Finite Group Databasebeta

← #29722550

Subgroups — #29722550

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×#16840037C211yesmaximal, characteristic, complemented
2#21455451C212yesmaximal
3C2×#16840003C211yesmaximal, characteristic, complemented
4C22×#14519713C211yesmaximal, characteristic, complemented
5#21447185C212yesmaximal, complemented
6#21447117C212yesmaximal, complemented
7#21446734C211yesmaximal, characteristic, complemented
8#21446604C211yesmaximal, characteristic, complemented
9#21446285C212yesmaximal, complemented
10#21445883C211yesmaximal, characteristic, complemented
11C61×#2929C211yesmaximal, characteristic
12C2×#14519807C2212yes—
13C22×Dic244C2211yescharacteristic
14C22×#11758194C2212yes—
15C2×#14519756C2212yes—
16#16840037—24no—
17#16840018—22no—
18#16840006—22no—
19#16840003—21no—
20#16840003—21no—
21#16840003—21no—
22#16840003—21no—
23#16840001—22no—
24#16839989—22no—
25C2×#14519713—24no—
26C2×#14519713C2212yes—
27C2×#14519713C2211yescharacteristic
28C2×#14519713C2211yescharacteristic
29#16839805C2212yes—
30C2×#14519711C2211yescharacteristic
31#16839770C2212yes—
32#16839763C2212yes—
33C2×#14519702C2212yes—
34C2×#14519695C2212yes—
35C2×#14519695C2212yes—
36#16839530C2212yes—
37#16839530C2212yes—
38C2×#131×C61C2211yescharacteristic
39C23×C8×C61C2211yescharacteristic
40C61×#525C2211yescharacteristic
41C61×#435—21no—
42C61×#435—21no—
43C8×D4×C61—21no—
44C8×D4×C61—21no—
45C61×#424C2211yescharacteristic
46C61×#408C2211yescharacteristic
47C61×#407C2211yescharacteristic
48C61×#405C2211yescharacteristic
49C22×Dic122C2312yes—
50#14519807—24no—
51#14519807—42no—
52C2×Dic244—21no—
53C2×Dic244—21no—
54C2×Dic244—22no—
55C2×Dic244—22no—
56C2×#11758202—22no—
57C2×#11758200—22no—
58C2×#11758194—22no—
59C2×#11758194—24no—
60C2×#11758194—42no—
61C2×#11758194C2312yes—
62C2×#11758194C2312yes—
63C22×#92418—22no—
64#14519756—22no—
65#14519756—22no—
66#14519756—22no—
67#14519756—22no—
68C2×C4⋊Dic61C2312yes—
69C2×#11758196—22no—
70C2×C22⋊Dic61—22no—
71#14519713—22no—
72#14519713—24no—
73#14519713D414yescomplemented
74#14519711—21no—
75#14519711—21no—
76#14519710—22no—
77#14519708—22no—
78#14519702—22no—
79#14519702—22no—
80#14519701—22no—
81#14519700—22no—
82#14519700—22no—
83#14519698—22no—
84#14519695—24no—
85#14519695—42no—
86C2×#41×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88#131×C61—22no—
89#131×C61—22no—
90C2×M16×C61—21no—
91C2×M16×C61C2311yescharacteristic
92C2×M16×C61C2311yescharacteristic
93C22×C8×C61—22no—
94C22×C8×C61—22no—
95C22×C8×C61—21no—
96C22×C8×C61D411yescharacteristic
97C22×C8×C61D411yescharacteristic
98C22×C8×C61D411yescharacteristic
99C22×C8×C61D411yescharacteristic
100C22×C8×C61C2311yescharacteristic
101C22×C8×C61C2311yescharacteristic
102#123×C61—21no—
103C22⋊Q8×C61D411yescharacteristic
104C22⋊Q8×C61D411yescharacteristic
105#121×C61D411yescharacteristic
106#121×C61D411yescharacteristic
107#120×C61—21no—
108C4×D4×C61—21no—
109C4×D4×C61—21no—
110#117×C61C2311yescharacteristic
111C4⋊C8×C61—21no—
112C4⋊C8×C61—21no—
113C22⋊C8×C61—21no—
114C22⋊C8×C61—21no—
115C22⋊C8×C61—21no—
116C22⋊C8×C61—21no—
117#97×C61—21no—
118C4×C8×C61—21no—
119C24×C61#4111yescharacteristic
120#41×C61—21no—
121#41×C61—21no—
122#41×C61—21no—
123#41×C61—21no—
124C2×Q8×C61C2×D411yescharacteristic
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61C2×D411yescharacteristic
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61#4111yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—22no—
141C22×Dic61—22no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148M16×C61—21no—
149M16×C61—21no—
150M16×C61—21no—
151M16×C61—21no—
152M16×C61—42no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—42no—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C2×C8×C61C2×D412yes—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61#4112yes—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61C2×D411yescharacteristic
171C4⋊C4×C61C2×D411yescharacteristic
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C42×C61—21no—
178C22⋊Dic61—42no—
179C22⋊Dic61—42no—
180#11758202—44no—
181#11758200—44no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184#11758196—42no—
185#11758196—42no—
186#11758194—22no—
187#11758194—22no—
188#11758194—22no—
189#11758194—44no—
190#11758194C2×D414yes—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—44no—
194Dic244—42no—
195Dic244—42no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—22no—
199C23×C61—21no—
200C23×C61—41no—
201C23×C61#12111yescharacteristic
202C23×C61#12111yescharacteristic
203C23×C61C2×#4111yescharacteristic
204Q8×C61—21no—
205Q8×C61—21no—
206D4×C61—21no—
207D4×C61—21no—
208D4×C61—42no—
209D4×C61—41no—
210D4×C61—41no—
211D4×C61—42no—
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—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61#12111yescharacteristic
228C2×C4×C61#12111yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61H(5,𝔽2)11yescharacteristic
234C2×C4×C612−1+411yescharacteristic
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—82no—
240Dic122—42no—
241Dic122—42no—
242Dic122—42no—
243Dic122—42no—
244C8×C61—21no—
245C8×C61—21no—
246C8×C61—41no—
247C8×C61—41no—
248C8×C61#12112yes—
249C8×C61#12112yes—
250#92418—24no—
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—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61#47312yescomplemented
266C22×C61C2×#12111yescharacteristic
267C22×C61#54811yescharacteristic
268C22×C61#55311yescharacteristic
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—41no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61C2×#12111yescharacteristic
275C4×C61#53711yescharacteristic
276C4×C61D4211yescharacteristic
277C4×C61#54711yescharacteristic
278Dic61—82no—
279Dic61—82no—
280C2×#473—611no—
281#3056—612no—
282C2×#467—611no—
283#2929D6111yescharacteristic
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—81no—
296C2×C61#241611yescharacteristic
297C2×C61C2×#47311yescharacteristic
298C2×C61#306711yescharacteristic
299C2×#137—612no—
300C22×Q16—611no—
301C2×#131C2×D6111yescharacteristic
302C22×M16—612no—
303C23×C8C2×D6111yescharacteristic
304#525C2×D6111yescharacteristic
305C2×C22⋊Q8—612no—
306#473—1224no—
307#470—1222no—
308#468—1222no—
309#467—1221no—
310#467—1221no—
311#467—1221no—
312#467—1221no—
313#467—1222no—
314#465—1222no—
315#435—21no—
316#435—21no—
317C8×D4—21no—
318C8×D4—21no—
319C2×#108—611no—
320C2×#108—612no—
321C2×#108—611no—
322C2×#108—1224no—
323#428—612no—
324C2×#107—611no—
325#424C2×D6111yescharacteristic
326#419—612no—
327#417—612no—
328C2×#103—612no—
329C2×#101—612no—
330C2×#101—612no—
331#408C2×D6111yescharacteristic
332#407C2×D6111yescharacteristic
333#405C2×D6111yescharacteristic
334#343—612no—
335#343—612no—
336C61#2480411yescharacteristic, complemented, Sylow
337C2×#41C22×D6111yescharacteristic
338C22×Q8—612no—
339C23×C4C22×D6111yescharacteristic
340#137—1224no—
341#137—2442no—
342C2×Q16—1222no—
343C2×Q16—1222no—
344C2×Q16—1222no—
345C2×Q16—1221no—
346C2×Q16—1221no—
347C2×SD16—1222no—
348#131—22no—
349#131—22no—
350C2×M16—21no—
351C2×M16—612no—
352C2×M16—612no—
353C2×M16—1222no—
354C2×M16—1224no—
355C2×M16—2442no—
356C2×M16C22×D6111yescharacteristic
357C2×M16C22×D6111yescharacteristic
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—21no—
361C22×C8—1222no—
362C22×C8C22⋊D6111yescharacteristic
363C22×C8C22⋊D6111yescharacteristic
364C22×C8C22⋊D6111yescharacteristic
365C22×C8C22⋊D6111yescharacteristic
366C22×C8C22×D6111yescharacteristic
367C22×C8C22×D6111yescharacteristic
368#123—21no—
369C22⋊Q8—1222no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8D24411yescharacteristic
374C22⋊Q8D24411yescharacteristic
375#121D24411yescharacteristic
376#121D24411yescharacteristic
377#120—21no—
378C4×D4—21no—
379C4×D4—21no—
380#117C22×D6111yescharacteristic
381C2×C4⋊C4—612no—
382C2×C4⋊C4—1222no—
383C2×C22⋊C4—1222no—
384#108—614no—
385#108—1224no—
386#108—1222no—
387#107—1221no—
388#107—1221no—
389#107—1222no—
390#106—1222no—
391C4⋊C8—21no—
392C4⋊C8—21no—
393#103—1222no—
394#103—1222no—
395#103—1222no—
396#103—1222no—
397#103—1222no—
398#102—1222no—
399#101—1224no—
400#101—2442no—
401C22⋊C8—21no—
402C22⋊C8—21no—
403C22⋊C8—21no—
404C22⋊C8—21no—
405#97—21no—
406C4×C8—21no—
407C24#1175822211yescharacteristic
408#41—21no—
409#41—21no—
410#41—21no—
411#41—21no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8C2×D24411yescharacteristic
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×D24411yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4C2×D24411yescharacteristic
431C22×C4C2×D24411yescharacteristic
432C22×C4#1175822211yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436Q16—2444no—
437Q16—2442no—
438Q16—2442no—
439SD16—2444no—
440M16—21no—
441M16—21no—
442M16—21no—
443M16—21no—
444M16—42no—
445M16—614no—
446M16—1222no—
447M16—1222no—
448M16—1222no—
449M16—2444no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—42no—
459C2×C8—1222no—
460C2×C8—1222no—
461C2×C8—2444no—
462C2×C8D4×D6112yes—
463C2×C8#1175822412yes—
464C2×C8C2×C22⋊D6111yescharacteristic
465C2×C8C2×C22⋊D6111yescharacteristic
466C2×C8C2×C22⋊D6111yescharacteristic
467C2×C8C2×C22⋊D6111yescharacteristic
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—2442no—
473C4⋊C4—2442no—
474C4⋊C4C2×D24411yescharacteristic
475C4⋊C4C2×D24411yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C22⋊C4—2442no—
482C22⋊C4—2442no—
483C42—21no—
484C23—21no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—41no—
489C23#1451975111yescharacteristic
490C23#1451975111yescharacteristic
491C23C2×#1175822211yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—2442no—
495Q8—2442no—
496Q8—2442no—
497Q8—2442no—
498D4—21no—
499D4—21no—
500D4—42no—
501D4—41no—
502D4—42no—
503D4—41no—
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—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—4882no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4#1451975111yescharacteristic
525C2×C4#1451975111yescharacteristic
526C2×C4C2×D4×D6111yescharacteristic
527C2×C4C22×C22⋊D6111yescharacteristic
528C2×C4C22×D24411yescharacteristic
529C2×C4C2×#1175822411yescharacteristic
530C2×C4#1451982311yescharacteristic
531C2×C4#1451982411yescharacteristic
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—1224no—
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—41no—
550C22—41no—
551C22—41no—
552C22#1684003712yescomplemented
553C22C2×#1451975111yescharacteristic
554C22#1684042111yescharacteristic
555C22#1684045811yescharacteristic
556C4—21no—
557C4—21no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—4882no—
562C4—4882no—
563C4C2×#1451974911yescharacteristic
564C4#1684034011yescharacteristic
565C4D4×D24411yescharacteristic
566C4#1684041611yescharacteristic
567C2—21no—
568C2—22no—
569C2—21no—
570C2—41no—
571C2—81no—
572C2#2144925911yescharacteristic
573C2C2×#1684003711yescharacteristic
574C2#2145553911yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2929C2
socleC22×C61C2×#121
prime radicalC61×#525C22
hypercenter#2929D61
hypocenterC61#24804
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. C2×#16840037C2semidirect
3. C2×#16840003C2semidirect
4. C22×#14519713C2semidirect
5. #21447185C2semidirect
6. #21447117C2semidirect
7. #21446734C2semidirect
8. #21446604C2semidirect
9. #21446285C2semidirect
10. #21445883C2semidirect
73. #14519713D4semidirect
265. C22×C61#473semidirect
336. C61#24804semidirect
552. C22#16840037semidirect

Maximal subgroups

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

subgroupindex
C2×#168400372
#214554512
C2×#168400032
C22×#145197132
#214471852
#214471172
#214467342
#214466042
#214462852
#214458832
C61×#29292
#2480461