Finite Group Databasebeta

← #35218126

Subgroups — #35218126

Order 20864.

832 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104731C211yesmaximal, characteristic, complemented
2C2×#19104709C211yesmaximal, characteristic, complemented
3C2×#19104707C212yesmaximal, complemented
4C2×D4×Dic163C211yesmaximal, characteristic, complemented
5#23781580C212yesmaximal, complemented
6#23781573C212yesmaximal, complemented
7#23781566C211yesmaximal, characteristic, complemented
8#23781561C211yesmaximal, characteristic, complemented
9#23781542C212yesmaximal, complemented
10C4×C22⋊Dic163C211yesmaximal, characteristic, complemented
11C2×#121×C163C211yesmaximal, characteristic, complemented
12C22×D652C2211yescharacteristic
13C22×C22⋊D163C2212yes—
14C22×C22⋊D163C2212yes—
15C22×C22⋊D163C2212yes—
16C23×Dic163C2212yes—
17#19104731—24no—
18#19104709—24no—
19#19104707—28no—
20D4×Dic163—24no—
21C2×C4⋊Dic163C2211yescharacteristic
22C2×Dic163⋊C4C2212yes—
23C2×C22⋊Dic163C2211yescharacteristic
24C2×C22⋊Dic163C2212yes—
25C2×C22⋊Dic163C2211yescharacteristic
26C2×(C2×D163)⋊C4C2212yes—
27C2×(C2×D163)⋊C4C2212yes—
28C2×(C2×D163)⋊C4C2212yes—
29C2×C4×Dic163C2211yescharacteristic
30C2×C4×Dic163C2211yescharacteristic
31C163⋊#95C2211yescharacteristic
32C163⋊#95C2211yescharacteristic
33C163⋊#95C2212yes—
34C22×D4×C163C2212yes—
35C22×D4×C163C2211yescharacteristic
36C23×C4×C163C2211yescharacteristic
37#121×C163—24no—
38C2×C4⋊C4×C163C2211yescharacteristic
39C2×C22⋊C4×C163C2212yes—
40C23×D163C2312yes—
41C2×D652—22no—
42C2×D652—22no—
43C2×D652—22no—
44C2×C22⋊D163—24no—
45C2×C22⋊D163—24no—
46C2×C22⋊D163—24no—
47C2×C22⋊D163—24no—
48C2×C22⋊D163—24no—
49C2×C22⋊D163—24no—
50C2×C22⋊D163—42no—
51C2×C22⋊D163—42no—
52C2×C22⋊D163—42no—
53C22×Dic163—24no—
54C22×Dic163—24no—
55C22×Dic163—42no—
56C22×Dic163C2311yescharacteristic
57C22×Dic163C2311yescharacteristic
58C22×Dic163C2312yes—
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C4⋊Dic163D414yescomplemented
62Dic163⋊C4—24no—
63C22⋊Dic163—24no—
64C22⋊Dic163—22no—
65C22⋊Dic163D414yescomplemented
66(C2×D163)⋊C4—24no—
67(C2×D163)⋊C4—24no—
68(C2×D163)⋊C4—42no—
69C4×Dic163—22no—
70C4×Dic163—22no—
71C24×C163C2312yes—
72C24×C163C2311yescharacteristic
73C2×D4×C163—24no—
74C2×D4×C163—24no—
75C2×D4×C163—22no—
76C2×D4×C163—22no—
77C2×D4×C163—42no—
78C2×D4×C163D414yescomplemented
79C22×C4×C163—22no—
80C22×C4×C163—22no—
81C22×C4×C163—22no—
82C22×C4×C163C2311yescharacteristic
83C22×C4×C163C2312yes—
84C22×C4×C163C2311yescharacteristic
85C4⋊C4×C163—22no—
86C22⋊C4×C163—24no—
87C22×D163—22no—
88C22×D163—22no—
89C22×D163—22no—
90C22×D163—22no—
91C22×D163—22no—
92C22×D163—22no—
93C22×D163—22no—
94D652—44no—
95C22⋊D163—48no—
96C22⋊D163—48no—
97C22⋊D163—48no—
98C2×Dic163—21no—
99C2×Dic163—21no—
100C2×Dic163—21no—
101C2×Dic163—22no—
102C2×Dic163—21no—
103C2×Dic163—22no—
104C2×Dic163—21no—
105C2×Dic163—21no—
106C2×Dic163—21no—
107C2×Dic163—21no—
108C2×Dic163—21no—
109C2×Dic163—48no—
110C2×Dic163C2×D412yes—
111C2×Dic163C2×D414yes—
112C2×Dic163C2×D412yes—
113C2×Dic163#4112yes—
114C23×C163—21no—
115C23×C163—21no—
116C23×C163—22no—
117C23×C163—22no—
118C23×C163—21no—
119C23×C163—22no—
120C23×C163—21no—
121C23×C163—22no—
122C23×C163—21no—
123C23×C163—22no—
124C23×C163—22no—
125C23×C163—21no—
126C23×C163C2×D412yes—
127C23×C163C2×D414yes—
128C23×C163C2411yescharacteristic
129D4×C163—28no—
130D4×C163—48no—
131C2×C4×C163—21no—
132C2×C4×C163—22no—
133C2×C4×C163—21no—
134C2×C4×C163—22no—
135C2×C4×C163—21no—
136C2×C4×C163—22no—
137C2×C4×C163—44no—
138C2×C4×C163C2×D412yes—
139C2×C4×C163C2×D412yes—
140C2×C4×C163#4112yes—
141C22×C163—24no—
142C22×C163—22no—
143C22×C163—22no—
144C22×C163—24no—
145C22×C163—22no—
146C22×C163—24no—
147C22×C163—41no—
148C22×C163—42no—
149C22×C163—41no—
150C22×C163—42no—
151C22×C163—42no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—42no—
155C22×C163C22×D411yescharacteristic
156C22×C163C22×D411yescharacteristic
157C22×C163C22×D411yescharacteristic
158C22×C163C2×#4111yescharacteristic
159C22×C163C2×#4111yescharacteristic
160C22×C163H(5,𝔽2)11yescharacteristic
161C22×C163H(5,𝔽2)11yescharacteristic
162C2×D163—42no—
163C2×D163—42no—
164C2×D163—42no—
165C2×D163—42no—
166C2×D163—42no—
167C2×D163—42no—
168C2×D163—42no—
169C4×C163—41no—
170C4×C163—42no—
171C4×C163#12114yescomplemented
172Dic163—22no—
173Dic163—24no—
174Dic163—22no—
175Dic163—22no—
176Dic163—41no—
177C2×C163—44no—
178C2×C163—42no—
179C2×C163C2×#12111yescharacteristic
180C2×C163#53211yescharacteristic
181C2×C163#54011yescharacteristic
182C2×C163D4211yescharacteristic
183C2×C163D4211yescharacteristic
184C2×C163#54611yescharacteristic
185C2×C163#55011yescharacteristic
186D163—82no—
187C163#242811yescharacteristic, complemented, Sylow
188#2428—1631nomaximal, Sylow
189C2×#127—1631no—
190C2×#121—1632no—
191C2×#121—1631no—
192C2×#121D16311yescharacteristic, complemented
193C2×C4×D4—1631no—
194#394—1632no—
195#392—1632no—
196#390—1631no—
197#389—1631no—
198#386—1632no—
199C4×C22⋊C4—1631no—
200C22×D4—1632no—
201C22×D4—1632no—
202C22×D4—1632no—
203C22×D4—1631no—
204C22×D4C2×D16312yes—
205C22×D4C2×D16311yescharacteristic
206C23×C4—1632no—
207C23×C4C2×D16311yescharacteristic
208#127—3264no—
209#121—24no—
210#121—3264no—
211#121—3268no—
212C4×D4—3264no—
213C2×C4⋊C4—1632no—
214C2×C4⋊C4—1631no—
215C2×C4⋊C4C2×D16311yescharacteristic
216C2×C22⋊C4—1632no—
217C2×C22⋊C4—1631no—
218C2×C22⋊C4—1632no—
219C2×C22⋊C4—1632no—
220C2×C22⋊C4—1632no—
221C2×C22⋊C4—1631no—
222C2×C22⋊C4C2×D16312yes—
223C2×C42—1631no—
224C2×C42—1631no—
225#95—1631no—
226#95—1631no—
227#95—1632no—
228C24—1632no—
229C24C22×D16311yescharacteristic
230C24C22×D16312yes—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—3264no—
237C2×D4—3264no—
238C2×D4—3264no—
239C2×D4—3264no—
240C2×D4—3264no—
241C2×D4—3264no—
242C2×D4—3262no—
243C2×D4—3262no—
244C2×D4—3262no—
245C2×D4—6522no—
246C2×D4—6522no—
247C2×D4—6522no—
248C2×D4C22⋊D16314yescomplemented
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—3264no—
253C22×C4—3264no—
254C22×C4—6522no—
255C22×C4—1631no—
256C22×C4—1631no—
257C22×C4—1631no—
258C22×C4—1631no—
259C22×C4—1632no—
260C22×C4C22×D16311yescharacteristic
261C22×C4C22×D16312yes—
262C22×C4C22×D16311yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—3264no—
265C4⋊C4—1634no—
266C22⋊C4—24no—
267C22⋊C4—3264no—
268C22⋊C4—3264no—
269C22⋊C4—3264no—
270C22⋊C4—3262no—
271C22⋊C4—6522no—
272C22⋊C4—1634no—
273C42—3262no—
274C42—3262no—
275C23—22no—
276C23—21no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—21no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23—22no—
286C23—21no—
287C23—3262no—
288C23—3262no—
289C23—3262no—
290C23—3262no—
291C23—3262no—
292C23—3262no—
293C23—3262no—
294C23D4×D16312yes—
295C23C2×C22⋊D16314yes—
296C23C23×D16311yescharacteristic
297D4—28no—
298D4—48no—
299D4—6528no—
300D4—6528no—
301D4—6528no—
302D4—6524no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—44no—
310C2×C4—3261no—
311C2×C4—3262no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3262no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4—6528no—
322C2×C4—1632no—
323C2×C4—1634no—
324C2×C4—1632no—
325C2×C4—1632no—
326C2×C4D4×D16312yes—
327C2×C4C2×C22⋊D16312yes—
328C2×C4#1457969812yes—
329C22—24no—
330C22—22no—
331C22—22no—
332C22—24no—
333C22—22no—
334C22—24no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—6522no—
344C22—6522no—
345C22—6522no—
346C22—6522no—
347C22—6522no—
348C22—6522no—
349C22—6522no—
350C22C2×D4×D16311yescharacteristic
351C22C2×D4×D16311yescharacteristic
352C22C22×C22⋊D16311yescharacteristic
353C22#41×D16311yescharacteristic
354C22C2×#1457969811yescharacteristic
355C22#1910478011yescharacteristic
356C22#1910478111yescharacteristic
357C4—41no—
358C4—42no—
359C4—3262no—
360C4—3262no—
361C4—3264no—
362C4—3262no—
363C4—6521no—
364C4#1910470714yescomplemented
365C2—44no—
366C2—42no—
367C2—13042no—
368C2C2×#1910470711yescharacteristic
369C2#2378221011yescharacteristic
370C2#2378226811yescharacteristic
371C2D4×C22⋊D16311yescharacteristic
372C2#2378229911yescharacteristic
373C2#2378230311yescharacteristic
374C2#2378233311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC2×#121×C163C2
hypercenterC2×#121D163
hypocenterC163#2428
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#19104731C2semidirect
2. C2×#19104709C2semidirect
3. C2×#19104707C2semidirect
4. C2×D4×Dic163C2semidirect
5. #23781580C2semidirect
6. #23781573C2semidirect
7. #23781566C2semidirect
8. #23781561C2semidirect
9. #23781542C2semidirect
10. C4×C22⋊Dic163C2semidirect
11. C2×#121×C163C2semidirect
61. C4⋊Dic163D4semidirect
65. C22⋊Dic163D4semidirect
78. C2×D4×C163D4semidirect
171. C4×C163#121semidirect
187. C163#2428semidirect
192. C2×#121D163semidirect
248. C2×D4C22⋊D163semidirect
364. C4#19104707semidirect

Maximal subgroups

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

subgroupindex
C2×#191047312
C2×#191047092
C2×#191047072
C2×D4×Dic1632
#237815802
#237815732
#237815662
#237815612
#237815422
C4×C22⋊Dic1632
C2×#121×C1632
#2428163