Finite Group Databasebeta

← #35227960

Subgroups — #35227960

Order 20864.

1012 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782293C211yesmaximal, characteristic, complemented
2#23782288C212yesmaximal, complemented
3#23782284C212yesmaximal, complemented
4#23782276C212yesmaximal, complemented
5#23782275C212yesmaximal, complemented
6#23782259C214yesmaximal, complemented
7#23782243C211yesmaximal, characteristic
8#23782239C211yesmaximal, characteristic, complemented
9#23782220C212yesmaximal, complemented
10#23782217C212yesmaximal, complemented
11#23782201C212yesmaximal, complemented
12#23782194C212yesmaximal, complemented
13#23782192C214yesmaximal, complemented
14#41×Dic163C211yesmaximal, characteristic, complemented
15#23782133C211yesmaximal, characteristic, complemented
16C2×Q8×Dic163C211yesmaximal, characteristic, complemented
17#536×C163C211yesmaximal, characteristic, complemented
18C2×#14579699C2211yescharacteristic
19#19104735C2212yes—
20#19104734C2212yes—
21#19104733C2212yes—
22#19104730C2214yes—
23#19104730C2214yes—
24#19104728C2214yes—
25#19104728C2214yes—
26#19104727C2212yes—
27#19104726C2212yes—
28#19104726C2214yes—
29#19104725C2214yes—
30#19104725C2214yes—
31#19104724C2212yes—
32#19104721C2214yes—
33#19104719C2214yes—
34#19104718C2214yes—
35#19104717C2214yes—
36#19104715C2214yes—
37#19104714C2212yes—
38#19104712C2214yes—
39#19104712C2214yes—
40#19104711C2212yes—
41#19104711C2212yes—
42#19104711C2212yes—
43#19104709C2212yes—
44#19104707C2214yes—
45#19104706C2212yes—
46Q8×Dic163C2212yes—
47Q8×Dic163C2212yes—
48Q8×Dic163C2211yescharacteristic
49Q8×Dic163C2212yes—
50Q8×Dic163C2211yescharacteristic
51Q8×Dic163C2212yes—
52#19104700C2214yes—
53#19104700C2211yescharacteristic
54#19104700C2211yescharacteristic
55D4×Dic163C2212yes—
56D4×Dic163C2211yescharacteristic
57D4×Dic163C2212yes—
58D4×Dic163C2211yescharacteristic
59#19104697C2211yescharacteristic
60#19104697C2211yescharacteristic
61(C22⋊D163)⋊C4C2212yes—
62(C22⋊D163)⋊C4C2214yes—
63(C22⋊D163)⋊C4C2212yes—
64#19104694C2211yescharacteristic
65#19104694C2212yes—
66#19104693C2214yes—
67#19104692C2212yes—
68#19104691C2214yes—
69C2×C4⋊Dic163C2211yescharacteristic
70C2×C4⋊Dic163C2212yes—
71C2×C4×Dic163C2212yes—
72C2×C4×Dic163C2212yes—
73C2×C4×Dic163C2211yescharacteristic
74C42×D163C2212yes—
75C2×#41×C163C2211yescharacteristic
76C22×Q8×C163C2211yescharacteristic
77C4⋊Q8×C163C2212yes—
78#124×C163C2212yes—
79#123×C163C2214yes—
80C22⋊Q8×C163C2212yes—
81C22⋊Q8×C163C2212yes—
82#117×C163C2211yescharacteristic
83#14579699—24no—
84C2×Dic326C2311yescharacteristic
85C2×Dic326C2314yes—
86C2×D652C2311yescharacteristic
87C2×C22⋊D163C2312yes—
88C2×C22⋊D163C2314yes—
89C22×Dic163C2311yescharacteristic
90C22×Dic163C2312yes—
91C22×Dic163C2311yescharacteristic
92C22×Dic163C2312yes—
93C22×Dic163C2312yes—
94C2×C4×D163C2312yes—
95C2×C4×D163C2312yes—
96C2×C4×D163C2312yes—
97C4⋊Dic163C2312yes—
98C4⋊Dic163C2312yes—
99C4⋊Dic163C2314yes—
100C4⋊Dic163C2314yes—
101C4⋊Dic163C2314yes—
102C4⋊Dic163C2311yescharacteristic
103C4⋊Dic163C2311yescharacteristic
104C4⋊Dic163C2312yes—
105C4⋊Dic163C2312yes—
106Dic163⋊C4C2314yes—
107Dic163⋊C4C2314yes—
108Dic163⋊C4C2312yes—
109Dic163⋊C4C2312yes—
110Dic163⋊C4C2314yes—
111Dic163⋊C4C2314yes—
112Dic163⋊C4C2314yes—
113C22⋊Dic163C2314yes—
114C22⋊Dic163C2312yes—
115C22⋊Dic163C2312yes—
116C22⋊Dic163C2312yes—
117(C2×D163)⋊C4C2312yes—
118(C2×D163)⋊C4C2312yes—
119(C2×D163)⋊C4C2314yes—
120(C2×D163)⋊C4C2314yes—
121(C2×D163)⋊C4C2314yes—
122C4×Dic163C2311yescharacteristic
123C4×Dic163C2312yes—
124C4×Dic163C2312yes—
125C4×Dic163C2311yescharacteristic
126C4×Dic163C2314yes—
127C4×Dic163C2311yescharacteristic
128C4×Dic163C2312yes—
129C4×Dic163C2311yescharacteristic
130C4×Dic163C2312yes—
131C4×Dic163C2312yes—
132C4×Dic163C2314yes—
133#41×C163—24no—
134C2×Q8×C163—24no—
135C2×Q8×C163C2312yes—
136C2×Q8×C163C2312yes—
137C2×Q8×C163C2311yescharacteristic
138C2×D4×C163C2312yes—
139C2×D4×C163C2311yescharacteristic
140C22×C4×C163C2312yes—
141C22×C4×C163C2312yes—
142C22×C4×C163C2311yescharacteristic
143C4⋊C4×C163C2311yescharacteristic
144C4⋊C4×C163C2311yescharacteristic
145C4⋊C4×C163C2314yes—
146C4⋊C4×C163C2314yes—
147C22⋊C4×C163C2312yes—
148C22⋊C4×C163C2314yes—
149C22⋊C4×C163C2314yes—
150C42×C163C2312yes—
151C22×D163C2412yes—
152Dic326—22no—
153Dic326—44no—
154D652—22no—
155C22⋊D163—24no—
156C22⋊D163—44no—
157C2×Dic163—22no—
158C2×Dic163—24no—
159C2×Dic163—41no—
160C2×Dic163—42no—
161C2×Dic163—42no—
162C2×Dic163C2411yescharacteristic
163C2×Dic163C2411yescharacteristic
164C2×Dic163C2411yescharacteristic
165C2×Dic163C2412yes—
166C2×Dic163C2412yes—
167C2×Dic163C2414yes—
168C2×Dic163C2412yes—
169C2×Dic163C2411yescharacteristic
170C4×D163—24no—
171C4×D163—24no—
172C4×D163—42no—
173C23×C163C2412yes—
174C23×C163C2411yescharacteristic
175Q8×C163—24no—
176Q8×C163—24no—
177Q8×C163—22no—
178D4×C163—24no—
179D4×C163—22no—
180C2×C4×C163—24no—
181C2×C4×C163—24no—
182C2×C4×C163#4114yes—
183C2×C4×C163C2412yes—
184C2×C4×C163C2412yes—
185C2×C4×C163C2414yes—
186C2×C4×C163C2412yes—
187C2×C4×C163C2411yescharacteristic
188C2×C4×C163C2411yescharacteristic
189C22×C163—21no—
190C22×C163—21no—
191C22×C163—22no—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163C2×#4112yes—
195C22×C163C2511yescharacteristic
196C2×D163—22no—
197C2×D163—22no—
198C2×D163—22no—
199C4×C163—22no—
200C4×C163—24no—
201C4×C163—22no—
202C4×C163—22no—
203C4×C163C2×#4112yes—
204C4×C163C2×#4112yes—
205Dic163—22no—
206Dic163—21no—
207Dic163—22no—
208Dic163—21no—
209Dic163—21no—
210Dic163—24no—
211Dic163—22no—
212Dic1632−1+412yes—
213C2×C163—22no—
214C2×C163—42no—
215C2×C163C22×#4111yescharacteristic
216C2×C163C2×2−1+411yescharacteristic
217C2×C163#58511yescharacteristic
218D163—42no—
219C163#346111yescharacteristic, complemented, Sylow
220#3461—1631nomaximal, Sylow
221#544—1631no—
222#542—1632no—
223#542—1632no—
224#541—1632no—
225#541—1632no—
226#539—1634no—
227#537—1631no—
228#536—1631no—
229#536D16311yescharacteristic, complemented
230#533—1632no—
231#533—1632no—
232#529—1632no—
233#529—1634no—
234#529—1632no—
235C4×#41—1631no—
236C4×#41—1631no—
237C2×C4×Q8—1631no—
238C2×#41—1631no—
239C2×#41C2×D16311yescharacteristic
240C22×Q8C2×D16311yescharacteristic
241C4⋊Q8—1632no—
242C4⋊Q8—1632no—
243C4⋊Q8—1632no—
244C4⋊Q8C2×D16312yes—
245#126—1634no—
246#126—1634no—
247#126—1634no—
248#126—1634no—
249#125—1634no—
250#125—1632no—
251#125—1634no—
252#125—1634no—
253#125—1632no—
254#124—1632no—
255#124—1634no—
256#124C2×D16312yes—
257#123—1634no—
258#123—1634no—
259#123—1634no—
260#123C2×D16314yes—
261C22⋊Q8—1632no—
262C22⋊Q8—1632no—
263C22⋊Q8—1632no—
264C22⋊Q8—1632no—
265C22⋊Q8—1634no—
266C22⋊Q8—1634no—
267C22⋊Q8—1634no—
268C22⋊Q8C2×D16312yes—
269C22⋊Q8C2×D16312yes—
270#121—1634no—
271#121—1632no—
272#121—1632no—
273C4×Q8—1632no—
274C4×Q8—1632no—
275C4×Q8—1632no—
276C4×Q8—1632no—
277C4×Q8—1631no—
278C4×Q8—1631no—
279C4×Q8—1631no—
280C4×Q8—1631no—
281C4×Q8—1634no—
282C4×D4—1631no—
283C4×D4—1632no—
284C4×D4—1632no—
285C4×D4—1631no—
286C4×D4—1631no—
287C4×D4—1631no—
288C4×D4—1632no—
289C4×D4—1632no—
290C4×D4—1634no—
291#117—1631no—
292#117—1632no—
293#117—1634no—
294#117—1634no—
295#117—1632no—
296#117C2×D16311yescharacteristic
297C2×C4⋊C4—1631no—
298C2×C4⋊C4—1632no—
299C2×C42—1631no—
300C2×C42—1632no—
301C2×C42—1632no—
302C2×C42—1632no—
303#41—24no—
304#41—3264no—
305C2×Q8—24no—
306C2×Q8—1631no—
307C2×Q8—1634no—
308C2×Q8C22×D16312yes—
309C2×Q8C22×D16311yescharacteristic
310C2×Q8C22×D16312yes—
311C2×D4—1632no—
312C2×D4—1634no—
313C2×D4—1631no—
314C2×D4C22×D16312yes—
315C2×D4C22×D16311yescharacteristic
316C22×C4—1631no—
317C22×C4—1632no—
318C22×C4—1632no—
319C22×C4—1632no—
320C22×C4—1631no—
321C22×C4—1632no—
322C22×C4—1632no—
323C22×C4—1632no—
324C22×C4C22×D16312yes—
325C22×C4C22×D16312yes—
326C22×C4C22×D16311yescharacteristic
327C4⋊C4—1632no—
328C4⋊C4—1632no—
329C4⋊C4—1634no—
330C4⋊C4—1634no—
331C4⋊C4—1632no—
332C4⋊C4—1632no—
333C4⋊C4—1631no—
334C4⋊C4—1631no—
335C4⋊C4—1632no—
336C4⋊C4—1632no—
337C4⋊C4—1634no—
338C4⋊C4—1634no—
339C4⋊C4—1634no—
340C4⋊C4—1634no—
341C4⋊C4—1634no—
342C4⋊C4—1634no—
343C4⋊C4C22×D16314yes—
344C4⋊C4C22×D16311yescharacteristic
345C4⋊C4C22×D16311yescharacteristic
346C4⋊C4C22×D16314yes—
347C22⋊C4—1632no—
348C22⋊C4—1632no—
349C22⋊C4—1632no—
350C22⋊C4—1634no—
351C22⋊C4—1632no—
352C22⋊C4—1632no—
353C22⋊C4—1634no—
354C22⋊C4—1634no—
355C22⋊C4—1634no—
356C22⋊C4C22×D16314yes—
357C22⋊C4C22×D16312yes—
358C22⋊C4C22×D16314yes—
359C42—1631no—
360C42—1631no—
361C42—1631no—
362C42—1631no—
363C42—1632no—
364C42—1634no—
365C42—1632no—
366C42—1632no—
367C42—1632no—
368C42—1632no—
369C42—1634no—
370C42C22×D16312yes—
371C23—1632no—
372C23C23×D16312yes—
373C23C23×D16311yescharacteristic
374Q8—24no—
375Q8—24no—
376Q8—22no—
377Q8—3262no—
378Q8—6524no—
379D4—24no—
380D4—22no—
381D4—3264no—
382D4—3262no—
383D4—6524no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—3264no—
387C2×C4—3262no—
388C2×C4—3264no—
389C2×C4—3264no—
390C2×C4—6522no—
391C2×C4—6521no—
392C2×C4—6522no—
393C2×C4—6522no—
394C2×C4—1631no—
395C2×C4—1631no—
396C2×C4—1631no—
397C2×C4—1632no—
398C2×C4—1632no—
399C2×C4—1632no—
400C2×C4—1634no—
401C2×C4—1631no—
402C2×C4#1457969814yes—
403C2×C4C23×D16314yes—
404C2×C4C23×D16312yes—
405C2×C4C23×D16312yes—
406C2×C4C23×D16312yes—
407C2×C4C23×D16311yescharacteristic
408C2×C4C23×D16311yescharacteristic
409C22—21no—
410C22—21no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—3262no—
415C22—3262no—
416C22—3262no—
417C22C2×#1457969812yes—
418C22C24×D16311yescharacteristic
419C4—22no—
420C4—24no—
421C4—22no—
422C4—22no—
423C4—3262no—
424C4—3261no—
425C4—3262no—
426C4—3261no—
427C4—3261no—
428C4—3264no—
429C4—3262no—
430C4—1632no—
431C4C2×#1457969812yes—
432C4C2×#1457969812yes—
433C2—22no—
434C2—42no—
435C2—6522no—
436C2C22×#1457969811yescharacteristic
437C22−1+4×D16311yescharacteristic
438C2#2378250211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#536×C163C2
prime radical#23782243C2
hypercenter#536D163
hypocenterC163#3461
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #536

Complements

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

NH ≅ G/Ntype
1. #23782293C2semidirect
2. #23782288C2semidirect
3. #23782284C2semidirect
4. #23782276C2semidirect
5. #23782275C2semidirect
6. #23782259C2semidirect
8. #23782239C2semidirect
9. #23782220C2semidirect
10. #23782217C2semidirect
11. #23782201C2semidirect
12. #23782194C2semidirect
13. #23782192C2semidirect
14. #41×Dic163C2semidirect
15. #23782133C2semidirect
16. C2×Q8×Dic163C2semidirect
17. #536×C163C2semidirect
219. C163#3461semidirect
229. #536D163semidirect

Maximal subgroups

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

subgroupindex
#237822932
#237822882
#237822842
#237822762
#237822752
#237822592
#237822432
#237822392
#237822202
#237822172
#237822012
#237821942
#237821922
#41×Dic1632
#237821332
C2×Q8×Dic1632
#536×C1632
#3461163