Finite Group Databasebeta

← #35226046

Subgroups — #35226046

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×#19104764C211yesmaximal, characteristic, complemented
2Q8×C22⋊D163C211yesmaximal, characteristic, complemented
3#23782309C211yesmaximal, characteristic
4#23782056C211yesmaximal, characteristic, complemented
5#23781908C211yesmaximal, characteristic, complemented
6#23781906C211yesmaximal, characteristic, complemented
7#23781885C211yesmaximal, characteristic, complemented
8#23781868C211yesmaximal, characteristic, complemented
9#23781824C211yesmaximal, characteristic, complemented
10#23781821C211yesmaximal, characteristic, complemented
11#23781816C211yesmaximal, characteristic, complemented
12#23781802C211yesmaximal, characteristic, complemented
13#23781766C211yesmaximal, characteristic, complemented
14#23781736C211yesmaximal, characteristic, complemented
15C22×SD16×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic, complemented
18#19104764—24no—
19C2×Dic652C2211yescharacteristic, complemented
20C2×#14579689C2211yescharacteristic, complemented
21C2×#14579689C2211yescharacteristic, complemented
22C2×#14579687C2211yescharacteristic
23C2×#14579687C2211yescharacteristic
24C2×#14579682C2211yescharacteristic, complemented
25#19104733—21no—
26#19104733C2211yescharacteristic, complemented
27#19104722C2211yescharacteristic
28#19104716—21no—
29#19104714—21no—
30#19104714C2211yescharacteristic
31#19104714C2211yescharacteristic
32#19104711—21no—
33#19104711C2211yescharacteristic, complemented
34#19104707—21no—
35#19104706C2211yescharacteristic
36#19104704—21no—
37Q8×Dic163C2211yescharacteristic, complemented
38D4×Dic163C2211yescharacteristic
39C4×C22⋊D163—21no—
40C4×C22⋊D163C2211yescharacteristic
41C2×C22⋊Dic163—21no—
42#19104669C2211yescharacteristic, complemented
43Dic163⋊C8C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104660C2211yescharacteristic, complemented
46#19104659C2211yescharacteristic, complemented
47#19104659C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic
50C22⋊(C163⋊C8)C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic, complemented
52#19104645C2211yescharacteristic, complemented
53C22×Q8×C163C2211yescharacteristic
54C22×D4×C163C2211yescharacteristic
55C2×SD16×C163—24no—
56C2×SD16×C163C2211yescharacteristic
57C2×SD16×C163C2211yescharacteristic
58C2×SD16×C163C2211yescharacteristic
59C2×SD16×C163C2211yescharacteristic
60C22×C8×C163C2211yescharacteristic
61#14579698—22no—
62#14579698—41no—
63C2×Dic326—21no—
64C2×Dic326C2311yescharacteristic
65C2×Dic326C2311yescharacteristic
66Q8×D163—22no—
67Q8×D163—41no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163D411yescharacteristic
71C22×Dic163—21no—
72C22×Dic163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2311yescharacteristic
75Dic652—22no—
76#14579689—22no—
77#14579689—41no—
78#14579687—22no—
79#14579687—41no—
80#14579682—22no—
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
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⋊Dic163—21no—
95C22⋊Dic163—21no—
96C22⋊Dic163—21no—
97C22⋊Dic163—21no—
98C22⋊Dic163D411yescharacteristic
99(C2×D163)⋊C4—21no—
100(C2×D163)⋊C4D411yescharacteristic
101C4×Dic163—21no—
102C4×Dic163C2311yescharacteristic
103C24×C163—21no—
104C2×Q8×C163—21no—
105C2×Q8×C163—22no—
106C2×Q8×C163—41no—
107C2×Q8×C163C2311yescharacteristic
108C2×Q8×C163C2311yescharacteristic
109C2×D4×C163—22no—
110C2×D4×C163—21no—
111C2×D4×C163—41no—
112C2×D4×C163C2311yescharacteristic
113C2×D4×C163C2311yescharacteristic
114C22×C4×C163—21no—
115C22×C4×C163C2311yescharacteristic
116SD16×C163—22no—
117SD16×C163—22no—
118SD16×C163—22no—
119SD16×C163D414yescomplemented
120C2×C8×C163—22no—
121C2×C8×C163C2311yescharacteristic
122C2×C8×C163C2311yescharacteristic
123C22×D163C2×D411yescharacteristic
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128Dic326—42no—
129C22⋊D163—22no—
130C22⋊D163—42no—
131C22⋊D163—81no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—42no—
137C2×Dic163—81no—
138C2×Dic163C2×D411yescharacteristic
139C2×Dic163C2×D411yescharacteristic
140C2×Dic163C2×D411yescharacteristic
141C4×D163—21no—
142C4×D163—21no—
143C4×D163—42no—
144C163⋊C8—21no—
145C163⋊C8—21no—
146C23×C163—21no—
147C23×C163—21no—
148C23×C163—42no—
149C23×C163—42no—
150C23×C163—81no—
151C23×C163C2×D411yescharacteristic
152Q8×C163—21no—
153Q8×C163—21no—
154Q8×C163—21no—
155Q8×C163—42no—
156Q8×C163C2×D412yes—
157D4×C163—21no—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—42no—
161D4×C163C2×D412yes—
162C2×C4×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—42no—
167C2×C4×C163C2×D411yescharacteristic
168C2×C4×C163C2411yescharacteristic
169C8×C163—21no—
170C8×C163C2×D412yes—
171C22×C163—21no—
172C22×C163—21no—
173C22×C163—22no—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—84no—
180C22×C163#13712yescomplemented
181C22×C163C22×D411yescharacteristic
182C2×D163—21no—
183C2×D163—21no—
184C2×D163—21no—
185C4×C163—21no—
186C4×C163—22no—
187C4×C163—41no—
188C4×C163C22×D411yescharacteristic
189C4×C163H(5,𝔽2)11yescharacteristic
190Dic163—21no—
191Dic163—21no—
192Dic163—21no—
193Dic163—41no—
194Dic163—41no—
195Dic163—41no—
196Dic163—41no—
197C2×C163—22no—
198C2×C163—42no—
199C2×C163—81no—
200C2×C163D4211yescharacteristic
201C2×C163C2×#13711yescharacteristic
202C2×C163#57711yescharacteristic
203D163—41no—
204C163#326611yescharacteristic, complemented, Sylow
205#3266—1631nomaximal, Sylow
206C2×#137—1631no—
207C22×SD16D16311yescharacteristic, complemented
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×#41—1631no—
222C22×Q8—1631no—
223C22×Q8C2×D16311yescharacteristic
224C22×D4C2×D16311yescharacteristic
225#137—3264no—
226C2×Q16—1631no—
227C2×Q16—1631no—
228C2×Q16—1631no—
229C2×SD16—24no—
230C2×SD16—1631no—
231C2×SD16—1631no—
232C2×SD16C2×D16311yescharacteristic
233C2×SD16C2×D16311yescharacteristic
234C2×SD16C2×D16311yescharacteristic
235C2×SD16C2×D16311yescharacteristic
236C2×M16—1631no—
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×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—21no—
267#41—3262no—
268#41—6521no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—3261no—
273C2×Q8—3262no—
274C2×Q8—6521no—
275C2×Q8—1631no—
276C2×Q8—1631no—
277C2×Q8C22×D16311yescharacteristic
278C2×Q8C22×D16311yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—3261no—
283C2×D4—3261no—
284C2×D4—1631no—
285C2×D4C22×D16311yescharacteristic
286C2×D4C22×D16311yescharacteristic
287C22×C4—21no—
288C22×C4—3261no—
289C22×C4—3261no—
290C22×C4—3261no—
291C22×C4—1631no—
292C22×C4C22×D16311yescharacteristic
293Q16—3262no—
294Q16—3262no—
295Q16—6521no—
296SD16—22no—
297SD16—22no—
298SD16—22no—
299SD16—3262no—
300SD16—6521no—
301SD16C22⋊D16314yescomplemented
302M16—3262no—
303C2×C8—22no—
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—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—1631no—
334C23D4×D16311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—3261no—
340Q8—3261no—
341Q8—3261no—
342Q8—3261no—
343Q8—6522no—
344Q8C2×C22⋊D16312yes—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—13041no—
350D4—3262no—
351D4—6522no—
352D4C2×C22⋊D16312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—13041no—
359C2×C4—3261no—
360C2×C4—3261no—
361C2×C4—3261no—
362C2×C4—3261no—
363C2×C4—3261no—
364C2×C4—3261no—
365C2×C4—6522no—
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—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—3261no—
386C22—3261no—
387C22—3261no—
388C22#1910476412yescomplemented
389C22C2×D4×D16311yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—3261no—
394C4—3261no—
395C4—3261no—
396C4—6521no—
397C4—6521no—
398C4—6521no—
399C4—6521no—
400C4C22×C22⋊D16311yescharacteristic
401C4#1910478011yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—6521no—
406C2D4×C22⋊D16311yescharacteristic
407C2C2×#1910476411yescharacteristic
408C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC22×SD16×C163C2
socleC22×C163C22×D4
prime radical#23782309C2
hypercenterC22×SD16D163
hypocenterC163#3266
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#19104764C2semidirect
2. Q8×C22⋊D163C2semidirect
4. #23782056C2semidirect
5. #23781908C2semidirect
6. #23781906C2semidirect
7. #23781885C2semidirect
8. #23781868C2semidirect
9. #23781824C2semidirect
10. #23781821C2semidirect
11. #23781816C2semidirect
12. #23781802C2semidirect
13. #23781766C2semidirect
14. #23781736C2semidirect
15. C22×SD16×C163C2semidirect
17. C2×Q8×D163C22semidirect
19. C2×Dic652C22semidirect
20. C2×#14579689C22semidirect
21. C2×#14579689C22semidirect
24. C2×#14579682C22semidirect
26. #19104733C22semidirect
33. #19104711C22semidirect
37. Q8×Dic163C22semidirect
42. #19104669C22semidirect
43. Dic163⋊C8C22semidirect
44. #19104660C22semidirect
45. #19104660C22semidirect
46. #19104659C22semidirect
47. #19104659C22semidirect
51. (C2×D163)⋊C8C22semidirect
52. #19104645C22semidirect
119. SD16×C163D4semidirect
180. C22×C163#137semidirect
204. C163#3266semidirect
207. C22×SD16D163semidirect
301. SD16C22⋊D163semidirect
388. C22#19104764semidirect

Maximal subgroups

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

subgroupindex
C2×#191047642
Q8×C22⋊D1632
#237823092
#237820562
#237819082
#237819062
#237818852
#237818682
#237818242
#237818212
#237818162
#237818022
#237817662
#237817362
C22×SD16×C1632
#3266163