Finite Group Databasebeta

← #29665251

Subgroups — #29665251

Order 15616.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456937C211yesmaximal, characteristic, complemented
2C2×#16839967C211yesmaximal, characteristic, complemented
3#21454843C211yesmaximal, characteristic, complemented
4#21445254C214yesmaximal, complemented
5#21445206C214yesmaximal, complemented
6#21444595C211yesmaximal, characteristic, complemented
7#21444492C211yesmaximal, characteristic, complemented
8#21444468C211yesmaximal, characteristic, complemented
9C61×#3036C211yesmaximal, characteristic
10C22×#11758200—21no—
11C2×#14519777C2211yescharacteristic
12#16840165C2212yescomplemented
13#16840108C2212yescomplemented
14#16840073—22no—
15#16840042—22no—
16#16839967—22no—
17#16839967C2214yes—
18D4×#92418—22no—
19D4×#92418C2214yes—
20#16839815C2211yescharacteristic
21#16839815C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519700—21no—
24C2×C22⋊#92418—22no—
25#16839514C2214yescomplemented
26#16839509C2214yescomplemented
27#16839508C2212yescomplemented
28#16839506C2212yescomplemented
29C2×AGL(1,ℤ/8)×C61—21no—
30C61×#532C2211yescharacteristic
31C2×C4×D4×C61C2211yescharacteristic
32C61×#463—22no—
33C61×#461C2212yes—
34C61×#459C2212yes—
35C61×#423C2211yescharacteristic
36C61×#417—21no—
37C22×Dic122—21no—
38C2×#11758200—24no—
39C2×#11758200—44no—
40C22×#92418—21no—
41C22×#92418—24no—
42#14519777—21no—
43#14519777—21no—
44#14519777—21no—
45#14519777C2312yes—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48#14519710—24no—
49C4⋊#92418—21no—
50C4⋊#92418D412yes—
51C4⋊#92418C2312yes—
52C4⋊#92418C2312yes—
53#14519700—24no—
54#14519698—24no—
55C22⋊#92418—21no—
56C22⋊#92418—24no—
57C22⋊#92418D412yes—
58C4×#92418—21no—
59C4×#92418C2312yes—
60C2×#41×C61—21no—
61C22×D4×C61D411yescharacteristic
62C23×C4×C61—21no—
63AGL(1,ℤ/8)×C61—44no—
64C2×SD16×C61—22no—
65C2×D8×C61—22no—
66C2×M16×C61—21no—
67C4⋊Q8×C61C2311yescharacteristic
68#127×C61C2311yescharacteristic
69#124×C61—21no—
70#121×C61—21no—
71#121×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74C4×D4×C61—21no—
75C4×D4×C61—21no—
76C4×D4×C61D412yes—
77C4×D4×C61C2312yes—
78C2×C4⋊C4×C61D411yescharacteristic
79C2×C22⋊C4×C61—21no—
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61—21no—
82C4⋊C8×C61C2312yes—
83#103×C61—22no—
84#102×C61—22no—
85C24×C61—21no—
86#41×C61—42no—
87#41×C61—42no—
88C2×Q8×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—22no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—41no—
95C2×D4×C61D814yes—
96C2×D4×C61C2×D412yes—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—41no—
101C22×C4×C61—82no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61#4111yescharacteristic
105C22×Dic61—41no—
106C2×Dic122—22no—
107C2×Dic122—42no—
108C2×Dic122—42no—
109C2×Dic122—41no—
110SD16×C61—44no—
111D8×C61—44no—
112M16×C61—42no—
113C2×C8×C61—22no—
114C4⋊C4×C61—21no—
115C4⋊C4×C61—21no—
116C4⋊C4×C61—21no—
117C4⋊C4×C61C2×D412yes—
118C22⋊C4×C61—22no—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C42×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61#4111yescharacteristic
125C42×C61C2411yescharacteristic
126#11758200—48no—
127C4⋊Dic61—22no—
128C4⋊Dic61—22no—
129C4⋊Dic61—41no—
130C4⋊Dic61—41no—
131C2×#92418—22no—
132C2×#92418—22no—
133C2×#92418—42no—
134C2×#92418—84no—
135C2×#92418C2×D414yes—
136C23×C61—22no—
137C23×C61—24no—
138C23×C61—41no—
139C23×C61—81no—
140C23×C61—81no—
141C23×C61#12111yescharacteristic
142Q8×C61—41no—
143Q8×C61—41no—
144D4×C61—22no—
145D4×C61—24no—
146D4×C61—41no—
147D4×C61—41no—
148D4×C61—42no—
149D4×C61—81no—
150D4×C61—82no—
151D4×C61C2×D814yes—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—44no—
156C2×C4×C61—41no—
157C2×C4×C61—82no—
158C2×C4×C61—81no—
159C2×C4×C61—82no—
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61C2×D812yes—
164C2×C4×C61#13512yes—
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C2×#4111yescharacteristic
168C2×Dic61—42no—
169C2×Dic61—82no—
170Dic122—42no—
171Dic122—42no—
172Dic122—82no—
173C8×C61—42no—
174#92418—24no—
175#92418—42no—
176#92418—42no—
177#19392—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—24no—
181C22×C61—42no—
182C22×C61—42no—
183C22×C61—84no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61#46612yes—
188C22×C61C2×#12111yescharacteristic
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61—81no—
194C4×C61#46612yes—
195C4×C61C22×D811yescharacteristic
196C4×C61C2×#13511yescharacteristic
197C4×C61#57712yes—
198Dic61—82no—
199#3226—611no—
200#3036D6111yescharacteristic
201C2×#461—611no—
202#2972—611no—
203#1681—614no—
204#1673—614no—
205#1571—611no—
206#1546—611no—
207#1540—611no—
208C2×C61—22no—
209C2×C61—44no—
210C2×C61—161no—
211C2×C61#304211yescharacteristic
212C2×C61C2×#46611yescharacteristic
213C2×C61#306911yescharacteristic
214C2×AGL(1,ℤ/8)—21no—
215C22×SD16—1221no—
216#532C2×D6111yescharacteristic
217C2×C4⋊Q8—611no—
218C2×C4×D4C2×D6111yescharacteristic
219#500—612no—
220#486—612no—
221#480—1222no—
222#474—1222no—
223#463—22no—
224#461—614no—
225#461—1222no—
226#461C2×D6112yes—
227#459C2×D6112yes—
228C8×D4—614no—
229C8×D4—1222no—
230#431—611no—
231#431—611no—
232#423C2×D6111yescharacteristic
233C2×C4⋊C8—611no—
234#417—21no—
235C2×#103—1221no—
236C2×C22⋊C8—1222no—
237#335—614no—
238#332—612no—
239#332—614no—
240#331—612no—
241C61#1939211yescharacteristic, complemented, Sylow
242C2×#41—21no—
243C22×Q8—1221no—
244C22×D4C22⋊D6111yescharacteristic
245C23×C4—21no—
246AGL(1,ℤ/8)—44no—
247C2×SD16—22no—
248C2×SD16—1224no—
249C2×SD16—2444no—
250C2×D8—22no—
251C2×M16—21no—
252C22×C8—1224no—
253C22×C8—1221no—
254C4⋊Q8—612no—
255C4⋊Q8—1221no—
256C4⋊Q8—1221no—
257C4⋊Q8—1221no—
258C4⋊Q8C22×D6111yescharacteristic
259#127C22×D6111yescharacteristic
260#124—21no—
261#121—21no—
262#121—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4C22⋊D6112yes—
268C4×D4C22×D6112yes—
269C2×C4⋊C4—1221no—
270C2×C4⋊C4—1221no—
271C2×C4⋊C4C22⋊D6111yescharacteristic
272C2×C22⋊C4—21no—
273C2×C42C22×D6111yescharacteristic
274#107—1224no—
275C4⋊C8—21no—
276C4⋊C8—612no—
277C4⋊C8—612no—
278C4⋊C8—612no—
279C4⋊C8—1221no—
280C4⋊C8C22×D6112yes—
281#103—22no—
282#103—1224no—
283#102—22no—
284#102—1224no—
285C22⋊C8—612no—
286C22⋊C8—1224no—
287C22⋊C8—1221no—
288C4×C8—612no—
289C4×C8—1221no—
290C24—21no—
291#41—42no—
292#41—42no—
293C2×Q8—21no—
294C2×Q8—1222no—
295C2×Q8—2442no—
296C2×Q8—2442no—
297C2×Q8—2441no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—41no—
304C2×D4#1175819914yes—
305C2×D4C2×C22⋊D6112yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4—2441no—
312C22×C4D4×D6111yescharacteristic
313C22×C4#1175822411yescharacteristic
314C22×C4C2×C22⋊D6111yescharacteristic
315SD16—44no—
316SD16—2448no—
317D8—44no—
318M16—42no—
319C2×C8—22no—
320C2×C8—614no—
321C2×C8—1222no—
322C2×C8—1222no—
323C2×C8—4884no—
324C2×C8—2442no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—1222no—
329C4⋊C4—1222no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C4⋊C4C2×C22⋊D6112yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C42D4×D6111yescharacteristic
338C42#1175822411yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C42C23×D6111yescharacteristic
341C23—22no—
342C23—24no—
343C23—41no—
344C23—81no—
345C23—81no—
346C23#1451974911yescharacteristic
347Q8—41no—
348Q8—41no—
349Q8—4882no—
350Q8—2442no—
351Q8—2442no—
352D4—24no—
353D4—22no—
354D4—41no—
355D4—41no—
356D4—42no—
357D4—81no—
358D4—82no—
359D4C2×#1175819914yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—4882no—
369C2×C4—2442no—
370C2×C4#1451974911yescharacteristic
371C2×C4#1451974911yescharacteristic
372C2×C4#1451974911yescharacteristic
373C2×C4C2×#1175819912yes—
374C2×C4#1451979712yes—
375C2×C4C2×D4×D6111yescharacteristic
376C2×C4C22×C22⋊D6111yescharacteristic
377C2×C4C2×#1175822411yescharacteristic
378C8—42no—
379C8—1224no—
380C8—2442no—
381C8—2442no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#1683999612yes—
392C22C2×#1451974911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—4882no—
399C4#1683999612yes—
400C4C22×#1175819911yescharacteristic
401C4C2×#1451979711yescharacteristic
402C4#1684057212yes—
403C2—22no—
404C2—44no—
405C2—161no—
406C2#2145533311yescharacteristic
407C2C2×#1683999611yescharacteristic
408C2#2145555811yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3036C2
socleC22×C61C2×#121
hypercenter#3036D61
hypocenterC61#19392
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #21456937C2semidirect
2. C2×#16839967C2semidirect
3. #21454843C2semidirect
4. #21445254C2semidirect
5. #21445206C2semidirect
6. #21444595C2semidirect
7. #21444492C2semidirect
8. #21444468C2semidirect
12. #16840165C22semidirect
13. #16840108C22semidirect
25. #16839514C22semidirect
26. #16839509C22semidirect
27. #16839508C22semidirect
28. #16839506C22semidirect
241. C61#19392semidirect

Maximal subgroups

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

subgroupindex
#214569372
C2×#168399672
#214548432
#214452542
#214452062
#214445952
#214444922
#214444682
C61×#30362
#1939261