Finite Group Databasebeta

← #35217794

Subgroups — #35217794

Order 20864.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic
2C2×#19104691C211yesmaximal, characteristic, complemented
3#23781606C211yesmaximal, characteristic, complemented
4#23781600C211yesmaximal, characteristic, complemented
5#23781599C211yesmaximal, characteristic, complemented
6#23781594C211yesmaximal, characteristic, complemented
7#23781593C211yesmaximal, characteristic, complemented
8#23781581C211yesmaximal, characteristic
9#23781580C211yesmaximal, characteristic
10C4⋊C4×Dic163C211yesmaximal, characteristic, complemented
11#23781516C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic163C211yesmaximal, characteristic, complemented
13#23781512C211yesmaximal, characteristic, complemented
14#23781512C211yesmaximal, characteristic, complemented
15#397×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163C2211yescharacteristic
17C23×Dic163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19(C22⋊D163)⋊C4—24no—
20#19104691—24no—
21C2×C4⋊Dic163C2211yescharacteristic
22C2×Dic163⋊C4C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
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
37C2×C4×Dic163C2211yescharacteristic
38C2×C4×Dic163C2211yescharacteristic
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
50#95×C163C2211yescharacteristic
51#95×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53C23×D163C2311yescharacteristic
54C2×C22⋊D163—22no—
55C2×C22⋊D163—22no—
56C2×C22⋊D163—22no—
57C22×Dic163—22no—
58C22×Dic163—22no—
59C22×Dic163D414yes—
60C22×Dic163C2311yescharacteristic
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C2×C4×D163—22no—
68C2×C4×D163—22no—
69C2×C4×D163—22no—
70C4⋊Dic163—22no—
71Dic163⋊C4—22no—
72Dic163⋊C4—22no—
73Dic163⋊C4—22no—
74Dic163⋊C4—22no—
75Dic163⋊C4—22no—
76C22⋊Dic163—22no—
77C22⋊Dic163—22no—
78(C2×D163)⋊C4—22no—
79(C2×D163)⋊C4—22no—
80(C2×D163)⋊C4—22no—
81(C2×D163)⋊C4—22no—
82(C2×D163)⋊C4—22no—
83C4×Dic163—22no—
84C4×Dic163—22no—
85C4×Dic163—22no—
86C4×Dic163—22no—
87C4×Dic163—22no—
88C24×C163C2311yescharacteristic
89C22×C4×C163C2311yescharacteristic
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C22×C4×C163C2311yescharacteristic
93C22×C4×C163C2311yescharacteristic
94C22×C4×C163C2311yescharacteristic
95C4⋊C4×C163—22no—
96C22⋊C4×C163—22no—
97C22⋊C4×C163—22no—
98C22⋊C4×C163—22no—
99C22×D163—21no—
100C22×D163—21no—
101C22×D163—21no—
102C22×D163—21no—
103C22×D163—21no—
104C22×D163—21no—
105C22×D163#4112yes—
106C22⋊D163—28no—
107C2×Dic163—21no—
108C2×Dic163—21no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—28no—
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×Dic163C2×D412yes—
124C2×Dic163C2×D412yes—
125C2×Dic163#4112yes—
126C2×Dic163#4112yes—
127C2×Dic163#4112yes—
128C2×Dic163#4112yes—
129C4×D163—44no—
130C23×C163—21no—
131C23×C163—21no—
132C23×C163—21no—
133C23×C163—21no—
134C23×C163—21no—
135C23×C163—21no—
136C23×C163C2×D412yes—
137C23×C163C2411yescharacteristic
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—21no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163#4112yes—
156C22×C163—22no—
157C22×C163—22no—
158C22×C163—22no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163C22×D411yescharacteristic
164C22×C163C2×#4111yescharacteristic
165C22×C163C2×#4111yescharacteristic
166C22×C163C2×#4111yescharacteristic
167C22×C163C2×#4111yescharacteristic
168C22×C163C2×#4111yescharacteristic
169C22×C163C2×#4111yescharacteristic
170C2×D163—22no—
171C2×D163—22no—
172C2×D163—22no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—41no—
176C2×D163—41no—
177C4×C163—22no—
178C4×C163—41no—
179C4×C163—41no—
180C4×C163—41no—
181C4×C163—41no—
182C4×C163—41no—
183Dic163—22no—
184Dic163—22no—
185Dic163—22no—
186Dic163—41no—
187Dic163—41no—
188Dic163—41no—
189Dic163#12314yes—
190C2×C163—42no—
191C2×C163C2×#12311yescharacteristic
192C2×C163#52511yescharacteristic
193C2×C163#52911yescharacteristic
194C2×C163#52911yescharacteristic
195C2×C163#52911yescharacteristic
196C2×C163#52911yescharacteristic
197C2×C163#53211yescharacteristic
198D163—42no—
199C163#239511yescharacteristic, complemented, Sylow
200#2395—1631nomaximal, Sylow
201C2×C4×D4—1631no—
202C2×#117—1631no—
203#400—1631no—
204#399—1631no—
205#398—1631no—
206#397—1631no—
207#397—1631no—
208#397D16311yescharacteristic, complemented
209#394—1631no—
210#394—1631no—
211C4×C4⋊C4—1631no—
212C4×C4⋊C4—1631no—
213C4×C22⋊C4—1631no—
214C4×C22⋊C4—1631no—
215C4×C22⋊C4—1631no—
216C22×D4—1631no—
217C23×C4—1631no—
218C23×C4—1631no—
219C4×D4—3264no—
220#117—3264no—
221C2×C4⋊C4—1631no—
222C2×C4⋊C4—1631no—
223C2×C4⋊C4—1631no—
224C2×C4⋊C4—1631no—
225C2×C4⋊C4—1631no—
226C2×C4⋊C4—1631no—
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⋊C4C2×D16311yescharacteristic
236C2×C22⋊C4C2×D16311yescharacteristic
237C2×C22⋊C4C2×D16311yescharacteristic
238C2×C42—1631no—
239C2×C42—1631no—
240C2×C42—1631no—
241C2×C42—1631no—
242C2×C42—1631no—
243#95—1631no—
244#95—1631no—
245#95—1631no—
246#95—1631no—
247#95—1631no—
248#95—1631no—
249#95—1631no—
250#95C2×D16311yescharacteristic
251#95C2×D16311yescharacteristic
252#95C2×D16311yescharacteristic
253C24—1631no—
254C24C22×D16311yescharacteristic
255C2×D4—3262no—
256C2×D4—3262no—
257C2×D4—3262no—
258C22×C4—3262no—
259C22×C4—3262no—
260C22×C4—3262no—
261C22×C4—3262no—
262C22×C4—3262no—
263C22×C4—1631no—
264C22×C4—1631no—
265C22×C4—1631no—
266C22×C4—1634no—
267C22×C4—1631no—
268C22×C4—1631no—
269C22×C4—1631no—
270C22×C4—1631no—
271C22×C4C22×D16311yescharacteristic
272C22×C4C22×D16311yescharacteristic
273C22×C4C22×D16311yescharacteristic
274C22×C4C22×D16311yescharacteristic
275C22×C4C22×D16311yescharacteristic
276C22×C4C22×D16311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—3262no—
279C4⋊C4—3262no—
280C4⋊C4—3262no—
281C4⋊C4—3262no—
282C4⋊C4—3262no—
283C4⋊C4—3262no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—3262no—
288C22⋊C4—3262no—
289C22⋊C4—3262no—
290C22⋊C4—3262no—
291C22⋊C4—3262no—
292C22⋊C4—3262no—
293C22⋊C4—3262no—
294C42—3262no—
295C42—3262no—
296C42—3262no—
297C42—3262no—
298C42—3262no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—3261no—
306C23—3261no—
307C23—3261no—
308C23—3261no—
309C23—3261no—
310C23—3261no—
311C23—1632no—
312C23D4×D16312yes—
313C23C23×D16311yescharacteristic
314D4—3268no—
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—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—3261no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—3261no—
338C2×C4—3268no—
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×C4—1632no—
353C2×C4—1632no—
354C2×C4—1632no—
355C2×C4#1457969812yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—3262no—
364C22—3262no—
365C22—3262no—
366C22—6521no—
367C22—6521no—
368C22—6521no—
369C22—6521no—
370C22C2×D4×D16311yescharacteristic
371C22#41×D16311yescharacteristic
372C22#41×D16311yescharacteristic
373C22#41×D16311yescharacteristic
374C22#41×D16311yescharacteristic
375C22#41×D16311yescharacteristic
376C22C2×#1457969811yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—3262no—
384C4—3262no—
385C4—3262no—
386C4—6521no—
387C4—6521no—
388C4—6521no—
389C4—1634no—
390C2—42no—
391C2—6522no—
392C2#123×D16311yescharacteristic
393C2#2378217311yescharacteristic
394C2#2378219211yescharacteristic
395C2#2378219411yescharacteristic
396C2#2378219811yescharacteristic
397C2#2378219811yescharacteristic
398C2#2378221011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#397×C163C2
prime radicalC22×C22⋊D163C22
hypercenter#397D163
hypocenterC163#2395
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#19104691C2semidirect
3. #23781606C2semidirect
4. #23781600C2semidirect
5. #23781599C2semidirect
6. #23781594C2semidirect
7. #23781593C2semidirect
10. C4⋊C4×Dic163C2semidirect
11. #23781516C2semidirect
12. C22⋊C4×Dic163C2semidirect
13. #23781512C2semidirect
14. #23781512C2semidirect
15. #397×C163C2semidirect
199. C163#2395semidirect
208. #397D163semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D163)⋊C42
C2×#191046912
#237816062
#237816002
#237815992
#237815942
#237815932
#237815812
#237815802
C4⋊C4×Dic1632
#237815162
C22⋊C4×Dic1632
#237815122
#397×C1632
#2395163