Finite Group Databasebeta

← #35298740

Subgroups — #35298740

Order 21248.

376 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
1#24916414C211yesmaximal, characteristic
2#24908239C211yesmaximal, characteristic
3#24906221C211yesmaximal, characteristic
4#24905893C211yesmaximal, characteristic
5C2×#19127431C211yesmaximal, characteristic
6#24905299C211yesmaximal, characteristic
7C83×#1301C211yesmaximal, characteristic
8C2×C4×Dic166C2211yescharacteristic
9#19127830—22no—
10#19127828—21no—
11#19127825—21no—
12C2×#14581728—21no—
13#19127725C2211yescharacteristic
14#19127724—21no—
15C4×#13050574C2211yescharacteristic
16M16×Dic83—21no—
17C2×C4×C83⋊C8C2211yescharacteristic
18#19127466—21no—
19#19127453—22no—
20#19127433—21no—
21#19127433—21no—
22#19127431C411yescharacteristic
23#19127431C411yescharacteristic
24#19127431C411yescharacteristic
25#19127431C411yescharacteristic
26#431×C83C2211yescharacteristic
27#423×C83C2211yescharacteristic
28C2×C4⋊C8×C83C2211yescharacteristic
29C22×Dic166Q811yescharacteristic
30C2×#13050574—21no—
31C2×#13050574D411yescharacteristic
32C2×#13050574D411yescharacteristic
33C22×C83⋊C8—21no—
34C4×Dic166—21no—
35C4×Dic166—21no—
36C4×Dic166C2×C411yescharacteristic
37C4×Dic166C2×C411yescharacteristic
38C4×Dic166D411yescharacteristic
39C4×Dic166D411yescharacteristic
40C2×C4⋊Dic83Q811yescharacteristic
41C2×Dic83⋊C4—21no—
42C2×C4×Dic83—21no—
43#14581728—42no—
44C4⋊(C83⋊C8)—21no—
45C4⋊(C83⋊C8)—21no—
46Dic83⋊C8—22no—
47Dic83⋊C8—22no—
48#14581703—21no—
49#14581702—22no—
50C4×C83⋊C8—21no—
51C4×C83⋊C8—21no—
52C4×C83⋊C8C2×C411yescharacteristic
53C4×C83⋊C8C2×C411yescharacteristic
54C8×Dic83—22no—
55C2×M16×C83—21no—
56C22×C8×C83—21no—
57C2×C42×C83C2311yescharacteristic
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83C2×C411yescharacteristic
62C4⋊C8×C83C2×C411yescharacteristic
63C4⋊C8×C83D412yes—
64C22⋊C8×C83—21no—
65C4×C8×C83—21no—
66C2×Dic166—22no—
67C2×Dic166—21no—
68C2×Dic166—41no—
69C2×Dic166C4⋊C411yescharacteristic
70C2×Dic166C4⋊C411yescharacteristic
71C22×Dic83—21no—
72C22×Dic83—21no—
73#13050574—21no—
74#13050574—21no—
75#13050574—21no—
76#13050574—41no—
77#13050574—41no—
78#13050574Q1612yes—
79C2×C83⋊C8—21no—
80C2×C83⋊C8—21no—
81C2×C83⋊C8—21no—
82C2×C83⋊C8—21no—
83C2×C83⋊C8—42no—
84C2×C83⋊C8—41no—
85C4⋊Dic83—21no—
86C4⋊Dic83C4⋊C411yescharacteristic
87C4⋊Dic83C4⋊C411yescharacteristic
88Dic83⋊C4—21no—
89Dic83⋊C4—21no—
90Dic83⋊C4—21no—
91Dic83⋊C4—21no—
92C4×Dic83—21no—
93C4×Dic83—21no—
94C4×Dic83—21no—
95C4×Dic83—21no—
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83C2×Q811yescharacteristic
98C22×C4×C83#4111yescharacteristic
99M16×C83—41no—
100M16×C83—41no—
101C2×C8×C83—21no—
102C2×C8×C83—21no—
103C2×C8×C83—22no—
104C2×C8×C83—41no—
105C2×C8×C83—41no—
106C2×C8×C83—41no—
107C42×C83C22×C411yescharacteristic
108C42×C83C2×D411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83#4111yescharacteristic
111C23×C83C22⋊Q811yescharacteristic
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—41no—
115C2×C4×C83C2×C4⋊C411yescharacteristic
116C2×C4×C83C4×D411yescharacteristic
117C2×C4×C83C4×D411yescharacteristic
118C2×C4×C83#12011yescharacteristic
119C2×C4×C83C22⋊Q811yescharacteristic
120C2×C4×C83#12311yescharacteristic
121C2×C4×C83C2×Q1611yescharacteristic
122C2×C4×C83#13511yescharacteristic
123C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C83#13711yescharacteristic
125C2×Dic83—21no—
126C2×Dic83—21no—
127C2×Dic83—21no—
128C2×Dic83—21no—
129C2×Dic83—42no—
130C2×Dic83—81no—
131Dic166—21no—
132Dic166—21no—
133Dic166—21no—
134Dic166—42no—
135Dic166#10812yescomplemented
136C8×C83—41no—
137C8×C83—41no—
138C8×C83—42no—
139C8×C83—81no—
140C83⋊C8—22no—
141C83⋊C8—22no—
142C83⋊C8—41no—
143C83⋊C8—41no—
144C83⋊C8—81no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83#38511yescharacteristic
148C22×C83#47711yescharacteristic
149C22×C83#47911yescharacteristic
150C4×C83—21no—
151C4×C83—21no—
152C4×C83—21no—
153C4×C83C2×#10811yescharacteristic
154C4×C83#43011yescharacteristic
155C4×C83C4×Q1611yescharacteristic
156C4×C83#44011yescharacteristic
157C4×C83#44911yescharacteristic
158C4×C83#45111yescharacteristic
159Dic83—22no—
160Dic83—41no—
161Dic83—41no—
162Dic83—41no—
163#8907—831nomaximal, Sylow
164C2×C83—21no—
165C2×C83—21no—
166C2×C83#185511yescharacteristic
167C2×C83#186111yescharacteristic
168C2×C83#186311yescharacteristic
169Q8×M16—831no—
170#1950—831no—
171#1567—831no—
172#1487—831no—
173C2×Q8⋊C8—831no—
174#1301D8311yescharacteristic
175#1278—831no—
176C83#890711yescharacteristic, complemented, Sylow
177C2×C4×Q8—831no—
178#446—1661no—
179#446—1662no—
180C8×Q8—1661no—
181#431C2×D8311yescharacteristic
182C2×#108—1661no—
183#423—831no—
184#423—1661no—
185#423C2×D8311yescharacteristic
186C2×C4⋊C8C2×D8311yescharacteristic
187C4×M16—831no—
188C4×M16—1661no—
189C2×C4×C8—831no—
190#341—1661no—
191#335—1662no—
192Q8⋊C8—831no—
193Q8⋊C8—831no—
194Q8⋊C8—831no—
195Q8⋊C8—831no—
196Q8⋊C8—1661no—
197Q8⋊C8—1661no—
198C22×Q8—831no—
199C2×M16—21no—
200C2×M16—831no—
201C2×M16—831no—
202C2×M16—1661no—
203C22×C8—21no—
204C22×C8—1661no—
205C4×Q8—831no—
206C4×Q8—831no—
207C4×Q8—831no—
208C4×Q8—831no—
209C4×Q8—1661no—
210C4×Q8—1661no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—1661no—
213C2×C42—1661no—
214C2×C42C22×D8311yescharacteristic
215#108—3322no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—1662no—
220C4⋊C8—1662no—
221C4⋊C8—1661no—
222C4⋊C8—1661no—
223C4⋊C8C4×D8311yescharacteristic
224C4⋊C8C4×D8311yescharacteristic
225C4⋊C8D33212yes—
226C22⋊C8—21no—
227#97—1662no—
228#97—1661no—
229C4×C8—21no—
230C4×C8—831no—
231C4×C8—831no—
232C4×C8—1662no—
233C4×C8—1661no—
234C4×C8—1661no—
235C2×Q8—3321no—
236C2×Q8—831no—
237C2×Q8—831no—
238C2×Q8—1662no—
239C2×Q8—1661no—
240C22×C4—1661no—
241C22×C4—1661no—
242C22×C4D4×D8311yescharacteristic
243C22×C4Q8×D8311yescharacteristic
244C22×C4#1305059011yescharacteristic
245M16—41no—
246M16—41no—
247M16—3321no—
248M16—3321no—
249M16—832no—
250M16—1661no—
251M16—1661no—
252M16—1661no—
253C2×C8—22no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—3322no—
260C2×C8—3321no—
261C2×C8—1661no—
262C2×C8—1661no—
263C2×C8—1661no—
264C2×C8—1661no—
265C4⋊C4—831no—
266C4⋊C4—831no—
267C4⋊C4—1661no—
268C4⋊C4—1661no—
269C4⋊C4—1661no—
270C4⋊C4—1661no—
271C4⋊C4—1661no—
272C42—1661no—
273C42—1661no—
274C42—1661no—
275C42—1661no—
276C42C2×C4×D8311yescharacteristic
277C42D4×D8311yescharacteristic
278C42C2×D33211yescharacteristic
279C42#1305058911yescharacteristic
280C23#1458177011yescharacteristic
281Q8—3322no—
282Q8—832no—
283Q8—1661no—
284Q8—1661no—
285Q8—1661no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—3322no—
290C2×C4—6641no—
291C2×C4—1661no—
292C2×C4—1661no—
293C2×C4—1661no—
294C2×C4—1661no—
295C2×C4C4⋊C4×D8311yescharacteristic
296C2×C4(C22⋊D83)⋊C411yescharacteristic
297C2×C4C4×D33211yescharacteristic
298C2×C4#1458176211yescharacteristic
299C2×C4#1458177211yescharacteristic
300C2×C4#1458177411yescharacteristic
301C2×C4Q16×D8311yescharacteristic
302C2×C4#1458181311yescharacteristic
303C2×C4#1458181611yescharacteristic
304C2×C4#1458182111yescharacteristic
305C8—42no—
306C8—41no—
307C8—41no—
308C8—81no—
309C8—3321no—
310C8—3321no—
311C8—6641no—
312C8—1662no—
313C8—1662no—
314C22—21no—
315C22—21no—
316C22#1912757511yescharacteristic
317C22#1912799511yescharacteristic
318C22#1912801111yescharacteristic
319C4—21no—
320C4—21no—
321C4—21no—
322C4—3321no—
323C4—3321no—
324C4—3321no—
325C4—1662no—
326C4#108×D8311yescharacteristic
327C4#1912774811yescharacteristic
328C4#1912779311yescharacteristic
329C4#1912779811yescharacteristic
330C4#1912784711yescharacteristic
331C4#1912785811yescharacteristic
332C2—21no—
333C2—21no—
334C2#2490766911yescharacteristic
335C2#2490770611yescharacteristic
336C2#2490771811yescharacteristic
subgroupG/H
centerC2×C4#14581762
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1301C2
socleC22×C83#385
prime radicalC23×C83C22⋊Q8
cosocleC2×C42×C83C23
hypercenter#1301D83
hypocenterC83#8907
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
135. Dic166#108semidirect
176. C83#8907semidirect

Maximal subgroups

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

subgroupindex
#249164142
#249082392
#249062212
#249058932
C2×#191274312
#249052992
C83×#13012
#890783