Finite Group Databasebeta

← #35217709

Subgroups — #35217709

Order 20864.

520 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104721C211yesmaximal, characteristic
2#23781570C211yesmaximal, characteristic, complemented
3#23781569C211yesmaximal, characteristic, complemented
4#23781559C211yesmaximal, characteristic, complemented
5#23781558C211yesmaximal, characteristic, complemented
6#23781557C211yesmaximal, characteristic, complemented
7#23781554C211yesmaximal, characteristic, complemented
8#23781544C211yesmaximal, characteristic
9#23781541C211yesmaximal, characteristic
10#23781538C211yesmaximal, characteristic, complemented
11#23781536C211yesmaximal, characteristic, complemented
12#23781536C211yesmaximal, characteristic, complemented
13#23781530C211yesmaximal, characteristic, complemented
14#23781529C211yesmaximal, characteristic, complemented
15#385×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19#19104721C418yescomplemented
20C2×C4⋊Dic163C2211yescharacteristic
21C2×Dic163⋊C4C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×C22⋊Dic163C2211yescharacteristic
25C2×C22⋊Dic163C2211yescharacteristic
26C2×C22⋊Dic163C2211yescharacteristic
27C2×(C2×D163)⋊C4C2211yescharacteristic
28C2×(C2×D163)⋊C4C2211yescharacteristic
29C2×(C2×D163)⋊C4C2211yescharacteristic
30C2×(C2×D163)⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×C4×Dic163C2211yescharacteristic
33C2×C4×Dic163C2211yescharacteristic
34C2×C4×Dic163C2211yescharacteristic
35C163⋊#95C2211yescharacteristic
36C163⋊#95C2211yescharacteristic
37C163⋊#95C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C23×C4×C163C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52C23×D163C2311yescharacteristic
53C2×Dic326—41no—
54C2×Dic326—41no—
55C2×Dic326C2×C414yescomplemented
56C2×C22⋊D163—41no—
57C2×C22⋊D163—41no—
58C2×C22⋊D163C2×C414yes—
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C2×C4×D163—41no—
67C2×C4×D163—41no—
68C2×C4×D163—41no—
69C4⋊Dic163—41no—
70Dic163⋊C4—41no—
71Dic163⋊C4—41no—
72Dic163⋊C4—41no—
73C22⋊Dic163—41no—
74C22⋊Dic163—41no—
75C22⋊Dic163C2×C414yescomplemented
76(C2×D163)⋊C4—41no—
77(C2×D163)⋊C4—41no—
78(C2×D163)⋊C4—41no—
79(C2×D163)⋊C4C2×C414yescomplemented
80(C2×D163)⋊C4C2×C414yescomplemented
81C4×Dic163—41no—
82C4×Dic163—41no—
83C4×Dic163C2×C414yescomplemented
84C24×C163C2311yescharacteristic
85C22×C4×C163—41no—
86C22×C4×C163—41no—
87C22×C4×C163—41no—
88C22×C4×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C22×C4×C163C2311yescharacteristic
94C4⋊C4×C163—41no—
95C4⋊C4×C163—41no—
96C22⋊C4×C163—41no—
97C22⋊C4×C163C2×C414yescomplemented
98C22×D163—21no—
99C22×D163—21no—
100C22×D163—21no—
101C22×D163—21no—
102C22×D163—21no—
103C22×D163—21no—
104C22×D163C22×C412yes—
105Dic326—44no—
106C22⋊D163—44no—
107C2×Dic163—21no—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163C22×C412yes—
126C2×Dic163C22×C412yes—
127C2×Dic163C22×C412yes—
128C4×D163—44no—
129C23×C163—21no—
130C23×C163—21no—
131C23×C163—21no—
132C23×C163—21no—
133C23×C163—21no—
134C23×C163—21no—
135C23×C163C22×C412yes—
136C23×C163C2411yescharacteristic
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—44no—
154C2×C4×C163C22×C412yes—
155C2×C4×C163C22×C412yes—
156C22×C163—22no—
157C22×C163—22no—
158C22×C163—22no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163C23×C411yescharacteristic
164C22×C163H(5,𝔽2)11yescharacteristic
165C22×C163H(5,𝔽2)11yescharacteristic
166C22×C163H(5,𝔽2)11yescharacteristic
167C22×C163H(5,𝔽2)11yescharacteristic
168C22×C1632−1+411yescharacteristic
169C22×C1632−1+411yescharacteristic
170C2×D163—22no—
171C2×D163—22no—
172C2×D163—22no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—41no—
176C2×D163—41no—
177C4×C163—22no—
178C4×C163—22no—
179C4×C163—41no—
180C4×C163—41no—
181C4×C163—41no—
182C4×C163—41no—
183Dic163—22no—
184Dic163—22no—
185Dic163—22no—
186Dic163—41no—
187Dic163—41no—
188Dic163—41no—
189Dic163—41no—
190C2×C163—42no—
191C2×C163#51811yescharacteristic
192C2×C163#52011yescharacteristic
193C2×C163#52011yescharacteristic
194C2×C163H(3,𝔽4)11yescharacteristic
195C2×C163#56211yescharacteristic
196C2×C163#56211yescharacteristic
197C2×C163#56311yescharacteristic
198D163—42no—
199C163#237911yescharacteristic, complemented, Sylow
200#2379—1631nomaximal, Sylow
201C2×#124—1631no—
202#391—1631no—
203#391—1631no—
204#388—1631no—
205#388—1631no—
206#388—1631no—
207#388—1631no—
208#386—1631no—
209#386—1631no—
210#385—1631no—
211#385—1631no—
212#385—1631no—
213#385D16311yescharacteristic, complemented
214#382—1631no—
215#382—1631no—
216C22×Q8—1631no—
217C22×D4—1631no—
218C23×C4—1631no—
219C23×C4C2×D16311yescharacteristic
220#124—1638no—
221C2×C4⋊C4—1631no—
222C2×C4⋊C4—1631no—
223C2×C4⋊C4—1631no—
224C2×C4⋊C4—1631no—
225C2×C4⋊C4C2×D16311yescharacteristic
226C2×C4⋊C4C2×D16311yescharacteristic
227C2×C22⋊C4—1631no—
228C2×C22⋊C4—1631no—
229C2×C22⋊C4—1631no—
230C2×C22⋊C4—1631no—
231C2×C22⋊C4—1631no—
232C2×C22⋊C4—1631no—
233C2×C22⋊C4—1631no—
234C2×C22⋊C4—1631no—
235C2×C22⋊C4C2×D16311yescharacteristic
236C2×C22⋊C4C2×D16311yescharacteristic
237C2×C42—1631no—
238C2×C42—1631no—
239C2×C42—1631no—
240#95—1631no—
241#95—1631no—
242#95—1631no—
243#95—1631no—
244#95—1631no—
245#95—1631no—
246#95—1631no—
247#95—1631no—
248#95—1631no—
249#95—1631no—
250#95C2×D16311yescharacteristic
251#95C2×D16311yescharacteristic
252C24—1631no—
253C24C22×D16311yescharacteristic
254C2×Q8—6521no—
255C2×Q8—6521no—
256C2×Q8—1634no—
257C2×D4—6521no—
258C2×D4—6521no—
259C2×D4—1634no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—6521no—
264C22×C4—6521no—
265C22×C4—6521no—
266C22×C4—1631no—
267C22×C4—1631no—
268C22×C4—1631no—
269C22×C4—1631no—
270C22×C4—1631no—
271C22×C4—1631no—
272C22×C4—1631no—
273C22×C4C22×D16311yescharacteristic
274C22×C4C22×D16311yescharacteristic
275C22×C4C22×D16311yescharacteristic
276C22×C4C22×D16311yescharacteristic
277C22×C4C22×D16311yescharacteristic
278C22×C4C22×D16311yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—6521no—
282C4⋊C4—6521no—
283C4⋊C4—6521no—
284C4⋊C4—6521no—
285C22⋊C4—41no—
286C22⋊C4—6521no—
287C22⋊C4—6521no—
288C22⋊C4—6521no—
289C22⋊C4—6521no—
290C22⋊C4—6521no—
291C22⋊C4—1634no—
292C22⋊C4—1634no—
293C22⋊C4—1634no—
294C22⋊C4C4×D16314yescomplemented
295C42—6521no—
296C42—6521no—
297C42—1634no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—3261no—
305C23—3261no—
306C23—3261no—
307C23—3261no—
308C23—3261no—
309C23—3261no—
310C23—1632no—
311C23C2×C4×D16312yes—
312C23C23×D16311yescharacteristic
313Q8—6524no—
314D4—6524no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—3261no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—3261no—
338C2×C4—3261no—
339C2×C4—3261no—
340C2×C4—3261no—
341C2×C4—3261no—
342C2×C4—3261no—
343C2×C4—3261no—
344C2×C4—3261no—
345C2×C4—3261no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—6524no—
351C2×C4—1632no—
352C2×C4—1632no—
353C2×C4—1632no—
354C2×C4C2×C4×D16312yes—
355C2×C4C2×C4×D16312yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3262no—
364C22—3262no—
365C22—3262no—
366C22—6521no—
367C22—6521no—
368C22—6521no—
369C22—6521no—
370C22C22×C4×D16311yescharacteristic
371C22#1910478011yescharacteristic
372C22#1910478011yescharacteristic
373C22#1910478011yescharacteristic
374C22#1910478111yescharacteristic
375C22#1910478211yescharacteristic
376C22#1910478311yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3262no—
384C4—3262no—
385C4—3262no—
386C4—6521no—
387C4—6521no—
388C4—6521no—
389C4—6521no—
390C2—42no—
391C2—6522no—
392C2#2378213711yescharacteristic
393C2#2378214411yescharacteristic
394C2#2378214611yescharacteristic
395C2#2378240811yescharacteristic
396C2#2378241111yescharacteristic
397C2#2378241211yescharacteristic
398C2#2378241611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#385×C163C2
socle, cosocleC23×C163C24
prime radicalC22×C22⋊D163C22
hypercenter#385D163
hypocenterC163#2379
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #23781570C2semidirect
3. #23781569C2semidirect
4. #23781559C2semidirect
5. #23781558C2semidirect
6. #23781557C2semidirect
7. #23781554C2semidirect
10. #23781538C2semidirect
11. #23781536C2semidirect
12. #23781536C2semidirect
13. #23781530C2semidirect
14. #23781529C2semidirect
15. #385×C163C2semidirect
19. #19104721C4semidirect
55. C2×Dic326C2×C4semidirect
75. C22⋊Dic163C2×C4semidirect
79. (C2×D163)⋊C4C2×C4semidirect
80. (C2×D163)⋊C4C2×C4semidirect
83. C4×Dic163C2×C4semidirect
97. C22⋊C4×C163C2×C4semidirect
199. C163#2379semidirect
213. #385D163semidirect
294. C22⋊C4C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047212
#237815702
#237815692
#237815592
#237815582
#237815572
#237815542
#237815442
#237815412
#237815382
#237815362
#237815302
#237815292
#385×C1632
#2379163