Finite Group Databasebeta

← #35228182

Subgroups — #35228182

Order 20864.

1492 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104782C211yesmaximal, characteristic, complemented
2C22×Q8×D163C211yesmaximal, characteristic, complemented
3Q8×D652C211yesmaximal, characteristic, complemented
4D4×Dic326C213yesmaximal, complemented
5Q8×C22⋊D163C211yesmaximal, characteristic, complemented
6#23782328C213yesmaximal, complemented
7#23782323C213yesmaximal, complemented
8#23782322C211yesmaximal, characteristic, complemented
9#23782321C213yesmaximal, complemented
10#23782238C213yesmaximal, complemented
11#23782235C213yesmaximal, complemented
12#23782175C213yesmaximal, complemented
13#23782174C213yesmaximal, complemented
14#23782141C211yesmaximal, characteristic, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16#19104782—28no—
17C2×#14579699C2213yes—
18C2×#14579698C2213yes—
19C2×#14579698C2213yescomplemented
20C22×Dic326C2213yes—
21C22×Dic326C2213yescomplemented
22C2×Q8×D163—28no—
23C2×Q8×D163C2213yes—
24C2×Q8×D163C2211yescharacteristic
25C2×Q8×D163C2213yes—
26C2×Q8×D163C2211yescharacteristic
27C2×Q8×D163C2211yescharacteristic, complemented
28C22×C4×D163C2213yes—
29#19104735C2213yescomplemented
30#19104734C2216yescomplemented
31#19104733C2213yescomplemented
32#19104722C2213yescomplemented
33#19104721C2216yescomplemented
34#19104720C2213yes—
35#19104717C2213yes—
36#19104717C2216yes—
37#19104715C2213yes—
38#19104715C2213yescomplemented
39#19104714C2213yes—
40#19104714C2213yescomplemented
41#19104713C2213yes—
42#19104712C2213yes—
43#19104712C2216yescomplemented
44#19104712C2216yescomplemented
45#19104711C2213yescomplemented
46#19104708C2213yes—
47#19104707C2213yes—
48#19104706C2213yes—
49#19104705C2211yescharacteristic
50#19104703C2213yes—
51Q8×Dic163C2211yescharacteristic, complemented
52C4×Dic326C2213yescomplemented
53#19104700C2213yescomplemented
54C4×D652C2213yes—
55D4×Dic163C2216yes—
56C4×C22⋊D163C2213yes—
57C4×C22⋊D163C2213yes—
58(C22⋊D163)⋊C4C2216yes—
59#19104693C2213yes—
60#19104692C2213yes—
61C2×#41×C163C2211yescharacteristic, complemented
62C22×Q8×C163C2211yescharacteristic
63#124×C163C2213yescomplemented
64C22⋊Q8×C163C2213yescomplemented
65#121×C163C2213yes—
66C4×Q8×C163C2211yescharacteristic, complemented
67C4×D4×C163C2213yes—
68C23×D163C2311yescharacteristic
69#14579699—212no—
70#14579698—212no—
71#14579698—212no—
72C2×Dic326—212no—
73C2×Dic326—212no—
74C2×Dic326C2316yes—
75C2×Dic326C2313yes—
76C2×Dic326C2313yes—
77C2×Dic326C2313yes—
78C2×Dic326C2313yes—
79Q8×D163—212no—
80Q8×D163—212no—
81Q8×D163—24no—
82Q8×D163—24no—
83Q8×D163D418yescomplemented
84C2×D652C2311yescharacteristic
85C2×D652C2311yescharacteristic
86C2×C22⋊D163C2313yes—
87C2×C22⋊D163C2313yes—
88C2×C22⋊D163C2313yes—
89C2×C22⋊D163C2311yescharacteristic
90C22×Dic163C2313yes—
91C22×Dic163C2316yes—
92C2×C4×D163—212no—
93C2×C4×D163C2313yes—
94C2×C4×D163C2313yes—
95C2×C4×D163C2313yes—
96C2×C4×D163C2313yes—
97C2×C4×D163C2313yes—
98C4⋊Dic163C2313yes—
99C4⋊Dic163C2316yes—
100C4⋊Dic163C2313yes—
101Dic163⋊C4C2313yes—
102Dic163⋊C4C2313yes—
103Dic163⋊C4C2316yes—
104Dic163⋊C4C2313yes—
105Dic163⋊C4C2313yes—
106C22⋊Dic163C2313yes—
107C22⋊Dic163C2313yes—
108C22⋊Dic163C2316yes—
109(C2×D163)⋊C4C2311yescharacteristic
110(C2×D163)⋊C4C2313yes—
111(C2×D163)⋊C4C2313yes—
112(C2×D163)⋊C4C2311yescharacteristic
113(C2×D163)⋊C4C2313yes—
114(C2×D163)⋊C4C2313yes—
115C4×Dic163C2313yes—
116C4×Dic163C2316yes—
117#41×C163—24no—
118C2×Q8×C163—24no—
119C2×Q8×C163C2313yes—
120C2×Q8×C163C2311yescharacteristic
121C2×D4×C163C2313yes—
122C2×D4×C163C2313yes—
123C22×C4×C163C2313yes—
124C22×C4×C163C2313yes—
125C4⋊C4×C163C2313yes—
126C4⋊C4×C163C2313yes—
127C22⋊C4×C163C2311yescharacteristic
128C22⋊C4×C163C2313yes—
129C22⋊C4×C163C2316yes—
130C42×C163C2313yes—
131C22×D163—22no—
132C22×D163—22no—
133C22×D163—22no—
134C22×D163C2411yescharacteristic
135C22×D163C2411yescharacteristic
136C22×D163C2411yescharacteristic
137Dic326—26no—
138Dic326—26no—
139Dic326—212no—
140Dic326—26no—
141Dic326C2×D4112yes—
142D652—22no—
143D652—22no—
144C22⋊D163—22no—
145C22⋊D163—26no—
146C22⋊D163—26no—
147C22⋊D163—26no—
148C2×Dic163—26no—
149C2×Dic163—212no—
150C2×Dic163C2411yescharacteristic
151C2×Dic163C2413yes—
152C2×Dic163C2413yes—
153C2×Dic163C2413yes—
154C2×Dic163C2413yes—
155C4×D163—26no—
156C4×D163—26no—
157C4×D163—26no—
158C4×D163—26no—
159C4×D163C2×D4112yes—
160C23×C163C2413yes—
161C23×C163C2411yescharacteristic
162Q8×C163—26no—
163Q8×C163C2×D414yes—
164D4×C163—26no—
165D4×C163—26no—
166C2×C4×C163—26no—
167C2×C4×C163—26no—
168C2×C4×C163C2411yescharacteristic
169C2×C4×C163C2413yes—
170C2×C4×C163C2413yes—
171C2×C4×C163C2411yescharacteristic
172C2×C4×C163C2413yes—
173C22×C163—21no—
174C22×C163—21no—
175C22×C163—23no—
176C22×C163—23no—
177C22×C163—23no—
178C22×C1632−1+412yescomplemented
179C22×C163C2511yescharacteristic
180C2×D163—28no—
181C2×D163—21no—
182C2×D163—21no—
183C2×D163—21no—
184C2×D163—21no—
185C2×D163—21no—
186C2×D163—21no—
187C2×D163—21no—
188C2×D163C22×D412yes—
189C2×D1632−1+412yescomplemented
190C4×C163—21no—
191C4×C163—23no—
192C4×C163—23no—
193C4×C163—21no—
194C4×C163C22×D416yes—
195Dic163—23no—
196Dic163—23no—
197Dic163—23no—
198Dic163—21no—
199Dic163C22×D416yes—
200C2×C163—22no—
201C2×C163—43no—
202C2×C163C23×D411yescharacteristic
203C2×C163C2×2−1+411yescharacteristic
204C2×C163C2×2−1+411yescharacteristic
205D163—22no—
206D163—22no—
207D163—41no—
208C163#348611yescharacteristic, complemented, Sylow
209#3486—1631nomaximal, Sylow
210C2×2−1+4—1631no—
211C23×Q8—1631no—
212D4×Q8—1631no—
213D4×Q8—1633no—
214D4×Q8—1631no—
215D4×Q8—1633no—
216#548—1631no—
217#548—1633no—
218#548—1633no—
219#548D16311yescharacteristic, complemented
220#536—1633no—
221#536—1633no—
222#525—1633no—
223#525—1633no—
224#519—1631no—
2252−1+4—3268no—
226C2×#41—1633no—
227C2×#41—1633no—
228C2×#41—1633no—
229C2×#41C2×D16311yescharacteristic, complemented
230C22×Q8—3268no—
231C22×Q8—1633no—
232C22×Q8—1633no—
233C22×Q8—1631no—
234C22×Q8—1631no—
235C22×Q8—1631no—
236C22×Q8—1633no—
237C22×Q8—1633no—
238C22×Q8C2×D16311yescharacteristic
239C23×C4—1633no—
240C4⋊Q8—1633no—
241C4⋊Q8—1636no—
242C4⋊Q8—1633no—
243#124—1633no—
244#124—1636no—
245#124C2×D16313yescomplemented
246#123—1633no—
247#123—1636no—
248#123—1633no—
249C22⋊Q8—1633no—
250C22⋊Q8—1633no—
251C22⋊Q8—1633no—
252C22⋊Q8—1633no—
253C22⋊Q8—1633no—
254C22⋊Q8—1633no—
255C22⋊Q8—1633no—
256C22⋊Q8—1636no—
257C22⋊Q8—1636no—
258C22⋊Q8C2×D16313yescomplemented
259#121—1633no—
260#121—1633no—
261#121—1633no—
262#121C2×D16313yes—
263#120—1633no—
264#120—1631no—
265C4×Q8—1631no—
266C4×Q8—1633no—
267C4×Q8—1633no—
268C4×Q8C2×D16311yescharacteristic, complemented
269C4×D4—1636no—
270C4×D4—1633no—
271C4×D4—1633no—
272C4×D4—1636no—
273C4×D4—1633no—
274C4×D4C2×D16313yes—
275#117—1633no—
276#117—1633no—
277C24—1631no—
278#41—24no—
279#41—32612no—
280#41—32612no—
281#41—32612no—
282C2×Q8—24no—
283C2×Q8—32612no—
284C2×Q8—32612no—
285C2×Q8—32612no—
286C2×Q8—3264no—
287C2×Q8—3264no—
288C2×Q8—32612no—
289C2×Q8—1638no—
290C2×Q8—1633no—
291C2×Q8—1636no—
292C2×Q8—1633no—
293C2×Q8—1633no—
294C2×Q8—1633no—
295C2×Q8C22×D16313yes—
296C2×Q8C22×D16311yescharacteristic
297C2×D4—1633no—
298C2×D4—1631no—
299C2×D4—1633no—
300C2×D4—1633no—
301C2×D4—1631no—
302C2×D4—1631no—
303C2×D4C22×D16313yes—
304C2×D4C22×D16313yes—
305C22×C4—32612no—
306C22×C4—1633no—
307C22×C4—1636no—
308C22×C4—1633no—
309C22×C4—1633no—
310C22×C4—1633no—
311C22×C4—1633no—
312C22×C4—1633no—
313C22×C4C22×D16313yes—
314C22×C4C22×D16313yes—
315C4⋊C4—1633no—
316C4⋊C4—1633no—
317C4⋊C4—1633no—
318C4⋊C4—1633no—
319C4⋊C4—1633no—
320C4⋊C4—1636no—
321C4⋊C4—1636no—
322C4⋊C4—1633no—
323C4⋊C4C22×D16313yes—
324C4⋊C4C22×D16313yes—
325C22⋊C4—1633no—
326C22⋊C4—1636no—
327C22⋊C4—1633no—
328C22⋊C4—1633no—
329C22⋊C4—1631no—
330C22⋊C4—1631no—
331C22⋊C4—1633no—
332C22⋊C4—1633no—
333C22⋊C4—1633no—
334C22⋊C4C22×D16316yes—
335C22⋊C4C22×D16311yescharacteristic
336C22⋊C4C22×D16313yes—
337C42—1633no—
338C42—1636no—
339C42C22×D16313yes—
340C23—3262no—
341C23—3262no—
342C23—3262no—
343C23—1631no—
344C23—1631no—
345C23—1631no—
346C23C23×D16313yes—
347C23C23×D16311yescharacteristic
348Q8—26no—
349Q8—32612no—
350Q8—3266no—
351Q8—3266no—
352Q8—3266no—
353Q8—16312no—
354Q8D4×D16314yes—
355D4—26no—
356D4—26no—
357D4—3262no—
358D4—3262no—
359D4—3266no—
360D4—3266no—
361D4—3262no—
362D4—3266no—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—3266no—
366C2×C4—3266no—
367C2×C4—3266no—
368C2×C4—32612no—
369C2×C4—3266no—
370C2×C4—3266no—
371C2×C4—1633no—
372C2×C4—1631no—
373C2×C4—1633no—
374C2×C4—1633no—
375C2×C4—1633no—
376C2×C4—16312no—
377C2×C4C23×D16313yes—
378C2×C4C23×D16311yescharacteristic
379C2×C4C23×D16313yes—
380C2×C4C23×D16311yescharacteristic
381C2×C4C23×D16313yes—
382C22—21no—
383C22—21no—
384C22—23no—
385C22—23no—
386C22—23no—
387C22—3268no—
388C22—3261no—
389C22—3261no—
390C22—3261no—
391C22—3261no—
392C22—3261no—
393C22—3261no—
394C22—3261no—
395C22—1632no—
396C22—1632no—
397C22#1910478212yescomplemented
398C22C24×D16311yescharacteristic
399C4—21no—
400C4—23no—
401C4—21no—
402C4—23no—
403C4—3263no—
404C4—3263no—
405C4—3263no—
406C4—3261no—
407C4—1636no—
408C4C2×D4×D16316yes—
409C2—22no—
410C2—43no—
411C2—3262no—
412C2—3262no—
413C2—6521no—
414C2C22×D4×D16311yescharacteristic
415C22−1+4×D16311yescharacteristic
416C2C2×#1910478211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D163
commutator, socle, cosocleC22×C163C25
Fitting#548×C163C2
hypercenter#548D163
hypocenterC163#3486
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#19104782C2semidirect
2. C22×Q8×D163C2semidirect
3. Q8×D652C2semidirect
4. D4×Dic326C2semidirect
5. Q8×C22⋊D163C2semidirect
6. #23782328C2semidirect
7. #23782323C2semidirect
8. #23782322C2semidirect
9. #23782321C2semidirect
10. #23782238C2semidirect
11. #23782235C2semidirect
12. #23782175C2semidirect
13. #23782174C2semidirect
14. #23782141C2semidirect
15. #548×C163C2semidirect
19. C2×#14579698C22semidirect
21. C22×Dic326C22semidirect
27. C2×Q8×D163C22semidirect
29. #19104735C22semidirect
30. #19104734C22semidirect
31. #19104733C22semidirect
32. #19104722C22semidirect
33. #19104721C22semidirect
38. #19104715C22semidirect
40. #19104714C22semidirect
43. #19104712C22semidirect
44. #19104712C22semidirect
45. #19104711C22semidirect
51. Q8×Dic163C22semidirect
52. C4×Dic326C22semidirect
53. #19104700C22semidirect
61. C2×#41×C163C22semidirect
63. #124×C163C22semidirect
64. C22⋊Q8×C163C22semidirect
66. C4×Q8×C163C22semidirect
83. Q8×D163D4semidirect
178. C22×C1632−1+4semidirect
189. C2×D1632−1+4semidirect
208. C163#3486semidirect
219. #548D163semidirect
229. C2×#41C2×D163semidirect
245. #124C2×D163semidirect
258. C22⋊Q8C2×D163semidirect
268. C4×Q8C2×D163semidirect
397. C22#19104782semidirect

Maximal subgroups

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

subgroupindex
C2×#191047822
C22×Q8×D1632
Q8×D6522
D4×Dic3262
Q8×C22⋊D1632
#237823282
#237823232
#237823222
#237823212
#237822382
#237822352
#237821752
#237821742
#237821412
#548×C1632
#3486163