Finite Group Databasebeta

← #35564465

Subgroups — #35564465

Order 21248.

2806 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910340C218yesmaximal, complemented
2#24910332C218yesmaximal, complemented
3#24910081C212yesmaximal, complemented
4#24910066C214yesmaximal, complemented
5#24909846C214yesmaximal
6#24909846C212yesmaximal
7#24909820C212yesmaximal, complemented
8C23×C4⋊C4×C83C211yesmaximal, characteristic
9C2×#14581751C2214yes—
10C2×#14581751C2218yes—
11#19127600C22116yescomplemented
12#19127587C22132yescomplemented
13#19127564C2214yes—
14#19127564C2218yes—
15#19127561C2218yes—
16#19127561C2214yes—
17#19127561C2218yes—
18#19127561C2214yes—
19#19127557C2214yes—
20C4⋊C4×Dic83C22116yescomplemented
21C4×C22⋊Dic83C2214yes—
22C4×C22⋊Dic83C2218yes—
23C4×C22⋊Dic83C2218yes—
24C4×C22⋊Dic83C2214yes—
25C24×C4×C83C2211yescharacteristic
26C24×C4×C83C2212yes—
27C22×C4⋊C4×C83C4116yescomplemented
28C22×C4⋊C4×C83C2211yescharacteristic
29C22×C4⋊C4×C83C2214yes—
30C22×C4⋊C4×C83C2212yes—
31C22×C4⋊C4×C83C2211yescharacteristic
32C22×C4⋊C4×C83C2214yes—
33#14581751—216no—
34#14581751—416no—
35C2×C4⋊Dic83C2318yes—
36C2×C4⋊Dic83C23116yes—
37C2×C22⋊Dic83C2318yes—
38C2×C22⋊Dic83C2318yes—
39C2×C22⋊Dic83C2318yes—
40C2×C4×Dic83C2318yes—
41C2×C4×Dic83C23116yes—
42C83⋊#95C23116yes—
43C83⋊#95C2318yes—
44C83⋊#95C2318yes—
45C83⋊#95C23116yes—
46C25×C83C2311yescharacteristic
47C23×C4×C83—28no—
48C23×C4×C83—48no—
49C23×C4×C83C2×C418yes—
50C23×C4×C83C2×C4116yes—
51C23×C4×C83C2312yes—
52C23×C4×C83C2312yes—
53C23×C4×C83C2311yescharacteristic
54C23×C4×C83C2314yes—
55C23×C4×C83C2312yes—
56C23×C4×C83C2314yes—
57C23×C4×C83C2312yes—
58C23×C4×C83C2311yescharacteristic
59C2×C4⋊C4×C83C2×C4132yes—
60C2×C4⋊C4×C83C2×C4132yes—
61C2×C4⋊C4×C83C2×C4116yes—
62C2×C4⋊C4×C83C2×C418yes—
63C2×C4⋊C4×C83C2×C418yes—
64C2×C4⋊C4×C83C2314yes—
65C2×C4⋊C4×C83C2318yes—
66C2×C4⋊C4×C83C2314yes—
67C22×Dic83C24116yes—
68C4⋊Dic83—216no—
69C4⋊Dic83—416no—
70C22⋊Dic83—216no—
71C22⋊Dic83—216no—
72C22⋊Dic83—216no—
73C4×Dic83—216no—
74C4×Dic83—416no—
75C24×C83—24no—
76C24×C83—24no—
77C24×C83—24no—
78C24×C83C22×C414yes—
79C24×C83C2411yescharacteristic
80C24×C83C2411yescharacteristic
81C24×C83C2411yescharacteristic
82C22×C4×C83—24no—
83C22×C4×C83—216no—
84C22×C4×C83—232no—
85C22×C4×C83—28no—
86C22×C4×C83—28no—
87C22×C4×C83—432no—
88C22×C4×C83—44no—
89C22×C4×C83—48no—
90C22×C4×C83—44no—
91C22×C4×C83C22×C414yes—
92C22×C4×C83C22×C4116yes—
93C22×C4×C83C22×C418yes—
94C22×C4×C83C22×C418yes—
95C22×C4×C83C22×C418yes—
96C22×C4×C83C22×C418yes—
97C22×C4×C83C22×C4116yes—
98C22×C4×C83C22×C414yes—
99C22×C4×C83#4118yes—
100C22×C4×C83C2412yes—
101C22×C4×C83C2414yes—
102C22×C4×C83C2414yes—
103C22×C4×C83C2412yes—
104C4⋊C4×C83C22×C4132yes—
105C4⋊C4×C83C22×C4116yes—
106C4⋊C4×C83C22×C4116yes—
107C23×C83—232no—
108C23×C83—28no—
109C23×C83—28no—
110C23×C83—28no—
111C23×C83—22no—
112C23×C83—22no—
113C23×C83—22no—
114C23×C83—22no—
115C23×C83—22no—
116C23×C83—22no—
117C23×C83C23×C412yes—
118C23×C83C23×C412yes—
119C23×C83C23×C412yes—
120C23×C83C2×#4114yes—
121C23×C83H(5,𝔽2)14yes—
122C23×C832−1+414yes—
123C23×C83C2511yescharacteristic
124C2×C4×C83—28no—
125C2×C4×C83—232no—
126C2×C4×C83—24no—
127C2×C4×C83—28no—
128C2×C4×C83—28no—
129C2×C4×C83—216no—
130C2×C4×C83—216no—
131C2×C4×C83—48no—
132C2×C4×C83—416no—
133C2×C4×C83—48no—
134C2×C4×C83#117116yescomplemented
135C2×C4×C83C23×C414yes—
136C2×C4×C83C23×C418yes—
137C2×C4×C83C23×C418yes—
138C2×C4×C83C23×C414yes—
139C2×C4×C83C2×#4118yes—
140C2×Dic83—216no—
141C2×Dic83—216no—
142C2×Dic83—216no—
143C22×C83—232no—
144C22×C83—22no—
145C22×C83—22no—
146C22×C83—22no—
147C22×C83—22no—
148C22×C83—22no—
149C22×C83—22no—
150C22×C83—28no—
151C22×C83—28no—
152C22×C83—28no—
153C22×C83C2×#11714yes—
154C22×C83#51814yes—
155C22×C83#51914yes—
156C22×C83C24×C411yescharacteristic
157C22×C83C22×#4112yes—
158C22×C83C2×H(5,𝔽2)12yes—
159C22×C83C2×2−1+412yes—
160C4×C83—24no—
161C4×C83—28no—
162C4×C83—28no—
163C4×C83C2×#11718yes—
164Dic83—416no—
165#34178—831nomaximal, Sylow
166C2×C83—24no—
167C2×C83—24no—
168C2×C83—24no—
169C2×C83C22×#11711yescharacteristic
170C2×C83C2×#51811yescharacteristic
171C2×C83C2×#51911yescharacteristic
172C2×C83#349214yes—
173C23×C4⋊C4D8311yescharacteristic
174#2345—838no—
175#2344—838no—
176#2316—832no—
177#2313—834no—
178#2280—834no—
179#2280—832no—
180#2274—832no—
181C83#3417811yescharacteristic, complemented, Sylow
182C24×C4C2×D8311yescharacteristic
183C24×C4C2×D8312yes—
184C2×#117—838no—
185C2×#117—834no—
186C22×C4⋊C4Dic83116yescomplemented
187C22×C4⋊C4C2×D8312yes—
188C22×C4⋊C4C2×D8311yescharacteristic
189C22×C4⋊C4C2×D8314yes—
190C22×C4⋊C4C2×D8314yes—
191C22×C4⋊C4C2×D8311yescharacteristic
192#389—8316no—
193#387—8332no—
194#381—834no—
195#381—838no—
196#380—838no—
197#380—834no—
198#380—838no—
199#380—834no—
200#379—834no—
201C4×C4⋊C4—8316no—
202C4×C22⋊C4—838no—
203C4×C22⋊C4—834no—
204C4×C22⋊C4—834no—
205C4×C22⋊C4—838no—
206C25C22×D8311yescharacteristic
207C23×C4—28no—
208C23×C4—48no—
209C23×C4C2×Dic8318yes—
210C23×C4C2×Dic83116yes—
211C23×C4C22×D8314yes—
212C23×C4C22×D8312yes—
213C23×C4C22×D8312yes—
214C23×C4C22×D8311yescharacteristic
215C23×C4C22×D8312yes—
216C23×C4C22×D8314yes—
217C23×C4C22×D8312yes—
218C23×C4C22×D8311yescharacteristic
219#117—33216no—
220#117—16616no—
221C2×C4⋊C4—8316no—
222C2×C4⋊C4—838no—
223C2×C4⋊C4C2×Dic83132yes—
224C2×C4⋊C4C2×Dic83116yes—
225C2×C4⋊C4C2×Dic83132yes—
226C2×C4⋊C4C2×Dic8318yes—
227C2×C4⋊C4C2×Dic8318yes—
228C2×C4⋊C4C22×D8314yes—
229C2×C4⋊C4C22×D8318yes—
230C2×C4⋊C4C22×D8314yes—
231C2×C22⋊C4—838no—
232C2×C22⋊C4—838no—
233C2×C22⋊C4—838no—
234C2×C42—8316no—
235C2×C42—838no—
236#95—8316no—
237#95—838no—
238#95—838no—
239#95—8316no—
240C24—24no—
241C24—24no—
242C24—24no—
243C24C22×Dic8314yes—
244C24C23×D8311yescharacteristic
245C24C23×D8311yescharacteristic
246C24C23×D8311yescharacteristic
247C22×C4—24no—
248C22×C4—216no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—232no—
252C22×C4—432no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—44no—
256C22×C4—8316no—
257C22×C4C22×Dic8314yes—
258C22×C4C22×Dic83116yes—
259C22×C4C22×Dic8318yes—
260C22×C4C22×Dic8318yes—
261C22×C4C22×Dic8318yes—
262C22×C4C22×Dic8318yes—
263C22×C4C22×Dic83116yes—
264C22×C4C22×Dic8314yes—
265C22×C4#1305059018yes—
266C22×C4C23×D8312yes—
267C22×C4C23×D8314yes—
268C22×C4C23×D8314yes—
269C22×C4C23×D8312yes—
270C4⋊C4—33216no—
271C4⋊C4—16616no—
272C4⋊C4C22×Dic83132yes—
273C4⋊C4C22×Dic83116yes—
274C4⋊C4C22×Dic83116yes—
275C22⋊C4—16616no—
276C22⋊C4—16616no—
277C22⋊C4—16616no—
278C42—33216no—
279C42—16616no—
280C23—232no—
281C23—28no—
282C23—28no—
283C23—28no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23C23×Dic8312yes—
291C23C23×Dic8312yes—
292C23C23×Dic8312yes—
293C23C2×#1305059014yes—
294C23#1458183714yes—
295C23#1458184014yes—
296C23C24×D8311yescharacteristic
297C2×C4—28no—
298C2×C4—232no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—216no—
303C2×C4—216no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—48no—
307C2×C4—16616no—
308C2×C4—16616no—
309C2×C4—16616no—
310C2×C4#14581751116yescomplemented
311C2×C4C23×Dic8314yes—
312C2×C4C23×Dic8318yes—
313C2×C4C23×Dic8318yes—
314C2×C4C23×Dic8314yes—
315C2×C4C2×#1305059018yes—
316C22—232no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—28no—
324C22—28no—
325C22—28no—
326C22C2×#1458175114yes—
327C22#1912817614yes—
328C22#1912817914yes—
329C22C24×Dic8311yescharacteristic
330C22C22×#1305059012yes—
331C22C2×#1458183712yes—
332C22C2×#1458184012yes—
333C4—24no—
334C4—28no—
335C4—28no—
336C4—33216no—
337C4C2×#1458175118yes—
338C2—24no—
339C2—24no—
340C2—24no—
341C2C22×#1458175111yescharacteristic
342C2C2×#1912817611yescharacteristic
343C2C2×#1912817911yescharacteristic
344C2#2492117214yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC23×C4⋊C4×C83C2
socle, cosocleC23×C83C25
prime radicalC25×C83C23
hypercenterC23×C4⋊C4D83
hypocenterC83#34178
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #24910340C2semidirect
2. #24910332C2semidirect
3. #24910081C2semidirect
4. #24910066C2semidirect
7. #24909820C2semidirect
11. #19127600C22semidirect
12. #19127587C22semidirect
20. C4⋊C4×Dic83C22semidirect
27. C22×C4⋊C4×C83C4semidirect
134. C2×C4×C83#117semidirect
181. C83#34178semidirect
186. C22×C4⋊C4Dic83semidirect
310. C2×C4#14581751semidirect

Maximal subgroups

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

subgroupindex
#249103402
#249103322
#249100812
#249100662
#249098462
#249098202
C23×C4⋊C4×C832
#3417883