Finite Group Databasebeta

← #35228631

Subgroups — #35228631

Order 20864.

992 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782393C211yesmaximal, characteristic, complemented
2#23782391C212yesmaximal, complemented
3#23782390C211yesmaximal, characteristic, complemented
4#23782384C212yesmaximal, complemented
5#23782354C212yesmaximal, complemented
6#23782351C214yesmaximal, complemented
7D4×Dic326C211yesmaximal, characteristic, complemented
8#23782328C212yesmaximal, complemented
9#23782275C212yesmaximal, complemented
10#23782272C214yesmaximal, complemented
11#23782220C211yesmaximal, characteristic, complemented
12#23782219C212yesmaximal, complemented
13#23782197C211yesmaximal, characteristic
14#23782194C212yesmaximal
15#23782142C212yesmaximal, complemented
16C4×Q8×D163C211yesmaximal, characteristic, complemented
17#556×C163C211yesmaximal, characteristic, complemented
18C22×Dic326C2212yes—
19C2×Q8×D163C2211yescharacteristic
20#19104735C2211yescharacteristic
21#19104734C2212yes—
22#19104734C2214yes—
23#19104733C2212yes—
24#19104733C2212yes—
25#19104728C2214yes—
26#19104728C2212yes—
27#19104728C2214yes—
28#19104728C2214yes—
29#19104727C2212yes—
30#19104727C2214yes—
31#19104726C2211yescharacteristic
32#19104726C2212yes—
33#19104726C2214yes—
34#19104726C2212yes—
35#19104725C2214yes—
36#19104725C2214yes—
37#19104725C2214yes—
38#19104722C2211yescharacteristic
39#19104717C2214yes—
40#19104717C2212yes—
41#19104715C2211yescharacteristic
42#19104715C2212yes—
43#19104714C2212yes—
44#19104713C2214yes—
45#19104712C2214yes—
46#19104712C2214yes—
47#19104711C2212yes—
48#19104706C2211yescharacteristic
49Q8×Dic163C2212yes—
50Q8×Dic163C2212yes—
51Q8×Dic163C2211yescharacteristic
52C4×Dic326C2211yescharacteristic
53C4×Dic326C2214yes—
54C4×Dic326C2212yes—
55C4×Dic326C2211yescharacteristic
56C4×Dic326C2214yes—
57C4×Dic326C2212yes—
58#19104700C2212yes—
59#19104700C2212yes—
60#19104700C2212yes—
61#19104700C2212yes—
62#19104700C2211yescharacteristic
63D4×Dic163C2212yes—
64C4×C22⋊D163C2212yes—
65(C22⋊D163)⋊C4C2214yes—
66#19104694C2212yes—
67#19104693C2212yes—
68#19104692C2214yes—
69#19104691C2214yes—
70#19104691C2214yes—
71C4⋊C4×D163C2212yes—
72C4⋊C4×D163C2211yescharacteristic
73C42×D163C2212yes—
74C42×D163C2211yescharacteristic
75C4⋊Q8×C163C2211yescharacteristic
76#126×C163C2214yes—
77#124×C163C2212yes—
78C22⋊Q8×C163C2212yes—
79C4×Q8×C163C2212yes—
80C4×Q8×C163C2211yescharacteristic
81C4×D4×C163C2211yescharacteristic
82#117×C163C2212yes—
83C2×Dic326—28no—
84C2×Dic326C2312yes—
85C2×Dic326C2312yes—
86C2×Dic326C2314yes—
87C2×Dic326C2311yescharacteristic
88C2×Dic326C2311yescharacteristic
89Q8×D163—24no—
90C2×C22⋊D163C2312yes—
91C22×Dic163C2314yes—
92C2×C4×D163C2312yes—
93C2×C4×D163C2311yescharacteristic
94C2×C4×D163C2311yescharacteristic
95C2×C4×D163C2312yes—
96C2×C4×D163C2311yescharacteristic
97C4⋊Dic163C2311yescharacteristic
98C4⋊Dic163C2312yes—
99C4⋊Dic163C2312yes—
100C4⋊Dic163C2312yes—
101C4⋊Dic163C2312yes—
102C4⋊Dic163C2312yes—
103C4⋊Dic163C2311yescharacteristic
104C4⋊Dic163C2314yes—
105C4⋊Dic163C2314yes—
106Dic163⋊C4C2312yes—
107Dic163⋊C4C2312yes—
108Dic163⋊C4C2312yes—
109Dic163⋊C4C2314yes—
110Dic163⋊C4C2314yes—
111Dic163⋊C4C2314yes—
112Dic163⋊C4C2314yes—
113Dic163⋊C4C2314yes—
114Dic163⋊C4C2314yes—
115Dic163⋊C4C2312yes—
116Dic163⋊C4C2314yes—
117Dic163⋊C4C2312yes—
118C22⋊Dic163C2312yes—
119C22⋊Dic163C2312yes—
120C22⋊Dic163C2312yes—
121C22⋊Dic163C2314yes—
122(C2×D163)⋊C4C2312yes—
123(C2×D163)⋊C4C2314yes—
124C4×Dic163C2312yes—
125C4×Dic163C2311yescharacteristic
126C4×Dic163C2312yes—
127C4×Dic163C2311yescharacteristic
128C4×Dic163C2312yes—
129C4×Dic163C2311yescharacteristic
130C4×Dic163C2314yes—
131C4×Dic163C2314yes—
132C4×Dic163C2314yes—
133C4×Dic163C2312yes—
134C2×Q8×C163C2311yescharacteristic
135C2×Q8×C163C2312yes—
136C2×D4×C163C2311yescharacteristic
137C22×C4×C163C2312yes—
138C4⋊C4×C163C2311yescharacteristic
139C4⋊C4×C163C2312yes—
140C4⋊C4×C163C2311yescharacteristic
141C4⋊C4×C163C2314yes—
142C4⋊C4×C163C2312yes—
143C4⋊C4×C163C2312yes—
144C22⋊C4×C163C2314yes—
145C22⋊C4×C163C2312yes—
146C22⋊C4×C163C2314yes—
147C42×C163C2314yes—
148C42×C163C2311yescharacteristic
149C42×C163C2312yes—
150C22×D163C2411yescharacteristic
151Dic326—24no—
152Dic326—22no—
153Dic326—24no—
154Dic326—28no—
155Dic326#4114yes—
156C22⋊D163—24no—
157C2×Dic163—28no—
158C2×Dic163C2411yescharacteristic
159C2×Dic163C2411yescharacteristic
160C2×Dic163C2412yes—
161C2×Dic163C2412yes—
162C2×Dic163C2414yes—
163C2×Dic163C2412yes—
164C2×Dic163C2411yescharacteristic
165C2×Dic163C2412yes—
166C4×D163—24no—
167C4×D163—22no—
168C4×D163—22no—
169C4×D163—22no—
170C4×D163—24no—
171C23×C163C2412yes—
172Q8×C163—22no—
173Q8×C163—24no—
174D4×C163—22no—
175C2×C4×C163—24no—
176C2×C4×C163C2412yes—
177C2×C4×C163C2412yes—
178C2×C4×C163C2411yescharacteristic
179C2×C4×C163C2414yes—
180C2×C4×C163C2412yes—
181C2×C4×C163C2411yescharacteristic
182C2×C4×C163C2411yescharacteristic
183C22×C163—22no—
184C22×C163—22no—
185C22×C163—22no—
186C22×C163C2511yescharacteristic
187C2×D163—21no—
188C2×D163—21no—
189C2×D1632−1+412yes—
190C4×C163—22no—
191C4×C163—22no—
192C4×C163—24no—
193C4×C163—22no—
194C4×C163—21no—
195C4×C163C2×#4112yes—
196C4×C1632−1+412yes—
197Dic163—21no—
198Dic163—24no—
199Dic163—21no—
200Dic163—22no—
201Dic163—21no—
202Dic163—22no—
203Dic163—22no—
204Dic163C2×#4114yes—
205C2×C163—42no—
206C2×C163C22×#4111yescharacteristic
207C2×C163C2×2−1+411yescharacteristic
208C2×C163C2×2−1+411yescharacteristic
209D163—22no—
210C163#352311yescharacteristic, complemented, Sylow
211#3523—1631nomaximal, Sylow
212#557—1631no—
213#557—1632no—
214#557—1631no—
215#556—1632no—
216#556D16311yescharacteristic, complemented
217#552—1634no—
218#552—1632no—
219D4×Q8—1632no—
220D4×Q8—1631no—
221#541—1634no—
222#541—1632no—
223#533—1631no—
224#533—1632no—
225#529—1632no—
226#529—1631no—
227#519—1632no—
228C2×C4×Q8—1631no—
229C22×Q8—1631no—
230C22×Q8—1632no—
231C4⋊Q8—1632no—
232C4⋊Q8—1632no—
233C4⋊Q8—1632no—
234C4⋊Q8—1634no—
235C4⋊Q8—1631no—
236C4⋊Q8C2×D16311yescharacteristic
237#126—1634no—
238#126—1632no—
239#126—1634no—
240#126—1634no—
241#126C2×D16314yes—
242#125—1631no—
243#125—1634no—
244#125—1632no—
245#125—1634no—
246#125—1634no—
247#125—1634no—
248#125—1632no—
249#125—1632no—
250#125—1634no—
251#124—1631no—
252#124C2×D16312yes—
253#123—1632no—
254#123—1634no—
255C22⋊Q8—1632no—
256C22⋊Q8—1632no—
257C22⋊Q8—1631no—
258C22⋊Q8—1634no—
259C22⋊Q8—1632no—
260C22⋊Q8—1634no—
261C22⋊Q8—1634no—
262C22⋊Q8C2×D16312yes—
263#121—1631no—
264C4×Q8—1631no—
265C4×Q8—1632no—
266C4×Q8—1632no—
267C4×Q8—1631no—
268C4×Q8—1632no—
269C4×Q8—1632no—
270C4×Q8—1632no—
271C4×Q8—1632no—
272C4×Q8—1631no—
273C4×Q8—1632no—
274C4×Q8—1634no—
275C4×Q8—1634no—
276C4×Q8—1631no—
277C4×Q8—1632no—
278C4×Q8C2×D16311yescharacteristic
279C4×Q8C2×D16312yes—
280C4×D4—1632no—
281C4×D4—1632no—
282C4×D4—1634no—
283C4×D4C2×D16311yescharacteristic
284#117—1632no—
285#117—1632no—
286#117—1634no—
287#117—1634no—
288#117—1634no—
289#117C2×D16312yes—
290C2×C4⋊C4—1632no—
291C2×C4⋊C4—1631no—
292C2×C42—1632no—
293C2×C42—1631no—
294C2×Q8—3268no—
295C2×Q8—3264no—
296C2×Q8—1631no—
297C2×Q8—1632no—
298C2×Q8—1632no—
299C2×Q8—1634no—
300C2×Q8—1631no—
301C2×Q8C22×D16312yes—
302C2×Q8C22×D16311yescharacteristic
303C2×D4—1632no—
304C2×D4C22×D16311yescharacteristic
305C22×C4—1634no—
306C22×C4—1631no—
307C22×C4—1631no—
308C22×C4—1632no—
309C22×C4—1632no—
310C22×C4—1631no—
311C22×C4C22×D16312yes—
312C4⋊C4—1632no—
313C4⋊C4—1632no—
314C4⋊C4—1631no—
315C4⋊C4—1632no—
316C4⋊C4—1632no—
317C4⋊C4—1634no—
318C4⋊C4—1634no—
319C4⋊C4—1632no—
320C4⋊C4—1634no—
321C4⋊C4—1632no—
322C4⋊C4—1634no—
323C4⋊C4—1632no—
324C4⋊C4—1634no—
325C4⋊C4—1632no—
326C4⋊C4—1634no—
327C4⋊C4—1632no—
328C4⋊C4—1631no—
329C4⋊C4—1634no—
330C4⋊C4—1634no—
331C4⋊C4—1632no—
332C4⋊C4—1634no—
333C4⋊C4C22×D16311yescharacteristic
334C4⋊C4C22×D16312yes—
335C4⋊C4C22×D16312yes—
336C4⋊C4C22×D16312yes—
337C4⋊C4C22×D16311yescharacteristic
338C4⋊C4C22×D16314yes—
339C22⋊C4—1632no—
340C22⋊C4—1632no—
341C22⋊C4—1634no—
342C22⋊C4—1632no—
343C22⋊C4—1632no—
344C22⋊C4—1634no—
345C22⋊C4C22×D16314yes—
346C22⋊C4C22×D16314yes—
347C22⋊C4C22×D16312yes—
348C42—1631no—
349C42—1632no—
350C42—1632no—
351C42—1631no—
352C42—1632no—
353C42—1631no—
354C42—1634no—
355C42—1634no—
356C42—1632no—
357C42—1634no—
358C42C22×D16311yescharacteristic
359C42C22×D16312yes—
360C42C22×D16314yes—
361C23—1631no—
362C23C23×D16312yes—
363Q8—22no—
364Q8—24no—
365Q8—3268no—
366Q8—3264no—
367Q8—3264no—
368Q8—3262no—
369Q8—1634no—
370D4—22no—
371D4—3264no—
372C2×C4—24no—
373C2×C4—3264no—
374C2×C4—3262no—
375C2×C4—3264no—
376C2×C4—3268no—
377C2×C4—3262no—
378C2×C4—3262no—
379C2×C4—1631no—
380C2×C4—1631no—
381C2×C4—1632no—
382C2×C4—1632no—
383C2×C4—1632no—
384C2×C4—1631no—
385C2×C4—1634no—
386C2×C4—1632no—
387C2×C4C23×D16312yes—
388C2×C4C23×D16314yes—
389C2×C4C23×D16311yescharacteristic
390C2×C4C23×D16312yes—
391C2×C4C23×D16312yes—
392C2×C4C23×D16311yescharacteristic
393C2×C4C23×D16311yescharacteristic
394C22—22no—
395C22—22no—
396C22—22no—
397C22—3261no—
398C22—3261no—
399C22—1632no—
400C22C24×D16311yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—21no—
405C4—24no—
406C4—3264no—
407C4—3261no—
408C4—3261no—
409C4—3262no—
410C4—3262no—
411C4—3261no—
412C4—3262no—
413C4—1634no—
414C4#41×D16312yes—
415C4#1910478212yes—
416C2—42no—
417C2—3262no—
418C2C2×#41×D16311yescharacteristic
419C22−1+4×D16311yescharacteristic
420C2C2×#1910478211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#556×C163C2
prime radical#19104706C22
hypercenter#556D163
hypocenterC163#3523
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #23782393C2semidirect
2. #23782391C2semidirect
3. #23782390C2semidirect
4. #23782384C2semidirect
5. #23782354C2semidirect
6. #23782351C2semidirect
7. D4×Dic326C2semidirect
8. #23782328C2semidirect
9. #23782275C2semidirect
10. #23782272C2semidirect
11. #23782220C2semidirect
12. #23782219C2semidirect
15. #23782142C2semidirect
16. C4×Q8×D163C2semidirect
17. #556×C163C2semidirect
210. C163#3523semidirect
216. #556D163semidirect

Maximal subgroups

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

subgroupindex
#237823932
#237823912
#237823902
#237823842
#237823542
#237823512
D4×Dic3262
#237823282
#237822752
#237822722
#237822202
#237822192
#237821972
#237821942
#237821422
C4×Q8×D1632
#556×C1632
#3523163