23×C61#13511yescharacteristic
183C23×C61AGL(1,ℤ/8)11yescharacteristic
184C23×C61#13711yescharacteristic
185C22×D61—21no—
186C22×D61—22no—
187C22×D61—21no—
188C22×D61—21no—
189C22×D61—42no—
190C22×D61—42no—
191C22×D61—41no—
192C22×D61—41no—
193C22×D61—41no—
194C22×D61—41no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—82no—
206C2×C4×C61—82no—
207C2×C4×C61—81no—
208C2×C4×C61—81no—
209C2×C4×C61—81no—
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61C2×C22⋊C411yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#12011yescharacteristic
214C22⋊D61—41no—
215C22⋊D61—41no—
216C22⋊D61—42no—
217C22⋊D61—42no—
218C22⋊D61—41no—
219C22⋊D61—41no—
220C22⋊D61—42no—
221C22⋊D61—42no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—22no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—81no—
233C2×Dic61—81no—
234C2×Dic61—81no—
235D244—42no—
236D244—42no—
237D244—84no—
238C4×D61—42no—
239C4×D61—42no—
240C4×D61—84no—
241C8×C61—81no—
242#92418—81no—
243#8378—611nomaximal, Sylow
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—41no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61#37911yescharacteristic
255C22×C61#41611yescharacteristic
256C22×C61#41711yescharacteristic
257C22×C61#44911yescharacteristic
258C22×C61#44911yescharacteristic
259C22×C61#45211yescharacteristic
260C22×C61#45211yescharacteristic
261C2×D61—42no—
262C2×D61—41no—
263C2×D61—41no—
264C2×D61—42no—
265C2×D61—41no—
266C2×D61—42no—
267C2×D61—82no—
268C2×D61—82no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—42no—
275C4×C61—42no—
276C4×C61—161no—
277C4×C61#41012yes—
278Dic61—41no—
279Dic61—41no—
280Dic61—41no—
281Dic61—42no—
282Dic61—161no—
283C2×#553—611no—
284#1824—611no—
285#1824D6111yescharacteristic, complemented
286#1799—611no—
287#1519—611no—
288#1519—611no—
289#1459—611no—
290C2×C61—21no—
291C2×C61—21no—
292C2×C61—21no—
293C2×C61—21no—
294C2×C61—21no—
295C2×C61—21no—
296C2×C61#178211yescharacteristic
297C2×C61#178811yescharacteristic
298C2×C61#179511yescharacteristic
299D61—81no—
300D61—82no—
301#553—1222no—
302#553—1222no—
303#553—1222no—
304#553—1222no—
305C2×#126—1221no—
306C2×#125—611no—
307C2×#123—1221no—
308C2×#121—611no—
309C2×#121—611no—
310C2×#121—1221no—
311C2×C4×D4—611no—
312C2×C4×D4—611no—
313C2×C4×D4—1221no—
314C2×#117—611no—
315C22×C4⋊C4C2×D6111yescharacteristic
316C2×#99—1221no—
317C2×C22⋊C8—611no—
318C2×C22⋊C8C2×D6111yescharacteristic
319#380—611no—
320#380C2×D6111yescharacteristic
321#342—1221no—
322#340—22no—
323#340—1222no—
324#327—1222no—
325#327—1222no—
326#324—1221no—
327#323—1221no—
328C61#837811yescharacteristic, complemented, Sylow
329C22×D4—611no—
330C22×D4—611no—
331C22×D4—1221no—
332C22×D4—1221no—
333C23×C4—21no—
334C23×C4—611no—
335C23×C4—1221no—
336C23×C4C22×D6111yescharacteristic
337C22×C8—21no—
338C22×C8—1221no—
339#126—1224no—
340#126—1224no—
341#125—1222no—
342#125—1222no—
343#123—1224no—
344#123—1224no—
345#121—612no—
346#121—612no—
347#121—1222no—
348#121—1224no—
349#121—1224no—
350#121—1221no—
351#121—1221no—
352#121—1222no—
353C4×D4—1222no—
354C4×D4—1224no—
355C4×D4—1224no—
356C4×D4—1221no—
357C4×D4—1221no—
358C4×D4—1221no—
359C4×D4—1221no—
360C4×D4—1222no—
361#117—1222no—
362#117—1222no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—1221no—
370C2×C4⋊C4—1221no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C4×D6111yescharacteristic
373C2×C4⋊C4C4×D6111yescharacteristic
374C2×C4⋊C4D24412yes—
375C2×C4⋊C4C22⋊D6112yes—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—611no—
378C2×C22⋊C4—611no—
379C2×C22⋊C4—1221no—
380C2×C22⋊C4—1221no—
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C42—611no—
385C2×C42—1221no—
386#99—2442no—
387C22⋊C8—21no—
388C22⋊C8—21no—
389C22⋊C8—1221no—
390C22⋊C8—1221no—
391#95—21no—
392#95—1221no—
393C24—611no—
394C24—1221no—
395C24D4×D6111yescharacteristic
396C2×D4—612no—
397C2×D4—1224no—
398C2×D4—1221no—
399C2×D4—1222no—
400C2×D4—1221no—
401C2×D4—1222no—
402C2×D4—1221no—
403C2×D4—1221no—
404C2×D4—1221no—
405C2×D4—1221no—
406C2×D4—1221no—
407C2×D4—4882no—
408C2×D4—2442no—
409C2×D4—2441no—
410C2×D4—2441no—
411C2×D4—2441no—
412C2×D4—2441no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—82no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—1224no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1221no—
427C22×C4—4882no—
428C22×C4—2442no—
429C22×C4—2441no—
430C22×C4C2×C4×D6111yescharacteristic
431C22×C4C2×D24411yescharacteristic
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4C2×C22⋊D6111yescharacteristic
436C2×C8—42no—
437C2×C8—41no—
438C2×C8—2442no—
439C2×C8—2441no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—612no—
446C4⋊C4—1224no—
447C4⋊C4—1224no—
448C4⋊C4—1221no—
449C4⋊C4—1221no—
450C4⋊C4—1221no—
451C4⋊C4—2442no—
452C4⋊C4#1175819812yes—
453C4⋊C4#1175819812yes—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—1224no—
457C22⋊C4—1224no—
458C22⋊C4—1224no—
459C22⋊C4—1221no—
460C22⋊C4—1221no—
461C22⋊C4—1221no—
462C22⋊C4—1222no—
463C22⋊C4—1221no—
464C22⋊C4—1222no—
465C22⋊C4—2441no—
466C22⋊C4—2441no—
467C42—1224no—
468C42—1221no—
469C42—1221no—
470C23—41no—
471C23—41no—
472C23—1221no—
473C23—1221no—
474C23—1222no—
475C23—1221no—
476C23—2441no—
477C23—2441no—
478C23—2442no—
479C23—2442no—
480C23—2441no—
481C23—2441no—
482C23C22⋊C4×D6111yescharacteristic
483C23#1451974511yescharacteristic
484C23#1451974711yescharacteristic
485C23#1451979811yescharacteristic
486C23#1451980011yescharacteristic
487C23#1451980411yescharacteristic
488C23#1451980911yescharacteristic
489D4—4884no—
490D4—2442no—
491D4—2441no—
492D4—2442no—
493D4—2441no—
494D4—2442no—
495D4—2441no—
496D4—2442no—
497D4—2441no—
498D4—2442no—
499D4—2442no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—1221no—
516C2×C4—1221no—
517C2×C4—1221no—
518C2×C4—1222no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—4884no—
522C2×C4—4881no—
523C2×C4—4881no—
524C2×C4—4881no—
525C2×C4—2442no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2442no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4C22⋊C4×D6111yescharacteristic
532C2×C4C2×#1175819811yescharacteristic
533C2×C4#1451974511yescharacteristic
534C2×C4#1451974711yescharacteristic
535C8—81no—
536C8—4881no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—4881no—
548C22—4881no—
549C22—4882no—
550C22—4882no—
551C22—2441no—
552C22—2442no—
553C22—2441no—
554C22—2442no—
555C22—2442no—
556C22—2441no—
557C22#1683961711yescharacteristic
558C22#1683975711yescharacteristic
559C22#1683976211yescharacteristic
560C22#1683990611yescharacteristic
561C22#1683991011yescharacteristic
562C22#1683992311yescharacteristic
563C22#1683992711yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—42no—
568C4—42no—
569C4—161no—
570C4—9761no—
571C4—2441no—
572C4—2441no—
573C4—2442no—
574C4—2441no—
575C4#1683973512yes—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2—4882no—
583C2—4881no—
584C2#2144573611yescharacteristic
585C2#2144575811yescharacteristic
586C2#2144578611yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1824C2
socleC22×C61#379
prime radicalC2×#16840460C2
cosocleC23×C4×C61C23
hypercenter#1824D61
hypocenterC61#8378
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
2. #21445909C2semidirect
3. #21445804C2semidirect
4. #21444388C2semidirect
5. #21444386C2semidirect
6. #21444135C2semidirect
7. C61×#1824C2semidirect
13. C2×#14519768C4semidirect
15. C2×#14519752C4semidirect
18. C2×#14519739C4semidirect
21. C2×#14519735C4semidirect
285. #1824D61semidirect
328. C61#8378semidirect

Maximal subgroups

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

subgroupindex
C2×#168404602
#214459092
#214458042
#214443882
#214443862
#214441352
C61×#18242
#837861