Finite Group Databasebeta

← #29835594

Subgroups — #29835594

Order 15616.

510 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#21456403C211yesmaximal, characteristic, complemented
2#21456386C211yesmaximal, characteristic
3#21448105C211yesmaximal, characteristic, complemented
4#21448064C211yesmaximal, characteristic, complemented
5#21448055C211yesmaximal, characteristic, complemented
6#21447929C212yesmaximal
7#21447924C212yesmaximal, complemented
8Q32×Dic61C212yesmaximal, complemented
9SD32×Dic61C212yesmaximal, complemented
10#21447412C211yesmaximal, characteristic, complemented
11C61×#2251C211yesmaximal, characteristic, complemented
12#16840317—21no—
13#16840291—21no—
14C2×#14519716C2212yes—
15#16840167C2211yescharacteristic
16#16840164C2211yescharacteristic
17#16840116—21no—
18#16840112—21no—
19#16840000—21no—
20#16840000C2211yescharacteristic
21#16840000C2212yes—
22#16839995C2211yescharacteristic
23#16839988—21no—
24Q16×Dic61—21no—
25Q16×Dic61C2211yescharacteristic
26Q16×Dic61C2212yes—
27SD16×Dic61—21no—
28D8×Dic61C2211yescharacteristic
29C4×C8×D61C2211yescharacteristic
30#16839590C2212yes—
31#16839588C2212yes—
32#16839566C2212yes—
33#16839566C2212yes—
34#16839566C2212yes—
35#16839563C2212yes—
36#16839538C2211yescharacteristic
37C16×Dic61C2212yes—
38C2×Q32×C61C2212yes—
39C2×SD32×C61C2212yes—
40C61×#495C2211yescharacteristic
41C61×#494C2211yescharacteristic
42C61×#346C2211yescharacteristic
43C2×C8×D61D411yescharacteristic
44C2×C8×D61C2311yescharacteristic
45#14519777—21no—
46#14519770—41no—
47#14519769—21no—
48#14519768—41no—
49#14519759—41no—
50#14519757—41no—
51#14519753—21no—
52#14519753—22no—
53#14519748—21no—
54Q8×Dic61—21no—
55Q8×Dic61—22no—
56#14519742—41no—
57D4×Dic61—21no—
58#14519737—41no—
59#14519735—41no—
60#14519733—41no—
61C42×D61D411yescharacteristic
62#14519716—24no—
63C2×#11758185C2312yes—
64#14519711—21no—
65#14519711C2311yescharacteristic
66#14519711C2312yes—
67#14519711C2311yescharacteristic
68#14519709—21no—
69#14519701—21no—
70#14519701—22no—
71#14519701—21no—
72#14519701—22no—
73#14519698—21no—
74#14519698—21no—
75C4×#92418D411yescharacteristic
76C8×Dic61D411yescharacteristic
77C8×Dic61C2311yescharacteristic
78C2×Q16×C61—21no—
79C2×Q16×C61C2312yes—
80C2×Q16×C61C2311yescharacteristic
81C2×SD16×C61—21no—
82C2×D8×C61C2311yescharacteristic
83C4⋊Q8×C61—21no—
84#124×C61—21no—
85Q32×C61—24no—
86SD32×C61—24no—
87C2×C16×C61C2312yes—
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—21no—
90C2×Q8×C61—22no—
91C2×D4×C61—21no—
92C2×C22⋊D61—41no—
93C22×Dic61—41no—
94C2×C4×D61D811yescharacteristic
95C2×C4×D61D811yescharacteristic
96C2×C4×D61C2×D411yescharacteristic
97C2×Dic122—41no—
98Q16×C61—22no—
99Q16×C61—22no—
100Q16×C61—22no—
101Q16×C61—42no—
102SD16×C61—42no—
103D8×C61—22no—
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61C2411yescharacteristic
106C4⋊C4×C61—41no—
107C4⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊C4×C61—41no—
110C42×C61C2×D411yescharacteristic
111C22⋊Dic61—41no—
112C22⋊Dic61—41no—
113#11758198—42no—
114#11758198—41no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—22no—
118C4⋊Dic61—41no—
119C4⋊Dic61—42no—
120C4⋊Dic61—41no—
121#11758196—41no—
122#11758196—41no—
123#11758196—41no—
124#11758196—42no—
125C4×Dic61—41no—
126C4×Dic61—42no—
127C4×Dic61D811yescharacteristic
128C4×Dic61D811yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418C2×D411yescharacteristic
131C2×#92418C2×D411yescharacteristic
132C8×D61—22no—
133C8×D61—22no—
134C16×C61#4114yes—
135#11758185—22no—
136C23×C61—41no—
137C22×D61C2×D811yescharacteristic
138Q8×C61—42no—
139Q8×C61—41no—
140Q8×C61—41no—
141Q8×C61—42no—
142D4×C61—41no—
143D4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61C2×D811yescharacteristic
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C22×D411yescharacteristic
149C22⋊D61—82no—
150C2×Dic61—41no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—82no—
154C2×Dic61C2×D811yescharacteristic
155C2×Dic61C2×D811yescharacteristic
156C2×Dic61C2×D811yescharacteristic
157C4×D61—21no—
158C4×D61—21no—
159C4×D61—22no—
160C4×D61—41no—
161Dic122—82no—
162C8×C61#13512yes—
163C8×C61C2×#4112yes—
164#92418—21no—
165#92418—21no—
166#33738—611nomaximal, Sylow
167C22×C61—81no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61C22×D811yescharacteristic
171C2×D61—21no—
172C2×D61—21no—
173C2×D61—21no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—81no—
177C4×C61—82no—
178C4×C61#53211yescharacteristic
179C4×C61C2×#13511yescharacteristic
180Dic61—21no—
181Dic61—21no—
182Dic61—81no—
183Dic61—82no—
184Dic61—81no—
185Dic61#51012yes—
186#3170—611no—
187#3168—611no—
188#2255—611no—
189#2251D6111yescharacteristic, complemented
190#2245—611no—
191#2243—611no—
192#2220—612no—
193#2219—612no—
194C4×Q32—612no—
195C4×SD32—612no—
196C4×M32—611no—
197C2×C61—161no—
198C2×C61#316911yescharacteristic
199C2×C61C2×#51011yescharacteristic
200C2×C61#341411yescharacteristic
201D61—41no—
202#533—1221no—
203#529—1221no—
204C2×Q32C2×D6112yes—
205C2×SD32C2×D6112yes—
206C2×M32—612no—
207#500—611no—
208#499—611no—
209#495C2×D6111yescharacteristic
210#494C2×D6111yescharacteristic
211#488—1221no—
212#487—1221no—
213#467—611no—
214#467—612no—
215#467—1221no—
216#466—611no—
217#465—1221no—
218C4×Q16—611no—
219C4×Q16—612no—
220C4×Q16—1221no—
221C4×SD16—1221no—
222C4×D8—611no—
223C2×C4×C8—611no—
224#367—612no—
225#366—612no—
226#358—612no—
227#358—612no—
228#358—612no—
229#357—612no—
230#346—611no—
231#346C2×D6111yescharacteristic
232C4×C16—612no—
233C61#3373811yescharacteristic, complemented, Sylow
234C2×Q16—21no—
235C2×Q16C22×D6112yes—
236C2×Q16C22×D6111yescharacteristic
237C2×SD16—21no—
238C2×D8C22×D6111yescharacteristic
239C22×C8—611no—
240C22×C8—611no—
241C4⋊Q8—21no—
242C4⋊Q8—1221no—
243#126—2441no—
244#125—1221no—
245#125—2441no—
246#124—21no—
247#123—2441no—
248C22⋊Q8—1221no—
249C22⋊Q8—1222no—
250C22⋊Q8—2441no—
251#121—1221no—
252C4×Q8—1221no—
253C4×Q8—1222no—
254C4×Q8—2441no—
255C4×D4—1221no—
256C4×D4—2441no—
257#117—2441no—
258#117—2441no—
259C2×C42—611no—
260Q32—24no—
261SD32—24no—
262M32—1224no—
263C2×C16—612no—
264C2×C16C22×D6112yes—
265#107—612no—
266#107—611no—
267#107—611no—
268#107—1221no—
269#106—1221no—
270#103—1221no—
271#103—1221no—
272#103—1222no—
273#103—1222no—
274#102—1221no—
275#102—1221no—
276C4×C8—611no—
277C4×C8—611no—
278C4×C8—611no—
279C4×C8C22×D6111yescharacteristic
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—2441no—
283C2×D4—21no—
284C2×D4—2441no—
285C22×C4—611no—
286C22×C4—611no—
287C22×C4—611no—
288C22×C4—2441no—
289Q16—22no—
290Q16—22no—
291Q16—22no—
292Q16—42no—
293SD16—42no—
294D8—22no—
295C2×C8—611no—
296C2×C8—611no—
297C2×C8—1222no—
298C2×C8—1222no—
299C2×C8D4×D6111yescharacteristic
300C2×C8C23×D6111yescharacteristic
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—1221no—
304C4⋊C4—1221no—
305C4⋊C4—1222no—
306C4⋊C4—2441no—
307C4⋊C4—2442no—
308C4⋊C4—2442no—
309C4⋊C4—2441no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—2442no—
316C22⋊C4—2441no—
317C22⋊C4—2441no—
318C22⋊C4—2441no—
319C42—611no—
320C42—611no—
321C42—611no—
322C42—2441no—
323C42—2442no—
324C42D4×D6111yescharacteristic
325C16—1222no—
326C16#1175822414yes—
327C23—41no—
328C23—611no—
329Q8—41no—
330Q8—42no—
331Q8—42no—
332Q8—41no—
333Q8—4882no—
334D4—41no—
335D4—41no—
336D4—4882no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—611no—
340C2×C4—611no—
341C2×C4—611no—
342C2×C4—1222no—
343C2×C4—1221no—
344C2×C4—1221no—
345C2×C4—4882no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2442no—
349C2×C4—2441no—
350C2×C4D8×D6111yescharacteristic
351C2×C4D8×D6111yescharacteristic
352C2×C4C2×D4×D6111yescharacteristic
353C8—1221no—
354C8—1221no—
355C8#1451979612yes—
356C8C2×#1175822412yes—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—1221no—
361C22—1221no—
362C22—1221no—
363C22C2×D8×D6111yescharacteristic
364C4—21no—
365C4—21no—
366C4—81no—
367C4—82no—
368C4—612no—
369C4—1221no—
370C4—1221no—
371C4—4881no—
372C4—4881no—
373C4—4882no—
374C4#1684031011yescharacteristic
375C4C2×#1451979611yescharacteristic
376C2—161no—
377C2—2441no—
378C2#2145639611yescharacteristic
379C2#510×D6111yescharacteristic
380C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2251C2
socleC22×C61C22×D8
prime radical#16839995C22
hypercenter#2251D61
hypocenterC61#33738
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2251

Complements

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

NH ≅ G/Ntype
1. #21456403C2semidirect
3. #21448105C2semidirect
4. #21448064C2semidirect
5. #21448055C2semidirect
7. #21447924C2semidirect
8. Q32×Dic61C2semidirect
9. SD32×Dic61C2semidirect
10. #21447412C2semidirect
11. C61×#2251C2semidirect
189. #2251D61semidirect
233. C61#33738semidirect

Maximal subgroups

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

subgroupindex
#214564032
#214563862
#214481052
#214480642
#214480552
#214479292
#214479242
Q32×Dic612
SD32×Dic612
#214474122
C61×#22512
#3373861