Finite Group Databasebeta

← C83×#36691 #35264515

Subgroups — C83×#36691 #35264515

Order 21248.

1474 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2845C212yesmaximal, complemented
2C83×#2844C212yesmaximal, complemented
3C83×#2843C214yesmaximal, complemented
4C83×#2841C218yesmaximal
5C83×#2402C214yesmaximal, complemented
6C83×#2399C214yesmaximal, complemented
7C83×#2396C214yesmaximal
8C83×#2285C211yesmaximal, characteristic
9C83×#2283C212yesmaximal
10C2×C4×Q8×C83C2214yes—
11C2×#117×C83C2212yes—
12C22×C42×C83C2211yescharacteristic
13#400×C83C2218yes—
14#400×C83C2218yes—
15#400×C83C2214yes—
16#399×C83C2214yes—
17#398×C83C2218yes—
18#398×C83C2218yes—
19#398×C83C2218yes—
20#398×C83C2218yes—
21#398×C83C2218yes—
22#396×C83C2218yes—
23#395×C83C2218yes—
24#395×C83C2218yes—
25#395×C83C2214yes—
26#393×C83C2214yes—
27#384×C83C2212yes—
28#383×C83C2214yes—
29#382×C83C2214yes—
30#382×C83C2211yescharacteristic
31#382×C83C2211yescharacteristic
32#381×C83C2214yes—
33#380×C83C2214yes—
34#380×C83C2214yes—
35C4×C4⋊C4×C83C2212yes—
36C4×C4⋊C4×C83C2212yes—
37C4×C4⋊C4×C83C2214yes—
38C4×C4⋊C4×C83C2218yes—
39C4×C4⋊C4×C83C2214yes—
40C4×C22⋊C4×C83C2214yes—
41#376×C83C2214yes—
42C22×Q8×C83C2314yes—
43C23×C4×C83C2311yescharacteristic
44C23×C4×C83C2312yes—
45C4×Q8×C83—48no—
46#117×C83—44no—
47C2×C4⋊C4×C83C2312yes—
48C2×C4⋊C4×C83C2312yes—
49C2×C4⋊C4×C83C2314yes—
50C2×C4⋊C4×C83C2318yes—
51C2×C4⋊C4×C83C2314yes—
52C2×C4⋊C4×C83C2314yes—
53C2×C4⋊C4×C83C2318yes—
54C2×C4⋊C4×C83C2318yes—
55C2×C4⋊C4×C83C2318yes—
56C2×C4⋊C4×C83C2318yes—
57C2×C22⋊C4×C83C2314yes—
58C2×C22⋊C4×C83C2318yes—
59C2×C42×C83Q818yescomplemented
60C2×C42×C83C2314yes—
61C2×C42×C83C2314yes—
62C2×C42×C83C2314yes—
63C2×C42×C83C2311yescharacteristic
64C2×C42×C83C2311yescharacteristic
65C2×C42×C83C2312yes—
66#95×C83C2318yes—
67#95×C83C2318yes—
68#95×C83C2318yes—
69#95×C83C2314yes—
70#95×C83C2314yes—
71#95×C83C2314yes—
72#95×C83C2314yes—
73#95×C83C2318yes—
74#95×C83C2314yes—
75#95×C83C2318yes—
76#95×C83C2314yes—
77C24×C83C2411yescharacteristic
78C2×Q8×C83—44no—
79C2×Q8×C83—48no—
80C22×C4×C83—22no—
81C22×C4×C83—22no—
82C22×C4×C83—44no—
83C22×C4×C83C2×Q814yes—
84C22×C4×C83C2×Q818yes—
85C22×C4×C83C2418yes—
86C22×C4×C83C2414yes—
87C22×C4×C83C2418yes—
88C22×C4×C83C2414yes—
89C22×C4×C83C2411yescharacteristic
90C22×C4×C83C2412yes—
91C22×C4×C83C2412yes—
92C22×C4×C83C2411yescharacteristic
93C4⋊C4×C83—44no—
94C4⋊C4×C83—48no—
95C4⋊C4×C83—44no—
96C4⋊C4×C83—48no—
97C4⋊C4×C83—48no—
98C4⋊C4×C83—48no—
99C4⋊C4×C83—42no—
100C4⋊C4×C83—42no—
101C4⋊C4×C83—48no—
102C4⋊C4×C83—44no—
103C22⋊C4×C83—48no—
104C22⋊C4×C83—44no—
105C42×C83—28no—
106C42×C83—28no—
107C42×C83—28no—
108C42×C83C2×Q814yes—
109C42×C83C2×Q814yes—
110C42×C83C2×Q818yes—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—24no—
114C23×C83C22×Q812yes—
115C23×C83C2511yescharacteristic
116Q8×C83—416no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—24no—
120C2×C4×C83—24no—
121C2×C4×C83—28no—
122C2×C4×C83—24no—
123C2×C4×C83—24no—
124C2×C4×C83—28no—
125C2×C4×C83—28no—
126C2×C4×C83—28no—
127C2×C4×C83—28no—
128C2×C4×C83—28no—
129C2×C4×C83—28no—
130C2×C4×C83—28no—
131C2×C4×C83—28no—
132C2×C4×C83—48no—
133C2×C4×C83C22×Q812yes—
134C2×C4×C83C22×Q814yes—
135C2×C4×C83C22×Q814yes—
136C2×C4×C83C22×Q812yes—
137C2×C4×C83H(5,𝔽2)12yes—
138C2×C4×C832−1+412yes—
139C22×C83—22no—
140C22×C83—24no—
141C22×C83—42no—
142C22×C83—42no—
143C22×C83C23×Q811yescharacteristic
144C22×C83C2×H(5,𝔽2)11yescharacteristic
145C22×C83C2×2−1+411yescharacteristic
146C22×C83#58514yes—
147C4×C83—22no—
148C4×C83—24no—
149C4×C83—24no—
150C4×C83—44no—
151C4×C83—44no—
152C4×C83—48no—
153C4×C83—48no—
154C4×C83#54414yes—
155#36691C8311yesmaximal, characteristic, direct product factor, Sylow
156C2×C83—42no—
157C2×C83C2×#54411yescharacteristic
158C2×C83#345812yes—
159C2×C83#352912yes—
160C2×C83#353112yes—
161#2845C2×C8312yescomplemented
162#2844C2×C8312yescomplemented
163#2843C2×C8314yescomplemented
164#2841C2×C8318yes—
165#2402C2×C8314yescomplemented
166#2399C2×C8314yescomplemented
167#2396C2×C8314yes—
168#2285C2×C8311yescharacteristic
169#2283C2×C8312yes—
170C83#3669111yescharacteristic, direct product factor, Sylow
171C2×C4×Q8C22×C8314yes—
172C2×#117C22×C8312yes—
173C22×C42C22×C8311yescharacteristic
174#400C22×C8318yes—
175#400C22×C8318yes—
176#400C22×C8314yes—
177#399C22×C8314yes—
178#398C22×C8318yes—
179#398C22×C8318yes—
180#398C22×C8318yes—
181#398C22×C8318yes—
182#398C22×C8318yes—
183#396C22×C8318yes—
184#395C22×C8318yes—
185#395C22×C8318yes—
186#395C22×C8314yes—
187#393C22×C8314yes—
188#384C22×C8312yes—
189#383C22×C8314yes—
190#382C22×C8311yescharacteristic
191#382C22×C8314yes—
192#382C22×C8311yescharacteristic
193#381C22×C8314yes—
194#380C22×C8314yes—
195#380C22×C8314yes—
196C4×C4⋊C4C22×C8318yes—
197C4×C4⋊C4C22×C8314yes—
198C4×C4⋊C4C22×C8312yes—
199C4×C4⋊C4C22×C8314yes—
200C4×C4⋊C4C22×C8312yes—
201C4×C22⋊C4C22×C8314yes—
202#376C22×C8314yes—
203C22×Q8C23×C8314yes—
204C23×C4C23×C8311yescharacteristic
205C23×C4C23×C8312yes—
206C4×Q8—48no—
207#117—44no—
208C2×C4⋊C4C23×C8312yes—
209C2×C4⋊C4C23×C8312yes—
210C2×C4⋊C4C23×C8314yes—
211C2×C4⋊C4C23×C8318yes—
212C2×C4⋊C4C23×C8314yes—
213C2×C4⋊C4C23×C8314yes—
214C2×C4⋊C4C23×C8318yes—
215C2×C4⋊C4C23×C8318yes—
216C2×C4⋊C4C23×C8318yes—
217C2×C4⋊C4C23×C8318yes—
218C2×C22⋊C4C23×C8314yes—
219C2×C22⋊C4C23×C8318yes—
220C2×C42Q8×C8318yescomplemented
221C2×C42C23×C8314yes—
222C2×C42C23×C8314yes—
223C2×C42C23×C8314yes—
224C2×C42C23×C8312yes—
225C2×C42C23×C8311yescharacteristic
226C2×C42C23×C8311yescharacteristic
227#95C23×C8318yes—
228#95C23×C8318yes—
229#95C23×C8318yes—
230#95C23×C8314yes—
231#95C23×C8314yes—
232#95C23×C8314yes—
233#95C23×C8314yes—
234#95C23×C8318yes—
235#95C23×C8314yes—
236#95C23×C8318yes—
237#95C23×C8314yes—
238C24C24×C8311yescharacteristic
239C2×Q8—44no—
240C2×Q8—48no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4C2×Q8×C8314yes—
245C22×C4C2×Q8×C8318yes—
246C22×C4C24×C8318yes—
247C22×C4C24×C8314yes—
248C22×C4C24×C8318yes—
249C22×C4C24×C8314yes—
250C22×C4C24×C8311yescharacteristic
251C22×C4C24×C8312yes—
252C22×C4C24×C8312yes—
253C22×C4C24×C8311yescharacteristic
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—48no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—48no—
260C4⋊C4—44no—
261C4⋊C4—48no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C22⋊C4—44no—
265C22⋊C4—48no—
266C42—28no—
267C42—28no—
268C42—28no—
269C42C2×Q8×C8314yes—
270C42C2×Q8×C8314yes—
271C42C2×Q8×C8318yes—
272C23—21no—
273C23—21no—
274C23—24no—
275C23C22×Q8×C8312yes—
276C23C25×C8311yescharacteristic
277Q8—416no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—28no—
286C2×C4—28no—
287C2×C4—28no—
288C2×C4—28no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—48no—
294C2×C4C22×Q8×C8312yes—
295C2×C4C22×Q8×C8314yes—
296C2×C4C22×Q8×C8314yes—
297C2×C4C22×Q8×C8312yes—
298C2×C4H(5,𝔽2)×C8312yes—
299C2×C42−1+4×C8312yes—
300C22—22no—
301C22—24no—
302C22—42no—
303C22—42no—
304C22C23×Q8×C8311yescharacteristic
305C22C2×H(5,𝔽2)×C8311yescharacteristic
306C22C2×2−1+4×C8311yescharacteristic
307C22#585×C8314yes—
308C4—22no—
309C4—24no—
310C4—24no—
311C4—48no—
312C4—44no—
313C4—48no—
314C4—44no—
315C4#544×C8314yes—
316C2—42no—
317C2C2×#544×C8311yescharacteristic
318C2C83×#345812yes—
319C2C83×#352912yes—
320C2C83×#353112yes—
subgroupG/H
centerC22×C4×C83C24
commutator, Frattini, cosocleC23C25×C83
socleC23×C83C25
prime radicalC24×C83C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#2845C2semidirect
2. C83×#2844C2semidirect
3. C83×#2843C2semidirect
5. C83×#2402C2semidirect
6. C83×#2399C2semidirect
59. C2×C42×C83Q8semidirect
155. #36691C83direct (N × H)
161. #2845C2×C83semidirect
162. #2844C2×C83semidirect
163. #2843C2×C83semidirect
165. #2402C2×C83semidirect
166. #2399C2×C83semidirect
170. C83#36691direct (N × H)
220. C2×C42Q8×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#28452
C83×#28442
C83×#28432
C83×#28412
C83×#24022
C83×#23992
C83×#23962
C83×#22852
C83×#22832
#3669183