Finite Group Databasebeta

← #35389747

Subgroups — #35389747

Order 21248.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918593C211yesmaximal, characteristic
2C2×#19127996C211yesmaximal, characteristic, complemented
3#24916935C211yesmaximal, characteristic
4#24906879C214yesmaximal, complemented
5#24906845C214yesmaximal, complemented
6#24906234C211yesmaximal, characteristic, complemented
7#24906143C211yesmaximal, characteristic, complemented
8#24906110C211yesmaximal, characteristic
9C83×#2982C211yesmaximal, characteristic, complemented
10C2×#14581823—21no—
11#19128251C2211yescharacteristic
12C2×#14581792C2211yescharacteristic
13#19128099C2212yes—
14#19128048C2212yes—
15#19128020—22no—
16#19127996—22no—
17#19127996C2214yes—
18#19127992—22no—
19#19127923—22no—
20#19127911C2212yes—
21#19127901C2212yes—
22C2×#14581725—21no—
23#19127725C2211yescharacteristic
24C2×C4⋊(C83⋊C8)C2211yescharacteristic
25#19127700—21no—
26C2×#14581716—21no—
27#19127452C2214yes—
28#19127449C2214yes—
29#19127446C2212yes—
30#19127444C2212yes—
31C2×C4×Q8×C83C2211yescharacteristic
32C8×Q8×C83—22no—
33C8×Q8×C83C2214yes—
34#431×C83—22no—
35#431×C83C2211yescharacteristic
36#431×C83C2211yescharacteristic
37C2×#13050590—21no—
38C22×Dic166—21no—
39#14581823—44no—
40C2×#13050580—22no—
41C2×#13050577—22no—
42C2×#13050574—21no—
43C22×C83⋊C8—21no—
44#14581792—21no—
45#14581792—21no—
46#14581792—21no—
47#14581792C2311yescharacteristic
48#14581792C2312yes—
49#14581789C2311yescharacteristic
50#14581780—21no—
51#14581766—21no—
52#14581766—21no—
53C4×D332—21no—
54C4×D332—21no—
55C2×C4⋊Dic83—21no—
56C2×C4⋊Dic83—21no—
57#14581725—24no—
58#14581724—24no—
59C4⋊(C83⋊C8)—21no—
60C4⋊(C83⋊C8)—21no—
61C4⋊(C83⋊C8)C2312yes—
62C4⋊(C83⋊C8)C2312yes—
63#14581716—24no—
64#14581715—22no—
65#14581714—24no—
66#14581712—22no—
67C22×Q8×C83D411yescharacteristic
68C4×Q8×C83—21no—
69C4×Q8×C83—21no—
70C4×Q8×C83D412yes—
71C4×Q8×C83C2312yes—
72C2×C4⋊C4×C83—21no—
73C2×C4⋊C4×C83D411yescharacteristic
74C2×C42×C83—21no—
75C2×C42×C83C2311yescharacteristic
76C4⋊C8×C83—24no—
77C4⋊C8×C83D412yes—
78C4⋊C8×C83C2312yes—
79C22⋊C8×C83—21no—
80C22⋊C8×C83D412yes—
81C4×C8×C83—24no—
82C4×C8×C83—21no—
83C4×C8×C83C2312yes—
84#13050590—42no—
85#13050590—42no—
86C2×Dic166—21no—
87C2×Dic166—22no—
88C2×Dic166—41no—
89C2×Dic166—82no—
90C2×D332—21no—
91C2×D332—21no—
92C2×D332—21no—
93C2×C22⋊D83—41no—
94C22×Dic83—41no—
95C2×C4×D83—41no—
96#13050580—44no—
97#13050577—44no—
98#13050574—42no—
99C2×C83⋊C8—22no—
100C2×C83⋊C8—22no—
101C2×C83⋊C8—42no—
102C4⋊Dic83—21no—
103C4⋊Dic83—22no—
104C4⋊Dic83—22no—
105C4⋊Dic83—21no—
106C4⋊Dic83—41no—
107C4⋊Dic83—41no—
108(C2×D83)⋊C4—41no—
109(C2×D83)⋊C4—41no—
110C2×Q8×C83—21no—
111C2×Q8×C83—22no—
112C2×Q8×C83SD1614yes—
113C2×Q8×C83C2×D412yes—
114C22×C4×C83—21no—
115C22×C4×C83—21no—
116C22×C4×C83C2×D411yescharacteristic
117C22×C4×C83C2×D411yescharacteristic
118C22×C4×C83#4111yescharacteristic
119C2×C8×C83—22no—
120C2×C8×C83C2×D414yes—
121C4⋊C4×C83—22no—
122C4⋊C4×C83—21no—
123C4⋊C4×C83—22no—
124C4⋊C4×C83C2×D412yes—
125C42×C83—22no—
126C42×C83—22no—
127C42×C83C2×D411yescharacteristic
128C42×C83C2×D411yescharacteristic
129C42×C83#4111yescharacteristic
130C42×C83C2411yescharacteristic
131C23×C83#12111yescharacteristic
132C22×D83—41no—
133Q8×C83—22no—
134Q8×C83—24no—
135Q8×C83C2×SD1614yes—
136C2×C4×C83—22no—
137C2×C4×C83—24no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—41no—
141C2×C4×C83—81no—
142C2×C4×C83#12111yescharacteristic
143C2×C4×C83#12111yescharacteristic
144C2×C4×C83#12111yescharacteristic
145C2×C4×C83C2×SD1612yes—
146C2×C4×C83#13512yes—
147C2×C4×C83C22×D411yescharacteristic
148C2×C4×C83C22×D411yescharacteristic
149C2×C4×C83C2×#4111yescharacteristic
150C22⋊D83—82no—
151C2×Dic83—41no—
152C2×Dic83—42no—
153C2×Dic83—82no—
154D332—41no—
155D332—41no—
156D332—42no—
157D332—81no—
158C4×D83—82no—
159Dic166—41no—
160Dic166—41no—
161Dic166—82no—
162Dic166—82no—
163C8×C83—24no—
164C8×C83—42no—
165C83⋊C8—42no—
166C83⋊C8—42no—
167C22×C83—21no—
168C22×C83—21no—
169C22×C83#46512yes—
170C22×C83C2×#12111yescharacteristic
171C2×D83—81no—
172C2×D83—81no—
173C2×D83—81no—
174C4×C83—21no—
175C4×C83—24no—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—42no—
179C4×C83#46512yes—
180C4×C83C22×SD1611yescharacteristic
181C4×C83C2×#13511yescharacteristic
182C4×C83#57812yes—
183Dic83—81no—
184Dic83—82no—
185#19403—831nomaximal, Sylow
186C2×C83—22no—
187C2×C83#304411yescharacteristic
188C2×C83C2×#46511yescharacteristic
189C2×C83#306811yescharacteristic
190D83—161no—
191#3227—831no—
192C2×#477—831no—
193#3037—831no—
194#2982D8311yescharacteristic, complemented
195#1678—834no—
196#1672—834no—
197#1570—831no—
198#1548—831no—
199#1540—831no—
200C83#1940311yescharacteristic, complemented, Sylow
201C2×#137—1661no—
202#532—831no—
203C2×C4⋊Q8—831no—
204C2×C4×Q8C2×D8311yescharacteristic
205#498—832no—
206#486—832no—
207#481—1662no—
208#477—834no—
209#477—1662no—
210#476—1662no—
211#464—1662no—
212#462—832no—
213#460—832no—
214C8×Q8—22no—
215C8×Q8C2×D8314yes—
216#431—22no—
217#431C2×D8311yescharacteristic
218#431C2×D8311yescharacteristic
219C2×#107—1661no—
220#423—831no—
221C2×C4⋊C8—831no—
222#417—1661no—
223C2×#103—1661no—
224#334—834no—
225#333—834no—
226#332—832no—
227#331—832no—
228C2×#41—1661no—
229C22×Q8—1661no—
230C22×Q8D33211yescharacteristic
231#137—3324no—
232C2×Q16—1662no—
233C2×SD16—1662no—
234C2×M16—1661no—
235C22×C8—1661no—
236C4⋊Q8—832no—
237C4⋊Q8—831no—
238C4⋊Q8—1661no—
239C4⋊Q8—1661no—
240C4⋊Q8—1661no—
241#127—831no—
242#124—1661no—
243#121—1661no—
244#121—1661no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8D33212yes—
248C4×Q8C22×D8312yes—
249C4×D4—1661no—
250C4×D4—1661no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—1661no—
253C2×C4⋊C4—1661no—
254C2×C4⋊C4D33211yescharacteristic
255C2×C42—21no—
256C2×C42C22×D8311yescharacteristic
257#107—1664no—
258#106—1664no—
259C4⋊C8—24no—
260C4⋊C8—832no—
261C4⋊C8—832no—
262C4⋊C8—1661no—
263C4⋊C8—1661no—
264C4⋊C8C22⋊D8312yes—
265C4⋊C8C22×D8312yes—
266#103—1664no—
267#103—1662no—
268#102—1664no—
269#102—1662no—
270C22⋊C8—21no—
271C22⋊C8C22⋊D8312yes—
272C4×C8—24no—
273C4×C8—21no—
274C4×C8C22×D8312yes—
275#41—3322no—
276#41—3322no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—3321no—
280C2×Q8—6642no—
281C2×Q8—1661no—
282C2×Q8—1662no—
283C2×Q8#1305057914yes—
284C2×Q8C2×D33212yes—
285C2×D4—3321no—
286C2×D4—1661no—
287C2×D4—1661no—
288C2×D4—1661no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—3321no—
292C22×C4—3321no—
293C22×C4C2×C22⋊D8311yescharacteristic
294C22×C4C2×D33211yescharacteristic
295C22×C4#1305059011yescharacteristic
296Q16—3324no—
297SD16—3324no—
298M16—3322no—
299C2×C8—22no—
300C2×C8—3322no—
301C2×C8—1662no—
302C2×C8—1662no—
303C2×C8C2×C22⋊D8314yes—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—3321no—
308C4⋊C4—3321no—
309C4⋊C4—1662no—
310C4⋊C4—1661no—
311C4⋊C4—1662no—
312C4⋊C4—1661no—
313C4⋊C4C2×D33212yes—
314C22⋊C4—3321no—
315C22⋊C4—3321no—
316C42—22no—
317C42—22no—
318C42C2×C22⋊D8311yescharacteristic
319C42C2×D33211yescharacteristic
320C42#1305059011yescharacteristic
321C42C23×D8311yescharacteristic
322C23—3321no—
323C23#1458176611yescharacteristic
324Q8—24no—
325Q8—22no—
326Q8—3321no—
327Q8—3321no—
328Q8—6642no—
329Q8—6642no—
330Q8C2×#1305057914yes—
331D4—3321no—
332D4—3321no—
333D4—3322no—
334D4—6641no—
335D4—6642no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—3321no—
343C2×C4—3322no—
344C2×C4—6642no—
345C2×C4—6642no—
346C2×C4#1458176611yescharacteristic
347C2×C4#1458176611yescharacteristic
348C2×C4#1458176611yescharacteristic
349C2×C4C2×#1305057912yes—
350C2×C4#1458181412yes—
351C2×C4C22×C22⋊D8311yescharacteristic
352C2×C4C22×D33211yescharacteristic
353C2×C4C2×#1305059011yescharacteristic
354C8—24no—
355C8—42no—
356C8—3322no—
357C8—3322no—
358C22—21no—
359C22—21no—
360C22—6641no—
361C22—6641no—
362C22—6641no—
363C22#1912793012yes—
364C22C2×#1458176611yescharacteristic
365C4—21no—
366C4—24no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—6641no—
371C4—6642no—
372C4#1912793012yes—
373C4C22×#1305057911yescharacteristic
374C4C2×#1458181411yescharacteristic
375C4#1912851612yes—
376C2—22no—
377C2—13281no—
378C2#2491699311yescharacteristic
379C2C2×#1912793011yescharacteristic
380C2#2491719111yescharacteristic
subgroupG/H
centerC22C2×#14581766
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC83×#2982C2
socleC22×C83C2×#121
prime radical#19128251C22
hypercenter#2982D83
hypocenterC83#19403
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. C2×#19127996C2semidirect
4. #24906879C2semidirect
5. #24906845C2semidirect
6. #24906234C2semidirect
7. #24906143C2semidirect
9. C83×#2982C2semidirect
194. #2982D83semidirect
200. C83#19403semidirect

Maximal subgroups

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

subgroupindex
#249185932
C2×#191279962
#249169352
#249068792
#249068452
#249062342
#249061432
#249061102
C83×#29822
#1940383