Finite Group Databasebeta

← #35226040

Subgroups — #35226040

Order 20864.

494 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
1C2×SD16×D163C211yesmaximal, characteristic, complemented
2Q8×C22⋊D163C211yesmaximal, characteristic, complemented
3#23782303C211yesmaximal, characteristic
4#23782058C211yesmaximal, characteristic, complemented
5#23781911C211yesmaximal, characteristic, complemented
6#23781903C211yesmaximal, characteristic, complemented
7#23781889C211yesmaximal, characteristic, complemented
8#23781868C211yesmaximal, characteristic, complemented
9#23781827C211yesmaximal, characteristic, complemented
10#23781819C211yesmaximal, characteristic, complemented
11#23781814C211yesmaximal, characteristic, complemented
12#23781804C211yesmaximal, characteristic, complemented
13#23781766C211yesmaximal, characteristic, complemented
14#23781734C211yesmaximal, characteristic, complemented
15C2×#137×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×#14579689C2211yescharacteristic, complemented
19C2×#14579688C2211yescharacteristic
20C2×#14579687C2211yescharacteristic
21C2×#14579686C2211yescharacteristic
22C2×#14579686C2211yescharacteristic
23SD16×D163—24no—
24C2×C8×D163C2211yescharacteristic
25#19104733—21no—
26#19104733C2211yescharacteristic, complemented
27#19104724C2211yescharacteristic
28#19104718—21no—
29#19104714—21no—
30#19104714C2211yescharacteristic, complemented
31#19104711—21no—
32#19104711C2211yescharacteristic
33#19104711C2211yescharacteristic
34#19104709C2211yescharacteristic
35#19104707—21no—
36#19104703—21no—
37Q8×Dic163C2211yescharacteristic, complemented
38D4×Dic163C2211yescharacteristic
39C4×C22⋊D163—21no—
40C4×C22⋊D163C2211yescharacteristic
41C2×(C2×D163)⋊C4—21no—
42#19104669C2211yescharacteristic, complemented
43Dic163⋊C8C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104659C2211yescharacteristic, complemented
46#19104659C2211yescharacteristic, complemented
47#19104659C2211yescharacteristic, complemented
48#19104657C2211yescharacteristic
49#19104656C2211yescharacteristic
50C22⋊(C163⋊C8)C2211yescharacteristic, complemented
51(C2×D163)⋊C8C2211yescharacteristic
52#19104645C2211yescharacteristic, complemented
53C2×#41×C163C2211yescharacteristic
54C22×Q8×C163C2211yescharacteristic, complemented
55#137×C163—24no—
56C2×Q16×C163C2211yescharacteristic, complemented
57C2×Q16×C163C2211yescharacteristic, complemented
58C2×SD16×C163C2211yescharacteristic
59C2×SD16×C163C2211yescharacteristic
60C2×M16×C163C2211yescharacteristic, complemented
61C23×D163—21no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64Q8×D163—22no—
65Q8×D163—41no—
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163D411yescharacteristic
70D4×D163—22no—
71D4×D163—41no—
72C22×Dic163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2311yescharacteristic
75#14579689—41no—
76#14579688—22no—
77#14579687—22no—
78#14579686—22no—
79#14579686—41no—
80C2×C163⋊C8C2311yescharacteristic
81C2×C163⋊C8C2311yescharacteristic
82C8×D163—22no—
83C4⋊Dic163—21no—
84C4⋊Dic163C2311yescharacteristic
85C4⋊Dic163C2311yescharacteristic
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91Dic163⋊C4D411yescharacteristic
92C22⋊Dic163—21no—
93C22⋊Dic163—21no—
94C22⋊Dic163D411yescharacteristic
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4—21no—
99(C2×D163)⋊C4—21no—
100(C2×D163)⋊C4D411yescharacteristic
101C4×Dic163—21no—
102C4×Dic163C2311yescharacteristic
103#41×C163—22no—
104#41×C163—41no—
105C2×Q8×C163—21no—
106C2×Q8×C163—22no—
107C2×Q8×C163—41no—
108C2×Q8×C163C2311yescharacteristic
109C2×Q8×C163C2311yescharacteristic
110C2×Q8×C163C2311yescharacteristic
111C2×D4×C163—21no—
112C2×D4×C163C2311yescharacteristic
113C22×C4×C163—21no—
114C22×C4×C163—21no—
115C22×C4×C163C2311yescharacteristic
116Q16×C163—22no—
117Q16×C163—22no—
118SD16×C163—22no—
119SD16×C163D414yescomplemented
120M16×C163—22no—
121C2×C8×C163C2311yescharacteristic
122C2×C8×C163C2311yescharacteristic
123C22×D163—21no—
124C22×D163—42no—
125C22×D163—42no—
126C22×D163—81no—
127C22×D163C2×D411yescharacteristic
128Dic326—21no—
129Dic326—21no—
130Dic326—42no—
131D652—21no—
132D652—21no—
133C22⋊D163—22no—
134C22⋊D163—42no—
135C22⋊D163—81no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—81no—
140C2×Dic163C2×D411yescharacteristic
141C2×Dic163C2×D411yescharacteristic
142C2×Dic163C2×D411yescharacteristic
143C4×D163—21no—
144C4×D163—21no—
145C4×D163—42no—
146C163⋊C8—21no—
147C163⋊C8—21no—
148C23×C163—21no—
149C23×C163C2×D411yescharacteristic
150Q8×C163—21no—
151Q8×C163—21no—
152Q8×C163—21no—
153Q8×C163—21no—
154Q8×C163—21no—
155Q8×C163—42no—
156Q8×C163C2×D412yes—
157D4×C163—21no—
158D4×C163—42no—
159D4×C163C2×D412yes—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—42no—
166C2×C4×C163—42no—
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2411yescharacteristic
169C8×C163—21no—
170C8×C163C2×D412yes—
171C22×C163—21no—
172C22×C163—22no—
173C22×C163—21no—
174C22×C163—21no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163C22×D411yescharacteristic
178C2×D163—21no—
179C2×D163—21no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
183C2×D163—84no—
184C2×D163#13712yescomplemented
185C4×C163—21no—
186C4×C163—22no—
187C4×C163—41no—
188C4×C163—41no—
189C4×C163C22×D411yescharacteristic
190C4×C163H(5,𝔽2)11yescharacteristic
191Dic163—21no—
192Dic163—21no—
193Dic163—21no—
194Dic163—41no—
195Dic163—41no—
196Dic163—41no—
197C2×C163—42no—
198C2×C163—41no—
199C2×C163D4211yescharacteristic
200C2×C163C2×#13711yescharacteristic
201C2×C163#57711yescharacteristic
202D163—22no—
203D163—81no—
204C163#326611yescharacteristic, complemented, Sylow
205#3266—1631nomaximal, Sylow
206C2×#137D16311yescharacteristic, complemented
207C22×SD16—1631no—
208D4×Q8—1631no—
209#546—1631no—
210#497—1631no—
211#468—1631no—
212#467—1631no—
213#464—1631no—
214#461—1631no—
215#452—1631no—
216#451—1631no—
217#450—1631no—
218#448—1631no—
219#440—1631no—
220#435—1631no—
221C2×#41C2×D16311yescharacteristic
222C22×Q8—1631no—
223C22×Q8C2×D16311yescharacteristic, complemented
224C22×D4—1631no—
225#137—24no—
226C2×Q16—1631no—
227C2×Q16C2×D16311yescharacteristic, complemented
228C2×Q16C2×D16311yescharacteristic, complemented
229C2×SD16—3264no—
230C2×SD16—1631no—
231C2×SD16—1631no—
232C2×SD16—1631no—
233C2×SD16—1631no—
234C2×SD16C2×D16311yescharacteristic
235C2×SD16C2×D16311yescharacteristic
236C2×M16C2×D16311yescharacteristic, complemented
237C22×C8—1631no—
238C4⋊Q8—3261no—
239C4⋊Q8—1631no—
240#124—1631no—
241#123—3261no—
242C22⋊Q8—3261no—
243C22⋊Q8—3261no—
244C22⋊Q8—1631no—
245C22⋊Q8—1631no—
246C22⋊Q8—1631no—
247#121—3261no—
248#121—1631no—
249#120—3261no—
250C4×Q8—1631no—
251C4×D4—3261no—
252C4×D4—1631no—
253C4×D4—1631no—
254C2×C22⋊C4—3261no—
255#107—1631no—
256C4⋊C8—1631no—
257#103—1631no—
258#103—1631no—
259#103—1631no—
260#103—1631no—
261#102—1631no—
262#102—1631no—
263C22⋊C8—1631no—
264C22⋊C8—1631no—
265#97—1631no—
266C24—3261no—
267#41—22no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3261no—
273C2×Q8—3262no—
274C2×Q8—6521no—
275C2×Q8—1631no—
276C2×Q8C22×D16311yescharacteristic
277C2×Q8C22×D16311yescharacteristic
278C2×Q8C22×D16311yescharacteristic
279C2×D4—21no—
280C2×D4—3261no—
281C2×D4—3262no—
282C2×D4—3261no—
283C2×D4—6521no—
284C2×D4—1631no—
285C2×D4—1631no—
286C2×D4C22×D16311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—3261no—
290C22×C4—3261no—
291C22×C4—1631no—
292C22×C4C22×D16311yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—6521no—
296SD16—22no—
297SD16—3262no—
298SD16—3262no—
299SD16—3262no—
300SD16—6521no—
301SD16C22⋊D16314yescomplemented
302M16—22no—
303C2×C8—3262no—
304C2×C8—1631no—
305C2×C8—1631no—
306C2×C8C22×D16311yescharacteristic
307C2×C8C22×D16311yescharacteristic
308C4⋊C4—3261no—
309C4⋊C4—3261no—
310C4⋊C4—3261no—
311C4⋊C4—3261no—
312C4⋊C4—3261no—
313C4⋊C4—3261no—
314C4⋊C4—1631no—
315C4⋊C4—1631no—
316C4⋊C4—1631no—
317C22⋊C4—3261no—
318C22⋊C4—3261no—
319C22⋊C4—3261no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—1631no—
325C22⋊C4—1631no—
326C42—3261no—
327C42—1631no—
328C23—21no—
329C23—13041no—
330C23—3261no—
331C23—6522no—
332C23—6522no—
333C23—1631no—
334C23D4×D16311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—3261no—
342Q8—3261no—
343Q8—6522no—
344Q8C2×C22⋊D16312yes—
345D4—21no—
346D4—42no—
347D4—13041no—
348D4—3262no—
349D4—3261no—
350D4—3261no—
351D4—6522no—
352D4C2×C22⋊D16312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—13041no—
361C2×C4—3261no—
362C2×C4—3261no—
363C2×C4—3261no—
364C2×C4—3261no—
365C2×C4—3261no—
366C2×C4—6522no—
367C2×C4—1631no—
368C2×C4—1631no—
369C2×C4—1631no—
370C2×C4D4×D16311yescharacteristic
371C2×C4C23×D16311yescharacteristic
372C8—21no—
373C8—3261no—
374C8—3261no—
375C8C2×C22⋊D16312yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—13044no—
383C22—3261no—
384C22—3261no—
385C22—6521no—
386C22—6521no—
387C22—6521no—
388C22—1632no—
389C22C2×D4×D16311yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—3261no—
395C4—3261no—
396C4—3261no—
397C4—6521no—
398C4—6521no—
399C4—6521no—
400C4C22×C22⋊D16311yescharacteristic
401C4#1910478011yescharacteristic
402C2—41no—
403C2—42no—
404C2—13041no—
405C2—3262no—
406C2D4×C22⋊D16311yescharacteristic
407C2#137×D16311yescharacteristic
408C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#137×C163C2
socleC22×C163C22×D4
prime radical#23782303C2
hypercenterC2×#137D163
hypocenterC163#3266
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D163C2semidirect
2. Q8×C22⋊D163C2semidirect
4. #23782058C2semidirect
5. #23781911C2semidirect
6. #23781903C2semidirect
7. #23781889C2semidirect
8. #23781868C2semidirect
9. #23781827C2semidirect
10. #23781819C2semidirect
11. #23781814C2semidirect
12. #23781804C2semidirect
13. #23781766C2semidirect
14. #23781734C2semidirect
15. C2×#137×C163C2semidirect
18. C2×#14579689C22semidirect
26. #19104733C22semidirect
30. #19104714C22semidirect
37. Q8×Dic163C22semidirect
42. #19104669C22semidirect
43. Dic163⋊C8C22semidirect
44. #19104660C22semidirect
45. #19104659C22semidirect
46. #19104659C22semidirect
47. #19104659C22semidirect
50. C22⋊(C163⋊C8)C22semidirect
52. #19104645C22semidirect
54. C22×Q8×C163C22semidirect
56. C2×Q16×C163C22semidirect
57. C2×Q16×C163C22semidirect
60. C2×M16×C163C22semidirect
119. SD16×C163D4semidirect
184. C2×D163#137semidirect
204. C163#3266semidirect
206. C2×#137D163semidirect
223. C22×Q8C2×D163semidirect
227. C2×Q16C2×D163semidirect
228. C2×Q16C2×D163semidirect
236. C2×M16C2×D163semidirect
301. SD16C22⋊D163semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1632
Q8×C22⋊D1632
#237823032
#237820582
#237819112
#237819032
#237818892
#237818682
#237818272
#237818192
#237818142
#237818042
#237817662
#237817342
C2×#137×C1632
#3266163