Finite Group Databasebeta

← #29665261

Subgroups — #29665261

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#21456967C211yesmaximal, characteristic, complemented
2#21455612C211yesmaximal, characteristic
3C2×#16840053C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445269C214yesmaximal, complemented
6#21445217C214yesmaximal, complemented
7#21444593C211yesmaximal, characteristic, complemented
8#21444499C211yesmaximal, characteristic, complemented
9C61×#1542C211yesmaximal, characteristic, complemented
10C2×#14519777C2211yescharacteristic
11C2×C4×D244C2211yescharacteristic
12#16840165C2212yescomplemented
13#16840112C2212yescomplemented
14#16840095—22no—
15#16840059—22no—
16#16840057—22no—
17#16840053—22no—
18#16840053C2214yes—
19#16840049C2212yes—
20#16840039C2212yes—
21C8×D244—22no—
22#16839827C2214yes—
23#16839815C2211yescharacteristic
24#16839805—21no—
25C2×#14519709—21no—
26#16839786C2211yescharacteristic
27#16839766—21no—
28C2×#14519699—21no—
29C22⋊C8×D61—22no—
30#16839514C2214yescomplemented
31#16839510C2214yescomplemented
32C61×#533C2211yescharacteristic
33C61×#431C2211yescharacteristic
34C2×C4⋊C8×C61C2211yescharacteristic
35C61×#333C2212yescomplemented
36C61×#332C2212yescomplemented
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40C2×#11758194—21no—
41C2×C8×D61—24no—
42#14519777—21no—
43#14519777—21no—
44#14519777—21no—
45#14519777C2312yes—
46C4×D244—21no—
47C4×D244—21no—
48C4×D244D412yes—
49C4×D244C2312yes—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61D411yescharacteristic
53C2×#11758198—21no—
54#14519710—22no—
55#14519710—24no—
56#14519709—24no—
57#14519708—22no—
58C4⋊#92418—21no—
59C4⋊#92418D412yes—
60C4⋊#92418C2312yes—
61#14519701—24no—
62#14519699—24no—
63#14519697—22no—
64#14519697—22no—
65C22⋊#92418D412yes—
66#14519689—24no—
67C4×#92418C2312yes—
68C22×C8×C61—21no—
69C4⋊Q8×C61C2311yescharacteristic
70C4⋊Q8×C61C2311yescharacteristic
71#125×C61—21no—
72C22⋊Q8×C61—21no—
73C22⋊Q8×C61—21no—
74C4×Q8×C61—21no—
75C4×Q8×C61—21no—
76#117×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—21no—
85C2×Q8×C61—21no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×Dic61—21no—
90C22×Dic61—41no—
91C2×D244—21no—
92C2×D244—22no—
93C2×D244Q1614yes—
94C2×D244C2×D412yes—
95C2×C4×D61—22no—
96C2×C4×D61—22no—
97C2×C4×D61—82no—
98C2×Dic122—22no—
99C2×Dic122—41no—
100C2×Dic122—82no—
101C2×C8×C61—22no—
102C2×C8×C61—22no—
103C2×C8×C61—42no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—21no—
107C4⋊C4×C61—21no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C22⋊C4×C61—41no—
111C42×C61—41no—
112C42×C61C2×D411yescharacteristic
113C42×C61C2×D411yescharacteristic
114C42×C61#4111yescharacteristic
115C42×C61C2411yescharacteristic
116#11758198—22no—
117#11758198—22no—
118C4⋊Dic61—21no—
119C4⋊Dic61—22no—
120C4⋊Dic61—22no—
121C4⋊Dic61—41no—
122C4⋊Dic61—41no—
123C4⋊Dic61C2×D412yes—
124#11758194—42no—
125C2×#92418—22no—
126C2×#92418C2×D414yes—
127C8×D61—84no—
128C23×C61#12111yescharacteristic
129C22×D61—22no—
130C22×D61—24no—
131C22×D61—81no—
132C22×D61—81no—
133Q8×C61—42no—
134Q8×C61—81no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
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—42no—
149C2×Dic61—81no—
150C2×Dic61—82no—
151D244—22no—
152D244—24no—
153D244C2×Q1614yes—
154C4×D61—44no—
155C4×D61—82no—
156Dic122—82no—
157Dic122—82no—
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—81no—
175C4×C61—81no—
176C4×C61#46712yes—
177C4×C61C22×Q1611yescharacteristic
178C4×C61C2×#13511yescharacteristic
179C4×C61#57712yes—
180Dic61—42no—
181Dic61—82no—
182#3230—611no—
183#3077—611no—
184C2×#476—611no—
185#2972—611no—
186#1683—614no—
187#1675—614no—
188#1571—611no—
189#1548—611no—
190#1542D6111yescharacteristic, complemented
191C2×C61—22no—
192C2×C61#304211yescharacteristic
193C2×C61C2×#46711yescharacteristic
194C2×C61#306811yescharacteristic
195D61—44no—
196#533C2×D6111yescharacteristic
197C2×C4⋊Q8—611no—
198C2×C4×D4—611no—
199#500—612no—
200#487—612no—
201#484—1222no—
202#478—1222no—
203#477—1222no—
204#476—612no—
205#476—614no—
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#423—611no—
215C2×C4⋊C8C2×D6111yescharacteristic
216#418—1221no—
217C2×#102—1221no—
218C2×C22⋊C8—1222no—
219#335—614no—
220#333—614no—
221#333C2×D6112yescomplemented
222#332C2×D6112yescomplemented
223C61#1939311yescharacteristic, complemented, Sylow
224C22×Q8—1221no—
225C22×D4—611no—
226C23×C4—1221no—
227C2×M16—1221no—
228C22×C8—21no—
229C22×C8—1224no—
230C4⋊Q8—612no—
231C4⋊Q8—1221no—
232C4⋊Q8—1221no—
233C4⋊Q8—1221no—
234C4⋊Q8C22×D6111yescharacteristic
235C4⋊Q8C22×D6111yescharacteristic
236#125—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×D4—612no—
242C4×D4—612no—
243C4×D4—1221no—
244C4×D4—1221no—
245#117—21no—
246C2×C4⋊C4—611no—
247C2×C4⋊C4—1221no—
248C2×C4⋊C4—1221no—
249C2×C22⋊C4—1221no—
250C2×C42C22×D6111yescharacteristic
251#107—1222no—
252#107—1224no—
253#106—1222no—
254#106—1224no—
255C4⋊C8—21no—
256C4⋊C8—612no—
257C4⋊C8—612no—
258C4⋊C8—1221no—
259C4⋊C8C22×D6112yes—
260C4⋊C8C22×D6112yes—
261#103—1224no—
262#102—1222no—
263#102—1222no—
264#102—1224no—
265C22⋊C8—21no—
266C22⋊C8—612no—
267C22⋊C8—1224no—
268C4×C8—21no—
269C4×C8—612no—
270C24—1221no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—1222no—
274C2×Q8—4882no—
275C2×Q8—2441no—
276C2×D4—612no—
277C2×D4—614no—
278C2×D4—1221no—
279C2×D4—1222no—
280C22×C4—1221no—
281C22×C4—1222no—
282C22×C4—1222no—
283C22×C4—4882no—
284C22×C4—2441no—
285C22×C4#1175822211yescharacteristic
286C22×C4D4×D6111yescharacteristic
287C22×C4D4×D6111yescharacteristic
288M16—2442no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—42no—
292C2×C8—614no—
293C2×C8—1222no—
294C2×C8—4884no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—612no—
302C4⋊C4—1221no—
303C4⋊C4—1222no—
304C4⋊C4—1222no—
305C4⋊C4—2441no—
306C4⋊C4—2441no—
307C22⋊C4—41no—
308C22⋊C4—1222no—
309C22⋊C4—1222no—
310C42—41no—
311C42#1175822211yescharacteristic
312C42D4×D6111yescharacteristic
313C42D4×D6111yescharacteristic
314C42C23×D6111yescharacteristic
315C23—1224no—
316C23—1222no—
317C23—4881no—
318C23—4881no—
319C23#1451975011yescharacteristic
320Q8—42no—
321Q8—81no—
322Q8—4882no—
323Q8—4882no—
324D4—614no—
325D4—1224no—
326D4—1222no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—1222no—
332C2×C4—4882no—
333C2×C4—4882no—
334C2×C4—4881no—
335C2×C4—2442no—
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—81no—
361C4—81no—
362C4—4882no—
363C4—2442no—
364C4#1684000212yes—
365C4C2×Q16×D6111yescharacteristic
366C4C2×#1451979811yescharacteristic
367C4#1684056912yes—
368C2—22no—
369C2—2444no—
370C2#2145533411yescharacteristic
371C2C2×#1684000211yescharacteristic
372C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1542D61
hypocenterC61#19393
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
1. #21456967C2semidirect
5. #21445269C2semidirect
6. #21445217C2semidirect
7. #21444593C2semidirect
8. #21444499C2semidirect
9. C61×#1542C2semidirect
12. #16840165C22semidirect
13. #16840112C22semidirect
30. #16839514C22semidirect
31. #16839510C22semidirect
35. C61×#333C22semidirect
36. C61×#332C22semidirect
190. #1542D61semidirect
221. #333C2×D61semidirect
222. #332C2×D61semidirect
223. C61#19393semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214556122
C2×#168400532
#214548422
#214452692
#214452172
#214445932
#214444992
C61×#15422
#1939361