Finite Group Databasebeta

← #35226243

Subgroups — #35226243

Order 20864.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104756C211yesmaximal, characteristic, complemented
2#23782326C211yesmaximal, characteristic, complemented
3#23782309C211yesmaximal, characteristic, complemented
4#23782046C211yesmaximal, characteristic, complemented
5#23781894C211yesmaximal, characteristic, complemented
6#23781891C211yesmaximal, characteristic, complemented
7#23781885C211yesmaximal, characteristic, complemented
8#23781882C211yesmaximal, characteristic, complemented
9#23781824C211yesmaximal, characteristic, complemented
10#23781821C211yesmaximal, characteristic, complemented
11#23781809C211yesmaximal, characteristic, complemented
12#23781799C211yesmaximal, characteristic, complemented
13SD16×Dic163C211yesmaximal, characteristic, complemented
14C8×C22⋊D163C211yesmaximal, characteristic, complemented
15C22×SD16×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic, complemented
17C2×#14579698C2211yescharacteristic
18#19104756—24no—
19C2×#14579689C2211yescharacteristic, complemented
20C2×#14579689C2211yescharacteristic, complemented
21C2×#14579688C2211yescharacteristic, complemented
22C2×#14579684C2211yescharacteristic
23C2×#14579684C2211yescharacteristic
24C2×C8×D163C2211yescharacteristic, complemented
25#19104724—21no—
26#19104724C2211yescharacteristic, complemented
27#19104722C2211yescharacteristic
28#19104716—21no—
29#19104714C2211yescharacteristic
30#19104711—21no—
31#19104711C2211yescharacteristic, complemented
32#19104709—21no—
33#19104709C2211yescharacteristic
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#19104667C2211yescharacteristic, complemented
43Dic163⋊C8C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104659C2211yescharacteristic, complemented
46#19104659C2211yescharacteristic, complemented
47#19104657C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic
50C22⋊(C163⋊C8)C2211yescharacteristic
51(C2×D163)⋊C8C2211yescharacteristic, complemented
52C8×Dic163C2211yescharacteristic, 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#14579700—22no—
62#14579700—41no—
63#14579698—22no—
64#14579698—41no—
65C2×Dic326C2311yescharacteristic
66C2×D652—21no—
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163D411yescharacteristic
72C22×Dic163—21no—
73C22×Dic163—21no—
74C2×C4×D163—21no—
75C2×C4×D163C2311yescharacteristic
76#14579689—22no—
77#14579689—41no—
78#14579688—22no—
79#14579684—22no—
80#14579684—41no—
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C8×D163—22no—
84C4⋊Dic163—21no—
85C4⋊Dic163C2311yescharacteristic
86C4⋊Dic163C2311yescharacteristic
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4D411yescharacteristic
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92C22⋊Dic163—21no—
93C22⋊Dic163—21no—
94C22⋊Dic163—21no—
95C22⋊Dic163D411yescharacteristic
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
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×D163—21no—
124C22×D163C2×D411yescharacteristic
125Dic326—21no—
126Dic326—21no—
127D652—21no—
128D652—21no—
129D652—42no—
130C22⋊D163—22no—
131C22⋊D163—42no—
132C22⋊D163—42no—
133C22⋊D163—81no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—42no—
138C2×Dic163—81no—
139C2×Dic163C2×D411yescharacteristic
140C2×Dic163C2×D411yescharacteristic
141C2×Dic163C2×D411yescharacteristic
142C4×D163—21no—
143C4×D163—21no—
144C4×D163—42no—
145C163⋊C8—21no—
146C163⋊C8—21no—
147C23×C163—21no—
148C23×C163—21no—
149C23×C163—42no—
150C23×C163—42no—
151C23×C163—81no—
152C23×C163C2×D411yescharacteristic
153Q8×C163—21no—
154Q8×C163—21no—
155Q8×C163—21no—
156Q8×C163—42no—
157Q8×C163C2×D412yes—
158D4×C163—21no—
159D4×C163—21no—
160D4×C163—21no—
161D4×C163—42no—
162D4×C163C2×D412yes—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—42no—
168C2×C4×C163C2×D411yescharacteristic
169C2×C4×C163C2411yescharacteristic
170C8×C163—21no—
171C8×C163C2×D412yes—
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—22no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—41no—
180C22×C163—84no—
181C22×C163#13512yescomplemented
182C22×C163C22×D411yescharacteristic
183C2×D163—21no—
184C2×D163—21no—
185C2×D163—21no—
186C2×D163—41no—
187C2×D163—41no—
188C2×D163—41no—
189C4×C163—21no—
190C4×C163—22no—
191C4×C163—41no—
192C4×C163C22×D411yescharacteristic
193C4×C163H(5,𝔽2)11yescharacteristic
194Dic163—21no—
195Dic163—21no—
196Dic163—21no—
197Dic163—41no—
198Dic163—41no—
199Dic163—41no—
200C2×C163—22no—
201C2×C163—42no—
202C2×C163—81no—
203C2×C163D4211yescharacteristic
204C2×C163C2×#13511yescharacteristic
205C2×C163#57711yescharacteristic
206D163—41no—
207D163—81no—
208C163#328211yescharacteristic, complemented, Sylow
209#3282—1631nomaximal, Sylow
210C2×#135—1631no—
211C22×SD16D16311yescharacteristic, complemented
212#548—1631no—
213#546—1631no—
214#495—1631no—
215#465—1631no—
216#465—1631no—
217#464—1631no—
218#463—1631no—
219#452—1631no—
220#451—1631no—
221#449—1631no—
222#447—1631no—
223C4×SD16—1631no—
224C8×D4—1631no—
225C2×#41—1631no—
226C2×#41—1631no—
227C22×Q8C2×D16311yescharacteristic
228C22×D4C2×D16311yescharacteristic
229#135—3264no—
230C2×Q16—1631no—
231C2×Q16—1631no—
232C2×SD16—24no—
233C2×SD16—1631no—
234C2×SD16C2×D16311yescharacteristic
235C2×SD16C2×D16311yescharacteristic
236C2×SD16C2×D16311yescharacteristic
237C2×SD16C2×D16311yescharacteristic
238C2×D8—1631no—
239C2×D8—1631no—
240C22×C8—1631no—
241C22×C8C2×D16311yescharacteristic
242#124—3261no—
243#124—1631no—
244#124—1631no—
245#123—3261no—
246C22⋊Q8—3261no—
247C22⋊Q8—1631no—
248C22⋊Q8—1631no—
249#121—3261no—
250#121—3261no—
251#121—1631no—
252#121—1631no—
253#120—3261no—
254C4×Q8—1631no—
255C4×D4—3261no—
256C4×D4—1631no—
257C4×D4—1631no—
258C2×C22⋊C4—3261no—
259#106—1631no—
260C4⋊C8—1631no—
261#103—1631no—
262#103—1631no—
263#103—1631no—
264#102—1631no—
265#102—1631no—
266#102—1631no—
267C22⋊C8—1631no—
268C22⋊C8—1631no—
269C4×C8—1631no—
270C24—21no—
271#41—3262no—
272#41—3262no—
273#41—6521no—
274#41—6521no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—1631no—
279C2×Q8C22×D16311yescharacteristic
280C2×Q8C22×D16311yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—3261no—
285C2×D4—3261no—
286C2×D4—3261no—
287C2×D4—3261no—
288C2×D4—1631no—
289C2×D4—1631no—
290C2×D4C22×D16311yescharacteristic
291C2×D4C22×D16311yescharacteristic
292C22×C4—21no—
293C22×C4—3261no—
294C22×C4—3261no—
295C22×C4—3261no—
296C22×C4—1631no—
297C22×C4C22×D16311yescharacteristic
298Q16—3262no—
299Q16—6521no—
300SD16—22no—
301SD16—22no—
302SD16—22no—
303SD16—3262no—
304SD16C22⋊D16314yescomplemented
305D8—3262no—
306D8—6521no—
307C2×C8—22no—
308C2×C8—3262no—
309C2×C8—1631no—
310C2×C8—1631no—
311C2×C8C22×D16311yescharacteristic
312C2×C8C22×D16311yescharacteristic
313C4⋊C4—3261no—
314C4⋊C4—3261no—
315C4⋊C4—3261no—
316C4⋊C4—1631no—
317C4⋊C4—1631no—
318C4⋊C4—1631no—
319C22⋊C4—3261no—
320C22⋊C4—3261no—
321C22⋊C4—3261no—
322C22⋊C4—3261no—
323C22⋊C4—3261no—
324C22⋊C4—3261no—
325C22⋊C4—3261no—
326C22⋊C4—3261no—
327C22⋊C4—3261no—
328C22⋊C4—1631no—
329C22⋊C4—1631no—
330C42—3261no—
331C42—1631no—
332C23—21no—
333C23—21no—
334C23—42no—
335C23—42no—
336C23—81no—
337C23—3261no—
338C23—1631no—
339C23D4×D16311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—42no—
344Q8—3261no—
345Q8—3261no—
346Q8C2×C22⋊D16312yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—13041no—
352D4—3262no—
353D4—3261no—
354D4—3261no—
355D4—6522no—
356D4—6522no—
357D4—6522no—
358D4C2×C22⋊D16312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—13041no—
365C2×C4—3261no—
366C2×C4—3261no—
367C2×C4—3261no—
368C2×C4—3261no—
369C2×C4—3261no—
370C2×C4—6522no—
371C2×C4—6522no—
372C2×C4—1631no—
373C2×C4—1631no—
374C2×C4—1631no—
375C2×C4D4×D16311yescharacteristic
376C2×C4C23×D16311yescharacteristic
377C8—21no—
378C8—3261no—
379C8—3261no—
380C8C2×C22⋊D16312yes—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—84no—
390C22—3261no—
391C22—3261no—
392C22—3261no—
393C22—6521no—
394C22—6521no—
395C22—6521no—
396C22#1910475612yescomplemented
397C22C2×D4×D16311yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—3261no—
402C4—3261no—
403C4—3261no—
404C4—6521no—
405C4—6521no—
406C4—6521no—
407C4C22×C22⋊D16311yescharacteristic
408C4#1910478011yescharacteristic
409C2—22no—
410C2—42no—
411C2—81no—
412C2—13041no—
413C2—6521no—
414C2D4×C22⋊D16311yescharacteristic
415C2C2×#1910475611yescharacteristic
416C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC22×SD16×C163C2
socleC22×C163C22×D4
hypercenterC22×SD16D163
hypocenterC163#3282
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×#19104756C2semidirect
2. #23782326C2semidirect
3. #23782309C2semidirect
4. #23782046C2semidirect
5. #23781894C2semidirect
6. #23781891C2semidirect
7. #23781885C2semidirect
8. #23781882C2semidirect
9. #23781824C2semidirect
10. #23781821C2semidirect
11. #23781809C2semidirect
12. #23781799C2semidirect
13. SD16×Dic163C2semidirect
14. C8×C22⋊D163C2semidirect
15. C22×SD16×C163C2semidirect
16. C2×#14579700C22semidirect
19. C2×#14579689C22semidirect
20. C2×#14579689C22semidirect
21. C2×#14579688C22semidirect
24. C2×C8×D163C22semidirect
26. #19104724C22semidirect
31. #19104711C22semidirect
37. Q8×Dic163C22semidirect
42. #19104667C22semidirect
43. Dic163⋊C8C22semidirect
44. #19104660C22semidirect
45. #19104659C22semidirect
46. #19104659C22semidirect
47. #19104657C22semidirect
51. (C2×D163)⋊C8C22semidirect
52. C8×Dic163C22semidirect
119. SD16×C163D4semidirect
181. C22×C163#135semidirect
208. C163#3282semidirect
211. C22×SD16D163semidirect
304. SD16C22⋊D163semidirect
396. C22#19104756semidirect

Maximal subgroups

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

subgroupindex
C2×#191047562
#237823262
#237823092
#237820462
#237818942
#237818912
#237818852
#237818822
#237818242
#237818212
#237818092
#237817992
SD16×Dic1632
C8×C22⋊D1632
C22×SD16×C1632
#3282163