Finite Group Databasebeta

← #35228545

Subgroups — #35228545

Order 20864.

1010 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782395C211yesmaximal, characteristic, complemented
2#23782389C211yesmaximal, characteristic, complemented
3#23782387C214yesmaximal, complemented
4#23782386C212yesmaximal, complemented
5#23782383C212yesmaximal, complemented
6#23782381C211yesmaximal, characteristic, complemented
7#23782378C212yesmaximal, complemented
8#23782330C211yesmaximal, characteristic, complemented
9#23782325C212yesmaximal, complemented
10#23782287C212yesmaximal, complemented
11#23782285C214yesmaximal, complemented
12#23782221C212yesmaximal, complemented
13#23782200C212yesmaximal, complemented
14#23782199C211yesmaximal, characteristic, complemented
15#23782141C212yesmaximal, complemented
16C4×#14579699C211yesmaximal, characteristic, complemented
17#533×C163C211yesmaximal, characteristic, complemented
18C2×#14579699C2211yescharacteristic
19C2×Q8×D163C2212yes—
20#19104734C2212yes—
21#19104734C2212yes—
22#19104734C2212yes—
23#19104733C2214yes—
24#19104730C2214yes—
25#19104730C2214yes—
26#19104729C2212yes—
27#19104729C2214yes—
28#19104728C2214yes—
29#19104727C2211yescharacteristic
30#19104724C2214yes—
31#19104724C2212yes—
32#19104723C2211yescharacteristic
33#19104723C2214yes—
34#19104721C2214yes—
35#19104721C2214yes—
36#19104720C2214yes—
37#19104718C2212yes—
38#19104715C2212yes—
39#19104714C2211yescharacteristic
40#19104713C2214yes—
41#19104713C2214yes—
42#19104713C2214yes—
43#19104711C2212yes—
44#19104710C2212yes—
45#19104710C2212yes—
46#19104709C2211yescharacteristic
47#19104708C2214yes—
48Q8×Dic163C2212yes—
49C4×Dic326C2211yescharacteristic
50C4×Dic326C2211yescharacteristic
51C4×Dic326C2214yes—
52C4×Dic326C2211yescharacteristic
53#19104700C2212yes—
54#19104700C2214yes—
55#19104700C2214yes—
56#19104700C2212yes—
57C4×D652C2211yescharacteristic
58C4×D652C2211yescharacteristic
59C4×D652C2211yescharacteristic
60#19104697C2214yes—
61#19104697C2212yes—
62C4×C22⋊D163C2212yes—
63C4×C22⋊D163C2212yes—
64C4×C22⋊D163C2212yes—
65#19104694C2211yescharacteristic
66#19104693C2214yes—
67#19104693C2212yes—
68#19104693C2212yes—
69#19104693C2212yes—
70#19104692C2212yes—
71#19104692C2214yes—
72#19104692C2212yes—
73C42×D163C2212yes—
74C4⋊Q8×C163C2211yescharacteristic
75C4⋊Q8×C163C2211yescharacteristic
76#125×C163C2212yes—
77C22⋊Q8×C163C2212yes—
78C22⋊Q8×C163C2212yes—
79C4×Q8×C163C2212yes—
80C4×Q8×C163C2212yes—
81#117×C163C2212yes—
82C2×C42×C163C2211yescharacteristic
83#14579699—24no—
84C2×Dic326C2314yes—
85C2×Dic326C2311yescharacteristic
86C2×Dic326C2314yes—
87C2×Dic326C2311yescharacteristic
88Q8×D163—28no—
89C2×D652C2311yescharacteristic
90C2×D652C2311yescharacteristic
91C2×D652C2314yes—
92C2×C22⋊D163C2312yes—
93C2×C22⋊D163C2312yes—
94C2×C4×D163C2312yes—
95C2×C4×D163C2314yes—
96C2×C4×D163C2312yes—
97C2×C4×D163C2312yes—
98C2×C4×D163C2312yes—
99C4⋊Dic163C2311yescharacteristic
100C4⋊Dic163C2311yescharacteristic
101C4⋊Dic163C2311yescharacteristic
102C4⋊Dic163C2314yes—
103C4⋊Dic163C2311yescharacteristic
104Dic163⋊C4C2312yes—
105Dic163⋊C4C2312yes—
106Dic163⋊C4C2314yes—
107Dic163⋊C4C2312yes—
108Dic163⋊C4C2312yes—
109Dic163⋊C4C2312yes—
110Dic163⋊C4C2314yes—
111Dic163⋊C4C2314yes—
112Dic163⋊C4C2312yes—
113C22⋊Dic163C2312yes—
114C22⋊Dic163C2312yes—
115(C2×D163)⋊C4C2312yes—
116(C2×D163)⋊C4C2314yes—
117(C2×D163)⋊C4C2312yes—
118(C2×D163)⋊C4C2312yes—
119(C2×D163)⋊C4C2312yes—
120(C2×D163)⋊C4C2314yes—
121(C2×D163)⋊C4C2314yes—
122(C2×D163)⋊C4C2312yes—
123(C2×D163)⋊C4C2314yes—
124(C2×D163)⋊C4C2314yes—
125(C2×D163)⋊C4C2312yes—
126C4×Dic163C2312yes—
127C4×Dic163C2312yes—
128C4×Dic163C2312yes—
129C4×Dic163C2314yes—
130C4×Dic163C2312yes—
131C4×Dic163C2314yes—
132C4×Dic163C2314yes—
133C2×Q8×C163C2312yes—
134C2×Q8×C163C2312yes—
135C22×C4×C163C2311yescharacteristic
136C22×C4×C163C2311yescharacteristic
137C22×C4×C163C2311yescharacteristic
138C4⋊C4×C163C2312yes—
139C4⋊C4×C163C2312yes—
140C4⋊C4×C163C2314yes—
141C4⋊C4×C163C2314yes—
142C4⋊C4×C163C2312yes—
143C4⋊C4×C163C2312yes—
144C22⋊C4×C163C2314yes—
145C42×C163C2311yescharacteristic
146C42×C163C2311yescharacteristic
147C42×C163C2311yescharacteristic
148C42×C163C2311yescharacteristic
149C42×C163C2314yes—
150C22×D163C2412yes—
151C22×D163C2412yes—
152Dic326—22no—
153Dic326—28no—
154Dic326—44no—
155Dic326#4114yes—
156D652—22no—
157D652—22no—
158D652—44no—
159C22⋊D163—24no—
160C22⋊D163—42no—
161C2×Dic163C2412yes—
162C2×Dic163C2412yes—
163C2×Dic163C2412yes—
164C2×Dic163C2412yes—
165C2×Dic163C2412yes—
166C2×Dic163C2412yes—
167C4×D163—24no—
168C4×D163—24no—
169C4×D163—28no—
170C4×D163—24no—
171C4×D163—42no—
172C23×C163C2411yescharacteristic
173Q8×C163—24no—
174Q8×C163—42no—
175C2×C4×C163—22no—
176C2×C4×C163—22no—
177C2×C4×C163—41no—
178C2×C4×C163C2411yescharacteristic
179C2×C4×C163C2411yescharacteristic
180C2×C4×C163C2411yescharacteristic
181C2×C4×C163C2411yescharacteristic
182C2×C4×C163C2414yes—
183C2×C4×C163C2414yes—
184C2×C4×C163C2411yescharacteristic
185C2×C4×C163C2411yescharacteristic
186C22×C163—21no—
187C22×C163—21no—
188C22×C163—21no—
189C22×C163C2511yescharacteristic
190C2×D163—22no—
191C2×D163—22no—
192C2×D163—22no—
193C2×D163—22no—
194C2×D163—22no—
195C2×D163—22no—
196C4×C163—21no—
197C4×C163—24no—
198C4×C163—21no—
199C4×C163—21no—
200C4×C163—21no—
201C4×C163—24no—
202C4×C163C2×#4112yes—
203C4×C1632−1+412yes—
204Dic163—22no—
205Dic163—22no—
206Dic163—22no—
207Dic163—22no—
208Dic163—22no—
209Dic163C2×#4114yes—
210C2×C163—41no—
211C2×C163C22×#4111yescharacteristic
212C2×C163C2×2−1+411yescharacteristic
213C2×C163#58511yescharacteristic
214D163—42no—
215D163—42no—
216C163#351811yescharacteristic, complemented, Sylow
217#3518—1631nomaximal, Sylow
218Q82—1631no—
219#556—1631no—
220#556—1632no—
221#556—1634no—
222#556—1632no—
223#555—1631no—
224#555—1632no—
225D4×Q8—1631no—
226#548—1632no—
227#542—1634no—
228#542—1632no—
229#533—1632no—
230#533D16311yescharacteristic, complemented
231#529—1632no—
232#529—1631no—
233#519—1632no—
234C4×#41—1631no—
235C2×#41—1631no—
236C22×Q8—1632no—
237C4⋊Q8—1634no—
238C4⋊Q8—1632no—
239C4⋊Q8—1632no—
240C4⋊Q8—1632no—
241C4⋊Q8C2×D16311yescharacteristic
242C4⋊Q8C2×D16311yescharacteristic
243#126—1634no—
244#126—1634no—
245#126—1634no—
246#126—1632no—
247#126—1634no—
248#125—1631no—
249#125C2×D16312yes—
250#124—1632no—
251#124—1634no—
252#124—1634no—
253#124—1634no—
254#124—1631no—
255#124—1634no—
256#123—1632no—
257#123—1634no—
258C22⋊Q8—1632no—
259C22⋊Q8—1631no—
260C22⋊Q8—1632no—
261C22⋊Q8—1634no—
262C22⋊Q8—1634no—
263C22⋊Q8—1634no—
264C22⋊Q8C2×D16312yes—
265C22⋊Q8C2×D16312yes—
266#121—1631no—
267#121—1632no—
268#121—1632no—
269#121—1634no—
270C4×Q8—1632no—
271C4×Q8—1634no—
272C4×Q8—1632no—
273C4×Q8—1632no—
274C4×Q8—1634no—
275C4×Q8—1631no—
276C4×Q8—1631no—
277C4×Q8—1634no—
278C4×Q8—1631no—
279C4×Q8C2×D16312yes—
280C4×Q8C2×D16312yes—
281C4×D4—1632no—
282C4×D4—1632no—
283C4×D4—1632no—
284C4×D4—1634no—
285C4×D4—1632no—
286C4×D4—1631no—
287C4×D4—1631no—
288C4×D4—1631no—
289#117—1631no—
290#117—1634no—
291#117—1632no—
292#117—1632no—
293#117—1632no—
294#117—1632no—
295#117—1632no—
296#117—1634no—
297#117C2×D16312yes—
298C2×C42—1632no—
299C2×C42C2×D16311yescharacteristic
300#41—3264no—
301C2×Q8—3268no—
302C2×Q8—1631no—
303C2×Q8—1634no—
304C2×Q8—1634no—
305C2×Q8—1631no—
306C2×Q8C22×D16312yes—
307C2×Q8C22×D16312yes—
308C2×D4—1632no—
309C2×D4—1632no—
310C2×D4—1631no—
311C2×D4—1631no—
312C2×D4—1634no—
313C22×C4—1632no—
314C22×C4—1632no—
315C22×C4—1634no—
316C22×C4—1632no—
317C22×C4—1632no—
318C22×C4C22×D16311yescharacteristic
319C22×C4C22×D16311yescharacteristic
320C22×C4C22×D16311yescharacteristic
321C4⋊C4—1632no—
322C4⋊C4—1631no—
323C4⋊C4—1631no—
324C4⋊C4—1631no—
325C4⋊C4—1632no—
326C4⋊C4—1631no—
327C4⋊C4—1634no—
328C4⋊C4—1632no—
329C4⋊C4—1632no—
330C4⋊C4—1634no—
331C4⋊C4—1632no—
332C4⋊C4—1634no—
333C4⋊C4—1632no—
334C4⋊C4—1634no—
335C4⋊C4C22×D16312yes—
336C4⋊C4C22×D16312yes—
337C4⋊C4C22×D16312yes—
338C4⋊C4C22×D16314yes—
339C4⋊C4C22×D16314yes—
340C4⋊C4C22×D16312yes—
341C22⋊C4—1632no—
342C22⋊C4—1632no—
343C22⋊C4—1634no—
344C22⋊C4—1632no—
345C22⋊C4—1632no—
346C22⋊C4—1632no—
347C22⋊C4—1632no—
348C22⋊C4—1634no—
349C22⋊C4—1634no—
350C22⋊C4—1632no—
351C22⋊C4—1634no—
352C22⋊C4—1632no—
353C22⋊C4—1634no—
354C22⋊C4C22×D16314yes—
355C42—1632no—
356C42—1632no—
357C42—1632no—
358C42—1634no—
359C42—1634no—
360C42—1632no—
361C42—1634no—
362C42C22×D16311yescharacteristic
363C42C22×D16311yescharacteristic
364C42C22×D16311yescharacteristic
365C42C22×D16311yescharacteristic
366C42C22×D16314yes—
367C23—1632no—
368C23—1632no—
369C23C23×D16311yescharacteristic
370Q8—24no—
371Q8—42no—
372Q8—3268no—
373Q8—3262no—
374Q8—6524no—
375Q8—1634no—
376D4—3264no—
377D4—3262no—
378D4—3262no—
379D4—6524no—
380D4—6522no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—3268no—
385C2×C4—3264no—
386C2×C4—3264no—
387C2×C4—3264no—
388C2×C4—6522no—
389C2×C4—1632no—
390C2×C4—1632no—
391C2×C4—1632no—
392C2×C4—1632no—
393C2×C4—1632no—
394C2×C4—1632no—
395C2×C4C23×D16311yescharacteristic
396C2×C4C23×D16311yescharacteristic
397C2×C4C23×D16314yes—
398C2×C4C23×D16314yes—
399C2×C4C23×D16311yescharacteristic
400C2×C4C23×D16311yescharacteristic
401C2×C4C23×D16311yescharacteristic
402C2×C4C23×D16311yescharacteristic
403C22—21no—
404C22—21no—
405C22—21no—
406C22—3262no—
407C22—3262no—
408C22—3262no—
409C22—3262no—
410C22—3262no—
411C22—3262no—
412C22C24×D16311yescharacteristic
413C4—21no—
414C4—24no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—24no—
419C4—3262no—
420C4—3262no—
421C4—3262no—
422C4—3262no—
423C4—3262no—
424C4—1634no—
425C4#41×D16312yes—
426C4#1910478312yes—
427C2—41no—
428C2—6522no—
429C2—6522no—
430C2C2×#41×D16311yescharacteristic
431C2C2×#1910478311yescharacteristic
432C2#2378250111yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#533×C163C2
hypercenter#533D163
hypocenterC163#3518
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #23782395C2semidirect
2. #23782389C2semidirect
3. #23782387C2semidirect
4. #23782386C2semidirect
5. #23782383C2semidirect
6. #23782381C2semidirect
7. #23782378C2semidirect
8. #23782330C2semidirect
9. #23782325C2semidirect
10. #23782287C2semidirect
11. #23782285C2semidirect
12. #23782221C2semidirect
13. #23782200C2semidirect
14. #23782199C2semidirect
15. #23782141C2semidirect
16. C4×#14579699C2semidirect
17. #533×C163C2semidirect
216. C163#3518semidirect
230. #533D163semidirect

Maximal subgroups

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

subgroupindex
#237823952
#237823892
#237823872
#237823862
#237823832
#237823812
#237823782
#237823302
#237823252
#237822872
#237822852
#237822212
#237822002
#237821992
#237821412
C4×#145796992
#533×C1632
#3518163