Finite Group Databasebeta

← #35389689

Subgroups — #35389689

Order 21248.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918390C211yesmaximal, characteristic
2C2×#19127986C211yesmaximal, characteristic, complemented
3#24916935C211yesmaximal, characteristic
4#24906893C214yesmaximal, complemented
5#24906811C214yesmaximal, complemented
6#24906234C211yesmaximal, characteristic, complemented
7#24906147C211yesmaximal, characteristic, complemented
8#24906110C211yesmaximal, characteristic
9C83×#2982C211yesmaximal, characteristic, complemented
10C2×#14581823—21no—
11#19128251C2211yescharacteristic
12C2×#14581792C2211yescharacteristic
13#19128074C2212yes—
14#19128052C2212yes—
15#19127986—22no—
16#19127986C2214yes—
17#19127923—22no—
18#19127911C2212yes—
19#19127907—22no—
20#19127901C2212yes—
21#19127842—22no—
22C2×#14581723—21no—
23#19127725C2211yescharacteristic
24C2×C4⋊(C83⋊C8)C2211yescharacteristic
25#19127700—21no—
26C2×#14581716—21no—
27#19127452C2214yes—
28#19127448C2214yes—
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×#13050579—24no—
42C2×#13050577—22no—
43C2×#13050574—21no—
44C22×C83⋊C8—21no—
45#14581792—21no—
46#14581792—21no—
47#14581792—21no—
48#14581792C2312yes—
49#14581792C2311yescharacteristic
50#14581789C2311yescharacteristic
51#14581780—21no—
52#14581766—21no—
53#14581766—21no—
54C4×D332—21no—
55C4×D332—21no—
56C2×C4⋊Dic83—21no—
57C2×C4⋊Dic83—21no—
58#14581723—24no—
59C4⋊(C83⋊C8)—21no—
60C4⋊(C83⋊C8)—21no—
61C4⋊(C83⋊C8)C2312yes—
62C4⋊(C83⋊C8)C2312yes—
63#14581717—24no—
64#14581716—24no—
65#14581715—22no—
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#13050579—84no—
98#13050577—44no—
99#13050574—42no—
100C2×C83⋊C8—22no—
101C2×C83⋊C8—22no—
102C2×C83⋊C8—42no—
103C4⋊Dic83—21no—
104C4⋊Dic83—22no—
105C4⋊Dic83—21no—
106C4⋊Dic83—22no—
107C4⋊Dic83—41no—
108C4⋊Dic83—41no—
109(C2×D83)⋊C4—41no—
110(C2×D83)⋊C4—41no—
111C2×Q8×C83—21no—
112C2×Q8×C83—22no—
113C2×Q8×C83SD1614yescomplemented
114C2×Q8×C83C2×D412yes—
115C22×C4×C83—21no—
116C22×C4×C83—21no—
117C22×C4×C83C2×D411yescharacteristic
118C22×C4×C83C2×D411yescharacteristic
119C22×C4×C83#4111yescharacteristic
120C2×C8×C83—22no—
121C2×C8×C83C2×D414yes—
122C4⋊C4×C83—22no—
123C4⋊C4×C83—21no—
124C4⋊C4×C83—22no—
125C4⋊C4×C83C2×D412yes—
126C42×C83—22no—
127C42×C83—22no—
128C42×C83C2×D411yescharacteristic
129C42×C83C2×D411yescharacteristic
130C42×C83#4111yescharacteristic
131C42×C83C2411yescharacteristic
132C23×C83#12111yescharacteristic
133C22×D83—41no—
134Q8×C83—22no—
135Q8×C83—24no—
136Q8×C83C2×SD1614yes—
137C2×C4×C83—22no—
138C2×C4×C83—24no—
139C2×C4×C83—22no—
140C2×C4×C83—22no—
141C2×C4×C83—41no—
142C2×C4×C83—81no—
143C2×C4×C83#12111yescharacteristic
144C2×C4×C83#12111yescharacteristic
145C2×C4×C83#12111yescharacteristic
146C2×C4×C83C2×SD1612yes—
147C2×C4×C83#13512yes—
148C2×C4×C83C22×D411yescharacteristic
149C2×C4×C83C22×D411yescharacteristic
150C2×C4×C83C2×#4111yescharacteristic
151C22⋊D83—82no—
152C2×Dic83—41no—
153C2×Dic83—42no—
154C2×Dic83—82no—
155D332—41no—
156D332—41no—
157D332—42no—
158D332—81no—
159C4×D83—82no—
160Dic166—41no—
161Dic166—41no—
162Dic166—44no—
163Dic166—82no—
164C8×C83—24no—
165C8×C83—42no—
166C83⋊C8—42no—
167C83⋊C8—42no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83#46512yes—
171C22×C83C2×#12111yescharacteristic
172C2×D83—81no—
173C2×D83—81no—
174C2×D83—81no—
175C4×C83—21no—
176C4×C83—24no—
177C4×C83—21no—
178C4×C83—21no—
179C4×C83—42no—
180C4×C83#46512yes—
181C4×C83C22×SD1611yescharacteristic
182C4×C83C2×#13511yescharacteristic
183C4×C83#57712yes—
184Dic83—82no—
185Dic83—81no—
186#19397—831nomaximal, Sylow
187C2×C83—22no—
188C2×C83#304311yescharacteristic
189C2×C83C2×#46511yescharacteristic
190C2×C83#306911yescharacteristic
191D83—161no—
192#3206—831no—
193C2×#475—831no—
194#3037—831no—
195#2982D8311yescharacteristic, complemented
196#1680—834no—
197#1666—834no—
198#1570—831no—
199#1549—831no—
200#1540—831no—
201C83#1939711yescharacteristic, complemented, Sylow
202C2×#137—1661no—
203#532—831no—
204C2×C4⋊Q8—831no—
205C2×C4×Q8C2×D8311yescharacteristic
206#492—832no—
207#487—832no—
208#475—834no—
209#475—1662no—
210#464—1662no—
211#462—832no—
212#461—1662no—
213#460—832no—
214#448—1662no—
215C8×Q8—22no—
216C8×Q8C2×D8314yes—
217#431—22no—
218#431C2×D8311yescharacteristic
219#431C2×D8311yescharacteristic
220C2×#106—1661no—
221#423—831no—
222C2×C4⋊C8—831no—
223#417—1661no—
224C2×#103—1661no—
225#334—834no—
226#332—832no—
227#332—834no—
228#331—832no—
229C2×#41—1661no—
230C22×Q8—1661no—
231C22×Q8D33211yescharacteristic
232#137—3324no—
233C2×Q16—1662no—
234C2×SD16—1662no—
235C2×SD16—1664no—
236C2×M16—1661no—
237C22×C8—1661no—
238C4⋊Q8—832no—
239C4⋊Q8—831no—
240C4⋊Q8—1661no—
241C4⋊Q8—1661no—
242C4⋊Q8—1661no—
243#127—831no—
244#124—1661no—
245#121—1661no—
246#121—1661no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D33212yes—
250C4×Q8C22×D8312yes—
251C4×D4—1661no—
252C4×D4—1661no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—1661no—
255C2×C4⋊C4—1661no—
256C2×C4⋊C4D33211yescharacteristic
257C2×C42—21no—
258C2×C42C22×D8311yescharacteristic
259#106—1664no—
260C4⋊C8—24no—
261C4⋊C8—832no—
262C4⋊C8—832no—
263C4⋊C8—1661no—
264C4⋊C8—1661no—
265C4⋊C8C22⋊D8312yes—
266C4⋊C8C22×D8312yes—
267#103—1664no—
268#103—1664no—
269#103—1662no—
270#102—1662no—
271C22⋊C8—21no—
272C22⋊C8C22⋊D8312yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C22×D8312yes—
276#41—3322no—
277#41—3322no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—3321no—
281C2×Q8—6642no—
282C2×Q8—1662no—
283C2×Q8—1661no—
284C2×Q8#1305057914yescomplemented
285C2×Q8C2×D33212yes—
286C2×D4—3321no—
287C2×D4—1661no—
288C2×D4—1661no—
289C2×D4—1661no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—3321no—
293C22×C4—3321no—
294C22×C4C2×C22⋊D8311yescharacteristic
295C22×C4C2×D33211yescharacteristic
296C22×C4#1305059011yescharacteristic
297Q16—3324no—
298SD16—3324no—
299SD16—6644no—
300M16—3322no—
301C2×C8—22no—
302C2×C8—3322no—
303C2×C8—1662no—
304C2×C8—1662no—
305C2×C8C2×C22⋊D8314yes—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—3321no—
310C4⋊C4—3321no—
311C4⋊C4—1661no—
312C4⋊C4—1662no—
313C4⋊C4—1662no—
314C4⋊C4—1661no—
315C4⋊C4C2×D33212yes—
316C22⋊C4—3321no—
317C22⋊C4—3321no—
318C42—22no—
319C42—22no—
320C42C2×C22⋊D8311yescharacteristic
321C42C2×D33211yescharacteristic
322C42#1305059011yescharacteristic
323C42C23×D8311yescharacteristic
324C23—3321no—
325C23#1458176611yescharacteristic
326Q8—24no—
327Q8—22no—
328Q8—3321no—
329Q8—3321no—
330Q8—3324no—
331Q8—6642no—
332Q8C2×#1305057914yes—
333D4—3321no—
334D4—3321no—
335D4—3322no—
336D4—6641no—
337D4—6642no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—3321no—
345C2×C4—3322no—
346C2×C4—6642no—
347C2×C4—6642no—
348C2×C4#1458176611yescharacteristic
349C2×C4#1458176611yescharacteristic
350C2×C4#1458176611yescharacteristic
351C2×C4C2×#1305057912yes—
352C2×C4#1458181412yes—
353C2×C4C22×C22⋊D8311yescharacteristic
354C2×C4C22×D33211yescharacteristic
355C2×C4C2×#1305059011yescharacteristic
356C8—24no—
357C8—42no—
358C8—3322no—
359C8—3322no—
360C22—21no—
361C22—21no—
362C22—6641no—
363C22—6641no—
364C22—6641no—
365C22#1912793012yes—
366C22C2×#1458176611yescharacteristic
367C4—24no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—6641no—
373C4—6642no—
374C4#1912793012yes—
375C4C22×#1305057911yescharacteristic
376C4C2×#1458181411yescharacteristic
377C4#1912850912yes—
378C2—22no—
379C2—13281no—
380C2#2491698411yescharacteristic
381C2C2×#1912793011yescharacteristic
382C2#2491720011yescharacteristic
subgroupG/H
centerC22C2×#14581766
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC83×#2982C2
socleC22×C83C2×#121
prime radical#19128251C22
hypercenter#2982D83
hypocenterC83#19397
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×#19127986C2semidirect
4. #24906893C2semidirect
5. #24906811C2semidirect
6. #24906234C2semidirect
7. #24906147C2semidirect
9. C83×#2982C2semidirect
113. C2×Q8×C83SD16semidirect
195. #2982D83semidirect
201. C83#19397semidirect
284. C2×Q8#13050579semidirect

Maximal subgroups

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

subgroupindex
#249183902
C2×#191279862
#249169352
#249068932
#249068112
#249062342
#249061472
#249061102
C83×#29822
#1939783