Finite Group Databasebeta

← #35218058

Subgroups — #35218058

Order 20864.

500 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104712C211yesmaximal, characteristic
2C2×C4×Dic326C211yesmaximal, characteristic, complemented
3#23781601C211yesmaximal, characteristic
4#23781601C211yesmaximal, characteristic
5#23781598C211yesmaximal, characteristic, complemented
6#23781597C211yesmaximal, characteristic, complemented
7#23781569C211yesmaximal, characteristic, complemented
8#23781562C211yesmaximal, characteristic, complemented
9#23781558C211yesmaximal, characteristic
10#23781552C211yesmaximal, characteristic, complemented
11#23781549C211yesmaximal, characteristic, complemented
12#23781534C211yesmaximal, characteristic, complemented
13#23781523C211yesmaximal, characteristic
14C22⋊C4×Dic163C211yesmaximal, characteristic
15#388×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18#19104712—24no—
19C4×Dic326—24no—
20C2×C4⋊Dic163C2211yescharacteristic
21C2×C4⋊Dic163C2211yescharacteristic
22C2×C4⋊Dic163C2211yescharacteristic
23C2×C4⋊Dic163C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×Dic163⋊C4C2211yescharacteristic
30C2×Dic163⋊C4C2211yescharacteristic
31C2×Dic163⋊C4C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C4×Dic163C2211yescharacteristic
36C2×C4×Dic163C2211yescharacteristic
37C2×C4×Dic163C2211yescharacteristic
38C163⋊#95C2211yescharacteristic
39C163⋊#95C2211yescharacteristic
40C163⋊#95C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C2×C4⋊C4×C163C2211yescharacteristic
47C2×C22⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C42×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C2×Dic326—22no—
54C2×Dic326—22no—
55C2×Dic326D414yes—
56C22×Dic163—22no—
57C22×Dic163—22no—
58C22×Dic163—22no—
59C22×Dic163C2311yescharacteristic
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C4⋊Dic163—22no—
68C4⋊Dic163—22no—
69C4⋊Dic163—41no—
70C4⋊Dic163—41no—
71Dic163⋊C4—22no—
72Dic163⋊C4—22no—
73Dic163⋊C4—22no—
74Dic163⋊C4—22no—
75Dic163⋊C4—22no—
76Dic163⋊C4—22no—
77Dic163⋊C4—22no—
78Dic163⋊C4—41no—
79C22⋊Dic163—22no—
80C22⋊Dic163—22no—
81C22⋊Dic163—41no—
82C4×Dic163—22no—
83C4×Dic163—22no—
84C4×Dic163—22no—
85C24×C163C2311yescharacteristic
86C22×C4×C163C2311yescharacteristic
87C22×C4×C163C2311yescharacteristic
88C22×C4×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C4⋊C4×C163—41no—
93C22⋊C4×C163—22no—
94C22⋊C4×C163—41no—
95C22⋊C4×C163Q814yescomplemented
96C42×C163—22no—
97Dic326—28no—
98C2×Dic163—21no—
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—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—44no—
118C2×Dic163C2×D412yes—
119C2×Dic163C2×D412yes—
120C2×Dic163#4112yes—
121C2×Dic163#4112yes—
122C2×Dic163#4112yes—
123C23×C163—21no—
124C23×C163—21no—
125C23×C163—21no—
126C23×C163—21no—
127C23×C163—21no—
128C23×C163—21no—
129C23×C163C2×Q812yes—
130C23×C163C2411yescharacteristic
131C2×C4×C163—21no—
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×C163C2×D412yes—
147C2×C4×C163C2×Q812yes—
148C2×C4×C163C2×Q812yes—
149C22×C163—22no—
150C22×C163—22no—
151C22×C163—22no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163C22×D411yescharacteristic
157C22×C163C22×Q811yescharacteristic
158C22×C163C2×#4111yescharacteristic
159C22×C163C2×#4111yescharacteristic
160C22×C163C2×#4111yescharacteristic
161C22×C163H(5,𝔽2)11yescharacteristic
162C22×C1632−1+411yescharacteristic
163C4×C163—22no—
164C4×C163—22no—
165C4×C163—22no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163—41no—
169Dic163—22no—
170Dic163—22no—
171Dic163—41no—
172Dic163—41no—
173Dic163—41no—
174Dic163—41no—
175Dic163—41no—
176Dic163#12314yes—
177C2×C163—42no—
178C2×C163C2×#12311yescharacteristic
179C2×C163#53311yescharacteristic
180C2×C163#54211yescharacteristic
181C2×C163D4×Q811yescharacteristic
182C2×C163#55011yescharacteristic
183C2×C163#55211yescharacteristic
184C2×C163#55411yescharacteristic
185C163#242311yescharacteristic, complemented, Sylow
186#2423—1631nomaximal, Sylow
187C2×C22⋊Q8—1631no—
188C2×C4×Q8—1631no—
189#399—1631no—
190#399—1631no—
191#398—1631no—
192#398—1631no—
193#391—1631no—
194#389—1631no—
195#388—1631no—
196#388D16311yescharacteristic
197#387—1631no—
198#387—1631no—
199#384—1631no—
200#380—1631no—
201C4×C22⋊C4—1631no—
202C22×Q8—1631no—
203C23×C4—1631no—
204C22⋊Q8—3264no—
205C4×Q8—3264no—
206C2×C4⋊C4—1631no—
207C2×C4⋊C4—1631no—
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×C42C2×D16311yescharacteristic
229#95—1631no—
230#95—1631no—
231#95—1631no—
232#95—1631no—
233#95—1631no—
234#95—1631no—
235#95—1631no—
236#95—1631no—
237#95C2×D16311yescharacteristic
238#95C2×D16311yescharacteristic
239C24C22×D16311yescharacteristic
240C2×Q8—3262no—
241C2×Q8—3262no—
242C2×Q8—1634no—
243C22×C4—3262no—
244C22×C4—3262no—
245C22×C4—3262no—
246C22×C4—1631no—
247C22×C4—1631no—
248C22×C4—1631no—
249C22×C4—1631no—
250C22×C4—1631no—
251C22×C4—1631no—
252C22×C4—1631no—
253C22×C4—1631no—
254C22×C4C22×D16311yescharacteristic
255C22×C4C22×D16311yescharacteristic
256C22×C4C22×D16311yescharacteristic
257C22×C4C22×D16311yescharacteristic
258C22×C4C22×D16311yescharacteristic
259C22×C4C22×D16311yescharacteristic
260C4⋊C4—41no—
261C4⋊C4—3262no—
262C4⋊C4—3262no—
263C4⋊C4—3262no—
264C4⋊C4—3262no—
265C4⋊C4—3262no—
266C4⋊C4—3262no—
267C4⋊C4—3262no—
268C4⋊C4—3262no—
269C4⋊C4—3262no—
270C4⋊C4—6521no—
271C4⋊C4—6521no—
272C4⋊C4—6521no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—3262no—
276C22⋊C4—3262no—
277C22⋊C4—6521no—
278C22⋊C4Dic32614yescomplemented
279C42—22no—
280C42—3262no—
281C42—3262no—
282C42—3262no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23C2×Dic32612yes—
290C23C23×D16311yescharacteristic
291Q8—3268no—
292C2×C4—21no—
293C2×C4—21no—
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—3261no—
308C2×C4—3261no—
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—3261no—
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—6524no—
327C2×C4—1632no—
328C2×C4—1632no—
329C2×C4—1632no—
330C2×C4—1632no—
331C2×C4—1632no—
332C2×C4D4×D16312yes—
333C2×C4C2×Dic32612yes—
334C2×C4C2×Dic32612yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C2×D4×D16311yescharacteristic
343C22C22×Dic32611yescharacteristic
344C22#41×D16311yescharacteristic
345C22#41×D16311yescharacteristic
346C22#41×D16311yescharacteristic
347C22#1910478011yescharacteristic
348C22#1910478211yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—3262no—
356C4—3262no—
357C4—6521no—
358C4—6521no—
359C4—6521no—
360C4—6521no—
361C4—6521no—
362C4—1634no—
363C2—42no—
364C2#123×D16311yescharacteristic
365C2#2378221811yescharacteristic
366C2#2378227911yescharacteristic
367C2D4×Dic32611yescharacteristic
368C2#2378233311yescharacteristic
369C2#2378235111yescharacteristic
370C2#2378237411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#388×C163C2
prime radicalC24×C163C23
hypercenter#388D163
hypocenterC163#2423
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic326C2semidirect
5. #23781598C2semidirect
6. #23781597C2semidirect
7. #23781569C2semidirect
8. #23781562C2semidirect
10. #23781552C2semidirect
11. #23781549C2semidirect
12. #23781534C2semidirect
95. C22⋊C4×C163Q8semidirect
185. C163#2423semidirect
278. C22⋊C4Dic326semidirect

Maximal subgroups

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

subgroupindex
C2×#191047122
C2×C4×Dic3262
#237816012
#237815982
#237815972
#237815692
#237815622
#237815582
#237815522
#237815492
#237815342
#237815232
C22⋊C4×Dic1632
#388×C1632
#2423163