Finite Group Databasebeta

← #29665511

Subgroups — #29665511

Order 15616.

608 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456967C211yesmaximal, characteristic, complemented
2#21455635C211yesmaximal, characteristic
3#21455601C211yesmaximal, characteristic
4#21454928C211yesmaximal, characteristic
5#21445276C212yesmaximal, complemented
6#21445269C212yesmaximal, complemented
7#21445222C212yesmaximal, complemented
8#21445217C212yesmaximal, complemented
9#21444595C211yesmaximal, characteristic, complemented
10C2×#16839510C211yesmaximal, characteristic, complemented
11C61×#1550C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840165C2212yes—
15#16840112C2212yes—
16#16840095—22no—
17#16840066—21no—
18#16840059—21no—
19#16840058C2212yes—
20#16840057—21no—
21#16840057—21no—
22#16840054C2211yescharacteristic
23#16840053C2212yes—
24#16840049C2211yescharacteristic
25#16840048—21no—
26#16840044C2211yescharacteristic
27#16840043—21no—
28#16840039C2211yescharacteristic
29#16839886C2212yes—
30#16839827C2212yes—
31C8×C22⋊D61—22no—
32#16839815C2211yescharacteristic
33#16839814—21no—
34#16839804—21no—
35#16839798—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839764—21no—
39#16839756—21no—
40#16839514C2212yes—
41#16839514C2212yes—
42#16839510C2212yes—
43#16839510C2212yes—
44C61×#533C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#333C2212yes—
48C61×#332C2212yes—
49C2×#11758222D411yescharacteristic
50C22×Dic122—21no—
51C22×#92418—21no—
52C2×C8×D61—22no—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777C2312yes—
57C4×Dic122D411yescharacteristic
58C4×Dic122C2311yescharacteristic
59C4×D244D411yescharacteristic
60C4×D244C2311yescharacteristic
61C4×C22⋊D61—21no—
62C4×C22⋊D61—21no—
63#14519736D411yescharacteristic
64#14519734—21no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C42×D61—21no—
68#14519711—22no—
69#14519710—21no—
70#14519710—21no—
71#14519710—22no—
72#14519710—22no—
73#14519709—22no—
74#14519708—21no—
75#14519708—21no—
76C4⋊#92418—21no—
77C4⋊#92418D411yescharacteristic
78C4⋊#92418D411yescharacteristic
79C4⋊#92418C2311yescharacteristic
80C4⋊#92418C2311yescharacteristic
81#14519706—22no—
82#14519702—22no—
83#14519701—22no—
84#14519701—22no—
85#14519700—21no—
86#14519700—21no—
87#14519699—22no—
88#14519697—21no—
89#14519697—21no—
90C22⋊#92418D412yes—
91#14519689—22no—
92C4×#92418C2312yes—
93C8×Dic61—22no—
94C2×M16×C61—21no—
95C22×C8×C61—21no—
96C4⋊Q8×C61C2311yescharacteristic
97C4⋊Q8×C61C2311yescharacteristic
98#125×C61—21no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103#117×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108C4⋊C8×C61C2312yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—21no—
115C22×Dic61—41no—
116#11758222—22no—
117#11758222Q1614yes—
118C2×D244C2×D411yescharacteristic
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×Dic122—22no—
123C2×Dic122—41no—
124C2×Dic122—82no—
125C2×Dic122C2×D411yescharacteristic
126M16×C61—42no—
127C2×C8×C61—22no—
128C2×C8×C61—22no—
129C2×C8×C61—81no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C42×C61—41no—
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61#4111yescharacteristic
141C42×C61C2411yescharacteristic
142C22⋊Dic61—21no—
143#11758198—21no—
144#11758198—21no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149C4⋊Dic61C2×D411yescharacteristic
150C4⋊Dic61C2×D411yescharacteristic
151#11758196—21no—
152#11758196—21no—
153C4×Dic61—21no—
154C4×Dic61—21no—
155C4×Dic61—21no—
156C2×#92418—21no—
157C2×#92418—21no—
158C2×#92418—42no—
159C2×#92418C2×D412yes—
160C2×#92418C2×D412yes—
161C8×D61—82no—
162C23×C61#12111yescharacteristic
163C22×D61—21no—
164Q8×C61—41no—
165Q8×C61—41no—
166Q8×C61—81no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61C2×Q1612yes—
176C2×C4×C61#13512yes—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C22⋊D61—24no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—42no—
185C2×Dic61—82no—
186D244—21no—
187D244C2×Q1612yes—
188C4×D61—24no—
189C4×D61—41no—
190C4×D61—41no—
191C4×D61—81no—
192Dic122—21no—
193Dic122—82no—
194Dic122—82no—
195Dic122C2×Q1612yes—
196C8×C61—42no—
197C8×C61—42no—
198#92418—22no—
199#92418—22no—
200#92418—41no—
201#92418—41no—
202#19417—611nomaximal, Sylow
203C22×C61—21no—
204C22×C61—21no—
205C22×C61#46712yes—
206C22×C61C2×#12111yescharacteristic
207C2×D61—22no—
208C2×D61—41no—
209C2×D61—41no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—81no—
214C4×C61—81no—
215C4×C61#46712yes—
216C4×C61C22×Q1611yescharacteristic
217C4×C61C2×#13511yescharacteristic
218C4×C61#57711yescharacteristic
219C4×C61#57811yescharacteristic
220Dic61—22no—
221Dic61—41no—
222Dic61—41no—
223Dic61—82no—
224#3230—611no—
225#3079—611no—
226#3076—611no—
227#2988—611no—
228#1684—612no—
229#1683—612no—
230#1676—612no—
231#1675—612no—
232#1571—611no—
233#1550D6111yescharacteristic, complemented
234C2×#333—611no—
235C2×C61—22no—
236C2×C61#304611yescharacteristic
237C2×C61C2×#46711yescharacteristic
238C2×C61#306811yescharacteristic
239D61—42no—
240#533C2×D6111yescharacteristic
241C2×C4⋊Q8—611no—
242C4×#41—611no—
243#500—612no—
244#487—612no—
245#484—1222no—
246#479—1221no—
247#478—1221no—
248#477—611no—
249#477—612no—
250#477—1221no—
251#477—1221no—
252#476—611no—
253#476—612no—
254#475—611no—
255#475—1221no—
256#474—611no—
257#474—1221no—
258C8×Q8—612no—
259C8×D4—612no—
260C8×D4—1222no—
261#431—611no—
262#431—1221no—
263#428—1221no—
264#427—1221no—
265#423C2×D6111yescharacteristic
266C2×C4⋊C8—611no—
267C2×C4⋊C8—1221no—
268C2×C4⋊C8C2×D6111yescharacteristic
269#417—1221no—
270#416—1221no—
271#335—612no—
272#335—612no—
273#333—612no—
274#333—612no—
275#333C2×D6112yes—
276#332C2×D6112yes—
277C61#1941711yescharacteristic, complemented, Sylow
278C2×#41—611no—
279C22×Q8—1221no—
280C2×M16—21no—
281C22×C8—21no—
282C22×C8—1222no—
283C22×C8—1221no—
284C4⋊Q8—612no—
285C4⋊Q8—1221no—
286C4⋊Q8—1221no—
287C4⋊Q8—1221no—
288C4⋊Q8C22×D6111yescharacteristic
289C4⋊Q8C22×D6111yescharacteristic
290#125—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—611no—
296C4×Q8—611no—
297C4×D4—611no—
298C4×D4—611no—
299C4×D4—1221no—
300C4×D4—1221no—
301#117—21no—
302#117—611no—
303#117—1221no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1221no—
306C2×C42—1221no—
307C2×C42C22×D6111yescharacteristic
308#107—1221no—
309#107—1221no—
310#107—1222no—
311#107—1222no—
312#107—1222no—
313#106—1221no—
314#106—1221no—
315#106—1222no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—611no—
319C4⋊C8—611no—
320C4⋊C8—611no—
321C4⋊C8—611no—
322C4⋊C8—1222no—
323C4⋊C8—1221no—
324C4⋊C8C22×D6112yes—
325C4⋊C8C22×D6112yes—
326#103—1221no—
327#103—1221no—
328#103—1222no—
329#103—1222no—
330#103—1222no—
331#102—1221no—
332#102—1221no—
333#102—1222no—
334C22⋊C8—612no—
335C22⋊C8—1222no—
336C4×C8—612no—
337C4×C8—1222no—
338#41—614no—
339#41—1222no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—611no—
343C2×Q8—1222no—
344C2×Q8—4882no—
345C2×Q8—2441no—
346C2×D4—611no—
347C2×D4—1221no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—2441no—
352C22×C4#1175822211yescharacteristic
353C22×C4D4×D6111yescharacteristic
354C22×C4D4×D6111yescharacteristic
355M16—42no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—81no—
359C2×C8—612no—
360C2×C8—612no—
361C2×C8—1221no—
362C2×C8—1221no—
363C2×C8—4882no—
364C2×C8—2442no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—611no—
372C4⋊C4—611no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4—1222no—
376C4⋊C4—1222no—
377C4⋊C4—2441no—
378C4⋊C4—2441no—
379C22⋊C4—41no—
380C22⋊C4—1221no—
381C22⋊C4—1221no—
382C22⋊C4—1221no—
383C42—41no—
384C42—1221no—
385C42—1221no—
386C42—1221no—
387C42#1175822211yescharacteristic
388C42D4×D6111yescharacteristic
389C42D4×D6111yescharacteristic
390C42C23×D6111yescharacteristic
391C23—1221no—
392C23#1451975011yescharacteristic
393Q8—41no—
394Q8—41no—
395Q8—81no—
396Q8—612no—
397Q8—1221no—
398Q8—4882no—
399Q8—4882no—
400D4—612no—
401D4—1224no—
402D4—1221no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—1221no—
409C2×C4—1224no—
410C2×C4—1221no—
411C2×C4—1221no—
412C2×C4—4881no—
413C2×C4—4882no—
414C2×C4—2442no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4#1451975011yescharacteristic
418C2×C4#1451975011yescharacteristic
419C2×C4#1451975011yescharacteristic
420C2×C4Q16×D6112yes—
421C2×C4#1451979812yes—
422C2×C4C2×D4×D6111yescharacteristic
423C2×C4C2×D4×D6111yescharacteristic
424C2×C4C2×#1175822211yescharacteristic
425C8—42no—
426C8—42no—
427C8—1222no—
428C8—1222no—
429C8—2441no—
430C8—2441no—
431C22—21no—
432C22—21no—
433C22—1222no—
434C22—2441no—
435C22—2441no—
436C22#1684000212yes—
437C22C2×#1451975011yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—1222no—
444C4—4882no—
445C4—2441no—
446C4—2441no—
447C4#1684000212yes—
448C4C2×Q16×D6111yescharacteristic
449C4C2×#1451979811yescharacteristic
450C4#1684057211yescharacteristic
451C4#1684057811yescharacteristic
452C2—22no—
453C2—2442no—
454C2#2145536811yescharacteristic
455C2C2×#1684000211yescharacteristic
456C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1550C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#1550D61
hypocenterC61#19417
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1550

Complements

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

NH ≅ G/Ntype
1. #21456967C2semidirect
5. #21445276C2semidirect
6. #21445269C2semidirect
7. #21445222C2semidirect
8. #21445217C2semidirect
9. #21444595C2semidirect
10. C2×#16839510C2semidirect
11. C61×#1550C2semidirect
233. #1550D61semidirect
277. C61#19417semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214556352
#214556012
#214549282
#214452762
#214452692
#214452222
#214452172
#214445952
C2×#168395102
C61×#15502
#1941761