Finite Group Databasebeta

← #35217604

Subgroups — #35217604

Order 20864.

632 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104721C211yesmaximal, characteristic
2C2×#19104700C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D163C211yesmaximal, characteristic
4#23781569C211yesmaximal, characteristic, complemented
5#23781565C211yesmaximal, characteristic
6#23781558C211yesmaximal, characteristic, complemented
7#23781557C211yesmaximal, characteristic, complemented
8#23781555C211yesmaximal, characteristic, complemented
9#23781553C211yesmaximal, characteristic, complemented
10#23781536C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic163C211yesmaximal, characteristic, complemented
12#23781516C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic163C211yesmaximal, characteristic, complemented
14#23781512C211yesmaximal, characteristic, complemented
15#385×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19#19104721C418yescomplemented
20#19104700—24no—
21C4×C22⋊D163—24no—
22C2×C4⋊Dic163C2211yescharacteristic
23C2×Dic163⋊C4C2211yescharacteristic
24C2×Dic163⋊C4C2211yescharacteristic
25C2×Dic163⋊C4C2211yescharacteristic
26C2×C22⋊Dic163C2211yescharacteristic
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
39C2×C4×Dic163C2211yescharacteristic
40C2×C4×Dic163C2211yescharacteristic
41C163⋊#95C2211yescharacteristic
42C163⋊#95C2211yescharacteristic
43C163⋊#95C2211yescharacteristic
44C163⋊#95C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C23×C4×C163C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic
49C2×C4⋊C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163C2211yescharacteristic
51C2×C22⋊C4×C163C2211yescharacteristic
52#95×C163C2211yescharacteristic
53#95×C163C2211yescharacteristic
54C23×D163C2311yescharacteristic
55C2×Dic326—22no—
56C2×Dic326—22no—
57C2×Dic326C2×C414yescomplemented
58C2×C22⋊D163—22no—
59C2×C22⋊D163—22no—
60C2×C22⋊D163C2×C414yes—
61C22×Dic163C2311yescharacteristic
62C22×Dic163C2311yescharacteristic
63C22×Dic163C2311yescharacteristic
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C2×C4×D163—22no—
69C2×C4×D163—22no—
70C2×C4×D163—41no—
71C4⋊Dic163—22no—
72Dic163⋊C4—22no—
73Dic163⋊C4—22no—
74Dic163⋊C4D414yes—
75C22⋊Dic163—22no—
76C22⋊Dic163—22no—
77C22⋊Dic163C2×C414yescomplemented
78(C2×D163)⋊C4—22no—
79(C2×D163)⋊C4—22no—
80(C2×D163)⋊C4—41no—
81(C2×D163)⋊C4C2×C414yescomplemented
82(C2×D163)⋊C4C2×C414yescomplemented
83C4×Dic163—22no—
84C4×Dic163—22no—
85C4×Dic163—22no—
86C4×Dic163—22no—
87C4×Dic163—41no—
88C4×Dic163—41no—
89C4×Dic163C2×C414yescomplemented
90C24×C163C2311yescharacteristic
91C22×C4×C163—22no—
92C22×C4×C163—22no—
93C22×C4×C163—41no—
94C22×C4×C163C2311yescharacteristic
95C22×C4×C163C2311yescharacteristic
96C22×C4×C163C2311yescharacteristic
97C22×C4×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C4⋊C4×C163—22no—
101C4⋊C4×C163—22no—
102C22⋊C4×C163—41no—
103C22⋊C4×C163C2×C414yescomplemented
104C22×D163—21no—
105C22×D163—21no—
106C22×D163—21no—
107C22×D163—21no—
108C22×D163—21no—
109C22×D163—21no—
110C22×D163C22×C412yes—
111Dic326—28no—
112C22⋊D163—28no—
113C2×Dic163—21no—
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×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163C22×C412yes—
128C2×Dic163C22×C412yes—
129C2×Dic163C22×C412yes—
130C2×Dic163C2×D412yes—
131C2×Dic163C2×D412yes—
132C2×Dic163#4112yes—
133C2×Dic163#4112yes—
134C4×D163—44no—
135C23×C163—21no—
136C23×C163—21no—
137C23×C163—21no—
138C23×C163—21no—
139C23×C163—21no—
140C23×C163—21no—
141C23×C163C22×C412yes—
142C23×C163C2411yescharacteristic
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—21no—
156C2×C4×C163—21no—
157C2×C4×C163—44no—
158C2×C4×C163C22×C412yes—
159C2×C4×C163C22×C412yes—
160C2×C4×C163C2×D412yes—
161C2×C4×C163#4112yes—
162C22×C163—22no—
163C22×C163—22no—
164C22×C163—22no—
165C22×C163—41no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163C23×C411yescharacteristic
170C22×C163C22×D411yescharacteristic
171C22×C163C2×#4111yescharacteristic
172C22×C163C2×#4111yescharacteristic
173C22×C163C2×#4111yescharacteristic
174C22×C163H(5,𝔽2)11yescharacteristic
175C22×C1632−1+411yescharacteristic
176C2×D163—22no—
177C2×D163—22no—
178C2×D163—22no—
179C2×D163—41no—
180C2×D163—41no—
181C2×D163—41no—
182C2×D163—41no—
183C4×C163—22no—
184C4×C163—22no—
185C4×C163—22no—
186C4×C163—22no—
187C4×C163—41no—
188C4×C163—41no—
189Dic163—22no—
190Dic163—22no—
191Dic163—22no—
192Dic163—22no—
193Dic163—41no—
194Dic163—41no—
195Dic163C4×D414yes—
196C2×C163—42no—
197C2×C163C2×C4×D411yescharacteristic
198C2×C163C4×#4111yescharacteristic
199C2×C163#52011yescharacteristic
200C2×C163#54611yescharacteristic
201C2×C163#54811yescharacteristic
202C2×C163#55611yescharacteristic
203C2×C163#55911yescharacteristic
204D163—42no—
205C163#236711yescharacteristic, complemented, Sylow
206#2367—1631nomaximal, Sylow
207C2×#124—1631no—
208C2×C4×Q8—1631no—
209C2×C4×D4—1631no—
210#391—1631no—
211#390—1631no—
212#388—1631no—
213#388—1631no—
214#388—1631no—
215#388—1631no—
216#385—1631no—
217#385D16311yescharacteristic, complemented
218C4×C4⋊C4—1631no—
219C4×C4⋊C4—1631no—
220C4×C22⋊C4—1631no—
221C4×C22⋊C4—1631no—
222C22×Q8—1631no—
223C22×D4—1631no—
224C23×C4—1631no—
225C23×C4C2×D16311yescharacteristic
226#124—1638no—
227C4×Q8—3264no—
228C4×D4—3264no—
229C2×C4⋊C4—1631no—
230C2×C4⋊C4—1631no—
231C2×C4⋊C4—1631no—
232C2×C4⋊C4—1631no—
233C2×C4⋊C4C2×D16311yescharacteristic
234C2×C4⋊C4C2×D16311yescharacteristic
235C2×C22⋊C4—1631no—
236C2×C22⋊C4—1631no—
237C2×C22⋊C4—1631no—
238C2×C22⋊C4—1631no—
239C2×C22⋊C4—1631no—
240C2×C22⋊C4—1631no—
241C2×C22⋊C4—1631no—
242C2×C22⋊C4—1631no—
243C2×C22⋊C4C2×D16311yescharacteristic
244C2×C22⋊C4C2×D16311yescharacteristic
245C2×C42—1631no—
246C2×C42—1631no—
247C2×C42—1631no—
248C2×C42—1631no—
249C2×C42—1631no—
250C2×C42—1631no—
251C2×C42—1631no—
252#95—1631no—
253#95—1631no—
254#95—1631no—
255#95—1631no—
256#95—1631no—
257#95—1631no—
258#95C2×D16311yescharacteristic
259#95C2×D16311yescharacteristic
260C24—1631no—
261C24C22×D16311yescharacteristic
262C2×Q8—3262no—
263C2×Q8—3262no—
264C2×Q8—1634no—
265C2×D4—3262no—
266C2×D4—3262no—
267C2×D4—1634no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—3262no—
272C22×C4—3262no—
273C22×C4—6521no—
274C22×C4—1631no—
275C22×C4—1631no—
276C22×C4—1631no—
277C22×C4—1631no—
278C22×C4—1631no—
279C22×C4—1631no—
280C22×C4—1631no—
281C22×C4C22×D16311yescharacteristic
282C22×C4C22×D16311yescharacteristic
283C22×C4C22×D16311yescharacteristic
284C22×C4C22×D16311yescharacteristic
285C22×C4C22×D16311yescharacteristic
286C22×C4C22×D16311yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—3262no—
290C4⋊C4—3262no—
291C4⋊C4—3262no—
292C4⋊C4—1634no—
293C22⋊C4—41no—
294C22⋊C4—3262no—
295C22⋊C4—3262no—
296C22⋊C4—3262no—
297C22⋊C4—3262no—
298C22⋊C4—6521no—
299C22⋊C4—1634no—
300C22⋊C4—1634no—
301C22⋊C4—1634no—
302C22⋊C4C4×D16314yescomplemented
303C42—3262no—
304C42—3262no—
305C42—3262no—
306C42—3262no—
307C42—6521no—
308C42—6521no—
309C42—1634no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—3261no—
317C23—3261no—
318C23—3261no—
319C23—3261no—
320C23—3261no—
321C23—3261no—
322C23—1632no—
323C23C2×C4×D16312yes—
324C23C23×D16311yescharacteristic
325Q8—3268no—
326D4—3268no—
327C2×C4—21no—
328C2×C4—21no—
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—44no—
342C2×C4—3261no—
343C2×C4—3261no—
344C2×C4—3261no—
345C2×C4—3261no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—3261no—
351C2×C4—3261no—
352C2×C4—3261no—
353C2×C4—3261no—
354C2×C4—3261no—
355C2×C4—3261no—
356C2×C4—6524no—
357C2×C4—1632no—
358C2×C4—1632no—
359C2×C4—1632no—
360C2×C4—1632no—
361C2×C4—1632no—
362C2×C4—1632no—
363C2×C4—1632no—
364C2×C4C2×C4×D16312yes—
365C2×C4C2×C4×D16312yes—
366C2×C4D4×D16312yes—
367C2×C4#1457969812yes—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—3262no—
376C22—3262no—
377C22—3262no—
378C22—6521no—
379C22—6521no—
380C22—6521no—
381C22—6521no—
382C22C22×C4×D16311yescharacteristic
383C22C2×D4×D16311yescharacteristic
384C22#41×D16311yescharacteristic
385C22#41×D16311yescharacteristic
386C22C2×#1457969811yescharacteristic
387C22#1910478011yescharacteristic
388C22#1910478311yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—3262no—
396C4—3262no—
397C4—3262no—
398C4—3262no—
399C4—6521no—
400C4—6521no—
401C4—1634no—
402C2—42no—
403C2—6522no—
404C2C4×D4×D16311yescharacteristic
405C2C4×#1457969811yescharacteristic
406C2#2378214611yescharacteristic
407C2#2378230211yescharacteristic
408C2#2378232511yescharacteristic
409C2#2378238611yescharacteristic
410C2#2378239911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutatorC22×C163C23×C4
Fitting#385×C163C2
socle, cosocleC23×C163C24
prime radicalC22×C22⋊D163C22
hypercenter#385D163
hypocenterC163#2367
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#19104700C2semidirect
4. #23781569C2semidirect
6. #23781558C2semidirect
7. #23781557C2semidirect
8. #23781555C2semidirect
9. #23781553C2semidirect
10. #23781536C2semidirect
11. C4⋊C4×Dic163C2semidirect
12. #23781516C2semidirect
13. C4×C22⋊Dic163C2semidirect
14. #23781512C2semidirect
15. #385×C163C2semidirect
19. #19104721C4semidirect
57. C2×Dic326C2×C4semidirect
77. C22⋊Dic163C2×C4semidirect
81. (C2×D163)⋊C4C2×C4semidirect
82. (C2×D163)⋊C4C2×C4semidirect
89. C4×Dic163C2×C4semidirect
103. C22⋊C4×C163C2×C4semidirect
205. C163#2367semidirect
217. #385D163semidirect
302. C22⋊C4C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047212
C2×#191047002
C2×C4×C22⋊D1632
#237815692
#237815652
#237815582
#237815572
#237815552
#237815532
#237815362
C4⋊C4×Dic1632
#237815162
C4×C22⋊Dic1632
#237815122
#385×C1632
#2367163