Finite Group Databasebeta

← #35299059

Subgroups — #35299059

Order 21248.

722 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14581788C211yesmaximal, characteristic
2#24905853C212yesmaximal, complemented
3#24905784C211yesmaximal, characteristic, complemented
4#24905295C212yesmaximal, complemented
5C83×#2972C211yesmaximal, characteristic, complemented
6C2×#14581788C411yescharacteristic
7C2×C4×D332C2211yescharacteristic
8C2×C4×C22⋊D83—22no—
9#19127754C2212yes—
10#19127604C411yescharacteristic
11#19127602C411yescharacteristic
12#19127601—22no—
13#19127570C411yescharacteristic
14C42×Dic83C2211yescharacteristic
15#19127428—24no—
16#19127425—22no—
17C2×C4×D4×C83C2211yescharacteristic
18C8×D4×C83—24no—
19#431×C83C2212yes—
20C2×C22⋊C8×C83—22no—
21C22×D332C2×C411yescharacteristic
22C22×C22⋊D83—22no—
23C22×C4×D83—21no—
24#14581788—22no—
25#14581788C814yescomplemented
26C4×D332—22no—
27C4×D332—22no—
28C4×C22⋊D83—48no—
29C2×C4⋊Dic83C2×C411yescharacteristic
30C2×Dic83⋊C4—22no—
31C2×C22⋊Dic83—22no—
32C2×(C2×D83)⋊C4—22no—
33C2×(C2×D83)⋊C4—21no—
34C2×C4×Dic83—21no—
35C2×C4×Dic83—21no—
36C2×C4×Dic83C2×C411yescharacteristic
37C2×C4×Dic83C2×C411yescharacteristic
38C4⋊(C83⋊C8)—22no—
39C22⋊(C83⋊C8)—22no—
40C4×C83⋊C8—22no—
41C22×D4×C83C2×C411yescharacteristic
42C23×C4×C83D412yes—
43C22×C8×C83—24no—
44C4×D4×C83—22no—
45C4×D4×C83—22no—
46C2×C4⋊C4×C83C2×C411yescharacteristic
47C2×C22⋊C4×C83D412yes—
48C2×C42×C83C2311yescharacteristic
49C4⋊C8×C83D414yescomplemented
50C22⋊C8×C83—24no—
51C22⋊C8×C83—22no—
52C4×C8×C83—22no—
53C23×D83—21no—
54C2×D332—22no—
55C2×D332—22no—
56C2×D332—21no—
57C2×D332C2×C812yescomplemented
58C2×C22⋊D83—28no—
59C2×C22⋊D83—44no—
60C2×C22⋊D83—44no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C2×C4×D83—44no—
65C2×C4×D83—41no—
66C2×C4×D83—41no—
67C2×C83⋊C8—42no—
68C4⋊Dic83—22no—
69Dic83⋊C4—44no—
70C22⋊Dic83—44no—
71(C2×D83)⋊C4—44no—
72(C2×D83)⋊C4—42no—
73C4×Dic83—24no—
74C4×Dic83—21no—
75C4×Dic83—22no—
76C4×Dic83—42no—
77C4×Dic83C2×C812yescomplemented
78C24×C83C22⋊C412yes—
79C2×D4×C83—22no—
80C2×D4×C83—22no—
81C2×D4×C83C2×C812yes—
82C2×D4×C83M1612yes—
83C22×C4×C83—22no—
84C22×C4×C83—22no—
85C22×C4×C83—44no—
86C22×C4×C83C22⋊C412yes—
87C22×C4×C83C22×C411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C2×C8×C83—24no—
91C2×C8×C83—84no—
92C4⋊C4×C83—22no—
93C22⋊C4×C83—24no—
94C42×C83C2×D412yes—
95C42×C83#4112yes—
96C23×C83—22no—
97C23×C83—22no—
98C23×C83—42no—
99C23×C83—42no—
100C23×C83C22⋊C814yes—
101C23×C83C2×C22⋊C411yescharacteristic
102C22×D83—22no—
103C22×D83—42no—
104C22×D83—42no—
105C22×D83—41no—
106C22×D83—41no—
107D4×C83—24no—
108D4×C83—24no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—41no—
112C2×C4×C83—44no—
113C2×C4×C83—42no—
114C2×C4×C83—82no—
115C2×C4×C83C4×D412yes—
116C2×C4×C83#12011yescharacteristic
117C2×C4×C83#12112yes—
118C2×C4×C83#12311yescharacteristic
119C2×C4×C83C22×C811yescharacteristic
120C2×C4×C83C2×M1611yescharacteristic
121C2×C4×C83#13112yes—
122C22⋊D83—416no—
123C2×Dic83—22no—
124C2×Dic83—22no—
125C2×Dic83—22no—
126C2×Dic83—22no—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—42no—
130C2×Dic83—41no—
131D332—24no—
132D332—42no—
133C4×D83—82no—
134C4×D83—82no—
135C8×C83—44no—
136C83⋊C8—82no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—24no—
140C22×C83—44no—
141C22×C83—42no—
142C22×C83—42no—
143C22×C83—82no—
144C22×C83#38611yescharacteristic
145C22×C83C2×C22⋊C811yescharacteristic
146C22×C83#40811yescharacteristic
147C2×D83—44no—
148C2×D83—42no—
149C2×D83—82no—
150C2×D83—81no—
151C2×D83—81no—
152C4×C83—22no—
153C4×C83—21no—
154C4×C83—42no—
155C4×C83C8×D412yes—
156C4×C83#43512yes—
157C4×C83#45312yes—
158Dic83—24no—
159Dic83—42no—
160Dic83—81no—
161#8959—831nomaximal, Sylow
162C2×C83—21no—
163C2×C83—21no—
164C2×C83—44no—
165C2×C83#186911yescharacteristic
166C2×C83#188411yescharacteristic
167C2×C83#188711yescharacteristic
168D83—82no—
169#2972D8311yescharacteristic, complemented
170C4×#127—831no—
171#1478—832no—
172#1460—831no—
173#1276—832no—
174C83#895911yescharacteristic, complemented, Sylow
175C2×#127—831no—
176C2×C4×D4—831no—
177C2×C4×D4—1662no—
178C2×C4×D4C2×D8311yescharacteristic
179C8×D4—24no—
180#431—832no—
181#431C2×D8312yes—
182C2×C22⋊C8—22no—
183#390—831no—
184#390—831no—
185#390—1662no—
186#384—831no—
187C43—831no—
188D4⋊C8—1664no—
189#323—1662no—
190C22×D4—831no—
191C22×D4—1662no—
192C22×D4C4×D8311yescharacteristic
193C23×C4—1661no—
194C23×C4C22⋊D8312yes—
195C22×C8—24no—
196#127—834no—
197#127—1662no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—3328no—
201C4×D4—1662no—
202C4×D4—1662no—
203C2×C4⋊C4—831no—
204C2×C4⋊C4—1662no—
205C2×C4⋊C4C4×D8311yescharacteristic
206C2×C22⋊C4—1662no—
207C2×C22⋊C4—1662no—
208C2×C22⋊C4—1661no—
209C2×C22⋊C4D33212yes—
210C2×C42—831no—
211C2×C42—831no—
212C2×C42—1661no—
213C2×C42—1661no—
214C2×C42C22×D8311yescharacteristic
215C4⋊C8—1662no—
216C4⋊C8C22⋊D8314yescomplemented
217C22⋊C8—24no—
218C22⋊C8—22no—
219C22⋊C8—1662no—
220C4×C8—22no—
221C4×C8—1662no—
222C24—1661no—
223C24(C2×D83)⋊C412yes—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—3324no—
227C2×D4—3324no—
228C2×D4—832no—
229C2×D4—1668no—
230C2×D4—1661no—
231C2×D4—1662no—
232C2×D4—1662no—
233C2×D4C8×D8312yes—
234C2×D4#1305057312yes—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—44no—
238C22×C4—3324no—
239C22×C4—3321no—
240C22×C4—3321no—
241C22×C4—1661no—
242C22×C4—1661no—
243C22×C4—1661no—
244C22×C4(C2×D83)⋊C412yes—
245C22×C4C2×C4×D8311yescharacteristic
246C22×C4C2×C22⋊D8311yescharacteristic
247C22×C4C2×D33211yescharacteristic
248C2×C8—24no—
249C2×C8—84no—
250C2×C8—3322no—
251C4⋊C4—22no—
252C4⋊C4—3324no—
253C4⋊C4—1662no—
254C22⋊C4—24no—
255C22⋊C4—3324no—
256C22⋊C4—3324no—
257C22⋊C4—3322no—
258C42—3322no—
259C42—832no—
260C42—1664no—
261C42—1662no—
262C42—1661no—
263C42C2×C22⋊D8312yes—
264C42#1305059012yes—
265C23—22no—
266C23—22no—
267C23—42no—
268C23—42no—
269C23—3322no—
270C23—3322no—
271C23—3321no—
272C23—3321no—
273C23—1662no—
274C23(C2×D83)⋊C814yes—
275C23C2×(C2×D83)⋊C411yescharacteristic
276D4—24no—
277D4—24no—
278D4—33216no—
279D4—3322no—
280D4—1664no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—42no—
286C2×C4—82no—
287C2×C4—3321no—
288C2×C4—3321no—
289C2×C4—3322no—
290C2×C4—3321no—
291C2×C4—6642no—
292C2×C4—6642no—
293C2×C4—1662no—
294C2×C4—1662no—
295C2×C4—1662no—
296C2×C4—1662no—
297C2×C4C4×C22⋊D8312yes—
298C2×C4#1458176111yescharacteristic
299C2×C4#1458176612yes—
300C2×C4#1458177511yescharacteristic
301C2×C4C2×C8×D8311yescharacteristic
302C2×C4C2×#1305057311yescharacteristic
303C2×C4#1458179812yes—
304C8—44no—
305C8—6642no—
306C22—21no—
307C22—21no—
308C22—24no—
309C22—44no—
310C22—42no—
311C22—42no—
312C22—82no—
313C22—3324no—
314C22—3322no—
315C22—6642no—
316C22—6641no—
317C22—6641no—
318C22#1912758011yescharacteristic
319C22C2×(C2×D83)⋊C811yescharacteristic
320C22#1912766711yescharacteristic
321C4—22no—
322C4—21no—
323C4—42no—
324C4—3322no—
325C4—6641no—
326C4—1664no—
327C4C8×C22⋊D8312yes—
328C4#1912777212yes—
329C4#1912786912yes—
330C2—21no—
331C2—21no—
332C2—44no—
333C2—6642no—
334C2#2490775511yescharacteristic
335C2#2490784611yescharacteristic
336C2#2490786111yescharacteristic
subgroupG/H
centerC2×C4#14581761
commutatorC2×C4×C83C22×C8
FrattiniC2×C42C22×D83
FittingC83×#2972C2
socleC22×C83#386
prime radicalC2×#14581788C4
cosocleC2×C42×C83C23
hypercenter#2972D83
hypocenterC83#8959
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #24905853C2semidirect
3. #24905784C2semidirect
4. #24905295C2semidirect
5. C83×#2972C2semidirect
25. #14581788C8semidirect
49. C4⋊C8×C83D4semidirect
57. C2×D332C2×C8semidirect
77. C4×Dic83C2×C8semidirect
169. #2972D83semidirect
174. C83#8959semidirect
216. C4⋊C8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C4×#145817882
#249058532
#249057842
#249052952
C83×#29722
#895983