Finite Group Databasebeta

← #35322782

Subgroups — #35322782

Order 21248.

474 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581737C211yesmaximal, characteristic
2C2×#19127504C211yesmaximal, characteristic, complemented
3C2×#19127501C211yesmaximal, characteristic
4C2×C16×Dic83C211yesmaximal, characteristic, complemented
5#24908327C211yesmaximal, characteristic
6#24908322C211yesmaximal, characteristic, complemented
7C2×#367×C83C211yesmaximal, characteristic, complemented
8C22×Dic332C2211yescharacteristic
9C22×D664C2211yescharacteristic
10C2×#14581737—22no—
11C2×#14581737—22no—
12C2×#14581737C414yescomplemented
13C22×C83⋊C16C2211yescharacteristic
14C2×#14581715—21no—
15C2×#14581712—21no—
16C2×C8×Dic83C2211yescharacteristic
17#19127606—21no—
18#19127602—21no—
19#19127504—22no—
20#19127501—22no—
21C16×Dic83—22no—
22C22×C16×C83C2211yescharacteristic
23C2×#107×C83C2211yescharacteristic
24C2×#107×C83C2211yescharacteristic
25#367×C83—22no—
26C22×Dic166—21no—
27C22×D332—21no—
28C2×Dic332—21no—
29C2×Dic332—21no—
30C2×Dic332—21no—
31C2×Dic332—21no—
32C2×Dic332—21no—
33C2×Dic332C2×C412yes—
34C2×D664—21no—
35C2×D664—21no—
36C2×D664—21no—
37C2×D664—21no—
38C2×D664—21no—
39C2×D664C2×C412yes—
40C22×C83⋊C8D411yescharacteristic
41C2×(C2×D83)⋊C4—42no—
42C2×C4×Dic83D411yescharacteristic
43#14581737—28no—
44C2×C83⋊C16—21no—
45C2×C83⋊C16—21no—
46C2×C83⋊C16D412yes—
47#14581715—42no—
48#14581712—42no—
49C8×Dic83—21no—
50C8×Dic83—21no—
51C83⋊#95—42no—
52C22×C8×C83C2311yescharacteristic
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C16×C83C2×C412yes—
56C2×C16×C83D412yes—
57C2×C16×C83D412yes—
58#107×C83—21no—
59#107×C83—21no—
60#107×C83—21no—
61#107×C83—21no—
62C23×D83—41no—
63C2×Dic166—22no—
64C2×Dic166—41no—
65C2×Dic166—41no—
66C2×Dic166—41no—
67C2×Dic166—41no—
68C2×Dic166—41no—
69C2×D332—22no—
70C2×D332—41no—
71C2×D332—41no—
72C2×D332—41no—
73C2×D332—41no—
74C2×D332—41no—
75C22×Dic83—41no—
76C22×Dic83D812yes—
77Dic332—22no—
78Dic332—22no—
79Dic332—41no—
80Dic332—41no—
81D664—22no—
82D664—22no—
83D664—41no—
84D664—41no—
85C2×C83⋊C8—21no—
86C2×C83⋊C8—21no—
87C2×C83⋊C8D812yes—
88(C2×D83)⋊C4—84no—
89C4×Dic83—21no—
90C4×Dic83—21no—
91C83⋊C16—22no—
92C22×C4×C83—41no—
93C22×C4×C83—41no—
94C22×C4×C83C2×D411yescharacteristic
95C2×C8×C83C22×C411yescharacteristic
96C2×C8×C83C2×D411yescharacteristic
97C2×C8×C83C2×D411yescharacteristic
98C2×C8×C83C2×D411yescharacteristic
99C2×C8×C83#4111yescharacteristic
100C2×C8×C83#4111yescharacteristic
101C4⋊C4×C83—41no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C16×C83C22⋊C414yescomplemented
106C23×C83C2×D811yescharacteristic
107C22×D83—42no—
108C22×D83—81no—
109C22×D83—81no—
110C22×D83—82no—
111C22×D83—82no—
112C2×C4×C83—81no—
113C2×C4×C83—81no—
114C2×C4×C83—82no—
115C2×C4×C83—82no—
116C2×C4×C83C4×D411yescharacteristic
117C2×C4×C83#12111yescharacteristic
118C2×C4×C83#12711yescharacteristic
119C2×C4×C83C2×D811yescharacteristic
120C2×C4×C83#13511yescharacteristic
121C2×C4×C83#13511yescharacteristic
122C2×Dic83—22no—
123C2×Dic83—22no—
124C2×Dic83—42no—
125C2×Dic83—81no—
126C2×Dic83—81no—
127C2×Dic83SD3214yescomplemented
128D332—42no—
129D332—42no—
130D332—81no—
131D332—81no—
132Dic166—42no—
133Dic166—42no—
134Dic166—81no—
135Dic166—81no—
136C8×C83C2×C22⋊C411yescharacteristic
137C8×C83C4×D411yescharacteristic
138C8×C83#12111yescharacteristic
139C8×C83#12411yescharacteristic
140C83⋊C8—22no—
141C22×C83C4×D811yescharacteristic
142C22×C83#46611yescharacteristic
143C22×C83#49311yescharacteristic
144C22×C83C2×SD3212yes—
145C22×C83#50812yes—
146C2×D83—84no—
147C2×D83—82no—
148C2×D83—162no—
149C2×D83—161no—
150C2×D83—161no—
151C4×C83—161no—
152C4×C83—161no—
153C4×C83#39011yescharacteristic
154C4×C83C4×D811yescharacteristic
155C4×C83#46611yescharacteristic
156C4×C83#49511yescharacteristic
157Dic83—24no—
158Dic83—82no—
159#13237—831nomaximal, Sylow
160C2×C83#197011yescharacteristic
161C2×C83C4×SD3212yes—
162C2×C83#221712yes—
163C2×C83#224811yescharacteristic
164C2×C83#224911yescharacteristic
165D83—162no—
166C22×SD32—831no—
167C2×#367D8311yescharacteristic, complemented
168C2×#358—831no—
169C2×#357—831no—
170C2×C4×C16—831no—
171#1970—831no—
172#1969—831no—
173C83#1323711yescharacteristic, complemented, Sylow
174C22×Q16—831no—
175C22×D8—831no—
176C2×SD32—834no—
177C2×SD32—1662no—
178C2×SD32—1662no—
179C22×C16—831no—
180C22×C16C2×D8311yescharacteristic
181C2×#107C2×D8311yescharacteristic
182C2×#107C2×D8311yescharacteristic
183C2×#103—1661no—
184C2×#102—1661no—
185C2×C4×C8—831no—
186#391—1661no—
187#390—1661no—
188#367—22no—
189#358—1662no—
190#357—1662no—
191C4×C16—1662no—
192C22×Q8—1661no—
193C22×D4—1661no—
194C2×Q16—832no—
195C2×Q16—1661no—
196C2×Q16—1661no—
197C2×Q16—1661no—
198C2×Q16—1661no—
199C2×Q16—1661no—
200C2×D8—832no—
201C2×D8—1661no—
202C2×D8—1661no—
203C2×D8—1661no—
204C2×D8—1661no—
205C2×D8—1661no—
206C22×C8—831no—
207C22×C8C22×D8311yescharacteristic
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C22⋊C4—3322no—
211C2×C42—831no—
212SD32—1668no—
213C2×C16—832no—
214C2×C16—1661no—
215C2×C16—1661no—
216C2×C16C4×D8312yes—
217C2×C16D33212yes—
218C2×C16C22⋊D8312yes—
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#103—3322no—
224#102—3322no—
225C4×C8—1661no—
226C4×C8—1661no—
227#95—3322no—
228C24—3321no—
229C2×Q8—3321no—
230C2×Q8—3321no—
231C2×Q8—3321no—
232C2×Q8—3321no—
233C2×Q8—3321no—
234C2×Q8—1662no—
235C2×D4—3321no—
236C2×D4—3321no—
237C2×D4—3321no—
238C2×D4—3321no—
239C2×D4—3321no—
240C2×D4—1662no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—3321no—
244C22×C4—832no—
245C22×C4D4×D8311yescharacteristic
246Q16—3321no—
247Q16—3321no—
248Q16—1662no—
249Q16—1662no—
250D8—3321no—
251D8—3321no—
252D8—1662no—
253D8—1662no—
254C2×C8—832no—
255C2×C8—1661no—
256C2×C8—1661no—
257C2×C8C2×C4×D8311yescharacteristic
258C2×C8D4×D8311yescharacteristic
259C2×C8C2×C22⋊D8311yescharacteristic
260C2×C8C2×D33211yescharacteristic
261C2×C8#1305059111yescharacteristic
262C2×C8#1305059111yescharacteristic
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C22⋊C4—6644no—
268C42—1661no—
269C42—1661no—
270C16—1662no—
271C16(C2×D83)⋊C414yescomplemented
272C23—3322no—
273C23—6641no—
274C23—6642no—
275C23—6642no—
276C23—6641no—
277C23D8×D8311yescharacteristic
278Q8—3322no—
279Q8—3322no—
280Q8—6641no—
281Q8—6641no—
282D4—3322no—
283D4—3322no—
284D4—6641no—
285D4—6641no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—3322no—
291C2×C4—834no—
292C2×C4—6641no—
293C2×C4—6641no—
294C2×C4—1662no—
295C2×C4—1662no—
296C2×C4#1458175411yescharacteristic
297C2×C4#1458176711yescharacteristic
298C2×C4#1458178811yescharacteristic
299C2×C4D8×D8311yescharacteristic
300C2×C4#1458181511yescharacteristic
301C2×C4#1458181511yescharacteristic
302C8—1662no—
303C8C2×(C2×D83)⋊C411yescharacteristic
304C8#1458175411yescharacteristic
305C8#1458176711yescharacteristic
306C8#1458178111yescharacteristic
307C22—13281no—
308C22—13281no—
309C22—13282no—
310C22—6644no—
311C22—6642no—
312C22#1912778311yescharacteristic
313C22#1912793811yescharacteristic
314C22#1912807611yescharacteristic
315C22SD32×D8312yes—
316C22#1912813212yes—
317C4—161no—
318C4—161no—
319C4—6642no—
320C4—1664no—
321C4#1912760211yescharacteristic
322C4#1912778311yescharacteristic
323C4#1912793811yescharacteristic
324C4#1912808611yescharacteristic
325C2—13282no—
326C2#2490832711yescharacteristic
327C2#2490931412yes—
328C2#2490956012yes—
329C2#2490971711yescharacteristic
330C2#2490972311yescharacteristic
subgroupG/H
centerC23D8×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC2×#367×C83C2
socleC23×C83C2×D8
prime radicalC22×D664C22
cosocleC22×C8×C83C23
hypercenterC2×#367D83
hypocenterC83#13237
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#367

Complements

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

NH ≅ G/Ntype
2. C2×#19127504C2semidirect
4. C2×C16×Dic83C2semidirect
6. #24908322C2semidirect
7. C2×#367×C83C2semidirect
12. C2×#14581737C4semidirect
105. C16×C83C22⋊C4semidirect
127. C2×Dic83SD32semidirect
167. C2×#367D83semidirect
173. C83#13237semidirect
271. C16(C2×D83)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#145817372
C2×#191275042
C2×#191275012
C2×C16×Dic832
#249083272
#249083222
C2×#367×C832
#1323783