Finite Group Databasebeta

← #29738824

Subgroups — #29738824

Order 15616.

774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840162C211yesmaximal, characteristic, complemented
2C2×#16839802C211yesmaximal, characteristic
3#21449537C211yesmaximal, characteristic
4#21449533C211yesmaximal, characteristic
5#21447305C211yesmaximal, characteristic, complemented
6#21447298C211yesmaximal, characteristic, complemented
7#21447160C211yesmaximal, characteristic, complemented
8#21447159C211yesmaximal, characteristic, complemented
9#21446556C211yesmaximal, characteristic, complemented
10#21446546C211yesmaximal, characteristic, complemented
11#21446020C211yesmaximal, characteristic
12#21445984C211yesmaximal, characteristic, complemented
13#21445850C211yesmaximal, characteristic
14#21445605C211yesmaximal, characteristic
15C61×#1853C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519768C2211yescharacteristic
19C2×#14519736C2211yescharacteristic
20#16840162—24no—
21#16839802—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839701—21no—
33#16839698—21no—
34#16839695—21no—
35#16839685—21no—
36#16839661C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839659—21no—
39#16839658C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839649—21no—
42#16839635—21no—
43#16839631C2211yescharacteristic
44#16839627C2211yescharacteristic
45#16839622—21no—
46#16839622C2211yescharacteristic
47C4×C22⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×C4⋊C4×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C61×#380C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194Q814yes—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519768—22no—
67#14519768—22no—
68#14519768—41no—
69#14519768—41no—
70#14519736—22no—
71#14519736—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61D411yescharacteristic
89C2×C22⋊Dic61D411yescharacteristic
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93#14519711—41no—
94#14519710—22no—
95#14519710—22no—
96#14519710D414yes—
97#14519709—22no—
98#14519708—22no—
99#14519708—22no—
100#14519708—22no—
101#14519687—22no—
102C8×Dic61—22no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684D411yescharacteristic
111#14519684D411yescharacteristic
112C23×C4×C61—21no—
113C23×C4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C4⋊C4×C61—22no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—41no—
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124#106×C61—22no—
125#106×C61—41no—
126C22⋊C8×C61Q814yes—
127#95×C61—21no—
128C24×C61C2×D411yescharacteristic
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×Q812yes—
142C22×C4×C61C2×Q812yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61C2×Q812yes—
157C2×C8×C61C2×Q812yes—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61C2×D412yes—
167C22⋊C4×C61—81no—
168C22⋊Dic61—41no—
169C22⋊Dic61—41no—
170C22⋊Dic61—42no—
171C22⋊Dic61—42no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—42no—
180C4⋊Dic61—42no—
181C4⋊Dic61—81no—
182C4⋊Dic61—81no—
183C4⋊Dic61C2×D412yes—
184#11758196—42no—
185#11758196—42no—
186#11758196—42no—
187#11758196—42no—
188#11758196—81no—
189#11758196—81no—
190C4×Dic61—21no—
191C4×Dic61—21no—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—42no—
195#11758194—28no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418C2×D412yes—
199C2×#92418C2×Q812yes—
200C2×#92418C2×Q812yes—
201C2×#92418#4112yes—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61C4⋊Q812yes—
209C23×C61C22×D411yescharacteristic
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—84no—
229C2×C4×C61C4⋊Q812yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×Q811yescharacteristic
232C2×C4×C61C22×Q811yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C8×C61—22no—
261C8×C61—22no—
262#92418—22no—
263#92418C22⋊Q814yes—
264#26075—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—41no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61C2×C4⋊Q811yescharacteristic
273C22×C61D4211yescharacteristic
274C22×C61#54711yescharacteristic
275C22×C61#57511yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57711yescharacteristic
278C22×C61#57711yescharacteristic
279C4×C61—22no—
280C4×C61—42no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61—81no—
284C4×C61C2×C22⋊Q811yescharacteristic
285C4×C61#54411yescharacteristic
286C4×C61D4×Q811yescharacteristic
287C4×C61#55411yescharacteristic
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—42no—
292Dic61—41no—
293Dic61—81no—
294Dic61—81no—
295Dic61—81no—
296C2×#499—611no—
297C2×#428—611no—
298#2442—611no—
299#2441—611no—
300#2085—611no—
301#2084—611no—
302#2059—611no—
303#2059—611no—
304#1943—611no—
305#1941—611no—
306#1854—611no—
307#1853D6111yescharacteristic
308#1847—611no—
309#1809—611no—
310#1753—611no—
311C2×C61—42no—
312C2×C61#244011yescharacteristic
313C2×C61#316311yescharacteristic
314C2×C61#316611yescharacteristic
315C2×C61#326711yescharacteristic
316C2×C61#327111yescharacteristic
317C2×C61#335111yescharacteristic
318C2×C61#335111yescharacteristic
319C22×M16—611no—
320C2×#125—611no—
321C2×#125—611no—
322C2×#117—611no—
323C22×C4⋊C4C2×D6111yescharacteristic
324#499—1224no—
325#428—1224no—
326C2×#107—611no—
327C2×#107—611no—
328C2×#107—611no—
329C2×#107—611no—
330C2×#106—611no—
331C2×#106—611no—
332C2×#106—611no—
333C2×#106—611no—
334C2×#106C2×D6111yescharacteristic
335C2×#106C2×D6111yescharacteristic
336C2×C22⋊C8C2×D6111yescharacteristic
337C2×#97—611no—
338C2×C4×C8—611no—
339#400—1221no—
340#399—1221no—
341#398—1221no—
342#396—1221no—
343#389—611no—
344#389—611no—
345#389—611no—
346#389—611no—
347#389—1221no—
348#387—1221no—
349#385—1221no—
350#384—611no—
351#382—611no—
352#380—611no—
353#380—1221no—
354#380C2×D6111yescharacteristic
355C4×C22⋊C4—611no—
356#340—611no—
357#340—611no—
358#340—611no—
359#340—611no—
360#340C2×D6111yescharacteristic
361#340C2×D6111yescharacteristic
362#336—611no—
363#336—611no—
364C61#2607511yescharacteristic, complemented, Sylow
365C23×C4—21no—
366C23×C4C22×D6111yescharacteristic
367C2×M16—614no—
368C2×M16—1222no—
369C2×M16—1222no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374#125—1222no—
375#125—1222no—
376#125—2441no—
377#125—2441no—
378#117—1222no—
379#117—2441no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—611no—
403C2×C22⋊C4—611no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—1221no—
409#107—614no—
410#107—1222no—
411#107—1222no—
412#107—2441no—
413#106—22no—
414#106—41no—
415#106—1222no—
416#106—1222no—
417#106—1222no—
418#106—1222no—
419C22⋊C8Dic12214yes—
420#97—1222no—
421C4×C8—1222no—
422#95—21no—
423#95—611no—
424#95—611no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432C24D4×D6111yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4C2×Dic12212yes—
453C22×C4D4×D6111yescharacteristic
454C22×C4Q8×D6112yes—
455C22×C4C23×D6111yescharacteristic
456M16—1228no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—612no—
462C2×C8—612no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8C2×Dic12212yes—
468C2×C8C2×Dic12212yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—612no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—4881no—
486C4⋊C4—4881no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—2442no—
490C4⋊C4—2442no—
491C4⋊C4—2442no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4D4×D6112yes—
496C22⋊C4—81no—
497C22⋊C4—2441no—
498C22⋊C4—2441no—
499C22⋊C4—2442no—
500C22⋊C4—2442no—
501C42—1221no—
502C42—1221no—
503C42—1221no—
504C42—1221no—
505C42—2442no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#1451977612yes—
513C23C2×D4×D6111yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—84no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1222no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4#1451977612yes—
558C2×C4C2×D4×D6111yescharacteristic
559C2×C4C2×Q8×D6111yescharacteristic
560C2×C4C22×Dic12211yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4#1451982211yescharacteristic
563C2×C4#1451982411yescharacteristic
564C8—22no—
565C8—22no—
566C8—614no—
567C8—1222no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#1451977611yescharacteristic
576C22#1684039411yescharacteristic
577C22#1684041011yescharacteristic
578C22#1684055811yescharacteristic
579C22#1684056111yescharacteristic
580C22#1684056811yescharacteristic
581C22#1684056811yescharacteristic
582C4—22no—
583C4—42no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—4881no—
588C4—4881no—
589C4—4881no—
590C4—2441no—
591C4—2442no—
592C4—2441no—
593C4—2441no—
594C4—2441no—
595C4C22⋊Q8×D6111yescharacteristic
596C4#1684039111yescharacteristic
597C4D4×Dic12211yescharacteristic
598C4#1684047111yescharacteristic
599C2—42no—
600C2#2144952011yescharacteristic
601C2#2145634711yescharacteristic
602C2#2145636911yescharacteristic
603C2#2145734211yescharacteristic
604C2#2145740711yescharacteristic
605C2#2145838211yescharacteristic
606C2#2145838311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1853C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1853D61
hypocenterC61#26075
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
1. C2×#16840162C2semidirect
5. #21447305C2semidirect
6. #21447298C2semidirect
7. #21447160C2semidirect
8. #21447159C2semidirect
9. #21446556C2semidirect
10. #21446546C2semidirect
12. #21445984C2semidirect
364. C61#26075semidirect

Maximal subgroups

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

subgroupindex
C2×#168401622
C2×#168398022
#214495372
#214495332
#214473052
#214472982
#214471602
#214471592
#214465562
#214465462
#214460202
#214459842
#214458502
#214456052
C61×#18532
#2607561