Finite Group Databasebeta

← #35414848

Subgroups — #35414848

Order 21248.

946 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127754C211yesmaximal, characteristic
2C22×C4⋊(C83⋊C8)C211yesmaximal, characteristic
3#24908182C214yesmaximal, complemented
4#24907643C214yesmaximal
5#24907615C212yesmaximal, complemented
6#24907580C212yesmaximal, complemented
7C4×#117×C83C211yesmaximal, characteristic
8C23×C83⋊C8C2212yes—
9#19127754—24no—
10C2×C4⋊(C83⋊C8)—24no—
11C2×C4⋊(C83⋊C8)C2212yes—
12C2×C4⋊(C83⋊C8)C2212yes—
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×C4⋊(C83⋊C8)C2211yescharacteristic
15C2×C4⋊(C83⋊C8)C2218yes—
16C2×C22⋊(C83⋊C8)C2212yes—
17C2×C4×C83⋊C8C2212yes—
18#19127456C2218yes—
19C2×#117×C83C411yescharacteristic
20C2×#117×C83C411yescharacteristic
21C22×C42×C83C2211yescharacteristic
22C4×C4⋊C4×C83C412yes—
23C4×C4⋊C4×C83C2212yes—
24C4×C22⋊C4×C83C412yes—
25C4×C22⋊C4×C83C2212yes—
26#376×C83C412yes—
27C43×C83C2212yes—
28C22×C83⋊C8—24no—
29C22×C83⋊C8—44no—
30C22×C83⋊C8C2312yes—
31C22×C83⋊C8C2312yes—
32C22×C83⋊C8C2314yes—
33C4⋊(C83⋊C8)—24no—
34C4⋊(C83⋊C8)—24no—
35C4⋊(C83⋊C8)—22no—
36C4⋊(C83⋊C8)—22no—
37C4⋊(C83⋊C8)—48no—
38C22⋊(C83⋊C8)—24no—
39C4×C83⋊C8—24no—
40C23×C4×C83C2×C411yescharacteristic
41C23×C4×C83C2×C411yescharacteristic
42C23×C4×C83C2311yescharacteristic
43#117×C83—24no—
44#117×C83C818yescomplemented
45C2×C4⋊C4×C83C2×C412yes—
46C2×C4⋊C4×C83C2×C412yes—
47C2×C22⋊C4×C83C2×C412yes—
48C2×C22⋊C4×C83C2×C412yes—
49C2×C42×C83C2×C412yes—
50C2×C42×C83C2×C414yes—
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83C2×C411yescharacteristic
53C2×C42×C83C2×C412yes—
54C2×C42×C83D414yes—
55C2×C42×C83Q814yes—
56C2×C42×C83C2314yes—
57C2×C42×C83C2311yescharacteristic
58C2×C42×C83C2311yescharacteristic
59#95×C83C2×C414yes—
60#95×C83C2×C414yes—
61C2×C83⋊C8—24no—
62C2×C83⋊C8—22no—
63C2×C83⋊C8—22no—
64C2×C83⋊C8—28no—
65C2×C83⋊C8—24no—
66C2×C83⋊C8—24no—
67C2×C83⋊C8—48no—
68C24×C83C22×C411yescharacteristic
69C22×C4×C83—22no—
70C22×C4×C83—22no—
71C22×C4×C83—42no—
72C22×C4×C83C4⋊C414yes—
73C22×C4×C83C4⋊C414yes—
74C22×C4×C83C2×C814yes—
75C22×C4×C83M1614yes—
76C22×C4×C83C22×C414yes—
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C22×C411yescharacteristic
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C22×C414yes—
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D412yes—
84C22×C4×C83C2×Q812yes—
85C22×C4×C83C2411yescharacteristic
86C4⋊C4×C83—24no—
87C4⋊C4×C83C2×C818yescomplemented
88C22⋊C4×C83—24no—
89C22⋊C4×C83C2×C818yes—
90C42×C83—24no—
91C42×C83—24no—
92C42×C83—24no—
93C42×C83—44no—
94C42×C83C4⋊C414yes—
95C42×C83C4⋊C414yes—
96C42×C83C2×C818yescomplemented
97C42×C83C2×D412yes—
98C42×C83C2×D412yes—
99C42×C83C2×Q812yes—
100C42×C83C2×Q812yes—
101C23×C83—22no—
102C23×C83—22no—
103C23×C83C2×C4⋊C412yes—
104C23×C83C22×C812yes—
105C23×C83C2×M1612yes—
106C23×C83C23×C411yescharacteristic
107C2×C4×C83—24no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—24no—
111C2×C4×C83—22no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—24no—
115C2×C4×C83—24no—
116C2×C4×C83—24no—
117C2×C4×C83—24no—
118C2×C4×C83—44no—
119C2×C4×C83C4⋊C8116yescomplemented
120C2×C4×C83C2×C4⋊C412yes—
121C2×C4×C83C2×C4⋊C412yes—
122C2×C4×C83C2×C4⋊C412yes—
123C2×C4×C83C2×C4⋊C412yes—
124C2×C4×C83C2×C4⋊C412yes—
125C2×C4×C83C22×C818yes—
126C2×C4×C83C22×C812yes—
127C2×C4×C83C22×C812yes—
128C2×C4×C83C2×M1612yes—
129C2×C4×C83C2×M1612yes—
130C2×C4×C83C22×D411yescharacteristic
131C2×C4×C83C22×Q811yescharacteristic
132C2×C4×C83H(5,𝔽2)12yes—
133C2×C4×C832−1+412yes—
134C83⋊C8—42no—
135C83⋊C8—44no—
136C83⋊C8—42no—
137C22×C83—22no—
138C22×C83—22no—
139C22×C83—24no—
140C22×C83—24no—
141C22×C83C2×C4⋊C814yes—
142C22×C83C22×C4⋊C411yescharacteristic
143C22×C83#52012yes—
144C22×C83C23×C811yescharacteristic
145C22×C83C22×M1611yescharacteristic
146C22×C83#56812yes—
147C4×C83—22no—
148C4×C83—22no—
149C4×C83—22no—
150C4×C83—28no—
151C4×C83—44no—
152C4×C83C2×C4⋊C814yes—
153C4×C83C2×C4⋊C814yes—
154C4×C83#53511yescharacteristic
155C4×C83#53611yescharacteristic
156C4×C83#54412yes—
157#21972—831nomaximal, Sylow
158C2×C83—24no—
159C2×C83#231711yescharacteristic
160C2×C83C22×C4⋊C811yescharacteristic
161C2×C83#290511yescharacteristic
162C2×C83#298812yes—
163C2×C83#298912yes—
164C2×#431—831no—
165C22×C4⋊C8—831no—
166C4×#117D8311yescharacteristic
167#1939—834no—
168#1850—834no—
169#1842—832no—
170#1831—832no—
171C83#2197211yescharacteristic, complemented, Sylow
172C23×C8—832no—
173C2×#117Dic8311yescharacteristic
174C2×#117Dic8311yescharacteristic
175C22×C42C2×D8311yescharacteristic
176#431—1664no—
177C2×C4⋊C8—838no—
178C2×C4⋊C8—831no—
179C2×C4⋊C8—832no—
180C2×C4⋊C8—831no—
181C2×C4⋊C8—832no—
182C2×C4⋊C8—1664no—
183C2×C22⋊C8—832no—
184C2×C4×C8—832no—
185C4×C4⋊C4Dic8312yes—
186C4×C4⋊C4C2×D8312yes—
187C4×C22⋊C4Dic8312yes—
188C4×C22⋊C4C2×D8312yes—
189#376Dic8312yes—
190C43C2×D8312yes—
191#336—838no—
192C23×C4C2×Dic8311yescharacteristic
193C23×C4C2×Dic8311yescharacteristic
194C23×C4C22×D8311yescharacteristic
195C22×C8—3324no—
196C22×C8—834no—
197C22×C8—832no—
198C22×C8—832no—
199C22×C8—1664no—
200#117—24no—
201#117C83⋊C818yescomplemented
202C2×C4⋊C4C2×Dic8312yes—
203C2×C4⋊C4C2×Dic8312yes—
204C2×C22⋊C4C2×Dic8312yes—
205C2×C22⋊C4C2×Dic8312yes—
206C2×C42Dic16614yes—
207C2×C42D33214yes—
208C2×C42C2×Dic8312yes—
209C2×C42C2×Dic8314yes—
210C2×C42C2×Dic8311yescharacteristic
211C2×C42C2×Dic8311yescharacteristic
212C2×C42C2×Dic8312yes—
213C2×C42C22×D8314yes—
214C2×C42C22×D8311yescharacteristic
215C2×C42C22×D8311yescharacteristic
216C4⋊C8—3328no—
217C4⋊C8—1664no—
218C4⋊C8—1664no—
219C4⋊C8—1662no—
220C4⋊C8—1662no—
221C22⋊C8—1664no—
222C4×C8—1664no—
223#95C2×Dic8314yes—
224#95C2×Dic8314yes—
225C24C22×Dic8311yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4C4⋊Dic8314yes—
230C22×C4C4⋊Dic8314yes—
231C22×C4C2×C83⋊C814yes—
232C22×C4#1305057414yes—
233C22×C4C22×Dic8314yes—
234C22×C4C22×Dic8314yes—
235C22×C4C22×Dic8311yescharacteristic
236C22×C4C22×Dic8311yescharacteristic
237C22×C4C22×Dic8311yescharacteristic
238C22×C4C22×Dic8311yescharacteristic
239C22×C4C22×Dic8311yescharacteristic
240C22×C4C2×D33212yes—
241C22×C4C2×Dic16612yes—
242C22×C4C23×D8311yescharacteristic
243C2×C8—3328no—
244C2×C8—1662no—
245C2×C8—1662no—
246C2×C8—1664no—
247C2×C8—1664no—
248C2×C8—1668no—
249C2×C8—1664no—
250C4⋊C4—24no—
251C4⋊C4C2×C83⋊C818yescomplemented
252C22⋊C4—24no—
253C22⋊C4C2×C83⋊C818yes—
254C42—24no—
255C42—24no—
256C42—24no—
257C42—44no—
258C42C4⋊Dic8314yes—
259C42C4⋊Dic8314yes—
260C42C2×C83⋊C818yescomplemented
261C42C2×D33212yes—
262C42C2×D33212yes—
263C42C2×Dic16612yes—
264C42C2×Dic16612yes—
265C23—22no—
266C23—22no—
267C23C2×C4⋊Dic8312yes—
268C23C22×C83⋊C812yes—
269C23C2×#1305057412yes—
270C23C23×Dic8311yescharacteristic
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—44no—
283C2×C4C4⋊(C83⋊C8)116yescomplemented
284C2×C4C2×C4⋊Dic8312yes—
285C2×C4C2×C4⋊Dic8312yes—
286C2×C4C2×C4⋊Dic8312yes—
287C2×C4C2×C4⋊Dic8312yes—
288C2×C4C2×C4⋊Dic8312yes—
289C2×C4C22×C83⋊C818yes—
290C2×C4C22×C83⋊C812yes—
291C2×C4C22×C83⋊C812yes—
292C2×C4C2×#1305057412yes—
293C2×C4C2×#1305057412yes—
294C2×C4C22×D33211yescharacteristic
295C2×C4C22×Dic16611yescharacteristic
296C2×C4#1458183812yes—
297C2×C4#1458183912yes—
298C8—3324no—
299C8—3322no—
300C8—3322no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—24no—
305C22C2×C4⋊(C83⋊C8)14yes—
306C22C22×C4⋊Dic8311yescharacteristic
307C22#1912818412yes—
308C22C23×C83⋊C811yescharacteristic
309C22C22×#1305057411yescharacteristic
310C22#1912846812yes—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—28no—
315C4—44no—
316C4C2×C4⋊(C83⋊C8)14yes—
317C4C2×C4⋊(C83⋊C8)14yes—
318C4#1912827011yescharacteristic
319C4#1912827711yescharacteristic
320C4#1912832912yes—
321C2—24no—
322C2#2491008611yescharacteristic
323C2C22×C4⋊(C83⋊C8)11yescharacteristic
324C2#2491602711yescharacteristic
325C2#2491657412yes—
326C2#2491658012yes—
subgroupG/H
center, FrattiniC22×C4C23×D83
commutatorC22×C83C23×C8
FittingC4×#117×C83C2
socleC23×C83C23×C4
prime radicalC24×C83C22×C4
cosocleC22×C4×C83C24
hypercenterC4×#117D83
hypocenterC83#21972
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
3. #24908182C2semidirect
5. #24907615C2semidirect
6. #24907580C2semidirect
44. #117×C83C8semidirect
87. C4⋊C4×C83C2×C8semidirect
96. C42×C83C2×C8semidirect
119. C2×C4×C83C4⋊C8semidirect
171. C83#21972semidirect
201. #117C83⋊C8semidirect
251. C4⋊C4C2×C83⋊C8semidirect
260. C42C2×C83⋊C8semidirect
283. C2×C4C4⋊(C83⋊C8)semidirect

Maximal subgroups

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

subgroupindex
C2×#191277542
C22×C4⋊(C83⋊C8)2
#249081822
#249076432
#249076152
#249075802
C4×#117×C832
#2197283