Finite Group Databasebeta

← #29665257

Subgroups — #29665257

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456970C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16840049C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445273C214yesmaximal, complemented
6#21445217C214yesmaximal, complemented
7#21444595C211yesmaximal, characteristic, complemented
8#21444474C211yesmaximal, characteristic, complemented
9C61×#1548C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11C2×C4×D244C2211yescharacteristic
12#16840165C2212yescomplemented
13#16840114C2212yescomplemented
14#16840097—22no—
15#16840063—22no—
16#16840054—22no—
17#16840053C2212yes—
18#16840049—22no—
19#16840049C2214yes—
20#16840043C2212yes—
21C8×D244—22no—
22#16839827C2214yes—
23#16839815C2211yescharacteristic
24#16839804—21no—
25C2×#14519708—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×#333C2214yescomplemented
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39C22×#92418—21no—
40C2×C8×D61—24no—
41#14519777C2311yescharacteristic
42#14519777C2311yescharacteristic
43#14519769—21no—
44#14519756—21no—
45#14519756—21no—
46C4×Dic122—21no—
47C4×Dic122—21no—
48C4×D244—21no—
49C4×D244—21no—
50C4×D244D412yes—
51C4×D244C2312yes—
52#14519736—21no—
53C2×C4⋊Dic61D411yescharacteristic
54C2×#11758198—21no—
55#14519711—22no—
56#14519710—24no—
57#14519709—22no—
58#14519708—24no—
59C4⋊#92418—21no—
60C4⋊#92418D412yes—
61C4⋊#92418C2312yes—
62#14519700—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×D244Q1614yes—
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#11758198—22no—
114#11758198—22no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—21no—
120C4⋊Dic61C2×D412yes—
121#11758196—41no—
122#11758196—41no—
123C4×Dic61—41no—
124C2×#92418—22no—
125C2×#92418—42no—
126C2×#92418C2×D414yes—
127C8×D61—84no—
128C23×C61#12111yescharacteristic
129C22×D61—22no—
130C22×D61—24no—
131C22×D61—81no—
132C22×D61—81no—
133Q8×C61—82no—
134Q8×C61—82no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—42no—
138C2×C4×C61—82no—
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61C2×Q1612yes—
143C2×C4×C61#13512yes—
144C2×C4×C61C22×D411yescharacteristic
145C2×C4×C61C22×D411yescharacteristic
146C2×C4×C61C2×#4111yescharacteristic
147C2×Dic61—22no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—81no—
151D244—22no—
152D244—24no—
153D244C2×Q1614yes—
154C4×D61—44no—
155C4×D61—82no—
156Dic122—42no—
157Dic122—81no—
158C8×C61—42no—
159C8×C61—42no—
160#92418—24no—
161#92418—42no—
162#19393—611nomaximal, Sylow
163C22×C61—21no—
164C22×C61—21no—
165C22×C61#46712yes—
166C22×C61C2×#12111yescharacteristic
167C2×D61—24no—
168C2×D61—42no—
169C2×D61—42no—
170C2×D61—84no—
171C4×C61—21no—
172C4×C61—21no—
173C4×C61—21no—
174C4×C61—82no—
175C4×C61#46712yes—
176C4×C61C22×Q1611yescharacteristic
177C4×C61C2×#13511yescharacteristic
178C4×C61#57712yes—
179Dic61—42no—
180Dic61—81no—
181Dic61—81no—
182#3230—611no—
183#3077—611no—
184C2×#476—611no—
185#2972—611no—
186#1683—614no—
187#1675—614no—
188#1571—611no—
189#1548D6111yescharacteristic, complemented
190#1542—611no—
191C2×C61—22no—
192C2×C61#304211yescharacteristic
193C2×C61C2×#46711yescharacteristic
194C2×C61#306811yescharacteristic
195D61—44no—
196#533—611no—
197C2×C4⋊Q8C2×D6111yescharacteristic
198C2×C4×D4—611no—
199#500—612no—
200#487—612no—
201#484—1222no—
202#478—1222no—
203#477—1222no—
204#476—614no—
205#476—612no—
206#476—1222no—
207#474—612no—
208C8×D4—614no—
209C8×D4—1222no—
210#431—611no—
211#431C2×D6111yescharacteristic
212#428—1221no—
213C2×#106—1221no—
214#423C2×D6111yescharacteristic
215C2×C4⋊C8—611no—
216#418—1221no—
217C2×#102—1221no—
218C2×C22⋊C8—1222no—
219#335—614no—
220#333—612no—
221#333C2×D6114yescomplemented
222#332—612no—
223C61#1939311yescharacteristic, complemented, Sylow
224C22×Q8—21no—
225C22×D4—611no—
226C23×C4—1221no—
227C2×M16—21no—
228C22×C8—1224no—
229C22×C8—1221no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—611no—
234C4⋊Q8—611no—
235C4⋊Q8C22×D6112yes—
236#125—1221no—
237C22⋊Q8—1221no—
238C22⋊Q8—1221no—
239C4×Q8—1221no—
240C4×Q8—1221no—
241C4×D4—612no—
242C4×D4—612no—
243C4×D4—1221no—
244C4×D4—1221no—
245#117—1221no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—611no—
249C2×C22⋊C4—1221no—
250C2×C42C22×D6111yescharacteristic
251#107—1224no—
252#107—1222no—
253#106—1224no—
254#106—1222no—
255C4⋊C8—21no—
256C4⋊C8—612no—
257C4⋊C8—612no—
258C4⋊C8—1221no—
259C4⋊C8C22×D6112yes—
260C4⋊C8C22×D6112yes—
261#103—1224no—
262#102—1224no—
263#102—1222no—
264#102—1222no—
265C22⋊C8—21no—
266C22⋊C8—612no—
267C22⋊C8—1224no—
268C4×C8—21no—
269C4×C8—612no—
270C24—1221no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—82no—
274C2×Q8—1221no—
275C2×Q8—1221no—
276C2×D4—612no—
277C2×D4—614no—
278C2×D4—1221no—
279C2×D4—1222no—
280C22×C4—41no—
281C22×C4—1221no—
282C22×C4—1222no—
283C22×C4—1222no—
284C22×C4—4882no—
285C22×C4#1175822211yescharacteristic
286C22×C4D4×D6111yescharacteristic
287C22×C4D4×D6111yescharacteristic
288M16—42no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—614no—
292C2×C8—1222no—
293C2×C8—4884no—
294C2×C8—2442no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—612no—
300C4⋊C4—1221no—
301C4⋊C4—1221no—
302C4⋊C4—1221no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C22⋊C4—1222no—
308C22⋊C4—1222no—
309C22⋊C4—2441no—
310C42—2441no—
311C42#1175822211yescharacteristic
312C42D4×D6111yescharacteristic
313C42D4×D6111yescharacteristic
314C42C23×D6111yescharacteristic
315C23—1224no—
316C23—1222no—
317C23—4881no—
318C23—4881no—
319C23#1451975011yescharacteristic
320Q8—82no—
321Q8—82no—
322Q8—4881no—
323Q8—2442no—
324D4—614no—
325D4—1224no—
326D4—1222no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—1222no—
332C2×C4—4882no—
333C2×C4—4881no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2444no—
337C2×C4#1451975011yescharacteristic
338C2×C4#1451975011yescharacteristic
339C2×C4#1451975011yescharacteristic
340C2×C4Q16×D6112yes—
341C2×C4#1451979812yes—
342C2×C4C2×D4×D6111yescharacteristic
343C2×C4C2×D4×D6111yescharacteristic
344C2×C4C2×#1175822211yescharacteristic
345C8—42no—
346C8—42no—
347C8—1224no—
348C8—2442no—
349C22—21no—
350C22—21no—
351C22—1224no—
352C22—4884no—
353C22—2442no—
354C22—2442no—
355C22#1684000212yes—
356C22C2×#1451975011yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—82no—
361C4—4881no—
362C4—4881no—
363C4—2442no—
364C4#1684000212yes—
365C4C2×Q16×D6111yescharacteristic
366C4C2×#1451979811yescharacteristic
367C4#1684057112yes—
368C2—22no—
369C2—2444no—
370C2#2145533011yescharacteristic
371C2C2×#1684000211yescharacteristic
372C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1548C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1548D61
hypocenterC61#19393
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #21456970C2semidirect
5. #21445273C2semidirect
6. #21445217C2semidirect
7. #21444595C2semidirect
8. #21444474C2semidirect
9. C61×#1548C2semidirect
12. #16840165C22semidirect
13. #16840114C22semidirect
30. #16839514C22semidirect
31. #16839510C22semidirect
32. #16839509C22semidirect
36. C61×#333C22semidirect
189. #1548D61semidirect
221. #333C2×D61semidirect
223. C61#19393semidirect

Maximal subgroups

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

subgroupindex
#214569702
#214556182
C2×#168400492
#214548422
#214452732
#214452172
#214445952
#214444742
C61×#15482
#1939361