Finite Group Databasebeta

← #29729778

Subgroups — #29729778

Order 15616.

764 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456193C211yesmaximal, characteristic, complemented
2#21455616C211yesmaximal, characteristic, complemented
3#21454762C211yesmaximal, characteristic, complemented
4#21447177C212yesmaximal, complemented
5#21447072C212yesmaximal, complemented
6#21446730C212yesmaximal, complemented
7#21446660C211yesmaximal, characteristic, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446478C212yesmaximal, complemented
10C4×#14519694C211yesmaximal, characteristic, complemented
11C2×C61×#459C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840126—22no—
14#16840108—21no—
15#16840108—21no—
16#16840062—22no—
17#16840052—21no—
18#16840042—21no—
19#16839891—22no—
20#16839889—21no—
21#16839886—21no—
22C2×#14519713C2212yes—
23#16839802C2212yes—
24#16839788C2212yes—
25#16839786C2211yescharacteristic
26#16839767C2212yes—
27#16839767C2212yes—
28#16839767C2212yes—
29C2×#14519694C2212yes—
30C2×#14519694C2211yescharacteristic
31C2×#14519694C2211yescharacteristic
32C2×#14519694C2212yes—
33#16839717C2212yes—
34C4×#11758194C2211yescharacteristic
35C4×#11758194C2211yescharacteristic
36#16839531C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839527C2212yes—
39C22×D8×C61C2212yes—
40C2×#127×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C61×#459—21no—
43C61×#459—21no—
44C61×#459—22no—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#102×C61C2212yes—
47C2×#11758194C2311yescharacteristic
48C2×#11758194C2311yescharacteristic
49C2×#11758194C2311yescharacteristic
50C2×#11758194C2312yes—
51C2×#11758194C2311yescharacteristic
52#14519777D411yescharacteristic
53#14519777D411yescharacteristic
54#14519769D412yes—
55#14519756—21no—
56#14519756—21no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59#14519736C2312yes—
60#14519713—42no—
61#14519710—22no—
62#14519708—22no—
63C4⋊#92418—21no—
64C4⋊#92418—21no—
65#14519706—22no—
66#14519706—22no—
67#14519698—22no—
68#14519698—22no—
69#14519698—22no—
70#14519698—22no—
71#14519698—22no—
72#14519698—22no—
73#14519694—22no—
74#14519694—22no—
75#14519694—24no—
76#14519694—42no—
77#14519688—21no—
78#14519688—21no—
79#14519687—22no—
80C4×#92418—21no—
81C4×#92418—21no—
82C8×Dic61—22no—
83C22×D4×C61—21no—
84C22×D4×C61—21no—
85C22×D4×C61C2312yes—
86C22×D4×C61C2311yescharacteristic
87C23×C4×C61—21no—
88C2×D8×C61—24no—
89C2×D8×C61—22no—
90C2×D8×C61—22no—
91C2×D8×C61—42no—
92C22×C8×C61C2312yes—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61—21no—
96#127×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61D411yescharacteristic
105C4⋊C8×C61D411yescharacteristic
106C4⋊C8×C61D412yes—
107#102×C61—22no—
108#102×C61—22no—
109C24×C61—21no—
110C24×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—21no—
119C2×D4×C61—22no—
120C2×D4×C61—21no—
121C2×D4×C61—22no—
122C2×D4×C61—41no—
123C2×D4×C61—41no—
124C2×D4×C61—41no—
125C2×D4×C61—41no—
126C2×D4×C61—82no—
127C2×D4×C61—82no—
128C2×D4×C61#4112yes—
129C22×C4×C61—21no—
130C22×C4×C61—41no—
131C22×C4×C61—41no—
132C22×C4×C61—82no—
133C22×C4×C61#4111yescharacteristic
134C22×C4×C61#4111yescharacteristic
135C22×C4×C61C2411yescharacteristic
136C2×Dic122—21no—
137C2×Dic122—21no—
138D8×C61—44no—
139D8×C61—44no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61C2×D412yes—
143C2×C8×C61C2×D412yes—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C22⋊C4×C61—41no—
147C22⋊C4×C61—41no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C22⋊Dic61—22no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61C2×D412yes—
156C4⋊Dic61C2×D412yes—
157#11758196—22no—
158#11758196—22no—
159C4×Dic61—22no—
160#11758194—21no—
161#11758194—21no—
162#11758194—21no—
163#11758194—21no—
164#11758194—22no—
165#11758194—21no—
166#11758194—22no—
167#11758194—21no—
168#11758194—21no—
169#11758194#4112yes—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—42no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61—81no—
182C23×C61—81no—
183C23×C61—82no—
184C23×C61—82no—
185C23×C61C2×#4111yescharacteristic
186D4×C61—22no—
187D4×C61—22no—
188D4×C61—42no—
189D4×C61—41no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—41no—
193D4×C61—42no—
194D4×C61—82no—
195D4×C61—82no—
196D4×C61—82no—
197D4×C61—82no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—82no—
202C2×C4×C61—82no—
203C2×C4×C61—81no—
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61#12111yescharacteristic
207C2×C4×C61#12111yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4111yescharacteristic
211C2×C4×C61C2×#4111yescharacteristic
212C2×C4×C61H(5,𝔽2)11yescharacteristic
213C2×C4×C612−1+411yescharacteristic
214C2×Dic61—22no—
215C2×Dic61—22no—
216Dic122—42no—
217Dic122—81no—
218C8×C61—22no—
219C8×C61—22no—
220#92418—22no—
221#92418—41no—
222#92418—41no—
223#92418—41no—
224#92418—42no—
225#25312—611nomaximal, Sylow
226C22×C61—21no—
227C22×C61—21no—
228C22×C61—42no—
229C22×C61—82no—
230C22×C61—82no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61—162no—
237C22×C61—162no—
238C22×C61C2×#12111yescharacteristic
239C22×C61#52911yescharacteristic
240C22×C61#54211yescharacteristic
241C4×C61—21no—
242C4×C61—21no—
243C4×C61—21no—
244C4×C61—81no—
245C4×C61#47212yes—
246C4×C61#54611yescharacteristic
247C4×C61#54811yescharacteristic
248C4×C61#54811yescharacteristic
249C4×C61#55011yescharacteristic
250Dic61—42no—
251Dic61—42no—
252#3138—611no—
253#3077—611no—
254C2×#459D6111yescharacteristic
255#2962—611no—
256#2063—612no—
257#2045—612no—
258#1978—612no—
259#1965—611no—
260#1950—611no—
261#1931—612no—
262C4×#100—611no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61—82no—
266C2×C61—162no—
267C2×C61#242611yescharacteristic
268C2×C61C2×#47211yescharacteristic
269C2×C61#306811yescharacteristic
270C22×D8C2×D6112yes—
271#533—611no—
272C2×#127C2×D6111yescharacteristic
273C2×C4×D4C2×D6111yescharacteristic
274#490—1222no—
275#486—1221no—
276#486—1221no—
277#478—1222no—
278#476—1221no—
279#474—1221no—
280#459—21no—
281#459—22no—
282#459—21no—
283#446—1221no—
284#446—1222no—
285C8×Q8—1221no—
286C2×#108—612no—
287#428—612no—
288#424—612no—
289#423—611no—
290C2×C4⋊C8C2×D6111yescharacteristic
291#418—612no—
292#418—612no—
293#418—612no—
294C2×#102C2×D6112yes—
295C2×#100—612no—
296C2×#100—611no—
297C2×#100—611no—
298C2×#100—612no—
299#405—612no—
300C4×M16—611no—
301C4×M16—611no—
302#343—611no—
303#343—611no—
304#341—612no—
305C61#2531211yescharacteristic, complemented, Sylow
306C22×D4—21no—
307C22×D4—21no—
308C22×D4C22×D6112yes—
309C22×D4C22×D6111yescharacteristic
310C23×C4—21no—
311C2×D8—24no—
312C2×D8—22no—
313C2×D8—22no—
314C2×D8—42no—
315C2×M16—611no—
316C2×M16—611no—
317C2×M16—612no—
318C2×M16—611no—
319C2×M16—611no—
320C22×C8C22×D6112yes—
321C4⋊Q8—611no—
322C4⋊Q8—611no—
323#127—21no—
324#127—21no—
325#127—21no—
326#127—21no—
327#125—612no—
328C22⋊Q8—1221no—
329C22⋊Q8—1221no—
330C4×Q8—1221no—
331C4×Q8—1221no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336#117—612no—
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C22⋊C4—21no—
339C2×C42C22×D6111yescharacteristic
340#108—2442no—
341#107—1222no—
342#106—1222no—
343C4⋊C8—1222no—
344C4⋊C8—1222no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347C4⋊C8C22⋊D6111yescharacteristic
348C4⋊C8C22⋊D6111yescharacteristic
349C4⋊C8C22⋊D6112yes—
350#102—22no—
351#102—22no—
352#102—1222no—
353#102—1222no—
354#102—1222no—
355#102—1222no—
356#102—1222no—
357#102—1222no—
358#100—1222no—
359#100—1224no—
360#100—1222no—
361#100—2442no—
362#97—1222no—
363#97—1221no—
364#97—1221no—
365C4×C8—1222no—
366C4×C8—1221no—
367C4×C8—1221no—
368C24—21no—
369C24—22no—
370C2×Q8—1221no—
371C2×Q8—1221no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—82no—
388C2×D4—82no—
389C2×D4#1175822412yes—
390C22×C4—21no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—82no—
394C22×C4#1175822411yescharacteristic
395C22×C4#1175822411yescharacteristic
396C22×C4C23×D6111yescharacteristic
397D8—44no—
398D8—44no—
399M16—612no—
400M16—1221no—
401M16—1222no—
402M16—1221no—
403M16—1221no—
404M16—1221no—
405M16—1221no—
406M16—1222no—
407M16—1221no—
408M16—1221no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—1221no—
412C2×C8—1221no—
413C2×C8—1221no—
414C2×C8—1222no—
415C2×C8—1221no—
416C2×C8C2×C22⋊D6112yes—
417C2×C8C2×C22⋊D6112yes—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—612no—
421C4⋊C4—612no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C4⋊C4—1221no—
425C4⋊C4—1221no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—1222no—
429C42—1222no—
430C42D4×D6111yescharacteristic
431C42D4×D6111yescharacteristic
432C42C2×C22⋊D6111yescharacteristic
433C42C2×C22⋊D6111yescharacteristic
434C23—22no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—81no—
441C23—81no—
442C23—82no—
443C23—82no—
444C23C2×#1175822411yescharacteristic
445Q8—4881no—
446Q8—2442no—
447D4—22no—
448D4—22no—
449D4—42no—
450D4—42no—
451D4—41no—
452D4—42no—
453D4—41no—
454D4—42no—
455D4—82no—
456D4—82no—
457D4—82no—
458D4—82no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4#1451974811yescharacteristic
468C2×C4#1451974811yescharacteristic
469C2×C4#1451974811yescharacteristic
470C2×C4#1451974811yescharacteristic
471C2×C4C2×D4×D6111yescharacteristic
472C2×C4C22×C22⋊D6111yescharacteristic
473C2×C4#41×D6111yescharacteristic
474C2×C4C2×#1175822411yescharacteristic
475C2×C4#1451982211yescharacteristic
476C2×C4#1451982411yescharacteristic
477C8—22no—
478C8—22no—
479C8—1222no—
480C8—2441no—
481C8—2441no—
482C8—2441no—
483C8—2442no—
484C22—21no—
485C22—21no—
486C22—42no—
487C22—82no—
488C22—82no—
489C22—81no—
490C22—81no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—162no—
495C22—162no—
496C22C2×#1451974811yescharacteristic
497C22#1684029311yescharacteristic
498C22#1684037911yescharacteristic
499C4—21no—
500C4—21no—
501C4—21no—
502C4—81no—
503C4—2442no—
504C4—2442no—
505C4#1684002912yes—
506C4#1684040611yescharacteristic
507C4#1684041811yescharacteristic
508C4#1684041911yescharacteristic
509C4#1684043011yescharacteristic
510C2—21no—
511C2—21no—
512C2—82no—
513C2—162no—
514C2#2144938111yescharacteristic
515C2C2×#1684002911yescharacteristic
516C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#459C2
socleC22×C61C2×#121
hypercenterC2×#459D61
hypocenterC61#25312
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. #21456193C2semidirect
2. #21455616C2semidirect
3. #21454762C2semidirect
4. #21447177C2semidirect
5. #21447072C2semidirect
6. #21446730C2semidirect
7. #21446660C2semidirect
8. #21446595C2semidirect
9. #21446478C2semidirect
10. C4×#14519694C2semidirect
305. C61#25312semidirect

Maximal subgroups

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

subgroupindex
#214561932
#214556162
#214547622
#214471772
#214470722
#214467302
#214466602
#214465952
#214464782
C4×#145196942
C2×C61×#4592
#2531261