Finite Group Databasebeta

← #35217685

Subgroups — #35217685

Order 20864.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104717C211yesmaximal, characteristic
2#23781565C211yesmaximal, characteristic
3#23781561C211yesmaximal, characteristic, complemented
4#23781560C211yesmaximal, characteristic, complemented
5#23781558C211yesmaximal, characteristic, complemented
6#23781555C211yesmaximal, characteristic, complemented
7#23781554C211yesmaximal, characteristic, complemented
8#23781553C211yesmaximal, characteristic, complemented
9#23781552C211yesmaximal, characteristic, complemented
10#23781550C211yesmaximal, characteristic, complemented
11#23781544C211yesmaximal, characteristic
12#23781538C211yesmaximal, characteristic, complemented
13#23781524C211yesmaximal, characteristic, complemented
14#23781522C211yesmaximal, characteristic, complemented
15#385×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18#19104717C418yescomplemented
19C2×C4⋊Dic163C2211yescharacteristic
20C2×C4⋊Dic163C2211yescharacteristic
21C2×Dic163⋊C4C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×C22⋊Dic163C2211yescharacteristic
27C2×C22⋊Dic163C2211yescharacteristic
28C2×C22⋊Dic163C2211yescharacteristic
29C2×(C2×D163)⋊C4C2211yescharacteristic
30C2×(C2×D163)⋊C4C2211yescharacteristic
31C2×(C2×D163)⋊C4C2211yescharacteristic
32C2×(C2×D163)⋊C4C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×C4×Dic163C2211yescharacteristic
35C2×C4×Dic163C2211yescharacteristic
36C2×C4×Dic163C2211yescharacteristic
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×C22⋊D163—41no—
54C2×C22⋊D163—41no—
55C2×C22⋊D163C2×C414yes—
56C22×Dic163C2311yescharacteristic
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2311yescharacteristic
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C2×C4×D163—41no—
64C2×C4×D163—41no—
65C2×C4×D163C2×C414yescomplemented
66C4⋊Dic163—41no—
67C4⋊Dic163C2×C414yescomplemented
68Dic163⋊C4—41no—
69Dic163⋊C4—41no—
70Dic163⋊C4—41no—
71Dic163⋊C4—41no—
72Dic163⋊C4C2×C414yescomplemented
73C22⋊Dic163—41no—
74C22⋊Dic163—41no—
75C22⋊Dic163C2×C414yes—
76(C2×D163)⋊C4—41no—
77(C2×D163)⋊C4—41no—
78(C2×D163)⋊C4—41no—
79(C2×D163)⋊C4—41no—
80(C2×D163)⋊C4C2×C414yescomplemented
81C4×Dic163—41no—
82C4×Dic163—41no—
83C4×Dic163—41no—
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×C414yes—
98C22×D163—21no—
99C22×D163—21no—
100C22×D163—21no—
101C22×D163—21no—
102C22×D163—21no—
103C22×D163—21no—
104C22×D163C22×C412yes—
105C22⋊D163—44no—
106C2×Dic163—21no—
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×Dic163C22×C412yes—
125C2×Dic163C22×C412yes—
126C2×Dic163C22×C412yes—
127C4×D163—44no—
128C23×C163—21no—
129C23×C163—21no—
130C23×C163—21no—
131C23×C163—21no—
132C23×C163—21no—
133C23×C163—21no—
134C23×C163C22×C412yes—
135C23×C163C2411yescharacteristic
136C2×C4×C163—21no—
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—44no—
153C2×C4×C163C22×C412yes—
154C2×C4×C163C22×C412yes—
155C22×C163—22no—
156C22×C163—22no—
157C22×C163—22no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163C23×C411yescharacteristic
163C22×C163H(5,𝔽2)11yescharacteristic
164C22×C163H(5,𝔽2)11yescharacteristic
165C22×C163H(5,𝔽2)11yescharacteristic
166C22×C163H(5,𝔽2)11yescharacteristic
167C22×C1632−1+411yescharacteristic
168C22×C1632−1+411yescharacteristic
169C2×D163—22no—
170C2×D163—22no—
171C2×D163—22no—
172C2×D163—41no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—41no—
176C4×C163—22no—
177C4×C163—22no—
178C4×C163—41no—
179C4×C163—41no—
180C4×C163—41no—
181C4×C163—41no—
182Dic163—22no—
183Dic163—22no—
184Dic163—22no—
185Dic163—41no—
186Dic163—41no—
187Dic163—41no—
188Dic163—41no—
189C2×C163—42no—
190C2×C163#51811yescharacteristic
191C2×C163#52011yescharacteristic
192C2×C163#52011yescharacteristic
193C2×C163#56011yescharacteristic
194C2×C163#56211yescharacteristic
195C2×C163#56211yescharacteristic
196C2×C163#56311yescharacteristic
197D163—42no—
198C163#237611yescharacteristic, complemented, Sylow
199#2376—1631nomaximal, Sylow
200C2×#123—1631no—
201#390—1631no—
202#389—1631no—
203#389—1631no—
204#388—1631no—
205#388—1631no—
206#388—1631no—
207#388—1631no—
208#387—1631no—
209#387—1631no—
210#386—1631no—
211#385—1631no—
212#385D16311yescharacteristic, complemented
213#380—1631no—
214#380—1631no—
215C22×D4—1631no—
216C23×C4—1631no—
217C23×C4C2×D16311yescharacteristic
218#123—1638no—
219C2×C4⋊C4—1631no—
220C2×C4⋊C4—1631no—
221C2×C4⋊C4—1631no—
222C2×C4⋊C4—1631no—
223C2×C4⋊C4—1631no—
224C2×C4⋊C4—1631no—
225C2×C4⋊C4—1631no—
226C2×C4⋊C4C2×D16311yescharacteristic
227C2×C4⋊C4C2×D16311yescharacteristic
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⋊C4—1631no—
236C2×C22⋊C4C2×D16311yescharacteristic
237C2×C22⋊C4C2×D16311yescharacteristic
238C2×C42—1631no—
239C2×C42—1631no—
240C2×C42—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#95C2×D16311yescharacteristic
250#95C2×D16311yescharacteristic
251C24—1631no—
252C24C22×D16311yescharacteristic
253C2×D4—6521no—
254C2×D4—6521no—
255C2×D4—1634no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—6521no—
260C22×C4—6521no—
261C22×C4—1631no—
262C22×C4—1631no—
263C22×C4—1631no—
264C22×C4—1631no—
265C22×C4—1631no—
266C22×C4—1631no—
267C22×C4—1631no—
268C22×C4—1634no—
269C22×C4C22×D16311yescharacteristic
270C22×C4C22×D16311yescharacteristic
271C22×C4C22×D16311yescharacteristic
272C22×C4C22×D16311yescharacteristic
273C22×C4C22×D16311yescharacteristic
274C22×C4C22×D16311yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—6521no—
278C4⋊C4—6521no—
279C4⋊C4—6521no—
280C4⋊C4—6521no—
281C4⋊C4—6521no—
282C4⋊C4—1634no—
283C4⋊C4—1634no—
284C22⋊C4—41no—
285C22⋊C4—6521no—
286C22⋊C4—6521no—
287C22⋊C4—6521no—
288C22⋊C4—6521no—
289C22⋊C4—6521no—
290C22⋊C4—6521no—
291C22⋊C4—1634no—
292C22⋊C4—1634no—
293C22⋊C4C4×D16314yes—
294C42—6521no—
295C42—6521no—
296C42—6521no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—3261no—
304C23—3261no—
305C23—3261no—
306C23—3261no—
307C23—3261no—
308C23—3261no—
309C23—1632no—
310C23C2×C4×D16312yes—
311C23C23×D16311yescharacteristic
312D4—6524no—
313C2×C4—21no—
314C2×C4—21no—
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—44no—
330C2×C4—3261no—
331C2×C4—3261no—
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—6524no—
349C2×C4—1632no—
350C2×C4—1632no—
351C2×C4—1632no—
352C2×C4C2×C4×D16312yes—
353C2×C4C2×C4×D16312yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—3262no—
362C22—3262no—
363C22—3262no—
364C22—6521no—
365C22—6521no—
366C22—6521no—
367C22—6521no—
368C22C22×C4×D16311yescharacteristic
369C22#1910478011yescharacteristic
370C22#1910478011yescharacteristic
371C22#1910478011yescharacteristic
372C22#1910478111yescharacteristic
373C22#1910478211yescharacteristic
374C22#1910478311yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—3262no—
382C4—3262no—
383C4—3262no—
384C4—6521no—
385C4—6521no—
386C4—6521no—
387C4—6521no—
388C2—42no—
389C2—6522no—
390C2#2378213711yescharacteristic
391C2#2378214411yescharacteristic
392C2#2378214611yescharacteristic
393C2#2378240411yescharacteristic
394C2#2378241111yescharacteristic
395C2#2378241211yescharacteristic
396C2#2378241611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#385×C163C2
socle, cosocleC23×C163C24
prime radicalC22×C22⋊D163C22
hypercenter#385D163
hypocenterC163#2376
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. #23781561C2semidirect
4. #23781560C2semidirect
5. #23781558C2semidirect
6. #23781555C2semidirect
7. #23781554C2semidirect
8. #23781553C2semidirect
9. #23781552C2semidirect
10. #23781550C2semidirect
12. #23781538C2semidirect
13. #23781524C2semidirect
14. #23781522C2semidirect
15. #385×C163C2semidirect
18. #19104717C4semidirect
65. C2×C4×D163C2×C4semidirect
67. C4⋊Dic163C2×C4semidirect
72. Dic163⋊C4C2×C4semidirect
80. (C2×D163)⋊C4C2×C4semidirect
198. C163#2376semidirect
212. #385D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047172
#237815652
#237815612
#237815602
#237815582
#237815552
#237815542
#237815532
#237815522
#237815502
#237815442
#237815382
#237815242
#237815222
#385×C1632
#2376163