Finite Group Databasebeta

← #29665232

Subgroups — #29665232

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456965C211yesmaximal, characteristic, complemented
2#21455618C211yesmaximal, characteristic
3C2×#16839968C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445243C214yesmaximal, complemented
6#21445195C214yesmaximal, complemented
7#21444492C211yesmaximal, characteristic, complemented
8#21444477C211yesmaximal, characteristic, complemented
9C61×#1570C211yesmaximal, characteristic, complemented
10C22×#11758190—21no—
11#16840319C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13C2×C4×D244C2211yescharacteristic
14#16840160C2212yescomplemented
15#16840113C2212yescomplemented
16#16840093—22no—
17#16840063—22no—
18#16840053C2212yes—
19#16840048—22no—
20#16840043C2212yes—
21#16839968—22no—
22#16839968C2214yes—
23C8×D244C2214yes—
24#16839827—22no—
25#16839815C2211yescharacteristic
26#16839804—21no—
27#16839768—21no—
28C2×#14519702—21no—
29C22⋊C8×D61—22no—
30#16839510C2212yescomplemented
31#16839509C2214yescomplemented
32#16839507C2212yescomplemented
33C61×#431C2211yescharacteristic
34C61×#423C2211yescharacteristic
35C2×C4⋊C8×C61C2211yescharacteristic
36C61×#334C2214yescomplemented
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C4×D61—21no—
40C2×#11758190—24no—
41C2×#11758190—44no—
42C2×C8×D61—24no—
43#14519777—21no—
44#14519777—21no—
45#14519777—21no—
46#14519777C2312yes—
47#14519777C2311yescharacteristic
48#14519777C2311yescharacteristic
49#14519769—21no—
50#14519756—21no—
51#14519756—21no—
52C4×Dic122—21no—
53C4×Dic122—21no—
54C4×D244—21no—
55C4×D244—21no—
56C4×D244D412yes—
57C4×D244C2312yes—
58#14519736—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61D411yescharacteristic
62C2×#11758198—21no—
63#14519711—22no—
64#14519709—24no—
65#14519709—22no—
66C4⋊#92418C2312yes—
67#14519702—24no—
68#14519702—24no—
69#14519699—22no—
70#14519699—22no—
71C22⋊#92418—21no—
72#14519689—24no—
73C4×#92418—21no—
74C2×M16×C61—21no—
75C22×C8×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61—21no—
79C4⋊C8×C61D412yes—
80C4⋊C8×C61C2312yes—
81C4⋊C8×C61C2312yes—
82C22⋊C8×C61D412yes—
83C4×C8×C61C2312yes—
84C23×D61—21no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×Dic61—21no—
89C22×Dic61—41no—
90C2×D244—21no—
91C2×D244—22no—
92C2×D244SD1614yes—
93C2×D244C2×D412yes—
94C2×C4×D61—22no—
95C2×C4×D61—22no—
96C2×C4×D61—82no—
97C2×Dic122—22no—
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×Dic122—42no—
101C2×Dic122—42no—
102C2×Dic122—41no—
103M16×C61—42no—
104C2×C8×C61—22no—
105C2×C8×C61—22no—
106C2×C8×C61—42no—
107C2×C8×C61C2×D414yes—
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—22no—
119C4⋊Dic61—21no—
120C4⋊Dic61—22no—
121C4⋊Dic61—21no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124C4⋊Dic61C2×D412yes—
125#11758196—41no—
126#11758196—41no—
127C4×Dic61—41no—
128C2×#92418—22no—
129#11758190—48no—
130C8×D61—84no—
131C23×C61#12111yescharacteristic
132C22×D61—22no—
133C22×D61—24no—
134C22×D61—81no—
135C22×D61—81no—
136C2×C4×C61—22no—
137C2×C4×C61—41no—
138C2×C4×C61#12111yescharacteristic
139C2×C4×C61#12111yescharacteristic
140C2×C4×C61#12111yescharacteristic
141C2×C4×C61C2×SD1612yes—
142C2×C4×C61#13512yes—
143C2×C4×C61C22×D411yescharacteristic
144C2×C4×C61C22×D411yescharacteristic
145C2×C4×C61C2×#4111yescharacteristic
146C2×Dic61—22no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—42no—
150C2×Dic61—81no—
151C2×Dic61—82no—
152D244—22no—
153D244—24no—
154D244C2×SD1614yes—
155C4×D61—44no—
156C4×D61—82no—
157Dic122—42no—
158Dic122—42no—
159Dic122—42no—
160Dic122—82no—
161Dic122—81no—
162C8×C61—24no—
163C8×C61—42no—
164C8×C61—42no—
165#92418—42no—
166#19390—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61#46512yes—
170C22×C61C2×#12111yescharacteristic
171C2×D61—24no—
172C2×D61—42no—
173C2×D61—42no—
174C2×D61—84no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61#46512yes—
179C4×C61C22×SD1611yescharacteristic
180C4×C61C2×#13511yescharacteristic
181C4×C61#57712yes—
182Dic61—42no—
183Dic61—81no—
184Dic61—82no—
185Dic61—81no—
186#3229—611no—
187#3077—611no—
188C2×#461—611no—
189#2972—611no—
190#1679—614no—
191#1671—614no—
192#1570D6111yescharacteristic, complemented
193#1546—611no—
194#1542—611no—
195C2×C61—22no—
196C2×C61#304211yescharacteristic
197C2×C61C2×#46511yescharacteristic
198C2×C61#306911yescharacteristic
199D61—44no—
200C22×SD16—1221no—
201#533—611no—
202C2×C4⋊Q8—611no—
203C2×C4×D4—611no—
204#498—612no—
205#487—612no—
206#483—1222no—
207#478—1222no—
208#476—612no—
209#475—1222no—
210#474—612no—
211#461—614no—
212#461—1222no—
213C8×D4—614no—
214C8×D4—1222no—
215#431—611no—
216#431C2×D6111yescharacteristic
217#428—1221no—
218#423C2×D6111yescharacteristic
219C2×C4⋊C8C2×D6111yescharacteristic
220#418—1221no—
221C2×#103—1221no—
222C2×C22⋊C8—1222no—
223#334C2×D6114yescomplemented
224#333—612no—
225#332—612no—
226#332—614no—
227C61#1939011yescharacteristic, complemented, Sylow
228C22×Q8—1221no—
229C22×D4—611no—
230C23×C4—1221no—
231C2×SD16—1224no—
232C2×SD16—2444no—
233C2×M16—21no—
234C22×C8—21no—
235C22×C8—1224no—
236C4⋊Q8—612no—
237C4⋊Q8—611no—
238C4⋊Q8—611no—
239C4⋊Q8—1221no—
240C4⋊Q8—1221no—
241C4⋊Q8—1221no—
242#125—1221no—
243C22⋊Q8—1221no—
244C22⋊Q8—1221no—
245C4×Q8—1221no—
246C4×Q8—1221no—
247C4×D4—612no—
248C4×D4—612no—
249C4×D4—1221no—
250C4×D4—1221no—
251#117—1221no—
252C2×C4⋊C4—611no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C42C22×D6111yescharacteristic
257#107—1222no—
258#106—1222no—
259#106—1224no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—612no—
263C4⋊C8D24412yes—
264C4⋊C8C22×D6112yes—
265C4⋊C8C22×D6112yes—
266#103—1224no—
267#103—1224no—
268#102—1222no—
269#102—1222no—
270C22⋊C8—1221no—
271C22⋊C8—1224no—
272C22⋊C8D24412yes—
273C4×C8—1221no—
274C4×C8C22×D6112yes—
275C24—1221no—
276C2×Q8—1222no—
277C2×Q8—1221no—
278C2×Q8—1221no—
279C2×Q8—2442no—
280C2×Q8—2442no—
281C2×Q8—2441no—
282C2×D4—612no—
283C2×D4—614no—
284C2×D4—1221no—
285C2×D4—1222no—
286C22×C4—1221no—
287C22×C4—1222no—
288C22×C4—1222no—
289C22×C4—4882no—
290C22×C4—2441no—
291C22×C4C2×D24411yescharacteristic
292C22×C4D4×D6111yescharacteristic
293C22×C4#1175822611yescharacteristic
294SD16—2448no—
295M16—42no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—42no—
299C2×C8—1222no—
300C2×C8—4884no—
301C2×C8C2×D24414yes—
302C4⋊C4—612no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C4⋊C4—1221no—
306C4⋊C4—1222no—
307C4⋊C4—1221no—
308C4⋊C4—1222no—
309C4⋊C4—1221no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4—2441no—
314C22⋊C4—1222no—
315C22⋊C4—1222no—
316C22⋊C4—2441no—
317C42—2441no—
318C42C2×D24411yescharacteristic
319C42D4×D6111yescharacteristic
320C42#1175822611yescharacteristic
321C42C23×D6111yescharacteristic
322C23—1224no—
323C23—1222no—
324C23—4881no—
325C23—4881no—
326C23#1451975211yescharacteristic
327Q8—4881no—
328Q8—4882no—
329Q8—2442no—
330Q8—2442no—
331Q8—2442no—
332D4—614no—
333D4—1224no—
334D4—1222no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—1222no—
338C2×C4—4882no—
339C2×C4—4882no—
340C2×C4—4881no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2442no—
344C2×C4—2444no—
345C2×C4#1451975211yescharacteristic
346C2×C4#1451975211yescharacteristic
347C2×C4#1451975211yescharacteristic
348C2×C4SD16×D6112yes—
349C2×C4#1451979812yes—
350C2×C4C2×D4×D6111yescharacteristic
351C2×C4C22×D24411yescharacteristic
352C2×C4C2×#1175822611yescharacteristic
353C8—24no—
354C8—42no—
355C8—42no—
356C8—2442no—
357C22—21no—
358C22—21no—
359C22—1224no—
360C22—4884no—
361C22—2442no—
362C22—2442no—
363C22#1683999212yes—
364C22C2×#1451975211yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—4882no—
369C4—4881no—
370C4—4881no—
371C4—2442no—
372C4#1683999212yes—
373C4C2×SD16×D6111yescharacteristic
374C4C2×#1451979811yescharacteristic
375C4#1684057412yes—
376C2—22no—
377C2—2444no—
378C2#2145533211yescharacteristic
379C2C2×#1683999211yescharacteristic
380C2#2145555711yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1570C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1570D61
hypocenterC61#19390
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1570

Complements

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

NH ≅ G/Ntype
1. #21456965C2semidirect
5. #21445243C2semidirect
6. #21445195C2semidirect
7. #21444492C2semidirect
8. #21444477C2semidirect
9. C61×#1570C2semidirect
14. #16840160C22semidirect
15. #16840113C22semidirect
30. #16839510C22semidirect
31. #16839509C22semidirect
32. #16839507C22semidirect
36. C61×#334C22semidirect
192. #1570D61semidirect
223. #334C2×D61semidirect
227. C61#19390semidirect

Maximal subgroups

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

subgroupindex
#214569652
#214556182
C2×#168399682
#214548422
#214452432
#214451952
#214444922
#214444772
C61×#15702
#1939061