Finite Group Databasebeta

← #35217960

Subgroups — #35217960

Order 20864.

604 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104713C211yesmaximal, characteristic, complemented
2C2×#19104712C211yesmaximal, characteristic, complemented
3C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic
4C2×#19104691C211yesmaximal, characteristic, complemented
5#23781593C211yesmaximal, characteristic, complemented
6#23781581C211yesmaximal, characteristic
7#23781580C211yesmaximal, characteristic
8#23781569C211yesmaximal, characteristic, complemented
9#23781562C211yesmaximal, characteristic, complemented
10#23781562C211yesmaximal, characteristic, complemented
11#23781561C211yesmaximal, characteristic, complemented
12#23781557C211yesmaximal, characteristic, complemented
13#23781554C211yesmaximal, characteristic, complemented
14#23781524C211yesmaximal, characteristic, complemented
15#397×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C23×Dic163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic
20#19104713—24no—
21#19104712—24no—
22(C22⋊D163)⋊C4—24no—
23#19104691—24no—
24C2×C4⋊Dic163C2211yescharacteristic
25C2×C4⋊Dic163C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×Dic163⋊C4C2211yescharacteristic
30C2×Dic163⋊C4C2211yescharacteristic
31C2×Dic163⋊C4C2211yescharacteristic
32C2×Dic163⋊C4C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×(C2×D163)⋊C4C2211yescharacteristic
36C2×(C2×D163)⋊C4C2211yescharacteristic
37C2×(C2×D163)⋊C4C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×C4×Dic163C2211yescharacteristic
41C2×C4×Dic163C2211yescharacteristic
42C2×C4×Dic163C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C163⋊#95C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic
49C2×C22⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51C2×C22⋊C4×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53#95×C163C2211yescharacteristic
54#95×C163C2211yescharacteristic
55C23×D163C2311yescharacteristic
56C2×Dic326—22no—
57C2×Dic326—22no—
58C2×Dic326—22no—
59C2×C22⋊D163—22no—
60C2×C22⋊D163—22no—
61C2×C22⋊D163—22no—
62C22×Dic163—22no—
63C22×Dic163—22no—
64C22×Dic163D414yes—
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C2×C4×D163—22no—
73C2×C4×D163—22no—
74C2×C4×D163—22no—
75C4⋊Dic163—22no—
76C4⋊Dic163—22no—
77Dic163⋊C4—22no—
78Dic163⋊C4—22no—
79Dic163⋊C4—22no—
80Dic163⋊C4—22no—
81Dic163⋊C4—22no—
82Dic163⋊C4—22no—
83Dic163⋊C4D414yescomplemented
84C22⋊Dic163—22no—
85C22⋊Dic163—22no—
86(C2×D163)⋊C4—22no—
87(C2×D163)⋊C4—22no—
88(C2×D163)⋊C4—22no—
89(C2×D163)⋊C4—22no—
90(C2×D163)⋊C4—22no—
91C4×Dic163—22no—
92C4×Dic163—22no—
93C4×Dic163—22no—
94C24×C163C2311yescharacteristic
95C22×C4×C163C2311yescharacteristic
96C22×C4×C163C2311yescharacteristic
97C22×C4×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C4⋊C4×C163—22no—
102C22⋊C4×C163—22no—
103C22⋊C4×C163—22no—
104C22⋊C4×C163—22no—
105C22×D163—21no—
106C22×D163—21no—
107C22×D163—21no—
108C22×D163—21no—
109C22×D163—21no—
110C22×D163—21no—
111C22×D163#4112yes—
112Dic326—44no—
113C22⋊D163—28no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—28no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163C2×D412yes—
131C2×Dic163C2×D412yes—
132C2×Dic163C2×D412yes—
133C2×Dic163C2×D412yes—
134C2×Dic163#4112yes—
135C2×Dic163#4112yes—
136C4×D163—44no—
137C23×C163—21no—
138C23×C163—21no—
139C23×C163—21no—
140C23×C163—21no—
141C23×C163—21no—
142C23×C163—21no—
143C23×C163C2×D412yes—
144C23×C163C2411yescharacteristic
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—21no—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163C2×D412yes—
163C22×C163—22no—
164C22×C163—22no—
165C22×C163—22no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163—41no—
170C22×C163C22×D411yescharacteristic
171C22×C163C22×D411yescharacteristic
172C22×C163C2×#4111yescharacteristic
173C22×C163C2×#4111yescharacteristic
174C22×C163C2×#4111yescharacteristic
175C22×C1632−1+411yescharacteristic
176C22×C1632−1+411yescharacteristic
177C2×D163—22no—
178C2×D163—22no—
179C2×D163—22no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
183C2×D163—41no—
184C4×C163—22no—
185C4×C163—41no—
186C4×C163—41no—
187C4×C163—41no—
188C4×C163—41no—
189C4×C163—41no—
190Dic163—22no—
191Dic163—22no—
192Dic163—22no—
193Dic163—41no—
194Dic163—41no—
195Dic163—41no—
196Dic163#12114yes—
197C2×C163—42no—
198C2×C163C2×#12111yescharacteristic
199C2×C163#52511yescharacteristic
200C2×C163#53611yescharacteristic
201C2×C163#54711yescharacteristic
202C2×C163#54811yescharacteristic
203C2×C163#55211yescharacteristic
204C2×C163#55611yescharacteristic
205D163—42no—
206C163#241511yescharacteristic, complemented, Sylow
207#2415—1631nomaximal, Sylow
208C2×C22⋊Q8—1631no—
209C2×C22⋊Q8—1631no—
210C2×C4×D4—1631no—
211C2×#117—1631no—
212#397—1631no—
213#397D16311yescharacteristic, complemented
214#394—1631no—
215#394—1631no—
216#391—1631no—
217#389—1631no—
218#389—1631no—
219#389—1631no—
220#388—1631no—
221#388—1631no—
222#380—1631no—
223C22×Q8—1631no—
224C22×D4—1631no—
225C23×C4—1631no—
226C23×C4—1631no—
227C22⋊Q8—3264no—
228C22⋊Q8—3264no—
229C4×D4—3264no—
230#117—3264no—
231C2×C4⋊C4—1631no—
232C2×C4⋊C4—1631no—
233C2×C4⋊C4—1631no—
234C2×C4⋊C4—1631no—
235C2×C4⋊C4—1631no—
236C2×C4⋊C4—1631no—
237C2×C4⋊C4—1631no—
238C2×C4⋊C4—1631no—
239C2×C4⋊C4—1631no—
240C2×C4⋊C4C2×D16311yescharacteristic
241C2×C22⋊C4—1631no—
242C2×C22⋊C4—1631no—
243C2×C22⋊C4—1631no—
244C2×C22⋊C4—1631no—
245C2×C22⋊C4—1631no—
246C2×C22⋊C4—1631no—
247C2×C22⋊C4—1631no—
248C2×C22⋊C4C2×D16311yescharacteristic
249C2×C22⋊C4C2×D16311yescharacteristic
250C2×C22⋊C4C2×D16311yescharacteristic
251C2×C42—1631no—
252C2×C42—1631no—
253C2×C42—1631no—
254#95—1631no—
255#95—1631no—
256#95—1631no—
257#95—1631no—
258#95—1631no—
259#95C2×D16311yescharacteristic
260#95C2×D16311yescharacteristic
261#95C2×D16311yescharacteristic
262C24—1631no—
263C24C22×D16311yescharacteristic
264C2×Q8—3262no—
265C2×Q8—3262no—
266C2×Q8—3262no—
267C2×D4—3262no—
268C2×D4—3262no—
269C2×D4—3262no—
270C22×C4—3262no—
271C22×C4—3262no—
272C22×C4—3262no—
273C22×C4—3262no—
274C22×C4—3262no—
275C22×C4—1631no—
276C22×C4—1631no—
277C22×C4—1631no—
278C22×C4—1634no—
279C22×C4—1631no—
280C22×C4—1631no—
281C22×C4—1631no—
282C22×C4—1631no—
283C22×C4C22×D16311yescharacteristic
284C22×C4C22×D16311yescharacteristic
285C22×C4C22×D16311yescharacteristic
286C22×C4C22×D16311yescharacteristic
287C22×C4C22×D16311yescharacteristic
288C22×C4C22×D16311yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—3262no—
291C4⋊C4—3262no—
292C4⋊C4—3262no—
293C4⋊C4—3262no—
294C4⋊C4—3262no—
295C4⋊C4—3262no—
296C4⋊C4—3262no—
297C4⋊C4—3262no—
298C4⋊C4—1634no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—3262no—
303C22⋊C4—3262no—
304C22⋊C4—3262no—
305C22⋊C4—3262no—
306C22⋊C4—3262no—
307C22⋊C4—3262no—
308C22⋊C4—3262no—
309C42—3262no—
310C42—3262no—
311C42—3262no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—3261no—
319C23—3261no—
320C23—3261no—
321C23—3261no—
322C23—3261no—
323C23—3261no—
324C23—1632no—
325C23D4×D16312yes—
326C23C23×D16311yescharacteristic
327Q8—6524no—
328D4—3268no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—3261no—
351C2×C4—3261no—
352C2×C4—3268no—
353C2×C4—3261no—
354C2×C4—3261no—
355C2×C4—3261no—
356C2×C4—3261no—
357C2×C4—3261no—
358C2×C4—3261no—
359C2×C4—3261no—
360C2×C4—3261no—
361C2×C4—3261no—
362C2×C4—6524no—
363C2×C4—1632no—
364C2×C4—1632no—
365C2×C4—1632no—
366C2×C4—1632no—
367C2×C4—1632no—
368C2×C4—1632no—
369C2×C4D4×D16312yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—3262no—
378C22—3262no—
379C22—3262no—
380C22—6521no—
381C22—6521no—
382C22—6521no—
383C22—6521no—
384C22C2×D4×D16311yescharacteristic
385C22C2×D4×D16311yescharacteristic
386C22#41×D16311yescharacteristic
387C22#41×D16311yescharacteristic
388C22#41×D16311yescharacteristic
389C22#1910478211yescharacteristic
390C22#1910478211yescharacteristic
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—3262no—
398C4—3262no—
399C4—3262no—
400C4—6521no—
401C4—6521no—
402C4—6521no—
403C4—1634no—
404C2—42no—
405C2—6522no—
406C2#121×D16311yescharacteristic
407C2#2378217311yescharacteristic
408C2#2378223711yescharacteristic
409C2#2378231311yescharacteristic
410C2#2378232311yescharacteristic
411C2#2378235111yescharacteristic
412C2#2378238211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#397×C163C2
prime radicalC22×C22⋊D163C22
hypercenter#397D163
hypocenterC163#2415
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#19104713C2semidirect
2. C2×#19104712C2semidirect
4. C2×#19104691C2semidirect
5. #23781593C2semidirect
8. #23781569C2semidirect
9. #23781562C2semidirect
10. #23781562C2semidirect
11. #23781561C2semidirect
12. #23781557C2semidirect
13. #23781554C2semidirect
14. #23781524C2semidirect
15. #397×C163C2semidirect
83. Dic163⋊C4D4semidirect
206. C163#2415semidirect
213. #397D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047132
C2×#191047122
C2×(C22⋊D163)⋊C42
C2×#191046912
#237815932
#237815812
#237815802
#237815692
#237815622
#237815612
#237815572
#237815542
#237815242
#397×C1632
#2415163