Finite Group Databasebeta

← #29665246

Subgroups — #29665246

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#21456940C211yesmaximal, characteristic, complemented
2#21455283C211yesmaximal, characteristic
3#21454843C211yesmaximal, characteristic, complemented
4#21445258C214yesmaximal, complemented
5#21445206C214yesmaximal, complemented
6#21444593C211yesmaximal, characteristic, complemented
7#21444491C211yesmaximal, characteristic, complemented
8#21444469C211yesmaximal, characteristic, complemented
9C2×C61×#461C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11#16840312C2211yescharacteristic
12#16840165C2212yescomplemented
13#16840110C2212yescomplemented
14#16840075—22no—
15#16840039—22no—
16#16839977—22no—
17#16839967C2212yes—
18#16839957C2212yes—
19D4×#92418—22no—
20D4×#92418C2214yes—
21#16839815C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839786C2211yescharacteristic
24#16839761—21no—
25C2×C22⋊#92418—22no—
26#16839514C2214yescomplemented
27#16839509C2212yescomplemented
28#16839508C2214yescomplemented
29#16839505C2212yescomplemented
30C22×SD16×C61—21no—
31C2×C4⋊Q8×C61C2211yescharacteristic
32C2×C4×D4×C61C2211yescharacteristic
33C61×#461—22no—
34C61×#461C2214yes—
35C2×C4⋊C8×C61C2211yescharacteristic
36C2×#103×C61—21no—
37C2×#11758222—21no—
38#14519803—44no—
39C2×#11758200—22no—
40C2×#11758199—22no—
41C2×#11758194—21no—
42C22×#92418—24no—
43#14519777C2311yescharacteristic
44#14519774C2311yescharacteristic
45#14519765—21no—
46#14519751—21no—
47#14519751—21no—
48C4×D244—21no—
49C4×D244—21no—
50#14519710—24no—
51C4⋊#92418—21no—
52C4⋊#92418D412yes—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519700—22no—
56#14519697—22no—
57#14519697—24no—
58C22⋊#92418—21no—
59C22⋊#92418—24no—
60C22⋊#92418D412yes—
61C4×#92418—21no—
62C4×#92418C2312yes—
63C22×Q8×C61—21no—
64C22×D4×C61D411yescharacteristic
65C23×C4×C61—21no—
66C2×SD16×C61—24no—
67C2×SD16×C61—44no—
68C22×C8×C61—21no—
69C4⋊Q8×C61—21no—
70C4⋊Q8×C61—21no—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61C2312yes—
73C4×D4×C61—21no—
74C4×D4×C61—21no—
75C4×D4×C61D412yes—
76C4×D4×C61C2312yes—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61D411yescharacteristic
80C2×C22⋊C4×C61—21no—
81C2×C42×C61C2311yescharacteristic
82C4⋊C8×C61—21no—
83C4⋊C8×C61C2312yes—
84#103×C61—24no—
85C24×C61—21no—
86C2×Q8×C61—22no—
87C2×Q8×C61—41no—
88C2×Q8×C61—42no—
89C2×Q8×C61—42no—
90C2×D4×C61—21no—
91C2×D4×C61—22no—
92C2×D4×C61D814yes—
93C2×D4×C61C2×D412yes—
94C22×C4×C61—21no—
95C22×C4×C61—22no—
96C22×C4×C61—22no—
97C22×C4×C61—41no—
98C22×C4×C61—82no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61#4111yescharacteristic
102C2×C22⋊D61—41no—
103#11758222—42no—
104#11758222—42no—
105C2×D244—21no—
106C2×D244—21no—
107C2×D244—21no—
108C2×C4×D61—41no—
109C2×Dic122—21no—
110SD16×C61—48no—
111C2×C8×C61—22no—
112C2×C8×C61—42no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—41no—
117C4⋊C4×C61—41no—
118C4⋊C4×C61C2×D412yes—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—22no—
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124C42×C61C2411yescharacteristic
125#11758200—44no—
126#11758199—44no—
127#11758198—41no—
128#11758198—41no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131#11758194—42no—
132C2×#92418—22no—
133C2×#92418—22no—
134C2×#92418—84no—
135C2×#92418C2×D414yes—
136C23×C61—22no—
137C23×C61—24no—
138C23×C61—81no—
139C23×C61—81no—
140C23×C61#12111yescharacteristic
141C22×D61—41no—
142Q8×C61—42no—
143Q8×C61—42no—
144Q8×C61—82no—
145D4×C61—22no—
146D4×C61—24no—
147D4×C61C2×D814yes—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61—41no—
151C2×C4×C61—44no—
152C2×C4×C61—42no—
153C2×C4×C61—82no—
154C2×C4×C61—81no—
155C2×C4×C61—82no—
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61C2×D812yes—
160C2×C4×C61#13512yes—
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C22⋊D61—82no—
165C2×Dic61—41no—
166D244—41no—
167D244—41no—
168D244—42no—
169D244—81no—
170C4×D61—82no—
171Dic122—41no—
172Dic122—41no—
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#46612yes—
185C22×C61C2×#12111yescharacteristic
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61—82no—
194C4×C61#46612yes—
195C4×C61C22×D811yescharacteristic
196C4×C61C2×#13511yescharacteristic
197C4×C61#57712yes—
198Dic61—81no—
199#3226—611no—
200#3036—611no—
201C2×#461D6111yescharacteristic, complemented
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#304211yescharacteristic
211C2×C61C2×#46611yescharacteristic
212C2×C61#306911yescharacteristic
213D61—161no—
214C2×AGL(1,ℤ/8)—1221no—
215C22×SD16—21no—
216#532—611no—
217C2×C4⋊Q8C2×D6111yescharacteristic
218C2×C4×D4C2×D6111yescharacteristic
219#500—612no—
220#486—612no—
221#480—1222no—
222#474—1222no—
223#463—1222no—
224#461—22no—
225#461—612no—
226#461C2×D6114yes—
227#459—612no—
228C8×D4—614no—
229C8×D4—1222no—
230#431—611no—
231#431—611no—
232#423—611no—
233C2×C4⋊C8C2×D6111yescharacteristic
234#417—1221no—
235C2×#103—21no—
236C2×C22⋊C8—1222no—
237#335—614no—
238#332—614no—
239#332—612no—
240#331—612no—
241C61#1939211yescharacteristic, complemented, Sylow
242C2×#41—1221no—
243C22×Q8—21no—
244C22×D4C22⋊D6111yescharacteristic
245C23×C4—21no—
246AGL(1,ℤ/8)—2444no—
247C2×SD16—24no—
248C2×SD16—44no—
249C2×SD16—1222no—
250C2×D8—1222no—
251C2×M16—1221no—
252C22×C8—21no—
253C22×C8—1224no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—611no—
258C4⋊Q8C22×D6112yes—
259#127—611no—
260#124—1221no—
261#121—1221no—
262#121—1221no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—1221no—
266C4×D4—1221no—
267C4×D4C22⋊D6112yes—
268C4×D4C22×D6112yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
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—24no—
282#103—1222no—
283#102—1224no—
284#102—1222no—
285C22⋊C8—612no—
286C22⋊C8—1224no—
287C22⋊C8—1221no—
288C4×C8—612no—
289C4×C8—1221no—
290C24—21no—
291#41—2442no—
292#41—2442no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—1221no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—1221no—
301C2×D4—1221no—
302C2×D4—1221no—
303C2×D4—2441no—
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—48no—
316SD16—2444no—
317D8—2444no—
318M16—2442no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—614no—
322C2×C8—1222no—
323C2×C8—1222no—
324C2×C8—4884no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—1221no—
331C4⋊C4—1221no—
332C4⋊C4C2×C22⋊D6112yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—2441no—
336C22⋊C4—2441no—
337C42D4×D6111yescharacteristic
338C42#1175822411yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C42C23×D6111yescharacteristic
341C23—22no—
342C23—24no—
343C23—81no—
344C23—81no—
345C23—2441no—
346C23#1451974911yescharacteristic
347Q8—42no—
348Q8—42no—
349Q8—82no—
350Q8—2441no—
351Q8—2441no—
352D4—24no—
353D4—22no—
354D4—4881no—
355D4—4882no—
356D4—2441no—
357D4—2441no—
358D4—2442no—
359D4C2×#1175819914yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—4882no—
369C2×C4—2441no—
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—4881no—
389C22—4881no—
390C22—4881no—
391C22#1683999612yes—
392C22C2×#1451974911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—82no—
398C4—4881no—
399C4#1683999612yes—
400C4C22×#1175819911yescharacteristic
401C4C2×#1451979711yescharacteristic
402C4#1684056812yes—
403C2—22no—
404C2—44no—
405C2—9761no—
406C2#2145532811yescharacteristic
407C2C2×#1683999611yescharacteristic
408C2#2145555311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#461C2
socleC22×C61C2×#121
prime radical#21455283C2
hypercenterC2×#461D61
hypocenterC61#19392
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. #21456940C2semidirect
3. #21454843C2semidirect
4. #21445258C2semidirect
5. #21445206C2semidirect
6. #21444593C2semidirect
7. #21444491C2semidirect
8. #21444469C2semidirect
9. C2×C61×#461C2semidirect
12. #16840165C22semidirect
13. #16840110C22semidirect
26. #16839514C22semidirect
27. #16839509C22semidirect
28. #16839508C22semidirect
29. #16839505C22semidirect
201. C2×#461D61semidirect
241. C61#19392semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214552832
#214548432
#214452582
#214452062
#214445932
#214444912
#214444692
C2×C61×#4612
#1939261