Finite Group Databasebeta

← #29665199

Subgroups — #29665199

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456767C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16840039C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445269C214yesmaximal, complemented
6#21445183C214yesmaximal, complemented
7#21444595C211yesmaximal, characteristic, complemented
8#21444474C211yesmaximal, characteristic, complemented
9C61×#1549C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11C2×C4×D244C2211yescharacteristic
12#16840141C2212yescomplemented
13#16840112C2212yescomplemented
14#16840063—22no—
15#16840053C2212yes—
16#16840043C2212yes—
17#16840039—22no—
18#16840039C2214yes—
19#16839969—22no—
20#16839918—22no—
21C8×D244—22no—
22#16839827C2214yes—
23#16839815C2211yescharacteristic
24#16839804—21no—
25C2×#14519710—21no—
26C2×C4⋊#92418C2211yescharacteristic
27#16839768—21no—
28C2×#14519697—21no—
29C22⋊C8×D61—22no—
30#16839514C2214yescomplemented
31#16839510C2212yescomplemented
32#16839509C2212yescomplemented
33C2×C4⋊Q8×C61C2211yescharacteristic
34C61×#431C2211yescharacteristic
35C61×#423C2211yescharacteristic
36C61×#332C2214yescomplemented
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39C2×#11758202—24no—
40C22×#92418—21no—
41C2×C8×D61—24no—
42#14519777C2311yescharacteristic
43#14519777C2311yescharacteristic
44#14519769—21no—
45#14519756—21no—
46#14519756—21no—
47C4×Dic122—21no—
48C4×Dic122—21no—
49C4×D244—21no—
50C4×D244—21no—
51C4×D244D412yes—
52C4×D244C2312yes—
53#14519736—21no—
54C2×C4⋊Dic61D411yescharacteristic
55C2×#11758198—21no—
56#14519711—22no—
57#14519710—24no—
58#14519709—22no—
59C4⋊#92418—21no—
60C4⋊#92418D412yes—
61C4⋊#92418C2312yes—
62#14519701—24no—
63#14519699—22no—
64#14519699—22no—
65#14519697—24no—
66C22⋊#92418D412yes—
67#14519689—24no—
68C4×#92418C2312yes—
69C22×Q8×C61—21no—
70C2×M16×C61—21no—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61C2312yes—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C42×C61C2311yescharacteristic
78C4⋊C8×C61—21no—
79C4⋊C8×C61C2312yes—
80C4⋊C8×C61C2312yes—
81C22⋊C8×C61—21no—
82C4×C8×C61—21no—
83C23×D61—21no—
84C2×Q8×C61—22no—
85C2×Q8×C61—41no—
86C2×Q8×C61—82no—
87C22×C4×C61—41no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61#4111yescharacteristic
91C22×Dic61—21no—
92C2×D244—21no—
93C2×D244—22no—
94C2×D244Q1614yescomplemented
95C2×D244C2×D412yes—
96C2×C4×D61—22no—
97C2×C4×D61—22no—
98C2×C4×D61—82no—
99C2×Dic122—21no—
100C2×Dic122—21no—
101M16×C61—42no—
102C2×C8×C61—22no—
103C2×C8×C61—22no—
104C4⋊C4×C61—22no—
105C4⋊C4×C61—22no—
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C42×C61C2×D411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61#4111yescharacteristic
111C42×C61C2411yescharacteristic
112C22⋊Dic61—41no—
113#11758202—84no—
114#11758198—22no—
115#11758198—22no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61C2×D412yes—
122#11758196—41no—
123#11758196—41no—
124C4×Dic61—41no—
125C2×#92418—22no—
126C2×#92418—42no—
127C2×#92418C2×D414yes—
128C8×D61—84no—
129C23×C61#12111yescharacteristic
130C22×D61—22no—
131C22×D61—24no—
132C22×D61—81no—
133C22×D61—81no—
134Q8×C61—44no—
135Q8×C61—82no—
136C2×C4×C61—22no—
137C2×C4×C61—41no—
138C2×C4×C61—42no—
139C2×C4×C61—82no—
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61C2×Q1612yes—
144C2×C4×C61#13512yes—
145C2×C4×C61C22×D411yescharacteristic
146C2×C4×C61C22×D411yescharacteristic
147C2×C4×C61C2×#4111yescharacteristic
148C2×Dic61—22no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—81no—
152D244—22no—
153D244—24no—
154D244C2×Q1614yes—
155C4×D61—44no—
156C4×D61—82no—
157Dic122—42no—
158Dic122—81no—
159C8×C61—42no—
160C8×C61—42no—
161#92418—24no—
162#92418—42no—
163#19387—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61#46712yes—
167C22×C61C2×#12111yescharacteristic
168C2×D61—24no—
169C2×D61—42no—
170C2×D61—42no—
171C2×D61—84no—
172C4×C61—21no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—82no—
176C4×C61#46712yes—
177C4×C61C22×Q1611yescharacteristic
178C4×C61C2×#13511yescharacteristic
179C4×C61#57612yes—
180Dic61—42no—
181Dic61—81no—
182Dic61—81no—
183#3209—611no—
184#3077—611no—
185C2×#474—611no—
186#2972—611no—
187#1683—614no—
188#1669—614no—
189#1571—611no—
190#1549D6111yescharacteristic, complemented
191#1542—611no—
192C2×C61—22no—
193C2×C61#304111yescharacteristic
194C2×C61C2×#46711yescharacteristic
195C2×C61#306911yescharacteristic
196D61—44no—
197#533—611no—
198C2×C4⋊Q8C2×D6111yescharacteristic
199C2×C4×D4—611no—
200#494—612no—
201#487—612no—
202#478—1222no—
203#476—612no—
204#474—614no—
205#474—612no—
206#474—1222no—
207#462—1222no—
208#451—1222no—
209C8×D4—614no—
210C8×D4—1222no—
211#431—611no—
212#431C2×D6111yescharacteristic
213#428—1221no—
214C2×#107—1221no—
215#423C2×D6111yescharacteristic
216C2×C4⋊C8—611no—
217#418—1221no—
218C2×#102—1221no—
219C2×C22⋊C8—1222no—
220#335—614no—
221#333—612no—
222#332—612no—
223#332C2×D6114yescomplemented
224C61#1938711yescharacteristic, complemented, Sylow
225C22×Q8—21no—
226C22×D4—611no—
227C23×C4—1221no—
228C2×Q16—1224no—
229C2×M16—21no—
230C22×C8—1224no—
231C22×C8—1221no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—611no—
236C4⋊Q8—611no—
237C4⋊Q8C22×D6112yes—
238#125—1221no—
239C22⋊Q8—1221no—
240C22⋊Q8—1221no—
241C4×Q8—1221no—
242C4×Q8—1221no—
243C4×D4—612no—
244C4×D4—612no—
245C4×D4—1221no—
246C4×D4—1221no—
247#117—1221no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—611no—
251C2×C22⋊C4—1221no—
252C2×C42C22×D6111yescharacteristic
253#107—1224no—
254#107—1222no—
255#106—1222no—
256C4⋊C8—21no—
257C4⋊C8—612no—
258C4⋊C8—612no—
259C4⋊C8—1221no—
260C4⋊C8C22×D6112yes—
261C4⋊C8C22×D6112yes—
262#103—1224no—
263#102—1224no—
264#102—1222no—
265#102—1222no—
266C22⋊C8—21no—
267C22⋊C8—612no—
268C22⋊C8—1224no—
269C4×C8—21no—
270C4×C8—612no—
271C24—1221no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—82no—
275C2×Q8—1221no—
276C2×Q8—1221no—
277C2×D4—612no—
278C2×D4—614no—
279C2×D4—1221no—
280C2×D4—1222no—
281C22×C4—41no—
282C22×C4—1221no—
283C22×C4—1222no—
284C22×C4—1222no—
285C22×C4—4882no—
286C22×C4#1175822211yescharacteristic
287C22×C4D4×D6111yescharacteristic
288C22×C4D4×D6111yescharacteristic
289Q16—4884no—
290M16—42no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—614no—
294C2×C8—1222no—
295C2×C8—4884no—
296C2×C8—2442no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—612no—
302C4⋊C4—1221no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4—1221no—
307C4⋊C4—2441no—
308C4⋊C4—2441no—
309C22⋊C4—1222no—
310C22⋊C4—1222no—
311C22⋊C4—2441no—
312C42—2441no—
313C42#1175822211yescharacteristic
314C42D4×D6111yescharacteristic
315C42D4×D6111yescharacteristic
316C42C23×D6111yescharacteristic
317C23—1224no—
318C23—1222no—
319C23—4881no—
320C23—4881no—
321C23#1451975011yescharacteristic
322Q8—44no—
323Q8—82no—
324Q8—4881no—
325Q8—2442no—
326D4—614no—
327D4—1224no—
328D4—1222no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—82no—
333C2×C4—1222no—
334C2×C4—4882no—
335C2×C4—4881no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2444no—
339C2×C4#1451975011yescharacteristic
340C2×C4#1451975011yescharacteristic
341C2×C4#1451975011yescharacteristic
342C2×C4Q16×D6112yes—
343C2×C4#1451979812yes—
344C2×C4C2×D4×D6111yescharacteristic
345C2×C4C2×D4×D6111yescharacteristic
346C2×C4C2×#1175822211yescharacteristic
347C8—42no—
348C8—42no—
349C8—1224no—
350C8—2442no—
351C22—21no—
352C22—21no—
353C22—1224no—
354C22—4884no—
355C22—2442no—
356C22—2442no—
357C22#1684000212yes—
358C22C2×#1451975011yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—82no—
363C4—4881no—
364C4—4881no—
365C4—2442no—
366C4#1684000212yes—
367C4C2×Q16×D6111yescharacteristic
368C4C2×#1451979811yescharacteristic
369C4#1684056612yes—
370C2—22no—
371C2—2444no—
372C2#2145532111yescharacteristic
373C2C2×#1684000211yescharacteristic
374C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1549D61
hypocenterC61#19387
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. #21456767C2semidirect
5. #21445269C2semidirect
6. #21445183C2semidirect
7. #21444595C2semidirect
8. #21444474C2semidirect
9. C61×#1549C2semidirect
12. #16840141C22semidirect
13. #16840112C22semidirect
30. #16839514C22semidirect
31. #16839510C22semidirect
32. #16839509C22semidirect
36. C61×#332C22semidirect
94. C2×D244Q16semidirect
190. #1549D61semidirect
223. #332C2×D61semidirect
224. C61#19387semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214556182
C2×#168400392
#214548422
#214452692
#214451832
#214445952
#214444742
C61×#15492
#1938761