Finite Group Databasebeta

← #29722551

Subgroups — #29722551

Order 15616.

858 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840034C211yesmaximal, characteristic, complemented
2#21455445C212yesmaximal
3#21454513C211yesmaximal, characteristic
4C22×#14519713C211yesmaximal, characteristic, complemented
5#21447184C212yesmaximal, complemented
6#21447119C212yesmaximal, complemented
7#21446735C211yesmaximal, characteristic, complemented
8#21446602C211yesmaximal, characteristic, complemented
9#21446288C212yesmaximal, complemented
10#21445884C211yesmaximal, characteristic, complemented
11C2×C61×#467C211yesmaximal, characteristic, complemented
12C2×#14519808C2212yes—
13C2×#14519783C2211yescharacteristic
14C22×#11758194C2212yes—
15#16840273C2211yescharacteristic
16#16840034—24no—
17#16840018—22no—
18#16840010—22no—
19#16840001—22no—
20#16839993—22no—
21#16839833—21no—
22#16839833—21no—
23C8×C22⋊D61—21no—
24C8×C22⋊D61—21no—
25C2×#14519713—24no—
26C2×#14519713C2212yes—
27C2×#14519713C2211yescharacteristic
28C2×#14519713C2211yescharacteristic
29#16839802C2212yes—
30#16839788C2211yescharacteristic
31#16839771C2212yes—
32#16839761C2212yes—
33C2×#14519696C2212yes—
34C2×#14519696C2212yes—
35#16839728C2211yescharacteristic
36#16839726C2211yescharacteristic
37#16839717C2211yescharacteristic
38#16839531C2212yes—
39#16839531C2212yes—
40C22×Q16×C61C2211yescharacteristic
41C23×C8×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2212yes—
43C61×#467—21no—
44C61×#467—21no—
45C61×#467—21no—
46C61×#467—21no—
47C2×#107×C61C2211yescharacteristic
48C2×#103×C61C2212yes—
49C2×#11758222C2311yescharacteristic
50#14519808—24no—
51#14519808—42no—
52C2×#11758202—22no—
53C2×#11758201—22no—
54#14519783—22no—
55#14519783—22no—
56C2×#11758194—22no—
57C2×#11758194—24no—
58C2×#11758194—42no—
59C2×#11758194C2311yescharacteristic
60C2×#11758194C2311yescharacteristic
61C2×#11758194C2312yes—
62C2×#11758194C2312yes—
63C2×#11758189—21no—
64C22×#92418—22no—
65C2×C8×D61—21no—
66#14519760—21no—
67#14519756D411yescharacteristic
68#14519756D411yescharacteristic
69#14519751D411yescharacteristic
70#14519751D411yescharacteristic
71#14519746—21no—
72C4×C22⋊D61—21no—
73C4×C22⋊D61—21no—
74#14519736C2311yescharacteristic
75#14519713—22no—
76#14519713—24no—
77#14519713D414yescomplemented
78#14519710—22no—
79#14519708—22no—
80#14519706—21no—
81#14519706—21no—
82#14519701—22no—
83#14519701—22no—
84#14519700—22no—
85#14519697—22no—
86#14519696—24no—
87#14519696—42no—
88C22⋊#92418—21no—
89C22⋊#92418—21no—
90#14519689—21no—
91#14519689—21no—
92#14519687—21no—
93C8×Dic61—21no—
94C22×Q8×C61C2312yes—
95C23×C4×C61C2311yescharacteristic
96C2×Q16×C61—22no—
97C2×Q16×C61—21no—
98C2×Q16×C61—21no—
99C2×Q16×C61—22no—
100C22×C8×C61—22no—
101C22×C8×C61—22no—
102C22×C8×C61D411yescharacteristic
103C22×C8×C61D411yescharacteristic
104C22×C8×C61D411yescharacteristic
105C22×C8×C61D411yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61C2312yes—
114C2×C22⋊C4×C61—22no—
115#107×C61—21no—
116#107×C61—21no—
117#103×C61—22no—
118#103×C61—22no—
119C24×C61#4111yescharacteristic
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61#4111yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—21no—
139C2×C22⋊D61—21no—
140C2×C22⋊D61—21no—
141#11758222—21no—
142#11758222—21no—
143#11758222—21no—
144#11758222—21no—
145C2×D244C2×D411yescharacteristic
146C2×C4×D61—21no—
147C2×Dic122C2×D411yescharacteristic
148Q16×C61—42no—
149Q16×C61—42no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61C2×D411yescharacteristic
157C2×C8×C61C2×D411yescharacteristic
158C2×C8×C61C2×D412yes—
159C2×C8×C61C2×D411yescharacteristic
160C2×C8×C61C2×D411yescharacteristic
161C2×C8×C61#4112yes—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C22⋊C4×C61—42no—
167C22⋊C4×C61—42no—
168C22⋊Dic61—21no—
169C22⋊Dic61—21no—
170C22⋊Dic61—21no—
171#11758202—44no—
172#11758201—44no—
173#11758198—21no—
174#11758198—21no—
175C4⋊Dic61C2×D411yescharacteristic
176C4⋊Dic61C2×D411yescharacteristic
177#11758196—21no—
178#11758196—21no—
179C4×Dic61—21no—
180#11758194—22no—
181#11758194—21no—
182#11758194—22no—
183#11758194—21no—
184#11758194—21no—
185#11758194—22no—
186#11758194—21no—
187#11758194—44no—
188#11758194C2×D414yes—
189C2×#92418—22no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—22no—
193C2×#92418—44no—
194#11758189—42no—
195C8×D61—42no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—41no—
200C23×C61#12111yescharacteristic
201C23×C61#12111yescharacteristic
202C23×C61C2×#4111yescharacteristic
203C22×D61—21no—
204Q8×C61—42no—
205Q8×C61—42no—
206Q8×C61—42no—
207Q8×C61—42no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—82no—
223C2×C4×C61#12111yescharacteristic
224C2×C4×C61#12111yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C22⋊D61—42no—
232C22⋊D61—41no—
233C22⋊D61—41no—
234C22⋊D61—42no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238D244—21no—
239D244—21no—
240C4×D61—41no—
241C4×D61—41no—
242Dic122—21no—
243Dic122—21no—
244C8×C61—21no—
245C8×C61—21no—
246C8×C61#12112yes—
247C8×C61#12112yes—
248#92418—24no—
249#92418—41no—
250#92418—41no—
251#92418—42no—
252#24804—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—21no—
259C22×C61—21no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61#47312yescomplemented
263C22×C61C2×#12111yescharacteristic
264C22×C61#54811yescharacteristic
265C22×C61#55311yescharacteristic
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—82no—
272C4×C61—82no—
273C4×C61C2×#12111yescharacteristic
274C4×C61#53711yescharacteristic
275C4×C61D4211yescharacteristic
276C4×C61#54711yescharacteristic
277Dic61—41no—
278Dic61—41no—
279Dic61—41no—
280C2×#473—611no—
281#3056—612no—
282C2×#467D6111yescharacteristic, complemented
283#2929—611no—
284C22×#108—611no—
285#2064—612no—
286#2052—612no—
287#1979—611no—
288#1952—611no—
289#1901—612no—
290#1816—611no—
291C2×C61—21no—
292C2×C61—21no—
293C2×C61—22no—
294C2×C61—41no—
295C2×C61#241611yescharacteristic
296C2×C61C2×#47311yescharacteristic
297C2×C61#306711yescharacteristic
298D61—81no—
299C2×#137—612no—
300C22×Q16C2×D6111yescharacteristic
301C2×#131—611no—
302C22×M16—612no—
303C23×C8C2×D6111yescharacteristic
304#525—611no—
305C2×C22⋊Q8C2×D6112yes—
306#473—1224no—
307#470—1222no—
308#468—1222no—
309#467—21no—
310#467—21no—
311#467—21no—
312#467—21no—
313#467—1222no—
314#465—1222no—
315#435—1221no—
316#435—1221no—
317C8×D4—1221no—
318C8×D4—1221no—
319C2×#108—611no—
320C2×#108—612no—
321C2×#108—611no—
322C2×#108—1224no—
323#428—612no—
324C2×#107C2×D6111yescharacteristic
325#424—611no—
326#419—612no—
327#417—612no—
328C2×#103C2×D6112yes—
329C2×#101—612no—
330C2×#101—612no—
331#408—611no—
332#407—611no—
333#405—611no—
334#343—612no—
335#343—612no—
336C61#2480411yescharacteristic, complemented, Sylow
337C2×#41—611no—
338C22×Q8C22×D6112yes—
339C23×C4C22×D6111yescharacteristic
340#137—1224no—
341#137—2442no—
342C2×Q16—22no—
343C2×Q16—22no—
344C2×Q16—21no—
345C2×Q16—21no—
346C2×Q16—1222no—
347C2×SD16—1222no—
348#131—1222no—
349#131—1222no—
350C2×M16—612no—
351C2×M16—612no—
352C2×M16—611no—
353C2×M16—611no—
354C2×M16—1222no—
355C2×M16—1221no—
356C2×M16—1224no—
357C2×M16—2442no—
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—1222no—
361C22×C8—1221no—
362C22×C8C22⋊D6111yescharacteristic
363C22×C8C22⋊D6111yescharacteristic
364C22×C8C22⋊D6111yescharacteristic
365C22×C8C22⋊D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367C22×C8C22×D6111yescharacteristic
368#123—1221no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—611no—
374C22⋊Q8—611no—
375#121—611no—
376#121—611no—
377#120—1221no—
378C4×D4—1221no—
379C4×D4—1221no—
380#117—611no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4C22×D6112yes—
383C2×C22⋊C4—22no—
384#108—614no—
385#108—1224no—
386#108—1222no—
387#107—21no—
388#107—21no—
389#107—1222no—
390#106—1222no—
391C4⋊C8—1221no—
392C4⋊C8—1221no—
393#103—22no—
394#103—22no—
395#103—1222no—
396#103—1222no—
397#103—1222no—
398#102—1222no—
399#101—1224no—
400#101—2442no—
401C22⋊C8—1221no—
402C22⋊C8—1221no—
403C22⋊C8—1221no—
404C22⋊C8—1221no—
405#97—1221no—
406C4×C8—1221no—
407C24#1175822411yescharacteristic
408#41—1221no—
409#41—1221no—
410#41—1221no—
411#41—1221no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—611no—
419C2×D4—611no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—1221no—
430C22×C4D4×D6111yescharacteristic
431C22×C4D4×D6111yescharacteristic
432C22×C4#1175822411yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436Q16—42no—
437Q16—42no—
438Q16—2444no—
439SD16—2444no—
440M16—614no—
441M16—1222no—
442M16—1221no—
443M16—1222no—
444M16—1221no—
445M16—1221no—
446M16—1222no—
447M16—1221no—
448M16—2444no—
449M16—2442no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—1222no—
457C2×C8—1222no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—2444no—
461C2×C8—2442no—
462C2×C8D4×D6112yes—
463C2×C8#1175822412yes—
464C2×C8C2×C22⋊D6111yescharacteristic
465C2×C8C2×C22⋊D6111yescharacteristic
466C2×C8C2×C22⋊D6111yescharacteristic
467C2×C8C2×C22⋊D6111yescharacteristic
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—611no—
473C4⋊C4—611no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—1221no—
479C22⋊C4—1221no—
480C22⋊C4—1221no—
481C22⋊C4—1221no—
482C22⋊C4—1221no—
483C42—1221no—
484C23—21no—
485C23—22no—
486C23—21no—
487C23—41no—
488C23—1221no—
489C23#1451974811yescharacteristic
490C23#1451974811yescharacteristic
491C23C2×#1175822411yescharacteristic
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—1221no—
497Q8—1221no—
498D4—1221no—
499D4—1221no—
500D4—2442no—
501D4—2441no—
502D4—2442no—
503D4—2441no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—82no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4#1451974811yescharacteristic
525C2×C4#1451974811yescharacteristic
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4#1451982311yescharacteristic
531C2×C4#1451982411yescharacteristic
532C8—21no—
533C8—21no—
534C8—1224no—
535C8—2441no—
536C8—2441no—
537C8—2442no—
538C8#1451974912yes—
539C8#1451974912yes—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—42no—
548C22—42no—
549C22—2441no—
550C22—2441no—
551C22—2441no—
552C22#1684003412yescomplemented
553C22C2×#1451974811yescharacteristic
554C22#1684041811yescharacteristic
555C22#1684046411yescharacteristic
556C4—21no—
557C4—21no—
558C4—82no—
559C4—82no—
560C4—2441no—
561C4—2441no—
562C4—2441no—
563C4C2×#1451974911yescharacteristic
564C4#1684034011yescharacteristic
565C4#1684039611yescharacteristic
566C4#1684041211yescharacteristic
567C2—21no—
568C2—22no—
569C2—21no—
570C2—41no—
571C2—4881no—
572C2#2144926011yescharacteristic
573C2C2×#1684003411yescharacteristic
574C2#2145553511yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#467C2
socleC22×C61C2×#121
prime radical#16840273C22
hypercenterC2×#467D61
hypocenterC61#24804
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×#16840034C2semidirect
4. C22×#14519713C2semidirect
5. #21447184C2semidirect
6. #21447119C2semidirect
7. #21446735C2semidirect
8. #21446602C2semidirect
9. #21446288C2semidirect
10. #21445884C2semidirect
11. C2×C61×#467C2semidirect
77. #14519713D4semidirect
262. C22×C61#473semidirect
282. C2×#467D61semidirect
336. C61#24804semidirect
552. C22#16840034semidirect

Maximal subgroups

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

subgroupindex
C2×#168400342
#214554452
#214545132
C22×#145197132
#214471842
#214471192
#214467352
#214466022
#214462882
#214458842
C2×C61×#4672
#2480461