Finite Group Databasebeta

← #35300125

Subgroups — #35300125

Order 21248.

738 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Dic83C211yesmaximal, characteristic
2#24907531C211yesmaximal, characteristic
3C2×#19127438C212yesmaximal
4#24905341C212yesmaximal
5C83×#2855C211yesmaximal, characteristic
6C22×C4⋊Dic83—21no—
7C22×C4×Dic83—21no—
8C2×C22⋊(C83⋊C8)C2212yes—
9C4×C4⋊Dic83—24no—
10C2×C83⋊#95C2212yes—
11#19127456—22no—
12#19127438—24no—
13C2×C22⋊Q8×C83C2211yescharacteristic
14C22×C42×C83C2211yescharacteristic
15#391×C83C412yescomplemented
16#389×C83C412yescomplemented
17#387×C83—22no—
18#385×C83—21no—
19#381×C83—21no—
20#380×C83C2211yescharacteristic
21C23×Dic83—22no—
22C22×C83⋊C8—22no—
23C2×C4⋊Dic83—24no—
24C2×C4⋊Dic83—28no—
25C2×C4×Dic83—24no—
26C2×C4×Dic83—42no—
27C2×C4×Dic83—42no—
28C22⋊(C83⋊C8)—22no—
29C22⋊(C83⋊C8)—22no—
30C83⋊#95—22no—
31C83⋊#95D414yes—
32C22×Q8×C83C2×C411yescharacteristic
33C23×C4×C83D411yescharacteristic
34C23×C4×C83D411yescharacteristic
35C23×C4×C83C2311yescharacteristic
36C22⋊Q8×C83—22no—
37C22⋊Q8×C83—22no—
38C2×C4⋊C4×C83—22no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2×C412yes—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C42×C83—22no—
45C2×C42×C83—22no—
46C2×C42×C83C2×C412yes—
47C2×C42×C83D412yes—
48#95×C83—22no—
49#95×C83—22no—
50#95×C83—22no—
51#95×C83—21no—
52C22×Dic83—28no—
53C22×Dic83—24no—
54C22×Dic83—44no—
55C22×Dic83—42no—
56C22×Dic83—42no—
57C2×C83⋊C8—44no—
58C2×C83⋊C8—42no—
59C4⋊Dic83—216no—
60C4×Dic83—48no—
61C24×C83C2×D411yescharacteristic
62C2×Q8×C83—22no—
63C2×Q8×C83—42no—
64C22×C4×C83—21no—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—24no—
71C22×C4×C83—22no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—42no—
75C22×C4×C83C22⋊C412yes—
76C22×C4×C83C22⋊C412yes—
77C22×C4×C83SD1612yes—
78C22×C4×C83Q1612yes—
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D412yes—
81C22×C4×C83C2×D411yescharacteristic
82C22×C4×C83#4112yes—
83C4⋊C4×C83—22no—
84C4⋊C4×C83—42no—
85C4⋊C4×C83—44no—
86C4⋊C4×C83—42no—
87C22⋊C4×C83—41no—
88C22⋊C4×C83—41no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C42×C83—24no—
92C42×C83—24no—
93C23×C83—22no—
94C23×C83—22no—
95C23×C83C2×C22⋊C411yescharacteristic
96C23×C83#12011yescharacteristic
97C23×C83#12311yescharacteristic
98C23×C83C2×SD1611yescharacteristic
99C23×C83C2×Q1611yescharacteristic
100C23×C83AGL(1,ℤ/8)11yescharacteristic
101C23×C83#13711yescharacteristic
102Q8×C83—82no—
103C2×C4×C83—24no—
104C2×C4×C83—24no—
105C2×C4×C83—22no—
106C2×C4×C83—22no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—42no—
120C2×C4×C83—42no—
121C2×C4×C83—42no—
122C2×C4×C83—42no—
123C2×C4×C83—42no—
124C2×C4×C83#10314yes—
125C2×C4×C83#10414yes—
126C2×C4×C83C4×D412yes—
127C2×C4×C83#12112yes—
128C2×Dic83—28no—
129C2×Dic83—44no—
130C2×Dic83—48no—
131C2×Dic83—48no—
132C2×Dic83—44no—
133C83⋊C8—82no—
134C22×C83—21no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—22no—
139C22×C83—22no—
140C22×C83—22no—
141C22×C83—22no—
142C22×C83#38611yescharacteristic
143C22×C83#41012yes—
144C22×C83C2×#10311yescharacteristic
145C22×C83#41711yescharacteristic
146C22×C83C2×#10412yes—
147C22×C83#45011yescharacteristic
148C22×C83#45111yescharacteristic
149C22×C83#48311yescharacteristic
150C22×C83#48411yescharacteristic
151C4×C83—24no—
152C4×C83—24no—
153C4×C83—22no—
154C4×C83—22no—
155C4×C83—81no—
156C4×C83—81no—
157C4×C83—82no—
158Dic83—48no—
159#9144—831nomaximal, Sylow
160C2×C83—21no—
161C2×C83—21no—
162C2×C83—21no—
163C2×C83—21no—
164C2×C83#150312yes—
165C2×C83#150912yes—
166C2×C83#187211yescharacteristic
167C2×C83#189611yescharacteristic
168C2×C83#189911yescharacteristic
169#2855D8311yescharacteristic
170C2×C4×C4⋊C4—831no—
171#1817—831no—
172C2×#328—832no—
173#1294—832no—
174C83#914411yescharacteristic, complemented, Sylow
175C2×C22⋊Q8C2×D8311yescharacteristic
176C22×C4⋊C4—1661no—
177C22×C42—1661no—
178C22×C42C2×D8311yescharacteristic
179C2×C22⋊C8—832no—
180#391Dic8312yescomplemented
181#389Dic8312yescomplemented
182#387—22no—
183#385—21no—
184#381—21no—
185#380C2×D8311yescharacteristic
186C4×C4⋊C4—1664no—
187C2×#95—832no—
188#336—1662no—
189#328—1664no—
190C22×Q8C2×Dic8311yescharacteristic
191C23×C4—1662no—
192C23×C4C22⋊D8311yescharacteristic
193C23×C4C22⋊D8311yescharacteristic
194C23×C4C22×D8311yescharacteristic
195C22×C8—1662no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—1664no—
201C2×C4⋊C4—1668no—
202C2×C4⋊C4C2×Dic8311yescharacteristic
203C2×C4⋊C4C2×Dic8312yes—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—3322no—
209C2×C42—3322no—
210C2×C42—1664no—
211C2×C42C2×Dic8312yes—
212C2×C42C22⋊D8312yes—
213C22⋊C8—1662no—
214C22⋊C8—1662no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—834no—
220#95—1662no—
221C24C2×C22⋊D8311yescharacteristic
222C2×Q8—22no—
223C2×Q8—42no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—3324no—
236C22×C4—3322no—
237C22×C4—3322no—
238C22×C4—1668no—
239C22×C4—1664no—
240C22×C4C22⋊Dic8312yes—
241C22×C4C22⋊Dic8312yes—
242C22×C4#1305057712yes—
243C22×C4#1305058012yes—
244C22×C4C22×Dic8311yescharacteristic
245C22×C4D4×D8312yes—
246C22×C4C2×C22⋊D8311yescharacteristic
247C22×C4#1305058912yes—
248C2×C8—3324no—
249C2×C8—3322no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—16616no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C42—24no—
260C42—24no—
261C42—3328no—
262C23—22no—
263C23—22no—
264C23C2×C22⋊Dic8311yescharacteristic
265C23#1458176011yescharacteristic
266C23#1458177311yescharacteristic
267C23C2×#1305057711yescharacteristic
268C23C2×#1305058011yescharacteristic
269C23#1458181711yescharacteristic
270C23#1458182211yescharacteristic
271Q8—82no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—3328no—
294C2×C4—3324no—
295C2×C4—3328no—
296C2×C4—3324no—
297C2×C4—1668no—
298C2×C4#1458171614yes—
299C2×C4#1458171914yes—
300C2×C4D4×Dic8312yes—
301C2×C4#1458176412yes—
302C8—6642no—
303C22—22no—
304C22—22no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—22no—
311C22#1912758111yescharacteristic
312C22#1912767512yes—
313C22C2×#1458171611yescharacteristic
314C22#1912770211yescharacteristic
315C22C2×#1458171912yes—
316C22#1912785111yescharacteristic
317C22#1912785611yescharacteristic
318C22#1912802911yescharacteristic
319C22#1912803411yescharacteristic
320C4—24no—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—82no—
325C4—81no—
326C4—81no—
327C4—3328no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2490596212yes—
333C2#2490598812yes—
334C2#2490777111yescharacteristic
335C2#2490790711yescharacteristic
336C2#2490792211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2855C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#2855D83
hypocenterC83#9144
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
15. #391×C83C4semidirect
16. #389×C83C4semidirect
174. C83#9144semidirect
180. #391Dic83semidirect
181. #389Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Dic832
#249075312
C2×#191274382
#249053412
C83×#28552
#914483