Finite Group Databasebeta

← #29836788

Subgroups — #29836788

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458961C212yesmaximal, complemented
2#21457623C211yesmaximal, characteristic, complemented
3#21457618C211yesmaximal, characteristic, complemented
4#21447946C212yesmaximal, complemented
5#21447897C211yesmaximal, characteristic, complemented
6#21447892C211yesmaximal, characteristic, complemented
7#21447789C212yesmaximal, complemented
8#21447782C212yesmaximal, complemented
9#21447701C211yesmaximal, characteristic, complemented
10#21447657C211yesmaximal, characteristic, complemented
11C61×#2248C211yesmaximal, characteristic, complemented
12#16840575—21no—
13#16840575C2212yescomplemented
14#16840563—21no—
15#16840563C2212yes—
16#16840203C2212yes—
17#16840201—22no—
18#16840199—22no—
19#16840198C2212yescomplemented
20#16840194C2211yescharacteristic
21#16840193—22no—
22SD32×D61—22no—
23#16840178C2212yescomplemented
24#16840034—21no—
25#16840034C2212yescomplemented
26#16840029—21no—
27#16840029C2212yes—
28#16839944—22no—
29#16839929—22no—
30#16839880C2211yescharacteristic
31#16839880C2211yescharacteristic
32#16839880C2211yescharacteristic, complemented
33#16839592C2211yescharacteristic, complemented
34#16839583C2211yescharacteristic, complemented
35#16839578C2212yescomplemented
36#16839575C2212yescomplemented
37#16839570C2211yescharacteristic
38#16839570C2211yescharacteristic, complemented
39#16839545C2212yescomplemented
40C2×SD32×C61C2212yescomplemented
41C2×SD32×C61C2212yescomplemented
42C61×#494C2211yescharacteristic, complemented
43C61×#493C2211yescharacteristic
44C4×C16×C61C2211yescharacteristic, complemented
45#14519825—22no—
46#14519822—22no—
47#14519809—42no—
48#14519809—42no—
49#14519808—22no—
50#14519808—22no—
51#14519803—22no—
52#14519803—22no—
53#14519801—42no—
54#14519801—42no—
55#14519800—22no—
56#14519800—41no—
57#14519799D411yescharacteristic
58#14519799C2311yescharacteristic
59#14519799C2311yescharacteristic
60#14519796—22no—
61#14519796—41no—
62Q16×D61—22no—
63Q16×D61—41no—
64D8×D61—22no—
65D8×D61—41no—
66#14519783—21no—
67#14519783—21no—
68#14519783C2312yes—
69Dic488D411yescharacteristic, complemented
70#14519723—22no—
71#14519722D411yescharacteristic, complemented
72#14519722D411yescharacteristic, complemented
73#14519721—22no—
74#14519720—22no—
75D976D411yescharacteristic, complemented
76#14519718—22no—
77#14519717C2312yes—
78#14519716—22no—
79C16×D61—22no—
80#14519713D411yescharacteristic
81#14519713C2311yescharacteristic
82#14519713C2311yescharacteristic
83#14519705—21no—
84#14519705—21no—
85#14519705D412yes—
86#14519696—22no—
87#14519694—22no—
88C2×Q16×C61—21no—
89C2×Q16×C61C2312yes—
90C2×D8×C61—21no—
91C2×D8×C61C2312yes—
92C4⋊Q8×C61—21no—
93#127×C61—21no—
94SD32×C61—22no—
95SD32×C61—22no—
96SD32×C61—42no—
97C2×C16×C61C2311yescharacteristic
98C2×C16×C61C2311yescharacteristic
99C4×C8×C61C2311yescharacteristic
100C2×Q8×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—81no—
103C2×C22⋊D61—82no—
104#11758226—42no—
105#11758226—42no—
106Q8×D61—42no—
107Q8×D61—42no—
108#11758224—42no—
109#11758224—42no—
110D4×D61—42no—
111D4×D61—42no—
112#11758222—21no—
113#11758222—21no—
114#11758222—82no—
115#11758222C2×D412yes—
116Q16×C61—22no—
117Q16×C61—22no—
118Q16×C61—41no—
119Q16×C61—41no—
120D8×C61—22no—
121D8×C61—22no—
122D8×C61—41no—
123D8×C61—41no—
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2411yescharacteristic
126C4⋊C4×C61—81no—
127C42×C61C2×D411yescharacteristic
128#11758202—42no—
129#11758202—42no—
130#11758201—42no—
131#11758201—42no—
132#11758200—42no—
133#11758200—42no—
134#11758199—42no—
135#11758199—42no—
136#11758194—21no—
137#11758194—21no—
138#11758194C2×D412yes—
139Dic244D811yescharacteristic
140Dic244C2×D411yescharacteristic
141Dic244C2×D411yescharacteristic
142D488D811yescharacteristic
143D488C2×D411yescharacteristic
144D488C2×D411yescharacteristic
145#11758190—21no—
146#11758190—21no—
147#11758190D812yes—
148#11758189—22no—
149#11758189—41no—
150#11758189—41no—
151C8×D61—22no—
152C8×D61—41no—
153C8×D61—41no—
154C16×C61—21no—
155C16×C61C2×D411yescharacteristic
156C16×C61C2×D411yescharacteristic
157#11758185—22no—
158C23×C61—42no—
159C22×D61—82no—
160Q8×C61—42no—
161Q8×C61—42no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—162no—
165C2×C4×C61—21no—
166C2×C4×C61—42no—
167C2×C4×C61C22×D411yescharacteristic
168C22⋊D61—22no—
169C22⋊D61—41no—
170C22⋊D61—41no—
171C22⋊D61—82no—
172C22⋊D61—82no—
173C2×Dic61—82no—
174D244—21no—
175D244—21no—
176D244—82no—
177D244C2×D812yes—
178C4×D61—22no—
179C4×D61—41no—
180C4×D61—41no—
181C4×D61—82no—
182C4×D61—82no—
183Dic122—21no—
184Dic122—21no—
185Dic122—82no—
186Dic122C2×D812yes—
187C8×C61C2×D811yescharacteristic
188C8×C61C2×D811yescharacteristic
189C8×C61C22×D411yescharacteristic
190C8×C61H(5,𝔽2)11yescharacteristic
191#92418—22no—
192#92418—41no—
193#92418—41no—
194#33856—611nomaximal, Sylow
195C22×C61—82no—
196C22×C61—162no—
197C22×C61#57611yescharacteristic
198C2×D61—22no—
199C2×D61—41no—
200C2×D61—41no—
201C2×D61—322no—
202C4×C61—22no—
203C4×C61—82no—
204C4×C61D4211yescharacteristic
205C4×C61C22×D811yescharacteristic
206Dic61—22no—
207Dic61—41no—
208Dic61—41no—
209#3416—612no—
210#3293—611no—
211#3292—611no—
212#2248D6111yescharacteristic, complemented
213#2223—612no—
214#2214—611no—
215#2213—611no—
216#2194—612no—
217#2193—612no—
218#2178—611no—
219#2170—611no—
220C2×C61—21no—
221C2×C61—162no—
222C2×C61D4×D811yescharacteristic
223D61—42no—
224D61—81no—
225D61—81no—
226#578—612no—
227#578—1221no—
228#576—612no—
229#576—1221no—
230#510—612no—
231#510—1222no—
232#509—612no—
233#509—1222no—
234#508—611no—
235#508—1222no—
236C2×SD32—1222no—
237C2×SD32C2×D6112yescomplemented
238C2×SD32C2×D6112yescomplemented
239#504—612no—
240#494C2×D6111yescharacteristic, complemented
241#493C2×D6111yescharacteristic
242#473—612no—
243#473—1221no—
244#472—612no—
245#472—1221no—
246#456—1222no—
247#453—1222no—
248#443—611no—
249#443—611no—
250#443—611no—
251#368—611no—
252#364—611no—
253#362—612no—
254#361—612no—
255#359—611no—
256#359—611no—
257#350—612no—
258C4×C16C2×D6111yescharacteristic, complemented
259C61#3385611yescharacteristic, complemented, Sylow
2602−1+4—1222no—
261H(5,𝔽2)—1222no—
262#137—1222no—
263#137—1222no—
264#137—2442no—
265#137—2442no—
266AGL(1,ℤ/8)—1222no—
267AGL(1,ℤ/8)—1222no—
268AGL(1,ℤ/8)—2442no—
269AGL(1,ℤ/8)—2442no—
270#135—611no—
271#135—611no—
272#135—611no—
273#135—1222no—
274#135—1222no—
275#135—2441no—
276#135—2441no—
277C2×Q16—21no—
278C2×Q16—1222no—
279C2×Q16—2441no—
280C2×Q16C22×D6112yes—
281C2×D8—21no—
282C2×D8—1222no—
283C2×D8—2441no—
284C2×D8C22×D6112yes—
285#131—612no—
286#131—1221no—
287#131—1221no—
288C4⋊Q8—21no—
289#127—21no—
290Q32—611no—
291Q32—1222no—
292SD32—22no—
293SD32—22no—
294SD32—42no—
295SD32—611no—
296SD32—611no—
297SD32—1222no—
298SD32—1222no—
299D16—611no—
300D16—1222no—
301M32—612no—
302M32—1222no—
303C2×C16—1222no—
304C2×C16C22×D6111yescharacteristic
305C2×C16C22×D6111yescharacteristic
306#108—611no—
307#108—611no—
308#108—611no—
309#104—612no—
310#104—1221no—
311#104—1221no—
312#101—1222no—
313#100—1222no—
314C4×C8C22×D6111yescharacteristic
315#41—612no—
316#41—1221no—
317#41—1221no—
318#41—4882no—
319#41—2442no—
320#41—2442no—
321#41—2442no—
322#41—2442no—
323C2×Q8—22no—
324C2×Q8—2442no—
325C2×Q8—2442no—
326C2×D4—22no—
327C2×D4—81no—
328C2×D4—4882no—
329C2×D4—2442no—
330C2×D4—2442no—
331Q16—22no—
332Q16—22no—
333Q16—41no—
334Q16—41no—
335Q16—611no—
336Q16—611no—
337Q16—611no—
338Q16—2442no—
339Q16—2442no—
340SD16—612no—
341SD16—1221no—
342SD16—1221no—
343SD16—2442no—
344SD16—2442no—
345SD16—2442no—
346SD16—2442no—
347D8—22no—
348D8—22no—
349D8—41no—
350D8—41no—
351D8—611no—
352D8—611no—
353D8—611no—
354D8—2442no—
355D8—2442no—
356M16—612no—
357M16—1222no—
358M16—1221no—
359M16—1221no—
360M16—2441no—
361M16—2441no—
362C2×C8—1222no—
363C2×C8—2441no—
364C2×C8—2441no—
365C2×C8D4×D6111yescharacteristic
366C2×C8C23×D6111yescharacteristic
367C4⋊C4—81no—
368C42D4×D6111yescharacteristic
369C16—21no—
370C16—1222no—
371C16D4×D6111yescharacteristic
372C16D4×D6111yescharacteristic
373C23—42no—
374C23—4882no—
375Q8—42no—
376Q8—42no—
377Q8—612no—
378Q8—1221no—
379Q8—1221no—
380Q8—4882no—
381D4—42no—
382D4—42no—
383D4—162no—
384D4—612no—
385D4—1222no—
386D4—1221no—
387D4—1221no—
388D4—4882no—
389D4—4882no—
390D4—4882no—
391D4—2441no—
392D4—2441no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—1222no—
396C2×C4—4882no—
397C2×C4—4882no—
398C2×C4—4882no—
399C2×C4—2441no—
400C2×C4—2441no—
401C2×C4C2×D4×D6111yescharacteristic
402C8—1222no—
403C8—2441no—
404C8—2441no—
405C8D8×D6111yescharacteristic
406C8D8×D6111yescharacteristic
407C8C2×D4×D6111yescharacteristic
408C8#1451982211yescharacteristic
409C22—82no—
410C22—162no—
411C22—1222no—
412C22—19522no—
413C22—2441no—
414C22—2441no—
415C22#1684056311yescharacteristic
416C4—22no—
417C4—82no—
418C4—1222no—
419C4—2441no—
420C4—2441no—
421C4#1684039711yescharacteristic
422C4C2×D8×D6111yescharacteristic
423C2—21no—
424C2—162no—
425C2—4881no—
426C2—4881no—
427C2—2442no—
428C2#2145749411yescharacteristic
subgroupG/H
centerC2#21457494
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2248C2
socleC2×C61D4×D8
hypercenter#2248D61
hypocenterC61#33856
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. #21458961C2semidirect
2. #21457623C2semidirect
3. #21457618C2semidirect
4. #21447946C2semidirect
5. #21447897C2semidirect
6. #21447892C2semidirect
7. #21447789C2semidirect
8. #21447782C2semidirect
9. #21447701C2semidirect
10. #21447657C2semidirect
11. C61×#2248C2semidirect
13. #16840575C22semidirect
19. #16840198C22semidirect
23. #16840178C22semidirect
25. #16840034C22semidirect
32. #16839880C22semidirect
33. #16839592C22semidirect
34. #16839583C22semidirect
35. #16839578C22semidirect
36. #16839575C22semidirect
38. #16839570C22semidirect
39. #16839545C22semidirect
40. C2×SD32×C61C22semidirect
41. C2×SD32×C61C22semidirect
42. C61×#494C22semidirect
44. C4×C16×C61C22semidirect
69. Dic488D4semidirect
71. #14519722D4semidirect
72. #14519722D4semidirect
75. D976D4semidirect
212. #2248D61semidirect
237. C2×SD32C2×D61semidirect
238. C2×SD32C2×D61semidirect
240. #494C2×D61semidirect
258. C4×C16C2×D61semidirect
259. C61#33856semidirect

Maximal subgroups

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

subgroupindex
#214589612
#214576232
#214576182
#214479462
#214478972
#214478922
#214477892
#214477822
#214477012
#214476572
C61×#22482
#3385661