Finite Group Databasebeta

← #29663074

Subgroups — #29663074

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456942C211yesmaximal, characteristic, complemented
2#21456737C211yesmaximal, characteristic
3C2×#16840134C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic, complemented
5#21445283C214yesmaximal, complemented
6#21445159C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444492C211yesmaximal, characteristic, complemented
9C61×#1545C211yesmaximal, characteristic, complemented
10C22×#11758200—21no—
11C22×#11758200—21no—
12C2×#14519785—21no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840167—21no—
16#16840138—21no—
17#16840134—21no—
18#16840134—21no—
19#16840134C2212yes—
20#16840134C2212yes—
21#16840109—21no—
22#16840108—21no—
23#16840076—22no—
24#16840043C2214yescomplemented
25#16839954C2214yes—
26#16839896—22no—
27#16839839—21no—
28#16839839C2212yescomplemented
29#16839827—21no—
30#16839827C2212yescomplemented
31#16839815C2211yescharacteristic
32#16839730—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839509C2214yescomplemented
37C2×#127×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#331C2214yes—
41C2×#11758222—21no—
42C22×Dic122—21no—
43C2×#11758200—22no—
44C2×#11758200—22no—
45C2×#11758200—22no—
46C2×#11758200—22no—
47C2×#11758200—44no—
48C2×#11758200—44no—
49C2×#11758199—24no—
50#14519785—44no—
51C2×#11758189—22no—
52C22×#92418D411yescharacteristic
53C22×#92418D411yescharacteristic
54C2×C8×D61—22no—
55#14519777—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777D411yescharacteristic
59#14519777C2312yes—
60#14519774D411yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751D412yes—
64C4×D244—21no—
65C4×D244C2312yes—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68#14519711—24no—
69C4⋊#92418C2312yes—
70#14519699—24no—
71#14519698—24no—
72C22⋊#92418—21no—
73#14519689—22no—
74#14519689—22no—
75C4×#92418—21no—
76C4×#92418D411yescharacteristic
77C4×#92418D411yescharacteristic
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C22×D4×C61—21no—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C2×M16×C61—21no—
84#127×C61—21no—
85#127×C61—21no—
86#127×C61—21no—
87#127×C61C2312yes—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2312yes—
91C4⋊C8×C61C2312yes—
92C22⋊C8×C61—21no—
93C4×C8×C61—21no—
94C24×C61—41no—
95C2×D4×C61—22no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C2×D4×C61—42no—
102C2×D4×C61—42no—
103C2×D4×C61—82no—
104C2×D4×C61—82no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C2×C22⋊D61—22no—
109C22×Dic61—41no—
110#11758222—42no—
111#11758222—42no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244C2×D412yes—
115C2×C4×D61—22no—
116C2×Dic122—21no—
117C2×Dic122—22no—
118C2×Dic122—42no—
119C2×Dic122—42no—
120C2×Dic122—41no—
121M16×C61—42no—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C42×C61C2411yescharacteristic
128#11758200—44no—
129#11758200—44no—
130#11758200—44no—
131#11758200—44no—
132#11758199—84no—
133#11758198—22no—
134#11758198—22no—
135C4⋊Dic61—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—22no—
138C4⋊Dic61—41no—
139C4⋊Dic61—41no—
140C4⋊Dic61C2×D412yes—
141C2×#92418—22no—
142C2×#92418—22no—
143C2×#92418D814yescomplemented
144C2×#92418C2×D411yescharacteristic
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147C2×#92418C2×D411yescharacteristic
148#11758189—44no—
149C8×D61—44no—
150C23×C61—42no—
151C23×C61—82no—
152C23×C61—82no—
153C23×C61—82no—
154C23×C61#12111yescharacteristic
155C22×D61—22no—
156D4×C61—42no—
157D4×C61—42no—
158D4×C61—44no—
159D4×C61—82no—
160D4×C61—82no—
161D4×C61—82no—
162D4×C61—82no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61C2×D812yes—
169C2×C4×C61AGL(1,ℤ/8)12yes—
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C22⋊D61—44no—
174C2×Dic61—22no—
175C2×Dic61—42no—
176C2×Dic61—82no—
177D244—24no—
178D244—42no—
179D244—42no—
180C4×D61—42no—
181C4×D61—42no—
182Dic122—42no—
183Dic122—42no—
184Dic122—42no—
185Dic122—82no—
186C8×C61—42no—
187C8×C61—42no—
188#92418—21no—
189#92418—21no—
190#92418—42no—
191#92418C2×D812yes—
192#92418C2×D812yes—
193#19186—611nomaximal, Sylow
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—82no—
197C22×C61—82no—
198C22×C61—82no—
199C22×C61—164no—
200C22×C61#45912yes—
201C22×C61C2×#12111yescharacteristic
202C2×D61—42no—
203C2×D61—42no—
204C2×D61—42no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61#45912yes—
208C4×C61#47112yes—
209C4×C61C22×D811yescharacteristic
210C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
211C4×C61#57712yes—
212Dic61—42no—
213Dic61—82no—
214#3226—611no—
215#3205—611no—
216C2×#492—611no—
217#2931—611no—
218#1685—614no—
219#1665—614no—
220#1564—611no—
221#1546—611no—
222#1545D6111yescharacteristic, complemented
223C2×C61—22no—
224C2×C61—162no—
225C2×C61C2×#45911yescharacteristic
226C2×C61#305711yescharacteristic
227C2×C61C2×#47111yescharacteristic
228D61—82no—
229C22×SD16—1221no—
230C22×SD16—1221no—
231C2×#131—1221no—
232#532—611no—
233C2×C4⋊Q8—611no—
234C2×#127C2×D6111yescharacteristic
235#500—1221no—
236#493—1221no—
237#492—612no—
238#492—612no—
239#492—1221no—
240#492—1221no—
241#486—1221no—
242#486—1221no—
243#480—1222no—
244#474—614no—
245#459—614no—
246#447—1222no—
247#436—612no—
248#436—1221no—
249C8×D4—612no—
250C8×D4—1221no—
251#431—611no—
252#431C2×D6111yescharacteristic
253#423C2×D6111yescharacteristic
254#408—1221no—
255C2×C4×C8—611no—
256#335—612no—
257#334—612no—
258#332—614no—
259#331C2×D6114yes—
260C61#1918611yescharacteristic, complemented, Sylow
261C2×#41—1221no—
262C22×Q8—1221no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4—21no—
266C2×SD16—1222no—
267C2×SD16—1222no—
268C2×SD16—1222no—
269C2×SD16—1222no—
270C2×SD16—2444no—
271C2×SD16—2444no—
272C2×D8—1224no—
273#131—2444no—
274C2×M16—21no—
275C2×M16—1222no—
276C22×C8—611no—
277C22×C8—611no—
278C22×C8—1222no—
279C4⋊Q8—611no—
280C4⋊Q8—612no—
281C4⋊Q8—1221no—
282C4⋊Q8—1221no—
283C4⋊Q8—1221no—
284#127—21no—
285#127—21no—
286#127—21no—
287#127—611no—
288#127C22×D6112yes—
289#124—1221no—
290#121—612no—
291#121—1221no—
292C4×D4—612no—
293C4×D4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C42C22×D6111yescharacteristic
297#107—1224no—
298C4⋊C8—21no—
299C4⋊C8—612no—
300C4⋊C8C22×D6112yes—
301C4⋊C8C22×D6112yes—
302#102—1224no—
303#102—1224no—
304C22⋊C8—21no—
305C22⋊C8—1222no—
306C22⋊C8—1222no—
307C22⋊C8—1221no—
308C4×C8—21no—
309C4×C8—611no—
310C4×C8—611no—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—1221no—
314C24—41no—
315#41—2442no—
316#41—2442no—
317C2×Q8—1221no—
318C2×Q8—1222no—
319C2×Q8—2442no—
320C2×Q8—2442no—
321C2×Q8—2441no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—612no—
333C2×D4—1222no—
334C2×D4—1221no—
335C2×D4—1221no—
336C22×C4—1222no—
337C22×C4—2441no—
338C22×C4#1175822211yescharacteristic
339C22×C4D4×D6111yescharacteristic
340C22×C4D4×D6111yescharacteristic
341SD16—2444no—
342SD16—2444no—
343SD16—2444no—
344SD16—2444no—
345D8—4884no—
346M16—42no—
347M16—2444no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—611no—
351C2×C8—611no—
352C2×C8—614no—
353C2×C8—611no—
354C2×C8—611no—
355C2×C8—1222no—
356C2×C8—1222no—
357C2×C8—2444no—
358C4⋊C4—612no—
359C4⋊C4—1221no—
360C4⋊C4—1222no—
361C4⋊C4—1222no—
362C4⋊C4—2441no—
363C4⋊C4—2441no—
364C22⋊C4—1222no—
365C22⋊C4—1222no—
366C42#1175822211yescharacteristic
367C42D4×D6111yescharacteristic
368C42D4×D6111yescharacteristic
369C42C23×D6111yescharacteristic
370C23—42no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—1222no—
375C23#1451975011yescharacteristic
376Q8—4882no—
377Q8—2442no—
378Q8—2442no—
379Q8—2442no—
380D4—42no—
381D4—42no—
382D4—44no—
383D4—82no—
384D4—82no—
385D4—82no—
386D4—82no—
387D4—1224no—
388D4—2444no—
389D4—2442no—
390D4—2442no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—1222no—
394C2×C4—4882no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4#1451975011yescharacteristic
399C2×C4#1451975011yescharacteristic
400C2×C4#1451975011yescharacteristic
401C2×C4D8×D6112yes—
402C2×C4#1451980412yes—
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C2×D4×D6111yescharacteristic
405C2×C4C2×#1175822211yescharacteristic
406C8—42no—
407C8—42no—
408C8—612no—
409C8—612no—
410C8—1221no—
411C8—1221no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—164no—
419C22—2442no—
420C22—2442no—
421C22—2442no—
422C22#1683995612yes—
423C22C2×#1451975011yescharacteristic
424C4—21no—
425C4—21no—
426C4—4882no—
427C4—2442no—
428C4#1683995612yes—
429C4#1684002412yes—
430C4C2×D8×D6111yescharacteristic
431C4C2×#1451980411yescharacteristic
432C4#1684057212yes—
433C2—22no—
434C2—162no—
435C2—4882no—
436C2C2×#1683995611yescharacteristic
437C2#2145545711yescharacteristic
438C2C2×#1684002411yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C2×#121
prime radical#21456737C2
hypercenter#1545D61
hypocenterC61#19186
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
1. #21456942C2semidirect
3. C2×#16840134C2semidirect
4. #21454530C2semidirect
5. #21445283C2semidirect
6. #21445159C2semidirect
7. #21444565C2semidirect
8. #21444492C2semidirect
9. C61×#1545C2semidirect
24. #16840043C22semidirect
28. #16839839C22semidirect
30. #16839827C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
36. #16839509C22semidirect
143. C2×#92418D8semidirect
222. #1545D61semidirect
260. C61#19186semidirect

Maximal subgroups

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

subgroupindex
#214569422
#214567372
C2×#168401342
#214545302
#214452832
#214451592
#214445652
#214444922
C61×#15452
#1918661