Finite Group Databasebeta

← #35218197

Subgroups — #35218197

Order 20864.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104714C211yesmaximal, characteristic, complemented
2C2×#19104712C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic163C211yesmaximal, characteristic
4#23781601C211yesmaximal, characteristic, complemented
5#23781588C211yesmaximal, characteristic, complemented
6#23781588C211yesmaximal, characteristic, complemented
7#23781577C211yesmaximal, characteristic, complemented
8#23781540C211yesmaximal, characteristic, complemented
9#23781538C211yesmaximal, characteristic, complemented
10#23781538C211yesmaximal, characteristic, complemented
11#23781537C211yesmaximal, characteristic, complemented
12#23781525C211yesmaximal, characteristic, complemented
13#23781523C211yesmaximal, characteristic, complemented
14#23781521C211yesmaximal, characteristic
15C22×C22⋊C4×C163C211yesmaximal, characteristic
16C22×Dic326C2211yescharacteristic, complemented
17C23×Dic163C2211yescharacteristic
18C23×Dic163C2211yescharacteristic
19#19104714—24no—
20#19104712—24no—
21C2×C4⋊Dic163C2211yescharacteristic, complemented
22C2×C4⋊Dic163C2211yescharacteristic, complemented
23C2×C4⋊Dic163C2211yescharacteristic, complemented
24C2×Dic163⋊C4C2211yescharacteristic, complemented
25C2×Dic163⋊C4C2211yescharacteristic, complemented
26C2×Dic163⋊C4C2211yescharacteristic, complemented
27C2×Dic163⋊C4C2211yescharacteristic, complemented
28C2×Dic163⋊C4C2211yescharacteristic, complemented
29C2×Dic163⋊C4C2211yescharacteristic, complemented
30C2×C22⋊Dic163—24no—
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163C2211yescharacteristic
38C2×C22⋊Dic163C2211yescharacteristic
39C2×C22⋊Dic163C2211yescharacteristic
40C2×C22⋊Dic163C2211yescharacteristic
41C163⋊#95C2211yescharacteristic, complemented
42C163⋊#95C2211yescharacteristic, complemented
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C25×C163C2211yescharacteristic
48C23×C4×C163C2211yescharacteristic
49C23×C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163—24no—
51C2×C22⋊C4×C163C2211yescharacteristic
52C2×C22⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54C2×C22⋊C4×C163C2211yescharacteristic
55C2×Dic326—22no—
56C2×Dic326—22no—
57C2×Dic326—41no—
58C22×Dic163—22no—
59C22×Dic163—22no—
60C22×Dic163—22no—
61C22×Dic163—22no—
62C22×Dic163—41no—
63C22×Dic163—41no—
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C4⋊Dic163—22no—
73C4⋊Dic163—22no—
74C4⋊Dic163—41no—
75Dic163⋊C4—22no—
76Dic163⋊C4—22no—
77Dic163⋊C4—22no—
78Dic163⋊C4—22no—
79Dic163⋊C4—41no—
80Dic163⋊C4—41no—
81C22⋊Dic163—22no—
82C22⋊Dic163—22no—
83C22⋊Dic163—22no—
84C22⋊Dic163—22no—
85C22⋊Dic163—22no—
86C22⋊Dic163—22no—
87C22⋊Dic163—22no—
88C22⋊Dic163—41no—
89C22⋊Dic163—41no—
90C22⋊Dic163D414yes—
91C24×C163—22no—
92C24×C163—22no—
93C24×C163—22no—
94C24×C163—22no—
95C24×C163—41no—
96C24×C163—41no—
97C24×C163Q814yescomplemented
98C24×C163C2311yescharacteristic
99C24×C163C2311yescharacteristic
100C24×C163C2311yescharacteristic
101C22×C4×C163—22no—
102C22×C4×C163—22no—
103C22×C4×C163—22no—
104C22×C4×C163—22no—
105C22×C4×C163—41no—
106C22×C4×C163—41no—
107C22×C4×C163C2311yescharacteristic
108C22×C4×C163C2311yescharacteristic
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111C22⋊C4×C163—22no—
112C22⋊C4×C163—22no—
113C22⋊C4×C163—22no—
114C22⋊C4×C163D414yes—
115Dic326—44no—
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×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163—21no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—44no—
139C2×Dic163—44no—
140C2×Dic163C2×D412yes—
141C2×Dic163C2×D412yes—
142C23×C163—21no—
143C23×C163—21no—
144C23×C163—21no—
145C23×C163—21no—
146C23×C163—21no—
147C23×C163—21no—
148C23×C163—21no—
149C23×C163—21no—
150C23×C163—28no—
151C23×C163—28no—
152C23×C163—21no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163—21no—
156C23×C163—21no—
157C23×C163—21no—
158C23×C163—44no—
159C23×C163—44no—
160C23×C163—44no—
161C23×C163—44no—
162C23×C163—44no—
163C23×C163C2×D412yes—
164C23×C163C2×D412yes—
165C23×C163C2×Q812yes—
166C23×C163C2×Q812yes—
167C23×C163C2×Q812yes—
168C23×C163#4112yes—
169C23×C163#4112yes—
170C23×C163C2411yescharacteristic
171C2×C4×C163—21no—
172C2×C4×C163—21no—
173C2×C4×C163—21no—
174C2×C4×C163—21no—
175C2×C4×C163—21no—
176C2×C4×C163—21no—
177C2×C4×C163—21no—
178C2×C4×C163—21no—
179C2×C4×C163—21no—
180C2×C4×C163—21no—
181C2×C4×C163—44no—
182C2×C4×C163—44no—
183C2×C4×C163C2×D412yes—
184C2×C4×C163C2×D412yes—
185C22×C163—22no—
186C22×C163—22no—
187C22×C163—22no—
188C22×C163—22no—
189C22×C163—22no—
190C22×C163—22no—
191C22×C163—22no—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—22no—
195C22×C163—22no—
196C22×C163—22no—
197C22×C163—22no—
198C22×C163—22no—
199C22×C163—22no—
200C22×C163—416no—
201C22×C163—41no—
202C22×C163—41no—
203C22×C163—41no—
204C22×C163—41no—
205C22×C163C22⋊Q814yescomplemented
206C22×C163C22⋊Q814yescomplemented
207C22×C163C22×D411yescharacteristic
208C22×C163C22×D411yescharacteristic
209C22×C163C22×Q811yescharacteristic
210C22×C163C2×#4111yescharacteristic
211C22×C163C2×#4111yescharacteristic
212C22×C163H(5,𝔽2)11yescharacteristic
213C22×C163H(5,𝔽2)11yescharacteristic
214C4×C163—22no—
215C4×C163—22no—
216C4×C163—41no—
217C4×C163—41no—
218Dic163—22no—
219Dic163—22no—
220Dic163—41no—
221Dic163—41no—
222Dic163—41no—
223Dic163—41no—
224Dic163—41no—
225Dic163—41no—
226C2×C163—24no—
227C2×C163—24no—
228C2×C163—42no—
229C2×C163C2×C22⋊Q811yescharacteristic
230C2×C163C2×C22⋊Q811yescharacteristic
231C2×C163#54311yescharacteristic
232C2×C163D4211yescharacteristic
233C2×C163#54611yescharacteristic
234C2×C163#54611yescharacteristic
235C2×C163#55111yescharacteristic
236C163#243711yescharacteristic, complemented, Sylow
237#2437—1631nomaximal, Sylow
238C2×C22⋊Q8—1631no—
239C2×C22⋊Q8—1631no—
240C22×C22⋊C4—1631no—
241C22×C22⋊C4D16311yescharacteristic
242#399—1631no—
243#396—1631no—
244#396—1631no—
245#393—1631no—
246#385—1631no—
247#385—1631no—
248#385—1631no—
249#385—1631no—
250#380—1631no—
251#380—1631no—
252#379—1631no—
253C25C2×D16311yescharacteristic
254C22×Q8—1631no—
255C23×C4—1631no—
256C23×C4—1631no—
257C23×C4C2×D16311yescharacteristic
258C23×C4C2×D16311yescharacteristic
259C22⋊Q8—3264no—
260C22⋊Q8—3264no—
261C2×C4⋊C4—1631no—
262C2×C4⋊C4—1631no—
263C2×C4⋊C4—1631no—
264C2×C4⋊C4—1631no—
265C2×C4⋊C4—1631no—
266C2×C4⋊C4—1631no—
267C2×C4⋊C4—1631no—
268C2×C4⋊C4—1631no—
269C2×C4⋊C4—1631no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—3264no—
272C2×C22⋊C4—1631no—
273C2×C22⋊C4—1631no—
274C2×C22⋊C4—1631no—
275C2×C22⋊C4—1631no—
276C2×C22⋊C4—1631no—
277C2×C22⋊C4—1631no—
278C2×C22⋊C4—1631no—
279C2×C22⋊C4—1631no—
280C2×C22⋊C4—1631no—
281C2×C22⋊C4—1631no—
282C2×C22⋊C4C2×D16311yescharacteristic
283C2×C22⋊C4C2×D16311yescharacteristic
284C2×C22⋊C4C2×D16311yescharacteristic
285C2×C22⋊C4C2×D16311yescharacteristic
286#95—1631no—
287#95—1631no—
288#95—1631no—
289#95—1631no—
290#95—1631no—
291#95—1631no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—41no—
297C24—41no—
298C24Dic32614yescomplemented
299C24C22×D16311yescharacteristic
300C24C22×D16311yescharacteristic
301C24C22×D16311yescharacteristic
302C2×Q8—3262no—
303C2×Q8—3262no—
304C2×Q8—6521no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—3262no—
312C22×C4—3262no—
313C22×C4—3262no—
314C22×C4—3262no—
315C22×C4—6521no—
316C22×C4—6521no—
317C22×C4—1631no—
318C22×C4—1631no—
319C22×C4—1631no—
320C22×C4—1631no—
321C22×C4—1631no—
322C22×C4—1631no—
323C22×C4—1631no—
324C22×C4—1631no—
325C22×C4C22×D16311yescharacteristic
326C22×C4C22×D16311yescharacteristic
327C22×C4C22×D16311yescharacteristic
328C22×C4C22×D16311yescharacteristic
329C4⋊C4—3262no—
330C4⋊C4—3262no—
331C4⋊C4—3262no—
332C4⋊C4—3262no—
333C4⋊C4—3262no—
334C4⋊C4—3262no—
335C4⋊C4—6521no—
336C4⋊C4—6521no—
337C4⋊C4—6521no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—3262no—
342C22⋊C4—3262no—
343C22⋊C4—3262no—
344C22⋊C4—3262no—
345C22⋊C4—3262no—
346C22⋊C4—3262no—
347C22⋊C4—3262no—
348C22⋊C4—6521no—
349C22⋊C4—6521no—
350C22⋊C4—1634no—
351C22⋊C4C22⋊D16314yes—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—28no—
367C23—28no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23D4×D16312yes—
374C23C2×C22⋊D16312yes—
375C23C2×Dic32612yes—
376C23C2×Dic32612yes—
377C23C2×Dic32612yes—
378C23#1457969812yes—
379C23#1457969912yes—
380C23C23×D16311yescharacteristic
381Q8—6524no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—3261no—
395C2×C4—3261no—
396C2×C4—3261no—
397C2×C4—3261no—
398C2×C4—3261no—
399C2×C4—3261no—
400C2×C4—3261no—
401C2×C4—3261no—
402C2×C4—3261no—
403C2×C4—3261no—
404C2×C4—3261no—
405C2×C4—3261no—
406C2×C4—3261no—
407C2×C4—3261no—
408C2×C4—3261no—
409C2×C4—3261no—
410C2×C4—3261no—
411C2×C4—3261no—
412C2×C4—3261no—
413C2×C4—3261no—
414C2×C4—3261no—
415C2×C4—3261no—
416C2×C4—6524no—
417C2×C4—6524no—
418C2×C4—1632no—
419C2×C4—1632no—
420C2×C4C2×C22⋊D16312yes—
421C2×C4C2×C22⋊D16312yes—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—416no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22#1910471214yescomplemented
443C22#1910471414yescomplemented
444C22C2×D4×D16311yescharacteristic
445C22C22×C22⋊D16311yescharacteristic
446C22C22×Dic32611yescharacteristic
447C22C2×#1457969811yescharacteristic
448C22C2×#1457969911yescharacteristic
449C22#1910478011yescharacteristic
450C22#1910478011yescharacteristic
451C4—22no—
452C4—22no—
453C4—41no—
454C4—41no—
455C4—3262no—
456C4—3262no—
457C4—6521no—
458C4—6521no—
459C4—6521no—
460C4—6521no—
461C4—6521no—
462C4—6521no—
463C2—24no—
464C2—24no—
465C2—42no—
466C2C2×#1910471211yescharacteristic
467C2C2×#1910471411yescharacteristic
468C2#2378229111yescharacteristic
469C2D4×C22⋊D16311yescharacteristic
470C2#2378230511yescharacteristic
471C2#2378230911yescharacteristic
472C2#2378234111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC22×C22⋊C4×C163C2
prime radicalC25×C163C22
hypercenterC22×C22⋊C4D163
hypocenterC163#2437
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19104714C2semidirect
2. C2×#19104712C2semidirect
4. #23781601C2semidirect
5. #23781588C2semidirect
6. #23781588C2semidirect
7. #23781577C2semidirect
8. #23781540C2semidirect
9. #23781538C2semidirect
10. #23781538C2semidirect
11. #23781537C2semidirect
12. #23781525C2semidirect
13. #23781523C2semidirect
16. C22×Dic326C22semidirect
21. C2×C4⋊Dic163C22semidirect
22. C2×C4⋊Dic163C22semidirect
23. C2×C4⋊Dic163C22semidirect
24. C2×Dic163⋊C4C22semidirect
25. C2×Dic163⋊C4C22semidirect
26. C2×Dic163⋊C4C22semidirect
27. C2×Dic163⋊C4C22semidirect
28. C2×Dic163⋊C4C22semidirect
29. C2×Dic163⋊C4C22semidirect
41. C163⋊#95C22semidirect
42. C163⋊#95C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
97. C24×C163Q8semidirect
205. C22×C163C22⋊Q8semidirect
206. C22×C163C22⋊Q8semidirect
236. C163#2437semidirect
298. C24Dic326semidirect
442. C22#19104712semidirect
443. C22#19104714semidirect

Maximal subgroups

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

subgroupindex
C2×#191047142
C2×#191047122
C22×C22⋊Dic1632
#237816012
#237815882
#237815772
#237815402
#237815382
#237815372
#237815252
#237815232
#237815212
C22×C22⋊C4×C1632
#2437163