Finite Group Databasebeta

← #35226081

Subgroups — #35226081

Order 20864.

510 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
1C2×#19104763C211yesmaximal, characteristic, complemented
2C2×#19104757C211yesmaximal, characteristic, complemented
3#23782336C211yesmaximal, characteristic, complemented
4#23782311C211yesmaximal, characteristic, complemented
5#23782054C211yesmaximal, characteristic, complemented
6#23781911C211yesmaximal, characteristic, complemented
7#23781901C211yesmaximal, characteristic, complemented
8#23781888C211yesmaximal, characteristic, complemented
9#23781861C211yesmaximal, characteristic, complemented
10#23781829C211yesmaximal, characteristic, complemented
11#23781812C211yesmaximal, characteristic, complemented
12#23781812C211yesmaximal, characteristic, complemented
13#23781801C211yesmaximal, characteristic, complemented
14#23781767C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579699C2211yescharacteristic
18C2×#14579699C2211yescharacteristic, complemented
19C22×D652C2211yescharacteristic
20#19104763—24no—
21#19104757—24no—
22C2×Dic652C2211yescharacteristic, complemented
23C2×#14579688C2211yescharacteristic
24C2×#14579688C2211yescharacteristic
25C2×#14579688C2211yescharacteristic, complemented
26C2×#14579688C2211yescharacteristic, complemented
27C2×D1304C2211yescharacteristic
28C2×D1304C2211yescharacteristic
29C2×D1304C2211yescharacteristic
30C2×D1304C2211yescharacteristic
31#19104732C2211yescharacteristic, complemented
32#19104731—21no—
33#19104723C2211yescharacteristic
34#19104720—21no—
35#19104714C2211yescharacteristic
36#19104709C2211yescharacteristic
37#19104709C2211yescharacteristic
38#19104709C2211yescharacteristic, complemented
39#19104708—21no—
40#19104708—21no—
41#19104708—21no—
42#19104703—21no—
43C4×Dic326C2211yescharacteristic, complemented
44C4×D652C2211yescharacteristic
45#19104697—21no—
46C22⋊C4×D163—21no—
47#19104669C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104660C2211yescharacteristic, complemented
50#19104657C2211yescharacteristic
51#19104657C2211yescharacteristic
52#19104657C2211yescharacteristic, complemented
53#19104657C2211yescharacteristic, complemented
54C2×M16×C163C2211yescharacteristic
55C22×C8×C163C2211yescharacteristic, complemented
56C4×D4×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic
59C22⋊C8×C163C2211yescharacteristic, complemented
60#97×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62#14579699—22no—
63#14579699—22no—
64#14579699—22no—
65#14579699—41no—
66#14579699—41no—
67#14579699—41no—
68C2×Dic326C2311yescharacteristic
69C2×Dic326C2311yescharacteristic
70C2×D652—22no—
71C2×D652—21no—
72C2×D652—21no—
73C2×D652—21no—
74C2×D652—41no—
75C2×D652C2311yescharacteristic
76C2×D652C2311yescharacteristic
77C2×D652C2311yescharacteristic
78C2×D652C2311yescharacteristic
79C2×C22⋊D163—21no—
80C2×C22⋊D163—21no—
81C2×C22⋊D163—21no—
82C2×C22⋊D163—21no—
83C2×C22⋊D163—21no—
84C2×C22⋊D163—21no—
85C2×C4×D163—21no—
86C2×C4×D163—21no—
87C2×C4×D163—21no—
88C2×C4×D163—21no—
89Dic652—22no—
90#14579688—22no—
91#14579688—22no—
92#14579688—41no—
93#14579688D414yescomplemented
94D1304—22no—
95D1304—22no—
96D1304—22no—
97D1304—41no—
98C4⋊Dic163C2311yescharacteristic
99C4⋊Dic163C2311yescharacteristic
100Dic163⋊C4—21no—
101Dic163⋊C4—21no—
102C22⋊Dic163—21no—
103(C2×D163)⋊C4—21no—
104(C2×D163)⋊C4—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108(C2×D163)⋊C4—21no—
109C4×Dic163—21no—
110C2×D4×C163D411yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C22×C4×C163C2311yescharacteristic
113M16×C163—22no—
114C2×C8×C163—22no—
115C2×C8×C163C2311yescharacteristic
116C2×C8×C163C2311yescharacteristic
117C2×C8×C163C2311yescharacteristic
118C2×C8×C163C2311yescharacteristic
119C4⋊C4×C163D411yescharacteristic
120C22⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163—21no—
125C22×D163—21no—
126C22×D163—21no—
127C22×D163—42no—
128C22×D163—42no—
129C22×D163—81no—
130Dic326—21no—
131Dic326—21no—
132Dic326—21no—
133Dic326C2×D412yes—
134D652—21no—
135D652—21no—
136D652—21no—
137D652—21no—
138D652—21no—
139D652—21no—
140D652—21no—
141D652—42no—
142D652—42no—
143D652—81no—
144D652C2×D412yes—
145C22⋊D163—42no—
146C22⋊D163—42no—
147C22⋊D163—42no—
148C22⋊D163—81no—
149C22⋊D163—81no—
150C22⋊D163—81no—
151C2×Dic163—21no—
152C2×Dic163—21no—
153C2×Dic163—21no—
154C2×Dic163—21no—
155C4×D163—42no—
156C4×D163—42no—
157C4×D163—42no—
158C4×D163—81no—
159C23×C163C2×D411yescharacteristic
160C23×C163C2×D411yescharacteristic
161D4×C163—41no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163C2×D411yescharacteristic
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2×D411yescharacteristic
170C2×C4×C163C2411yescharacteristic
171C8×C163—21no—
172C8×C163—21no—
173C8×C163—21no—
174C8×C163C2×D412yes—
175C22×C163—21no—
176C22×C163—21no—
177C22×C163—21no—
178C22×C163—21no—
179C22×C163—21no—
180C22×C163—21no—
181C22×C163C22×D411yescharacteristic
182C2×D163—22no—
183C2×D163—41no—
184C2×D163—41no—
185C2×D163—41no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—41no—
189C2×D163—41no—
190C2×D163—41no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—84no—
195C4×C163—21no—
196C4×C163—21no—
197C4×C163—21no—
198C4×C163—21no—
199C4×C163C22×D411yescharacteristic
200C4×C163H(5,𝔽2)11yescharacteristic
201Dic163—22no—
202Dic163—41no—
203Dic163—41no—
204Dic163—41no—
205C2×C163—41no—
206C2×C163—41no—
207C2×C163D4211yescharacteristic
208C2×C163#57511yescharacteristic
209C2×C163#57611yescharacteristic
210D163—42no—
211D163—81no—
212D163—81no—
213D163—81no—
214C163#326811yescharacteristic, complemented, Sylow
215#3268—1631nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1631no—
217C2×#135—1631no—
218#550—1631no—
219#546—1631no—
220#496—1631no—
221#468—1631no—
222#466—1631no—
223#464—1631no—
224#459—1631no—
225#452—1631no—
226#449—1631no—
227#449—1631no—
228#447—1631no—
229#440—1631no—
230#435D16311yescharacteristic, complemented
231C2×#41—1631no—
232C2×#41—1631no—
233C2×#41—1631no—
234C22×D4—1631no—
235AGL(1,ℤ/8)—3264no—
236#135—3264no—
237C2×Q16—1631no—
238C2×SD16—1631no—
239C2×SD16—1631no—
240C2×SD16—1631no—
241C2×SD16—1631no—
242C2×D8—1631no—
243C2×D8—1631no—
244C2×D8—1631no—
245C2×D8—1631no—
246C2×M16C2×D16311yescharacteristic
247C22×C8C2×D16311yescharacteristic, complemented
248#127—3261no—
249#127—1631no—
250#124—1631no—
251#123—3261no—
252C22⋊Q8—1631no—
253#121—3261no—
254#121—3261no—
255#121—3261no—
256#121—1631no—
257#121—1631no—
258#121—1631no—
259#120—3261no—
260C4×Q8—1631no—
261C4×D4—3261no—
262C4×D4—1631no—
263C4×D4C2×D16311yescharacteristic
264C2×C22⋊C4—3261no—
265#107—1631no—
266C4⋊C8C2×D16311yescharacteristic, complemented
267#103—1631no—
268#103—1631no—
269#102—1631no—
270#102—1631no—
271#102—1631no—
272#102—1631no—
273C22⋊C8C2×D16311yescharacteristic
274C22⋊C8C2×D16311yescharacteristic, complemented
275#97C2×D16311yescharacteristic, complemented
276C24—3261no—
277#41—3262no—
278#41—3262no—
279#41—3262no—
280#41—6521no—
281#41—6521no—
282#41—6521no—
283C2×Q8—1631no—
284C2×Q8—1631no—
285C2×D4—3262no—
286C2×D4—3261no—
287C2×D4—3261no—
288C2×D4—3261no—
289C2×D4—3261no—
290C2×D4—3261no—
291C2×D4—3261no—
292C2×D4—3261no—
293C2×D4—3261no—
294C2×D4—3261no—
295C2×D4—6521no—
296C2×D4—1631no—
297C2×D4—1631no—
298C2×D4—1631no—
299C2×D4—1631no—
300C2×D4D65211yescharacteristic
301C22×C4—3261no—
302C22×C4—3261no—
303C22×C4—3261no—
304C22×C4—3261no—
305C22×C4C22×D16311yescharacteristic
306C22×C4C22×D16311yescharacteristic
307Q16—3262no—
308SD16—3262no—
309SD16—3262no—
310SD16—6521no—
311SD16—1634no—
312D8—3262no—
313D8—3262no—
314D8—3262no—
315D8—6521no—
316M16—22no—
317C2×C8—22no—
318C2×C8C22×D16311yescharacteristic
319C2×C8C22×D16311yescharacteristic
320C2×C8C22×D16311yescharacteristic
321C2×C8C22×D16311yescharacteristic
322C4⋊C4—3261no—
323C4⋊C4—3261no—
324C4⋊C4—1631no—
325C4⋊C4—1631no—
326C4⋊C4D65211yescharacteristic
327C22⋊C4—3261no—
328C22⋊C4—3261no—
329C22⋊C4—3261no—
330C22⋊C4—3261no—
331C22⋊C4—3261no—
332C22⋊C4—3261no—
333C22⋊C4—3261no—
334C22⋊C4D65211yescharacteristic
335C22⋊C4D65211yescharacteristic
336C42—3261no—
337C42C22×D16311yescharacteristic
338C23—13041no—
339C23—3261no—
340C23—3261no—
341C23—3261no—
342C23—3261no—
343C23—6522no—
344C23—6522no—
345C23C2×D65211yescharacteristic
346C23C2×D65211yescharacteristic
347Q8—3261no—
348Q8—3261no—
349Q8—3261no—
350Q8—1632no—
351D4—41no—
352D4—13041no—
353D4—13041no—
354D4—13041no—
355D4—13041no—
356D4—3261no—
357D4—3261no—
358D4—3261no—
359D4—3261no—
360D4—3261no—
361D4—3261no—
362D4—3261no—
363D4—6522no—
364D4—6522no—
365D4—6522no—
366D4—6522no—
367D4—6522no—
368D4—1632no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—13041no—
374C2×C4—3261no—
375C2×C4—3261no—
376C2×C4—3261no—
377C2×C4—3261no—
378C2×C4—6522no—
379C2×C4—6522no—
380C2×C4—6522no—
381C2×C4C2×D65211yescharacteristic
382C2×C4C2×D65211yescharacteristic
383C2×C4C2×D65211yescharacteristic
384C2×C4C2×D65211yescharacteristic
385C2×C4C23×D16311yescharacteristic
386C8—21no—
387C8—21no—
388C8—21no—
389C8D4×D16312yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—13044no—
397C22—3262no—
398C22—6521no—
399C22—6521no—
400C22—6521no—
401C22—6521no—
402C22—6521no—
403C22—6521no—
404C22—6521no—
405C22—6521no—
406C22—6521no—
407C22—6521no—
408C22—6521no—
409C22C22×D65211yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—3262no—
415C4—6521no—
416C4—6521no—
417C4—6521no—
418C4C2×D4×D16311yescharacteristic
419C4#1910478111yescharacteristic
420C2—41no—
421C2—41no—
422C2—13041no—
423C2—13041no—
424C2—13041no—
425C2—6522no—
426C2D4×D65211yescharacteristic
427C2#2378246511yescharacteristic
428C2#2378247011yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#435×C163C2
socleC22×C163C22×D4
hypercenter#435D163
hypocenterC163#3268
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19104763C2semidirect
2. C2×#19104757C2semidirect
3. #23782336C2semidirect
4. #23782311C2semidirect
5. #23782054C2semidirect
6. #23781911C2semidirect
7. #23781901C2semidirect
8. #23781888C2semidirect
9. #23781861C2semidirect
10. #23781829C2semidirect
11. #23781812C2semidirect
12. #23781812C2semidirect
13. #23781801C2semidirect
14. #23781767C2semidirect
15. #435×C163C2semidirect
18. C2×#14579699C22semidirect
22. C2×Dic652C22semidirect
25. C2×#14579688C22semidirect
26. C2×#14579688C22semidirect
31. #19104732C22semidirect
38. #19104709C22semidirect
43. C4×Dic326C22semidirect
47. #19104669C22semidirect
48. #19104660C22semidirect
49. #19104660C22semidirect
52. #19104657C22semidirect
53. #19104657C22semidirect
55. C22×C8×C163C22semidirect
57. C4⋊C8×C163C22semidirect
59. C22⋊C8×C163C22semidirect
60. #97×C163C22semidirect
93. #14579688D4semidirect
214. C163#3268semidirect
230. #435D163semidirect
247. C22×C8C2×D163semidirect
266. C4⋊C8C2×D163semidirect
274. C22⋊C8C2×D163semidirect
275. #97C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047632
C2×#191047572
#237823362
#237823112
#237820542
#237819112
#237819012
#237818882
#237818612
#237818292
#237818122
#237818012
#237817672
#435×C1632
#3268163