Finite Group Databasebeta

← #35322833

Subgroups — #35322833

Order 21248.

422 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581737C211yesmaximal, characteristic
2C2×#19127531C211yesmaximal, characteristic, complemented
3C2×#19127518C211yesmaximal, characteristic, complemented
4C2×#19127515C211yesmaximal, characteristic
5#24908378C211yesmaximal, characteristic, complemented
6#24908373C211yesmaximal, characteristic
7C83×#2108C211yesmaximal, characteristic, complemented
8C22×Dic332C2211yescharacteristic
9C22×D664C2211yescharacteristic
10C2×#14581737—22no—
11C2×#14581737—21no—
12C2×#14581737—21no—
13C2×#14581737C414yescomplemented
14C2×#14581728C2211yescharacteristic
15C2×#14581728C2211yescharacteristic
16#19127709—21no—
17#19127705—21no—
18C2×#14581710—21no—
19C2×#14581708—21no—
20#19127531—22no—
21#19127518—22no—
22#19127515—22no—
23C2×M32×C83C2211yescharacteristic
24C22×C16×C83C2211yescharacteristic
25#405×C83C2211yescharacteristic
26#346×C83—21no—
27C4×C16×C83—21no—
28C22×Dic166—21no—
29C22×D332—21no—
30C2×Dic332—21no—
31C2×Dic332—21no—
32C2×Dic332—21no—
33C2×Dic332—21no—
34C2×Dic332—21no—
35C2×Dic332C2×C412yes—
36C2×D664—21no—
37C2×D664—21no—
38C2×D664—21no—
39C2×D664—21no—
40C2×D664—21no—
41C2×D664C2×C412yes—
42C2×#13050574—21no—
43C2×#13050574—21no—
44#14581737—24no—
45#14581737—24no—
46#14581728—21no—
47#14581728—21no—
48#14581728—21no—
49#14581728—21no—
50#14581715—42no—
51#14581712—42no—
52#14581710—42no—
53#14581708—42no—
54C2×M16×C83D411yescharacteristic
55C22×C8×C83C2311yescharacteristic
56#117×C83D411yescharacteristic
57M32×C83—21no—
58M32×C83D412yes—
59C2×C16×C83—21no—
60C2×C16×C83C2×C412yes—
61C2×C16×C83D412yes—
62C2×C16×C83D411yescharacteristic
63C2×C16×C83D411yescharacteristic
64#97×C83—21no—
65C4×C8×C83—21no—
66C23×D83—41no—
67C2×Dic166—22no—
68C2×Dic166—41no—
69C2×Dic166—41no—
70C2×Dic166—41no—
71C2×Dic166—41no—
72C2×Dic166—41no—
73C2×D332—22no—
74C2×D332—41no—
75C2×D332—41no—
76C2×D332—41no—
77C2×D332—41no—
78C2×D332—41no—
79C22×Dic83—41no—
80Dic332—22no—
81Dic332—22no—
82Dic332—41no—
83Dic332—41no—
84D664—22no—
85D664—22no—
86D664—41no—
87D664—41no—
88#13050574—41no—
89#13050574—41no—
90#13050574—41no—
91#13050574—41no—
92C2×C83⋊C8—41no—
93C2×C83⋊C8—41no—
94C22×C4×C83C2×D411yescharacteristic
95M16×C83—21no—
96M16×C83D812yes—
97C2×C8×C83—21no—
98C2×C8×C83C22×C411yescharacteristic
99C2×C8×C83C2×D411yescharacteristic
100C2×C8×C83C2×D411yescharacteristic
101C2×C8×C83C2×D411yescharacteristic
102C2×C8×C83#4111yescharacteristic
103C2×C8×C83#4111yescharacteristic
104C4⋊C4×C83D812yes—
105C22⋊C4×C83—21no—
106C42×C83—21no—
107C16×C83—22no—
108C16×C83C22⋊C412yes—
109C16×C83C22⋊C412yes—
110C23×C83#13511yescharacteristic
111C22×D83—42no—
112C22×D83—81no—
113C22×D83—81no—
114C22×D83—162no—
115C2×C4×C83—22no—
116C2×C4×C83C4×D411yescharacteristic
117C2×C4×C83#12111yescharacteristic
118C2×C4×C83#12711yescharacteristic
119C2×C4×C83C2×D811yescharacteristic
120C2×C4×C83C2×D811yescharacteristic
121C2×C4×C83#13511yescharacteristic
122C2×Dic83—42no—
123C2×Dic83—81no—
124C2×Dic83—81no—
125D332—42no—
126D332—42no—
127D332—81no—
128D332—81no—
129Dic166—42no—
130Dic166—42no—
131Dic166—81no—
132Dic166—81no—
133C8×C83—22no—
134C8×C83C2×C22⋊C411yescharacteristic
135C8×C83C4×D411yescharacteristic
136C8×C83#12111yescharacteristic
137C8×C83#12411yescharacteristic
138C83⋊C8—81no—
139C83⋊C8—81no—
140C22×C83—22no—
141C22×C83C4×D811yescharacteristic
142C22×C83#46611yescharacteristic
143C22×C83#49511yescharacteristic
144C2×D83—82no—
145C2×D83—162no—
146C2×D83—162no—
147C2×D83—162no—
148C4×C83—42no—
149C4×C83#39011yescharacteristic
150C4×C83C4×D811yescharacteristic
151C4×C83#46611yescharacteristic
152C4×C83#49311yescharacteristic
153Dic83—82no—
154#13246—831nomaximal, Sylow
155C2×C83—21no—
156C2×C83—21no—
157C2×C83#197011yescharacteristic
158C2×C83#222312yes—
159D83—162no—
160C22×SD32—831no—
161C2×#368—831no—
162C2×#362—831no—
163C2×#361—831no—
164#2108D8311yescharacteristic, complemented
165#1979—831no—
166#1978—831no—
167C83#1324611yescharacteristic, complemented, Sylow
168C22×Q16—831no—
169C22×D8—831no—
170C2×SD32—834no—
171C2×SD32—1662no—
172C2×SD32—1661no—
173C2×SD32—1661no—
174C2×M32C2×D8311yescharacteristic
175C22×C16C2×D8311yescharacteristic
176C2×#108—831no—
177C2×#108—831no—
178#419—1661no—
179#418—1661no—
180C2×#101—1661no—
181C2×#100—1661no—
182#405C2×D8311yescharacteristic
183#368—1662no—
184#362—1662no—
185#361—1662no—
186#346—21no—
187C4×C16—21no—
188C22×Q8—1661no—
189C22×D4—1661no—
190C2×Q16—832no—
191C2×Q16—1661no—
192C2×Q16—1661no—
193C2×Q16—1661no—
194C2×Q16—1661no—
195C2×Q16—1661no—
196C2×D8—832no—
197C2×D8—1661no—
198C2×D8—1661no—
199C2×D8—1661no—
200C2×D8—1661no—
201C2×D8—1661no—
202C2×M16—1661no—
203C2×M16—1661no—
204C2×M16D33211yescharacteristic
205C22×C8C22×D8311yescharacteristic
206#117D33211yescharacteristic
207SD32—1664no—
208SD32—1664no—
209M32—21no—
210M32D33212yes—
211C2×C16—21no—
212C2×C16C4×D8312yes—
213C2×C16D33212yes—
214C2×C16C22⋊D8311yescharacteristic
215C2×C16C22⋊D8311yescharacteristic
216#108—1661no—
217#108—1661no—
218#108—1661no—
219#108—1661no—
220#103—3322no—
221#102—3322no—
222#101—3322no—
223#100—3322no—
224#97—21no—
225C4×C8—21no—
226C24—3321no—
227C2×Q8—3321no—
228C2×Q8—3321no—
229C2×Q8—3321no—
230C2×Q8—3321no—
231C2×Q8—3321no—
232C2×Q8—1662no—
233C2×D4—3321no—
234C2×D4—3321no—
235C2×D4—3321no—
236C2×D4—3321no—
237C2×D4—3321no—
238C2×D4—1662no—
239C22×C4—3321no—
240C22×C4C2×D33211yescharacteristic
241Q16—3321no—
242Q16—3321no—
243Q16—1662no—
244Q16—1662no—
245D8—3321no—
246D8—3321no—
247D8—1662no—
248D8—1662no—
249M16—21no—
250M16—3321no—
251M16—3321no—
252M16—3321no—
253M16—3321no—
254M16D66412yes—
255C2×C8—21no—
256C2×C8—3321no—
257C2×C8—3321no—
258C2×C8C2×C4×D8311yescharacteristic
259C2×C8C2×C22⋊D8311yescharacteristic
260C2×C8C2×D33211yescharacteristic
261C2×C8C2×D33211yescharacteristic
262C2×C8#1305059011yescharacteristic
263C2×C8#1305059011yescharacteristic
264C4⋊C4D66412yes—
265C22⋊C4—21no—
266C42—21no—
267C16—22no—
268C16(C2×D83)⋊C412yes—
269C16(C2×D83)⋊C412yes—
270C23—13282no—
271C23—3322no—
272C23—6641no—
273C23—6641no—
274C23#1458181411yescharacteristic
275Q8—3322no—
276Q8—3322no—
277Q8—6641no—
278Q8—6641no—
279D4—3322no—
280D4—3322no—
281D4—6641no—
282D4—6641no—
283C2×C4—22no—
284C2×C4—3322no—
285C2×C4—6641no—
286C2×C4—6641no—
287C2×C4C4×D33211yescharacteristic
288C2×C4#1458176611yescharacteristic
289C2×C4#1458178911yescharacteristic
290C2×C4C2×D66411yescharacteristic
291C2×C4C2×D66411yescharacteristic
292C2×C4#1458181411yescharacteristic
293C8—22no—
294C8—6641no—
295C8—6641no—
296C8C2×(C2×D83)⋊C411yescharacteristic
297C8C4×D33211yescharacteristic
298C8#1458176611yescharacteristic
299C8#1458178011yescharacteristic
300C22—22no—
301C22—13282no—
302C22—13282no—
303C22—13282no—
304C22—6642no—
305C22C4×D66411yescharacteristic
306C22#1912793711yescharacteristic
307C22#1912808511yescharacteristic
308C4—42no—
309C4—6642no—
310C4#1912760311yescharacteristic
311C4C4×D66411yescharacteristic
312C4#1912793711yescharacteristic
313C4#1912807811yescharacteristic
314C2—21no—
315C2—21no—
316C2—13282no—
317C2#2490832911yescharacteristic
318C2#2490959312yes—
subgroupG/H
centerC22#19127937
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#2108C2
socleC22×C83#466
prime radicalC22×D664C22
cosocleC22×C8×C83C23
hypercenter#2108D83
hypocenterC83#13246
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
2. C2×#19127531C2semidirect
3. C2×#19127518C2semidirect
5. #24908378C2semidirect
7. C83×#2108C2semidirect
13. C2×#14581737C4semidirect
164. #2108D83semidirect
167. C83#13246semidirect

Maximal subgroups

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

subgroupindex
C22×#145817372
C2×#191275312
C2×#191275182
C2×#191275152
#249083782
#249083732
C83×#21082
#1324683