Finite Group Databasebeta

← #29665443

Subgroups — #29665443

Order 15616.

620 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456767C211yesmaximal, characteristic, complemented
2#21455628C211yesmaximal, characteristic
3#21455609C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic
5#21445276C212yesmaximal, complemented
6#21445269C212yesmaximal, complemented
7#21445188C212yesmaximal, complemented
8#21445183C212yesmaximal, complemented
9#21444593C211yesmaximal, characteristic, complemented
10#21444507C211yesmaximal, characteristic, complemented
11C2×C61×#332C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840141C2212yes—
15#16840112C2212yes—
16#16840070—21no—
17#16840063—21no—
18#16840058C2211yescharacteristic
19#16840054—21no—
20#16840053C2211yescharacteristic
21#16840049—21no—
22#16840048C2211yescharacteristic
23#16840044C2212yes—
24#16840043C2211yescharacteristic
25#16840039C2212yes—
26#16839969—21no—
27#16839969—21no—
28#16839918—22no—
29#16839886C2212yes—
30#16839827C2212yes—
31C8×C22⋊D61—22no—
32#16839815C2211yescharacteristic
33#16839814—21no—
34#16839802—21no—
35#16839800—21no—
36#16839786C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839761—21no—
39#16839759—21no—
40#16839514C2212yes—
41#16839514C2212yes—
42#16839510C2212yes—
43#16839509C2212yes—
44C2×C4⋊Q8×C61C2211yescharacteristic
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yes—
48C61×#332C2212yes—
49C2×#11758222D411yescharacteristic
50C2×#11758202—22no—
51C2×#11758202—22no—
52C2×#11758194—21no—
53C2×C8×D61—22no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519769—21no—
57#14519756—21no—
58#14519756—21no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61C4×Dic122D411yescharacteristic
62C4×Dic122C2311yescharacteristic
63C4×D244D411yescharacteristic
64C4×D244C2311yescharacteristic
65C4×C22⋊D61—21no—
66C4×C22⋊D61—21no—
67#14519736—21no—
68#14519736D411yescharacteristic
69#14519734—21no—
70C42×D61—21no—
71#14519711—21no—
72#14519711—21no—
73#14519710—22no—
74#14519709—21no—
75#14519709—21no—
76#14519708—22no—
77C4⋊#92418—21no—
78C4⋊#92418D411yescharacteristic
79C4⋊#92418D411yescharacteristic
80C4⋊#92418C2312yes—
81#14519706—22no—
82#14519702—21no—
83#14519702—21no—
84#14519701—22no—
85#14519701—22no—
86#14519700—22no—
87#14519699—21no—
88#14519699—21no—
89#14519697—22no—
90C22⋊#92418D412yes—
91#14519689—22no—
92C4×#92418C2312yes—
93C8×Dic61—22no—
94C22×Q8×C61—21no—
95C22×C8×C61—21no—
96C22×C8×C61—21no—
97C4⋊Q8×C61—21no—
98C4⋊Q8×C61—21no—
99C4⋊Q8×C61—21no—
100C4⋊Q8×C61C2312yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104C4⋊C8×C61—21no—
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107C4⋊C8×C61C2311yescharacteristic
108C4⋊C8×C61C2311yescharacteristic
109C2×Q8×C61—22no—
110C2×Q8×C61—41no—
111C2×Q8×C61—81no—
112C2×Q8×C61—81no—
113C22×C4×C61—41no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C2×C22⋊D61—21no—
118#11758222—22no—
119#11758222Q1614yescomplemented
120C2×D244C2×D411yescharacteristic
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122C2×D411yescharacteristic
127C2×C8×C61—22no—
128C2×C8×C61—21no—
129C2×C8×C61—21no—
130C2×C8×C61—42no—
131C2×C8×C61—81no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61#4111yescharacteristic
139C42×C61C2411yescharacteristic
140C22⋊Dic61—21no—
141C22⋊Dic61—41no—
142#11758202—82no—
143#11758202—82no—
144#11758198—21no—
145#11758198—21no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61C2×D411yescharacteristic
151C4⋊Dic61C2×D411yescharacteristic
152#11758196—21no—
153#11758196—21no—
154#11758196—41no—
155#11758196—41no—
156C4×Dic61—21no—
157C4×Dic61—21no—
158C4×Dic61—21no—
159C4×Dic61—41no—
160#11758194—42no—
161C2×#92418—22no—
162C2×#92418C2×D412yes—
163C2×#92418C2×D412yes—
164C8×D61—82no—
165C23×C61#12111yescharacteristic
166C22×D61—21no—
167Q8×C61—42no—
168Q8×C61—42no—
169Q8×C61—82no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—41no—
173C2×C4×C61—42no—
174C2×C4×C61—82no—
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61C2×Q1612yes—
179C2×C4×C61#13512yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C22⋊D61—24no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189D244—21no—
190D244C2×Q1612yes—
191C4×D61—24no—
192C4×D61—41no—
193C4×D61—41no—
194C4×D61—81no—
195Dic122—21no—
196Dic122—41no—
197Dic122—41no—
198Dic122—81no—
199Dic122C2×Q1612yes—
200C8×C61—41no—
201C8×C61—41no—
202C8×C61—42no—
203#92418—22no—
204#92418—22no—
205#92418—42no—
206#19411—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61#46712yes—
210C22×C61C2×#12111yescharacteristic
211C2×D61—22no—
212C2×D61—41no—
213C2×D61—41no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—82no—
218C4×C61#46712yes—
219C4×C61C22×Q1611yescharacteristic
220C4×C61C2×#13511yescharacteristic
221C4×C61#57611yescharacteristic
222C4×C61#57711yescharacteristic
223Dic61—22no—
224Dic61—41no—
225Dic61—41no—
226Dic61—81no—
227Dic61—81no—
228#3209—611no—
229#3079—611no—
230#3076—611no—
231#2988—611no—
232#1684—612no—
233#1683—612no—
234#1670—612no—
235#1669—612no—
236#1571—611no—
237#1550—611no—
238C2×#332D6111yescharacteristic, complemented
239C2×C61—22no—
240C2×C61#304511yescharacteristic
241C2×C61C2×#46711yescharacteristic
242C2×C61#306911yescharacteristic
243D61—42no—
244#533—611no—
245C2×C4⋊Q8C2×D6111yescharacteristic
246C4×#41—611no—
247#494—612no—
248#487—612no—
249#479—1221no—
250#478—1221no—
251#477—611no—
252#477—1221no—
253#476—611no—
254#476—1221no—
255#475—612no—
256#475—611no—
257#474—612no—
258#474—611no—
259#462—1221no—
260#462—1221no—
261#451—1222no—
262C8×Q8—612no—
263C8×D4—612no—
264C8×D4—1222no—
265#431—611no—
266#431—1221no—
267#428—1221no—
268#427—1221no—
269#423—611no—
270C2×C4⋊C8—1221no—
271C2×C4⋊C8C2×D6111yescharacteristic
272C2×C4⋊C8C2×D6111yescharacteristic
273#417—1221no—
274#416—1221no—
275#335—612no—
276#335—612no—
277#333—612no—
278#332—612no—
279#332C2×D6112yes—
280#332C2×D6112yes—
281C61#1941111yescharacteristic, complemented, Sylow
282C2×#41—611no—
283C22×Q8—21no—
284C2×Q16—1222no—
285C2×Q16—1222no—
286C2×M16—1221no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—1222no—
290C4⋊Q8—21no—
291C4⋊Q8—21no—
292C4⋊Q8—21no—
293C4⋊Q8—611no—
294C4⋊Q8—611no—
295C4⋊Q8C22×D6112yes—
296#125—1221no—
297C22⋊Q8—1221no—
298C22⋊Q8—1221no—
299C4×Q8—611no—
300C4×Q8—611no—
301C4×Q8—1221no—
302C4×Q8—1221no—
303C4×D4—611no—
304C4×D4—611no—
305C4×D4—1221no—
306C4×D4—1221no—
307#117—611no—
308#117—1221no—
309#117—1221no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C42—1221no—
313C2×C42C22×D6111yescharacteristic
314#107—1222no—
315#107—1221no—
316#107—1221no—
317#106—1222no—
318#106—1221no—
319#106—1221no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—611no—
323C4⋊C8—611no—
324C4⋊C8—612no—
325C4⋊C8—1222no—
326C4⋊C8—1221no—
327C4⋊C8C22×D6112yes—
328C4⋊C8C22×D6111yescharacteristic
329C4⋊C8C22×D6111yescharacteristic
330#103—1222no—
331#103—1221no—
332#103—1221no—
333#103—1222no—
334#103—1222no—
335#102—1222no—
336#102—1221no—
337#102—1221no—
338C22⋊C8—612no—
339C22⋊C8—1222no—
340C4×C8—612no—
341C4×C8—1222no—
342#41—614no—
343#41—1222no—
344C2×Q8—22no—
345C2×Q8—41no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—611no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×D4—611no—
352C2×D4—1221no—
353C22×C4—41no—
354C22×C4—1221no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4#1175822211yescharacteristic
358C22×C4D4×D6111yescharacteristic
359C22×C4D4×D6111yescharacteristic
360Q16—4882no—
361Q16—4882no—
362M16—2442no—
363C2×C8—22no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—42no—
367C2×C8—81no—
368C2×C8—612no—
369C2×C8—612no—
370C2×C8—1222no—
371C2×C8—4882no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—611no—
377C4⋊C4—611no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C22⋊C4—1221no—
387C22⋊C4—1221no—
388C22⋊C4—1221no—
389C22⋊C4—2441no—
390C42—1221no—
391C42—1221no—
392C42—1221no—
393C42—2441no—
394C42#1175822211yescharacteristic
395C42D4×D6111yescharacteristic
396C42D4×D6111yescharacteristic
397C42C23×D6111yescharacteristic
398C23—1221no—
399C23#1451975011yescharacteristic
400Q8—42no—
401Q8—42no—
402Q8—82no—
403Q8—612no—
404Q8—1221no—
405Q8—4881no—
406Q8—2441no—
407Q8—2441no—
408D4—612no—
409D4—1224no—
410D4—1221no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—82no—
416C2×C4—1221no—
417C2×C4—1224no—
418C2×C4—1221no—
419C2×C4—1221no—
420C2×C4—4881no—
421C2×C4—2441no—
422C2×C4—2441no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4#1451975011yescharacteristic
426C2×C4#1451975011yescharacteristic
427C2×C4#1451975011yescharacteristic
428C2×C4Q16×D6112yes—
429C2×C4#1451979812yes—
430C2×C4C2×D4×D6111yescharacteristic
431C2×C4C2×D4×D6111yescharacteristic
432C2×C4C2×#1175822211yescharacteristic
433C8—41no—
434C8—41no—
435C8—42no—
436C8—1222no—
437C8—1222no—
438C8—2442no—
439C22—21no—
440C22—21no—
441C22—1222no—
442C22—2441no—
443C22—2441no—
444C22#1684000212yes—
445C22C2×#1451975011yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—82no—
450C4—1222no—
451C4—4881no—
452C4—4881no—
453C4—2441no—
454C4—2441no—
455C4#1684000212yes—
456C4C2×Q16×D6111yescharacteristic
457C4C2×#1451979811yescharacteristic
458C4#1684056311yescharacteristic
459C4#1684056811yescharacteristic
460C2—22no—
461C2—2442no—
462C2#2145536311yescharacteristic
463C2C2×#1684000211yescharacteristic
464C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#332C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenterC2×#332D61
hypocenterC61#19411
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
1. #21456767C2semidirect
5. #21445276C2semidirect
6. #21445269C2semidirect
7. #21445188C2semidirect
8. #21445183C2semidirect
9. #21444593C2semidirect
10. #21444507C2semidirect
11. C2×C61×#332C2semidirect
119. #11758222Q16semidirect
238. C2×#332D61semidirect
281. C61#19411semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214556282
#214556092
#214549282
#214452762
#214452692
#214451882
#214451832
#214445932
#214445072
C2×C61×#3322
#1941161