Finite Group Databasebeta

← #35217211

Subgroups — #35217211

Order 20864.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104712C211yesmaximal, characteristic
2C2×#19104700C211yesmaximal, characteristic, complemented
3C2×#19104691C211yesmaximal, characteristic
4#23781569C211yesmaximal, characteristic, complemented
5#23781563C211yesmaximal, characteristic, complemented
6#23781562C211yesmaximal, characteristic, complemented
7#23781560C211yesmaximal, characteristic, complemented
8#23781559C211yesmaximal, characteristic
9#23781559C211yesmaximal, characteristic
10#23781558C211yesmaximal, characteristic
11#23781548C211yesmaximal, characteristic, complemented
12#23781547C211yesmaximal, characteristic, complemented
13#23781523C211yesmaximal, characteristic
14C4×Dic163⋊C4C211yesmaximal, characteristic, complemented
15#388×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18#19104712C418yescomplemented
19#19104700—24no—
20#19104691—24no—
21C2×C4⋊Dic163C2211yescharacteristic
22C2×C4⋊Dic163C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×Dic163⋊C4C2211yescharacteristic
30C2×Dic163⋊C4C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C4×Dic163C2211yescharacteristic
35C2×C4×Dic163C2211yescharacteristic
36C2×C4×Dic163C2211yescharacteristic
37C2×C4×Dic163C2211yescharacteristic
38C2×C4×Dic163C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51C2×C42×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53#95×C163C2211yescharacteristic
54C2×Dic326—22no—
55C2×Dic326—22no—
56C2×Dic326C2×C414yes—
57C22×Dic163—22no—
58C22×Dic163—22no—
59C22×Dic163C2×C414yes—
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C4⋊Dic163—41no—
69C4⋊Dic163C2×C414yes—
70Dic163⋊C4—22no—
71Dic163⋊C4—22no—
72Dic163⋊C4—22no—
73Dic163⋊C4—22no—
74Dic163⋊C4—22no—
75Dic163⋊C4—41no—
76Dic163⋊C4C2×C414yes—
77Dic163⋊C4C2×C414yes—
78C22⋊Dic163—22no—
79C22⋊Dic163—41no—
80C22⋊Dic163C2×C414yes—
81C4×Dic163—22no—
82C4×Dic163—22no—
83C4×Dic163—22no—
84C4×Dic163—41no—
85C4×Dic163D414yes—
86C24×C163C2311yescharacteristic
87C22×C4×C163C2311yescharacteristic
88C22×C4×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C4⋊C4×C163—22no—
94C22⋊C4×C163—22no—
95C22⋊C4×C163—41no—
96C22⋊C4×C163C2×C414yes—
97C42×C163—41no—
98Dic326—28no—
99C2×Dic163—21no—
100C2×Dic163—21no—
101C2×Dic163—21no—
102C2×Dic163—21no—
103C2×Dic163—21no—
104C2×Dic163—21no—
105C2×Dic163—21no—
106C2×Dic163—21no—
107C2×Dic163—28no—
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×Dic163C22×C412yes—
118C2×Dic163C22×C412yes—
119C2×Dic163C22×C412yes—
120C2×Dic163C22×C412yes—
121C2×Dic163C2×D412yes—
122C2×Dic163C2×D412yes—
123C2×Dic163#4112yes—
124C23×C163—21no—
125C23×C163—21no—
126C23×C163—21no—
127C23×C163—21no—
128C23×C163—21no—
129C23×C163—21no—
130C23×C163C22×C412yes—
131C23×C163C2411yescharacteristic
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
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×C163C22×C412yes—
148C2×C4×C163C22×C412yes—
149C2×C4×C163C2×D412yes—
150C22×C163—22no—
151C22×C163—22no—
152C22×C163—22no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163C23×C411yescharacteristic
158C22×C163C22×D411yescharacteristic
159C22×C163C2×#4111yescharacteristic
160C22×C163H(5,𝔽2)11yescharacteristic
161C22×C1632−1+411yescharacteristic
162C22×C1632−1+411yescharacteristic
163C22×C1632−1+411yescharacteristic
164C4×C163—22no—
165C4×C163—22no—
166C4×C163—22no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163—41no—
170Dic163—22no—
171Dic163—22no—
172Dic163—22no—
173Dic163—22no—
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177Dic163C4×D414yes—
178C2×C163—42no—
179C2×C163C2×C4×D411yescharacteristic
180C2×C163#51911yescharacteristic
181C2×C163#52011yescharacteristic
182C2×C163#53611yescharacteristic
183C2×C163#53711yescharacteristic
184C2×C163#54111yescharacteristic
185C2×C163#54211yescharacteristic
186C163#232311yescharacteristic, complemented, Sylow
187#2323—1631nomaximal, Sylow
188C2×C22⋊Q8—1631no—
189C2×C4×Q8—1631no—
190C2×#117—1631no—
191#391—1631no—
192#389—1631no—
193#389—1631no—
194#389—1631no—
195#388—1631no—
196#388—1631no—
197#388—1631no—
198#388D16311yescharacteristic
199#387—1631no—
200#387—1631no—
201#380—1631no—
202C4×C4⋊C4—1631no—
203C22×Q8—1631no—
204C23×C4—1631no—
205C22⋊Q8—1638no—
206C4×Q8—3264no—
207#117—3264no—
208C2×C4⋊C4—1631no—
209C2×C4⋊C4—1631no—
210C2×C4⋊C4—1631no—
211C2×C4⋊C4—1631no—
212C2×C4⋊C4—1631no—
213C2×C4⋊C4—1631no—
214C2×C4⋊C4—1631no—
215C2×C4⋊C4—1631no—
216C2×C4⋊C4—1631no—
217C2×C4⋊C4—1631no—
218C2×C4⋊C4C2×D16311yescharacteristic
219C2×C22⋊C4—1631no—
220C2×C22⋊C4—1631no—
221C2×C22⋊C4—1631no—
222C2×C22⋊C4C2×D16311yescharacteristic
223C2×C22⋊C4C2×D16311yescharacteristic
224C2×C22⋊C4C2×D16311yescharacteristic
225C2×C42—1631no—
226C2×C42—1631no—
227C2×C42—1631no—
228C2×C42—1631no—
229C2×C42—1631no—
230C2×C42C2×D16311yescharacteristic
231#95—1631no—
232#95—1631no—
233#95—1631no—
234#95—1631no—
235#95—1631no—
236#95—1631no—
237#95—1631no—
238#95—1631no—
239#95C2×D16311yescharacteristic
240#95C2×D16311yescharacteristic
241C24C22×D16311yescharacteristic
242C2×Q8—3262no—
243C2×Q8—3262no—
244C2×Q8—1634no—
245C22×C4—3262no—
246C22×C4—3262no—
247C22×C4—1631no—
248C22×C4—1631no—
249C22×C4—1631no—
250C22×C4—1634no—
251C22×C4—1631no—
252C22×C4—1631no—
253C22×C4—1631no—
254C22×C4—1631no—
255C22×C4—1631no—
256C22×C4C22×D16311yescharacteristic
257C22×C4C22×D16311yescharacteristic
258C22×C4C22×D16311yescharacteristic
259C22×C4C22×D16311yescharacteristic
260C22×C4C22×D16311yescharacteristic
261C22×C4C22×D16311yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—3262no—
264C4⋊C4—3262no—
265C4⋊C4—3262no—
266C4⋊C4—3262no—
267C4⋊C4—3262no—
268C4⋊C4—6521no—
269C4⋊C4—6521no—
270C4⋊C4—1634no—
271C4⋊C4—1634no—
272C4⋊C4—1634no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—3262no—
276C22⋊C4—6521no—
277C22⋊C4—1634no—
278C22⋊C4C4×D16314yes—
279C42—41no—
280C42—3262no—
281C42—3262no—
282C42—3262no—
283C42—6521no—
284C42—1634no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23C2×C4×D16312yes—
292C23C23×D16311yescharacteristic
293Q8—3268no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3268no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—1632no—
328C2×C4—1632no—
329C2×C4—1632no—
330C2×C4—1632no—
331C2×C4—1632no—
332C2×C4—1632no—
333C2×C4—1632no—
334C2×C4C2×C4×D16312yes—
335C2×C4C2×C4×D16312yes—
336C2×C4D4×D16312yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22C22×C4×D16311yescharacteristic
345C22C2×D4×D16311yescharacteristic
346C22#41×D16311yescharacteristic
347C22#1910478011yescharacteristic
348C22#1910478211yescharacteristic
349C22#1910478211yescharacteristic
350C22#1910478211yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3262no—
358C4—3262no—
359C4—3262no—
360C4—3262no—
361C4—6521no—
362C4—6521no—
363C4—6521no—
364C4—1634no—
365C2—42no—
366C2C4×D4×D16311yescharacteristic
367C2#2378214211yescharacteristic
368C2#2378214411yescharacteristic
369C2#2378223511yescharacteristic
370C2#2378224211yescharacteristic
371C2#2378227211yescharacteristic
372C2#2378228311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#388×C163C2
socle, cosocleC23×C163C24
prime radicalC24×C163C23
hypercenter#388D163
hypocenterC163#2323
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#19104700C2semidirect
4. #23781569C2semidirect
5. #23781563C2semidirect
6. #23781562C2semidirect
7. #23781560C2semidirect
11. #23781548C2semidirect
12. #23781547C2semidirect
14. C4×Dic163⋊C4C2semidirect
18. #19104712C4semidirect
186. C163#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#191047122
C2×#191047002
C2×#191046912
#237815692
#237815632
#237815622
#237815602
#237815592
#237815582
#237815482
#237815472
#237815232
C4×Dic163⋊C42
#388×C1632
#2323163