Finite Group Databasebeta

← #35226051

Subgroups — #35226051

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×#19104768C211yesmaximal, characteristic, complemented
2C22×#14579688C211yesmaximal, characteristic, complemented
3D4×Dic326C211yesmaximal, characteristic, complemented
4#23782311C211yesmaximal, characteristic
5#23782059C211yesmaximal, characteristic, complemented
6#23781911C211yesmaximal, characteristic, complemented
7#23781906C211yesmaximal, characteristic, complemented
8#23781888C211yesmaximal, characteristic, complemented
9#23781871C211yesmaximal, characteristic, complemented
10#23781829C211yesmaximal, characteristic, complemented
11#23781823C211yesmaximal, characteristic, complemented
12#23781818C211yesmaximal, characteristic, complemented
13#23781806C211yesmaximal, characteristic, complemented
14#23781767C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18C22×Dic326C2211yescharacteristic, complemented
19C22×D652C2211yescharacteristic
20#19104768—24no—
21C2×Dic652C2211yescharacteristic, complemented
22C2×Dic652C2211yescharacteristic, complemented
23C2×Dic652C2211yescharacteristic, complemented
24C2×#14579688—24no—
25C2×#14579688C2211yescharacteristic
26C2×#14579688C2211yescharacteristic
27C2×#14579688C2211yescharacteristic
28C2×#14579688C2211yescharacteristic
29C2×#14579688C2211yescharacteristic
30C2×#14579688C2211yescharacteristic
31#19104735C2211yescharacteristic, complemented
32#19104734—21no—
33#19104723C2211yescharacteristic
34#19104720—21no—
35#19104714C2211yescharacteristic
36#19104714C2211yescharacteristic
37#19104714C2211yescharacteristic, complemented
38#19104712—21no—
39#19104712—21no—
40#19104709C2211yescharacteristic
41#19104708—21no—
42#19104703—21no—
43C4×Dic326C2211yescharacteristic, complemented
44C4×D652C2211yescharacteristic
45D4×Dic163—21no—
46C22⋊C4×D163—21no—
47#19104669C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104660C2211yescharacteristic, complemented
50#19104660C2211yescharacteristic, complemented
51#19104660C2211yescharacteristic, complemented
52#19104657C2211yescharacteristic
53#19104657C2211yescharacteristic
54C2×M16×C163C2211yescharacteristic, complemented
55C22×C8×C163C2211yescharacteristic
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—41no—
64C2×Dic326—22no—
65C2×Dic326—22no—
66C2×Dic326—21no—
67C2×Dic326—21no—
68C2×Dic326—41no—
69C2×Dic326—41no—
70C2×Dic326C2311yescharacteristic
71C2×Dic326C2311yescharacteristic
72C2×Dic326C2311yescharacteristic
73C2×Dic326C2311yescharacteristic
74C2×D652—22no—
75C2×D652—21no—
76C2×D652—41no—
77C2×D652C2311yescharacteristic
78C2×D652C2311yescharacteristic
79C2×C22⋊D163—21no—
80C2×C22⋊D163—21no—
81C22×Dic163—21no—
82C22×Dic163—21no—
83C2×C4×D163—21no—
84C2×C4×D163—21no—
85Dic652—22no—
86Dic652—22no—
87Dic652—41no—
88#14579688—22no—
89#14579688—22no—
90#14579688—22no—
91#14579688—22no—
92#14579688—41no—
93#14579688D414yescomplemented
94C4⋊Dic163—21no—
95C4⋊Dic163—21no—
96C4⋊Dic163C2311yescharacteristic
97C4⋊Dic163C2311yescharacteristic
98Dic163⋊C4—21no—
99Dic163⋊C4—21no—
100Dic163⋊C4—21no—
101Dic163⋊C4—21no—
102C22⋊Dic163—21no—
103C22⋊Dic163—21no—
104C22⋊Dic163—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—42no—
126C22×D163—42no—
127C22×D163—81no—
128Dic326—21no—
129Dic326—21no—
130Dic326—21no—
131Dic326—21no—
132Dic326—21no—
133Dic326—21no—
134Dic326—21no—
135Dic326—42no—
136Dic326—42no—
137Dic326C2×D412yes—
138D652—21no—
139D652—21no—
140D652—21no—
141D652—42no—
142D652C2×D412yes—
143C22⋊D163—42no—
144C22⋊D163—81no—
145C2×Dic163—21no—
146C2×Dic163—21no—
147C2×Dic163—21no—
148C2×Dic163—21no—
149C2×Dic163—21no—
150C2×Dic163—21no—
151C2×Dic163—42no—
152C2×Dic163—42no—
153C4×D163—42no—
154C4×D163—81no—
155C23×C163C2×D411yescharacteristic
156C23×C163C2×D411yescharacteristic
157D4×C163—22no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163C2×D411yescharacteristic
163C2×C4×C163C2×D411yescharacteristic
164C2×C4×C163C2×D411yescharacteristic
165C2×C4×C163C2×D411yescharacteristic
166C2×C4×C163C2411yescharacteristic
167C8×C163—21no—
168C8×C163—21no—
169C8×C163—21no—
170C8×C163C2×D412yes—
171C22×C163—21no—
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—21no—
175C22×C163—21no—
176C22×C163#13712yescomplemented
177C22×C163C22×D411yescharacteristic
178C2×D163—22no—
179C2×D163—41no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
183C2×D163—41no—
184C2×D163—84no—
185C4×C163—21no—
186C4×C163—21no—
187C4×C163—21no—
188C4×C163—21no—
189C4×C163C22×D411yescharacteristic
190C4×C163H(5,𝔽2)11yescharacteristic
191Dic163—22no—
192Dic163—41no—
193Dic163—41no—
194Dic163—41no—
195Dic163—41no—
196Dic163—41no—
197C2×C163—22no—
198C2×C163—41no—
199C2×C163D4211yescharacteristic
200C2×C163C2×#13711yescharacteristic
201C2×C163#57711yescharacteristic
202D163—42no—
203D163—81no—
204C163#326611yescharacteristic, complemented, Sylow
205#3266—1631nomaximal, Sylow
206C2×#137—1631no—
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#435D16311yescharacteristic, complemented
221C2×#41—1631no—
222C22×Q8—1631no—
223C22×Q8—1631no—
224C22×D4—1631no—
225#137—3264no—
226C2×Q16—1631no—
227C2×Q16—1631no—
228C2×Q16—1631no—
229C2×SD16—3264no—
230C2×SD16—1631no—
231C2×SD16—1631no—
232C2×SD16—1631no—
233C2×SD16—1631no—
234C2×SD16—1631no—
235C2×SD16—1631no—
236C2×M16C2×D16311yescharacteristic, complemented
237C22×C8C2×D16311yescharacteristic
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×D4C2×D16311yescharacteristic
254C2×C22⋊C4—3261no—
255#107—1631no—
256C4⋊C8C2×D16311yescharacteristic, complemented
257#103—1631no—
258#103—1631no—
259#103—1631no—
260#103—1631no—
261#102—1631no—
262#102—1631no—
263C22⋊C8C2×D16311yescharacteristic
264C22⋊C8C2×D16311yescharacteristic, complemented
265#97C2×D16311yescharacteristic, complemented
266C24—3261no—
267#41—3262no—
268#41—6521no—
269C2×Q8—3262no—
270C2×Q8—3262no—
271C2×Q8—3261no—
272C2×Q8—3261no—
273C2×Q8—6521no—
274C2×Q8—6521no—
275C2×Q8—1631no—
276C2×Q8—1631no—
277C2×Q8—1631no—
278C2×Q8—1631no—
279C2×D4—3262no—
280C2×D4—3261no—
281C2×D4—3261no—
282C2×D4—3261no—
283C2×D4—6521no—
284C2×D4—1631no—
285C2×D4—1631no—
286C2×D4D65211yescharacteristic
287C22×C4—3261no—
288C22×C4—3261no—
289C22×C4—3261no—
290C22×C4—3261no—
291C22×C4C22×D16311yescharacteristic
292C22×C4C22×D16311yescharacteristic
293Q16—3262no—
294Q16—3262no—
295Q16—6521no—
296SD16—3262no—
297SD16—3262no—
298SD16—3262no—
299SD16—3262no—
300SD16—6521no—
301SD16—1634no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D16311yescharacteristic
305C2×C8C22×D16311yescharacteristic
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⋊C4D65211yescharacteristic
317C22⋊C4—3261no—
318C22⋊C4—3261no—
319C22⋊C4—3261no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4D65211yescharacteristic
325C22⋊C4D65211yescharacteristic
326C42—3261no—
327C42C22×D16311yescharacteristic
328C23—13041no—
329C23—3261no—
330C23—3261no—
331C23—6522no—
332C23—6522no—
333C23C2×D65211yescharacteristic
334C23C2×D65211yescharacteristic
335Q8—3261no—
336Q8—3261no—
337Q8—3261no—
338Q8—3261no—
339Q8—3261no—
340Q8—3261no—
341Q8—3261no—
342Q8—6522no—
343Q8—6522no—
344Q8—1632no—
345D4—22no—
346D4—13041no—
347D4—3261no—
348D4—3261no—
349D4—3261no—
350D4—6522no—
351D4—6522no—
352D4—1632no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—13041no—
358C2×C4—3261no—
359C2×C4—3261no—
360C2×C4—3261no—
361C2×C4—3261no—
362C2×C4—3261no—
363C2×C4—3261no—
364C2×C4—6522no—
365C2×C4—6522no—
366C2×C4—6522no—
367C2×C4C2×D65211yescharacteristic
368C2×C4C2×D65211yescharacteristic
369C2×C4C2×D65211yescharacteristic
370C2×C4C2×D65211yescharacteristic
371C2×C4C23×D16311yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D16312yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—13044no—
382C22—3262no—
383C22—6521no—
384C22—6521no—
385C22—6521no—
386C22—6521no—
387C22—6521no—
388C22#1910476812yescomplemented
389C22C22×D65211yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—3262no—
395C4—6521no—
396C4—6521no—
397C4—6521no—
398C4—6521no—
399C4—6521no—
400C4C2×D4×D16311yescharacteristic
401C4#1910478111yescharacteristic
402C2—22no—
403C2—41no—
404C2—13041no—
405C2—6522no—
406C2D4×D65211yescharacteristic
407C2C2×#1910476811yescharacteristic
408C2#2378247711yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#435×C163C2
socleC22×C163C22×D4
prime radical#23782311C2
hypercenter#435D163
hypocenterC163#3266
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×#19104768C2semidirect
2. C22×#14579688C2semidirect
3. D4×Dic326C2semidirect
5. #23782059C2semidirect
6. #23781911C2semidirect
7. #23781906C2semidirect
8. #23781888C2semidirect
9. #23781871C2semidirect
10. #23781829C2semidirect
11. #23781823C2semidirect
12. #23781818C2semidirect
13. #23781806C2semidirect
14. #23781767C2semidirect
15. #435×C163C2semidirect
18. C22×Dic326C22semidirect
21. C2×Dic652C22semidirect
22. C2×Dic652C22semidirect
23. C2×Dic652C22semidirect
31. #19104735C22semidirect
37. #19104714C22semidirect
43. C4×Dic326C22semidirect
47. #19104669C22semidirect
48. #19104660C22semidirect
49. #19104660C22semidirect
50. #19104660C22semidirect
51. #19104660C22semidirect
54. C2×M16×C163C22semidirect
57. C4⋊C8×C163C22semidirect
59. C22⋊C8×C163C22semidirect
60. #97×C163C22semidirect
93. #14579688D4semidirect
176. C22×C163#137semidirect
204. C163#3266semidirect
220. #435D163semidirect
236. C2×M16C2×D163semidirect
256. C4⋊C8C2×D163semidirect
264. C22⋊C8C2×D163semidirect
265. #97C2×D163semidirect
388. C22#19104768semidirect

Maximal subgroups

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

subgroupindex
C2×#191047682
C22×#145796882
D4×Dic3262
#237823112
#237820592
#237819112
#237819062
#237818882
#237818712
#237818292
#237818232
#237818182
#237818062
#237817672
#435×C1632
#3266163