Finite Group Databasebeta

← #29718954

Subgroups — #29718954

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455755C211yesmaximal, characteristic
2C2×#16839900C211yesmaximal, characteristic
3#21454479C211yesmaximal, characteristic, complemented
4C2×#16839772C211yesmaximal, characteristic, complemented
5#21447187C212yesmaximal, complemented
6#21447080C212yesmaximal, complemented
7#21446719C212yesmaximal, complemented
8#21446472C212yesmaximal, complemented
9#21446024C211yesmaximal, characteristic, complemented
10#21445742C211yesmaximal, characteristic, complemented
11C61×#1900C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758200C2212yes—
14C2×#14519736C2211yescharacteristic
15#16840083—21no—
16#16840083—21no—
17#16840078—21no—
18#16840073—21no—
19#16839900—21no—
20#16839900—21no—
21#16839900—22no—
22#16839820—22no—
23#16839816—21no—
24#16839812—21no—
25C2×#14519712C2212yes—
26#16839802C2211yescharacteristic
27#16839802C2211yescharacteristic
28#16839790C2212yes—
29#16839772—21no—
30#16839772—21no—
31#16839772C2212yes—
32#16839772C2212yes—
33#16839767C2211yescharacteristic
34#16839767C2211yescharacteristic
35C2×#14519702—21no—
36C2×#14519702—21no—
37C2×#14519700C2212yes—
38C2×#14519695C2212yes—
39C2×#14519695C2212yes—
40#16839726C2211yescharacteristic
41C2×C22⋊#92418C2211yescharacteristic
42#16839720C2212yes—
43#16839526C2212yes—
44C22×M16×C61C2211yescharacteristic
45C2×#121×C61C2211yescharacteristic
46C2×#100×C61C2212yes—
47C61×#407C2211yescharacteristic
48C61×#343C2212yes—
49C22×Dic122—21no—
50C22×Dic122—21no—
51C22×Dic122—24no—
52C22×Dic122—21no—
53C22×Dic122—21no—
54C22×Dic122—21no—
55C22×Dic122—42no—
56C22×Dic122C2312yes—
57C23×Dic61—21no—
58C2×#11758200—24no—
59C2×#11758200—22no—
60C2×#11758200—22no—
61C2×#11758200—42no—
62C2×#11758194C2311yescharacteristic
63C2×#11758194C2311yescharacteristic
64C22×#92418C2312yes—
65#14519736—21no—
66#14519736—21no—
67#14519736D411yescharacteristic
68#14519736D411yescharacteristic
69#14519736C2311yescharacteristic
70#14519736C2311yescharacteristic
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D411yescharacteristic
73C2×C22⋊Dic61—21no—
74C2×C4×Dic61—21no—
75#14519712—42no—
76#14519710—21no—
77#14519710—21no—
78#14519708—21no—
79#14519708—21no—
80#14519706—22no—
81#14519706—22no—
82#14519702—22no—
83#14519702—22no—
84#14519702—22no—
85#14519702—22no—
86#14519700—22no—
87#14519700—22no—
88#14519698—21no—
89#14519698—21no—
90#14519698—21no—
91#14519698—21no—
92#14519695—24no—
93#14519695—42no—
94C22⋊#92418—21no—
95C22⋊#92418—21no—
96C22⋊#92418D411yescharacteristic
97C22⋊#92418D411yescharacteristic
98C22⋊#92418D412yes—
99#14519687—22no—
100C8×Dic61—22no—
101C22×D4×C61—21no—
102C22×D4×C61C2311yescharacteristic
103C23×C4×C61C2311yescharacteristic
104C2×M16×C61—21no—
105C2×M16×C61—42no—
106C2×M16×C61C2312yes—
107C2×M16×C61C2312yes—
108C22×C8×C61—21no—
109#121×C61—21no—
110#121×C61—21no—
111#121×C61—21no—
112#121×C61—21no—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115#100×C61—42no—
116C22⋊C8×C61—21no—
117C22⋊C8×C61—21no—
118C24×C61—21no—
119C24×C61#4111yescharacteristic
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—41no—
125C2×D4×C61—41no—
126C2×D4×C61—82no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61#4111yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—22no—
138C22×Dic61—21no—
139C22×Dic61—44no—
140C22×Dic61—41no—
141C22×Dic61—41no—
142C2×Dic122—24no—
143C2×Dic122—24no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—22no—
148C2×Dic122—22no—
149C2×Dic122—44no—
150C2×Dic122—44no—
151C2×Dic122—44no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—41no—
155C2×Dic122—41no—
156C2×Dic122#4114yes—
157M16×C61—22no—
158M16×C61—22no—
159M16×C61—22no—
160M16×C61—22no—
161M16×C61—42no—
162C2×C8×C61—22no—
163C2×C8×C61—22no—
164C2×C8×C61—42no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C22⋊C4×C61—41no—
168C22⋊C4×C61—41no—
169C22⋊Dic61—21no—
170C22⋊Dic61—21no—
171C22⋊Dic61—21no—
172C22⋊Dic61—21no—
173#11758200—44no—
174#11758200—44no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61C2×D411yescharacteristic
178C4⋊Dic61C2×D411yescharacteristic
179C4⋊Dic61C2×D411yescharacteristic
180C4⋊Dic61C2×D411yescharacteristic
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185#11758194—22no—
186#11758194—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418C2×D412yes—
192C2×#92418C2×D412yes—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—42no—
198C23×C61—81no—
199C23×C61—81no—
200C23×C61#12111yescharacteristic
201C23×C61#12111yescharacteristic
202C23×C61C2×#4111yescharacteristic
203D4×C61—41no—
204D4×C61—41no—
205D4×C61—42no—
206D4×C61—82no—
207D4×C61—82no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—81no—
217C2×C4×C61#12111yescharacteristic
218C2×C4×C61#12111yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×C4×C61C2×#4112yes—
222C2×C4×C61H(5,𝔽2)12yes—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—24no—
228C2×Dic61—44no—
229C2×Dic61—44no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—81no—
234Dic122—24no—
235Dic122—24no—
236Dic122—44no—
237Dic122—44no—
238Dic122—42no—
239Dic122—42no—
240C8×C61—42no—
241C8×C61—42no—
242#92418—22no—
243#92418—22no—
244#92418—41no—
245#92418—41no—
246#24523—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—44no—
255C22×C61—81no—
256C22×C61—81no—
257C22×C61—81no—
258C22×C61—162no—
259C22×C61#47112yes—
260C22×C61C2×#12111yescharacteristic
261C22×C61#55112yes—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—81no—
265C4×C61#52511yescharacteristic
266C4×C61#53511yescharacteristic
267C4×C61#54612yes—
268Dic61—44no—
269Dic61—41no—
270Dic61—41no—
271Dic61—41no—
272Dic61—41no—
273#3094—611no—
274C2×#448—611no—
275#2924—611no—
276C2×#419—611no—
277#2064—612no—
278#2046—612no—
279#1977—612no—
280#1930—612no—
281#1900D6111yescharacteristic
282#1855—611no—
283#1784—611no—
284C2×C61—22no—
285C2×C61—22no—
286C2×C61—41no—
287C2×C61—161no—
288C2×C61#241111yescharacteristic
289C2×C61C2×#47111yescharacteristic
290C2×C61#306911yescharacteristic
291C23×Q8—611no—
292C22×SD16—612no—
293C22×M16C2×D6111yescharacteristic
294C2×#121C2×D6111yescharacteristic
295C2×#117—611no—
296#482—1221no—
297#482—1221no—
298#481—1221no—
299#480—1221no—
300#448—1221no—
301#448—1221no—
302#448—1222no—
303#433—1222no—
304#432—1221no—
305#431—1221no—
306C2×#108—612no—
307#428—611no—
308#428—611no—
309#424—612no—
310#419—612no—
311#419—612no—
312#419—1221no—
313#419—1221no—
314#418—611no—
315#418—611no—
316C2×#103—612no—
317C2×#103—1221no—
318C2×#103—1221no—
319C2×#101—612no—
320C2×#101—612no—
321C2×#100C2×D6112yes—
322#407—611no—
323#407C2×D6111yescharacteristic
324C2×C22⋊C8—611no—
325#405—612no—
326#343C2×D6112yes—
327#341—612no—
328C61#2452311yescharacteristic, complemented, Sylow
329C22×Q8—612no—
330C22×Q8—1224no—
331C22×Q8—1221no—
332C22×Q8—1221no—
333C22×Q8—1221no—
334C22×Q8—1221no—
335C22×Q8—1221no—
336C22×Q8—2442no—
337C22×D4—21no—
338C22×D4C22×D6111yescharacteristic
339C23×C4—1221no—
340C23×C4C22×D6111yescharacteristic
341C2×SD16—1222no—
342C2×SD16—1222no—
343C2×SD16—1224no—
344C2×SD16—2442no—
345C2×M16—21no—
346C2×M16—42no—
347C2×M16—611no—
348C2×M16—611no—
349C2×M16C22×D6112yes—
350C2×M16C22×D6112yes—
351C22×C8—21no—
352C22×C8—612no—
353#121—21no—
354#121—21no—
355#121—21no—
356#121—21no—
357#117—611no—
358#117—611no—
359#117—611no—
360#117—611no—
361#117—1221no—
362#117—1221no—
363C2×C4⋊C4—611no—
364C2×C4⋊C4—611no—
365C2×C4⋊C4C22×D6111yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1221no—
368C2×C42—1221no—
369#108—2442no—
370#107—1221no—
371#107—1221no—
372#106—1221no—
373#106—1221no—
374C4⋊C8—1222no—
375C4⋊C8—1222no—
376#103—1222no—
377#103—1222no—
378#103—1222no—
379#103—1222no—
380#103—1222no—
381#103—1222no—
382#102—1221no—
383#102—1221no—
384#102—1221no—
385#102—1221no—
386#101—1224no—
387#101—2442no—
388#100—42no—
389C22⋊C8—21no—
390C22⋊C8—21no—
391C22⋊C8—611no—
392C22⋊C8—611no—
393C22⋊C8—612no—
394C22⋊C8—1221no—
395C22⋊C8—1221no—
396#97—1222no—
397C4×C8—1222no—
398C24—21no—
399C24#1175822211yescharacteristic
400C2×Q8—614no—
401C2×Q8—1224no—
402C2×Q8—1224no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—1222no—
406C2×Q8—1222no—
407C2×Q8—1222no—
408C2×Q8—2444no—
409C2×Q8—2444no—
410C2×Q8—2444no—
411C2×Q8—2442no—
412C2×Q8—2442no—
413C2×Q8—2441no—
414C2×Q8—2441no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—82no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—1221no—
426C22×C4—1222no—
427C22×C4—1221no—
428C22×C4—2444no—
429C22×C4—2441no—
430C22×C4—2441no—
431C22×C4#1175822211yescharacteristic
432C22×C4D4×D6111yescharacteristic
433C22×C4D4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4D4×D6111yescharacteristic
436C22×C4C23×D6111yescharacteristic
437SD16—2444no—
438SD16—2444no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—22no—
443M16—42no—
444M16—1222no—
445M16—1222no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—42no—
449C2×C8—612no—
450C2×C8—612no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1221no—
454C2×C8—1221no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—611no—
458C4⋊C4—611no—
459C4⋊C4—611no—
460C4⋊C4—611no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—1221no—
468C22⋊C4—1221no—
469C42—1221no—
470C42—1221no—
471C42—1221no—
472C42—1221no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—81no—
479C23—81no—
480C23#1451975011yescharacteristic
481C23#1451975011yescharacteristic
482C23C2×#1175822211yescharacteristic
483Q8—1224no—
484Q8—1224no—
485Q8—2444no—
486Q8—2444no—
487Q8—2442no—
488Q8—2442no—
489D4—41no—
490D4—41no—
491D4—42no—
492D4—82no—
493D4—82no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—81no—
503C2×C4—1221no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1224no—
508C2×C4—4881no—
509C2×C4—2444no—
510C2×C4—2444no—
511C2×C4—2442no—
512C2×C4—2442no—
513C2×C4—2442no—
514C2×C4#1451975011yescharacteristic
515C2×C4#1451975011yescharacteristic
516C2×C4C2×D4×D6111yescharacteristic
517C2×C4C2×D4×D6111yescharacteristic
518C2×C4#41×D6112yes—
519C2×C4#1451982212yes—
520C8—42no—
521C8—42no—
522C8—1222no—
523C8—1222no—
524C8—2441no—
525C8—2441no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—44no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—162no—
538C22#1684002412yes—
539C22C2×#1451975011yescharacteristic
540C22#1684044112yes—
541C4—21no—
542C4—21no—
543C4—81no—
544C4—2441no—
545C4—2441no—
546C4—2441no—
547C4—2441no—
548C4—2444no—
549C4#1684027011yescharacteristic
550C4#1684032711yescharacteristic
551C4#1684040112yes—
552C2—22no—
553C2—22no—
554C2—41no—
555C2—161no—
556C2#2144920711yescharacteristic
557C2C2×#1684002411yescharacteristic
558C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1900C2
socleC22×C61C2×#121
prime radicalC2×#121×C61C22
hypercenter#1900D61
hypocenterC61#24523
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1900

Complements

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

NH ≅ G/Ntype
3. #21454479C2semidirect
4. C2×#16839772C2semidirect
5. #21447187C2semidirect
6. #21447080C2semidirect
7. #21446719C2semidirect
8. #21446472C2semidirect
9. #21446024C2semidirect
10. #21445742C2semidirect
328. C61#24523semidirect

Maximal subgroups

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

subgroupindex
#214557552
C2×#168399002
#214544792
C2×#168397722
#214471872
#214470802
#214467192
#214464722
#214460242
#214457422
C61×#19002
#2452361