Finite Group Databasebeta

← #35218097

Subgroups — #35218097

Order 20864.

632 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104721C211yesmaximal, characteristic, complemented
2C2×#19104713C211yesmaximal, characteristic, complemented
3C2×#19104708C211yesmaximal, characteristic
4C2×#19104700C211yesmaximal, characteristic, complemented
5#23781589C211yesmaximal, characteristic, complemented
6#23781582C211yesmaximal, characteristic, complemented
7#23781580C211yesmaximal, characteristic, complemented
8#23781566C211yesmaximal, characteristic, complemented
9#23781565C211yesmaximal, characteristic, complemented
10#23781565C211yesmaximal, characteristic, complemented
11#23781560C211yesmaximal, characteristic, complemented
12#23781555C211yesmaximal, characteristic, complemented
13#23781553C211yesmaximal, characteristic, complemented
14#23781512C211yesmaximal, characteristic, complemented
15#396×C163C211yesmaximal, characteristic, complemented
16C22×Dic326C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C22×C4×D163C2211yescharacteristic
21#19104721—24no—
22#19104713—24no—
23#19104708—24no—
24#19104700—24no—
25C2×Dic163⋊C4C2211yescharacteristic
26C2×Dic163⋊C4C2211yescharacteristic
27C2×Dic163⋊C4C2211yescharacteristic
28C2×Dic163⋊C4C2211yescharacteristic
29C2×Dic163⋊C4C2211yescharacteristic
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
32C2×(C2×D163)⋊C4C2211yescharacteristic
33C2×(C2×D163)⋊C4C2211yescharacteristic
34C2×(C2×D163)⋊C4C2211yescharacteristic
35C2×(C2×D163)⋊C4C2211yescharacteristic
36C2×(C2×D163)⋊C4C2211yescharacteristic
37C2×(C2×D163)⋊C4C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×(C2×D163)⋊C4C2211yescharacteristic
41C2×C4×Dic163C2211yescharacteristic
42C2×C4×Dic163C2211yescharacteristic
43C2×C4×Dic163C2211yescharacteristic
44C2×C4×Dic163C2211yescharacteristic
45C163⋊#95C2211yescharacteristic
46C163⋊#95C2211yescharacteristic
47C163⋊#95C2211yescharacteristic
48C2×C4⋊C4×C163C2211yescharacteristic
49C2×C4⋊C4×C163C2211yescharacteristic
50C2×C4⋊C4×C163C2211yescharacteristic
51C2×C22⋊C4×C163C2211yescharacteristic
52C2×C22⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54#95×C163C2211yescharacteristic
55C23×D163C2311yescharacteristic
56C23×D163C2311yescharacteristic
57C2×Dic326—22no—
58C2×Dic326—22no—
59C2×Dic326D414yescomplemented
60C2×D652—22no—
61C2×D652—22no—
62C2×D652—41no—
63C2×C22⋊D163—22no—
64C2×C22⋊D163—22no—
65C2×C22⋊D163—22no—
66C2×C22⋊D163—22no—
67C2×C22⋊D163—22no—
68C2×C22⋊D163—41no—
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C22×Dic163C2311yescharacteristic
73C22×Dic163C2311yescharacteristic
74C22×Dic163C2311yescharacteristic
75C2×C4×D163—22no—
76C2×C4×D163—22no—
77C2×C4×D163—22no—
78Dic163⋊C4—22no—
79Dic163⋊C4—22no—
80Dic163⋊C4—22no—
81Dic163⋊C4—41no—
82Dic163⋊C4D414yes—
83C22⋊Dic163—22no—
84C22⋊Dic163—22no—
85(C2×D163)⋊C4—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—
91(C2×D163)⋊C4—22no—
92(C2×D163)⋊C4—22no—
93(C2×D163)⋊C4—41no—
94C4×Dic163—22no—
95C4×Dic163—22no—
96C4×Dic163—22no—
97C4×Dic163—22no—
98C24×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102C22×C4×C163C2311yescharacteristic
103C22×C4×C163C2311yescharacteristic
104C22×C4×C163C2311yescharacteristic
105C4⋊C4×C163—22no—
106C4⋊C4×C163—22no—
107C4⋊C4×C163—41no—
108C22⋊C4×C163—22no—
109C22⋊C4×C163—22no—
110C22⋊C4×C163—41no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163—21no—
115C22×D163—21no—
116C22×D163—21no—
117C22×D163—21no—
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—21no—
122C22×D163—21no—
123C22×D163—21no—
124C22×D163#4112yes—
125Dic326—28no—
126D652—44no—
127C22⋊D163—44no—
128C22⋊D163—44no—
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—21no—
139C2×Dic163—21no—
140C2×Dic163—21no—
141C2×Dic163C2×D412yes—
142C2×Dic163C2×D412yes—
143C2×Dic163C2×D412yes—
144C2×Dic163C2×D412yes—
145C2×Dic163#4112yes—
146C2×Dic163#4112yes—
147C4×D163—44no—
148C23×C163—21no—
149C23×C163—21no—
150C23×C163—21no—
151C23×C163—21no—
152C23×C163—21no—
153C23×C163—21no—
154C23×C163—21no—
155C23×C163C2411yescharacteristic
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×C163—21no—
163C2×C4×C163—21no—
164C2×C4×C163—21no—
165C2×C4×C163—21no—
166C2×C4×C163—21no—
167C2×C4×C163—21no—
168C2×C4×C163—21no—
169C2×C4×C163—21no—
170C2×C4×C163—21no—
171C2×C4×C163—21no—
172C2×C4×C163C2×D412yes—
173C2×C4×C163C2×D412yes—
174C22×C163—41no—
175C22×C163—41no—
176C22×C163—41no—
177C22×C163—41no—
178C22×C163—41no—
179C22×C163—41no—
180C22×C163—41no—
181C22×C163C22×D411yescharacteristic
182C22×C163C22×D411yescharacteristic
183C22×C163C2×#4111yescharacteristic
184C22×C163C2×#4111yescharacteristic
185C22×C163C2×#4111yescharacteristic
186C22×C163H(5,𝔽2)11yescharacteristic
187C22×C1632−1+411yescharacteristic
188C2×D163—22no—
189C2×D163—22no—
190C2×D163—22no—
191C2×D163—41no—
192C2×D163—41no—
193C2×D163—41no—
194C2×D163—41no—
195C2×D163—41no—
196C2×D163—41no—
197C2×D163—41no—
198C2×D163—41no—
199C2×D163—41no—
200C2×D163—41no—
201C2×D163—41no—
202C4×C163—22no—
203C4×C163—22no—
204C4×C163—41no—
205C4×C163—41no—
206C4×C163—41no—
207C4×C163—41no—
208Dic163—22no—
209Dic163—22no—
210Dic163—22no—
211Dic163—41no—
212Dic163—41no—
213Dic163#12114yes—
214C2×C163—81no—
215C2×C163C2×#12111yescharacteristic
216C2×C163#52911yescharacteristic
217C2×C163#54211yescharacteristic
218C2×C163#54611yescharacteristic
219C2×C163#54811yescharacteristic
220C2×C163#54811yescharacteristic
221C2×C163#55011yescharacteristic
222D163—42no—
223D163—81no—
224C163#242611yescharacteristic, complemented, Sylow
225#2426—1631nomaximal, Sylow
226C2×#124—1631no—
227C2×C22⋊Q8—1631no—
228C2×#121—1631no—
229C2×C4×Q8—1631no—
230#396—1631no—
231#396D16311yescharacteristic, complemented
232#394—1631no—
233#394—1631no—
234#390—1631no—
235#390—1631no—
236#390—1631no—
237#389—1631no—
238#388—1631no—
239#388—1631no—
240C4×C22⋊C4—1631no—
241C22×Q8—1631no—
242C22×D4—1631no—
243C22×D4—1631no—
244C22×D4—1631no—
245C23×C4—1631no—
246#124—3264no—
247C22⋊Q8—3264no—
248#121—3264no—
249C4×Q8—3264no—
250C2×C4⋊C4—1631no—
251C2×C4⋊C4—1631no—
252C2×C4⋊C4—1631no—
253C2×C4⋊C4—1631no—
254C2×C4⋊C4—1631no—
255C2×C4⋊C4C2×D16311yescharacteristic
256C2×C4⋊C4C2×D16311yescharacteristic
257C2×C4⋊C4C2×D16311yescharacteristic
258C2×C22⋊C4—1631no—
259C2×C22⋊C4—1631no—
260C2×C22⋊C4—1631no—
261C2×C22⋊C4—1631no—
262C2×C22⋊C4—1631no—
263C2×C22⋊C4—1631no—
264C2×C22⋊C4—1631no—
265C2×C22⋊C4—1631no—
266C2×C22⋊C4—1631no—
267C2×C22⋊C4—1631no—
268C2×C22⋊C4—1631no—
269C2×C22⋊C4C2×D16311yescharacteristic
270C2×C22⋊C4C2×D16311yescharacteristic
271C2×C22⋊C4C2×D16311yescharacteristic
272C2×C42—1631no—
273C2×C42—1631no—
274C2×C42—1631no—
275C2×C42—1631no—
276#95—1631no—
277#95—1631no—
278#95—1631no—
279#95C2×D16311yescharacteristic
280C24—1631no—
281C24—1631no—
282C24C22×D16311yescharacteristic
283C2×Q8—3262no—
284C2×Q8—3262no—
285C2×Q8—1634no—
286C2×D4—3262no—
287C2×D4—3262no—
288C2×D4—3262no—
289C2×D4—3262no—
290C2×D4—3262no—
291C2×D4—3262no—
292C2×D4—3262no—
293C2×D4—6521no—
294C2×D4—6521no—
295C22×C4—3262no—
296C22×C4—3262no—
297C22×C4—3262no—
298C22×C4—1631no—
299C22×C4—1631no—
300C22×C4—1631no—
301C22×C4—1631no—
302C22×C4—1631no—
303C22×C4—1631no—
304C22×C4C22×D16311yescharacteristic
305C22×C4C22×D16311yescharacteristic
306C22×C4C22×D16311yescharacteristic
307C22×C4C22×D16311yescharacteristic
308C22×C4C22×D16311yescharacteristic
309C22×C4C22×D16311yescharacteristic
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—3262no—
314C4⋊C4—3262no—
315C4⋊C4—3262no—
316C4⋊C4—6521no—
317C4⋊C4—1634no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—3262no—
322C22⋊C4—3262no—
323C22⋊C4—3262no—
324C22⋊C4—3262no—
325C22⋊C4—3262no—
326C22⋊C4—3262no—
327C22⋊C4—3262no—
328C22⋊C4—3262no—
329C22⋊C4—3262no—
330C22⋊C4—3262no—
331C22⋊C4—6521no—
332C42—3262no—
333C42—3262no—
334C42—3262no—
335C42—3262no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—3261no—
344C23—3261no—
345C23—3261no—
346C23—3261no—
347C23—3261no—
348C23—3261no—
349C23—3261no—
350C23—3261no—
351C23—3261no—
352C23—3261no—
353C23—3261no—
354C23—3261no—
355C23—3261no—
356C23—1632no—
357C23C23×D16311yescharacteristic
358Q8—3268no—
359D4—6524no—
360D4—6524no—
361D4—6524no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—3261no—
379C2×C4—3261no—
380C2×C4—3261no—
381C2×C4—3261no—
382C2×C4—3261no—
383C2×C4—3261no—
384C2×C4—3261no—
385C2×C4—3261no—
386C2×C4—3261no—
387C2×C4—3261no—
388C2×C4—3261no—
389C2×C4—3261no—
390C2×C4—6524no—
391C2×C4—1632no—
392C2×C4—1632no—
393C2×C4—1632no—
394C2×C4—1632no—
395C2×C4—1632no—
396C2×C4—1632no—
397C2×C4D4×D16312yes—
398C2×C4D4×D16312yes—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—3262no—
407C22—3262no—
408C22—3262no—
409C22—6521no—
410C22—6521no—
411C22—6521no—
412C22—6521no—
413C22—6521no—
414C22—6521no—
415C22—6521no—
416C22—6521no—
417C22—6521no—
418C22—6521no—
419C22—6521no—
420C22C2×D4×D16311yescharacteristic
421C22C2×D4×D16311yescharacteristic
422C22#41×D16311yescharacteristic
423C22#41×D16311yescharacteristic
424C22#41×D16311yescharacteristic
425C22#1910478011yescharacteristic
426C22#1910478311yescharacteristic
427C4—22no—
428C4—22no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—3262no—
434C4—3262no—
435C4—3262no—
436C4—6521no—
437C4—6521no—
438C4—1634no—
439C2—81no—
440C2—13041no—
441C2—6522no—
442C2#121×D16311yescharacteristic
443C2#2378219811yescharacteristic
444C2#2378228511yescharacteristic
445C2#2378230411yescharacteristic
446C2#2378232511yescharacteristic
447C2#2378232511yescharacteristic
448C2#2378233311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
Fitting#396×C163C2
prime radicalC2×#19104708C2
hypercenter#396D163
hypocenterC163#2426
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#19104721C2semidirect
2. C2×#19104713C2semidirect
4. C2×#19104700C2semidirect
5. #23781589C2semidirect
6. #23781582C2semidirect
7. #23781580C2semidirect
8. #23781566C2semidirect
9. #23781565C2semidirect
10. #23781565C2semidirect
11. #23781560C2semidirect
12. #23781555C2semidirect
13. #23781553C2semidirect
14. #23781512C2semidirect
15. #396×C163C2semidirect
59. C2×Dic326D4semidirect
224. C163#2426semidirect
231. #396D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047212
C2×#191047132
C2×#191047082
C2×#191047002
#237815892
#237815822
#237815802
#237815662
#237815652
#237815602
#237815552
#237815532
#237815122
#396×C1632
#2426163