Finite Group Databasebeta

← #35322231

Subgroups — #35322231

Order 21248.

324 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127531C211yesmaximal, characteristic, complemented
2C2×#19127508C211yesmaximal, characteristic
3#24908352C211yesmaximal, characteristic
4#24907344C211yesmaximal, characteristic, complemented
5#24905558C211yesmaximal, characteristic, complemented
6#24905540C211yesmaximal, characteristic, complemented
7C83×#2138C211yesmaximal, characteristic
8C22×C83⋊C16C2211yescharacteristic
9C2×#14581728C2211yescharacteristic
10C2×#14581728C2211yescharacteristic
11C2×#14581719—21no—
12#19127668—21no—
13#19127657—21no—
14C2×C8×Dic83C2211yescharacteristic
15#19127531—22no—
16#19127508—22no—
17#19127460—21no—
18C2×#135×C83C2211yescharacteristic
19C22×C16×C83C2211yescharacteristic
20#427×C83C2211yescharacteristic
21#358×C83C411yescharacteristic, complemented
22#358×C83C411yescharacteristic, complemented
23#357×C83C411yescharacteristic, complemented
24#357×C83C411yescharacteristic, complemented
25C2×#13050574—21no—
26C2×#13050574—21no—
27C22×C83⋊C8D411yescharacteristic
28C2×C4×Dic83D411yescharacteristic
29C2×C83⋊C16—21no—
30C2×C83⋊C16—21no—
31C2×C83⋊C16—21no—
32#14581728—21no—
33#14581728—21no—
34#14581728—21no—
35#14581728D412yes—
36#14581719—42no—
37C22⋊(C83⋊C8)—41no—
38C22⋊(C83⋊C8)—41no—
39#14581703—41no—
40C4×C83⋊C8—41no—
41C8×Dic83—21no—
42C8×Dic83—21no—
43C2×#41×C83—21no—
44#135×C83—21no—
45#135×C83—21no—
46#135×C83—21no—
47#135×C83—21no—
48C2×Q16×C83C2×C411yescharacteristic
49C2×SD16×C83—21no—
50C2×D8×C83C2×C411yescharacteristic
51C22×C8×C83C2311yescharacteristic
52#117×C83—21no—
53C2×C16×C83—21no—
54C2×C16×C83—21no—
55C2×C16×C83C2×C411yescharacteristic
56C2×C16×C83C2×C411yescharacteristic
57#107×C83C2×C411yescharacteristic
58#107×C83C2×C411yescharacteristic
59#106×C83—21no—
60C22×Dic83—21no—
61#13050574—22no—
62#13050574—41no—
63#13050574—41no—
64#13050574—41no—
65C2×C83⋊C8—21no—
66C2×C83⋊C8—21no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8—41no—
69C2×C83⋊C8D811yescharacteristic
70C2×C83⋊C8D811yescharacteristic
71C4×Dic83—21no—
72C4×Dic83—21no—
73C83⋊C16—41no—
74#41×C83—41no—
75#41×C83—41no—
76#41×C83—41no—
77#41×C83—41no—
78C2×Q8×C83—21no—
79C2×D4×C83—21no—
80C2×D4×C83—41no—
81C22×C4×C83—41no—
82C22×C4×C83C2×D411yescharacteristic
83Q16×C83—21no—
84Q16×C83—21no—
85SD16×C83—41no—
86SD16×C83—41no—
87D8×C83—21no—
88D8×C83—21no—
89C2×C8×C83C22×C411yescharacteristic
90C2×C8×C83C2×D411yescharacteristic
91C2×C8×C83#4111yescharacteristic
92C2×C8×C83#4111yescharacteristic
93C2×C8×C83#4111yescharacteristic
94C2×C8×C83#4111yescharacteristic
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C22⋊C4×C83—41no—
98C42×C83—41no—
99C16×C83—21no—
100C16×C83—21no—
101C23×C83—41no—
102C23×C83#13511yescharacteristic
103Q8×C83—41no—
104Q8×C83—41no—
105D4×C83—41no—
106D4×C83—41no—
107D4×C83—81no—
108D4×C83—81no—
109C2×C4×C83—41no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—81no—
113C2×C4×C83—81no—
114C2×C4×C83C4×D411yescharacteristic
115C2×C4×C83#12111yescharacteristic
116C2×C4×C83#12311yescharacteristic
117C2×C4×C83C2×D811yescharacteristic
118C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C83#13711yescharacteristic
120C2×Dic83—22no—
121C2×Dic83—41no—
122C2×Dic83—41no—
123C8×C83#11711yescharacteristic
124C8×C83C4×D411yescharacteristic
125C8×C83#12411yescharacteristic
126C8×C83#12611yescharacteristic
127C83⋊C8—21no—
128C83⋊C8—21no—
129C83⋊C8—42no—
130C83⋊C8—81no—
131C22×C83—21no—
132C22×C83—21no—
133C22×C83—81no—
134C22×C83—81no—
135C22×C83—81no—
136C22×C83C4×D811yescharacteristic
137C22×C83#46411yescharacteristic
138C22×C83#48211yescharacteristic
139C4×C83—81no—
140C4×C83—81no—
141C4×C83—81no—
142C4×C83#38811yescharacteristic
143C4×C83#44011yescharacteristic
144C4×C83#45911yescharacteristic
145C4×C83#48011yescharacteristic
146Dic83—42no—
147#13134—831nomaximal, Sylow
148C2×C83—21no—
149C2×C83—21no—
150C2×C83—161no—
151C2×C83#192511yescharacteristic
152C2×C83#217811yescharacteristic
153C2×C83#217911yescharacteristic
154C2×#368—831no—
155C2×#359—831no—
156#2138D8311yescharacteristic
157#1974—831no—
158#1772—831no—
159#1382—831no—
160#1374—831no—
161C83#1313411yescharacteristic, complemented, Sylow
162C2×#135C2×D8311yescharacteristic
163C22×C16—831no—
164C22×C16C2×D8311yescharacteristic
165C2×#108—831no—
166C2×#108—831no—
167#427C2×D8311yescharacteristic
168C2×#104—1661no—
169#408—1661no—
170#405—1661no—
171C2×C4×C8—831no—
172#368—1662no—
173#359—1662no—
174#358Dic8311yescharacteristic, complemented
175#358Dic8311yescharacteristic, complemented
176#357Dic8311yescharacteristic, complemented
177#357Dic8311yescharacteristic, complemented
178#339—1661no—
179C2×#41—21no—
180#135—21no—
181#135—21no—
182#135—21no—
183#135—21no—
184C2×Q16C2×Dic8311yescharacteristic
185C2×SD16—21no—
186C2×D8C2×Dic8311yescharacteristic
187C2×M16—1661no—
188C2×M16—1661no—
189C22×C8—831no—
190C22×C8C22×D8311yescharacteristic
191#117—21no—
192C2×C42—831no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—1661no—
196C2×C16—1661no—
197C2×C16—1661no—
198C2×C16C2×Dic8311yescharacteristic
199C2×C16C2×Dic8311yescharacteristic
200#108—832no—
201#108—1661no—
202#108—1661no—
203#108—1661no—
204#107C2×Dic8311yescharacteristic
205#107C2×Dic8311yescharacteristic
206#106—21no—
207#104—3322no—
208C22⋊C8—3321no—
209C22⋊C8—3321no—
210#97—3321no—
211C4×C8—3321no—
212C4×C8—1661no—
213C4×C8—1661no—
214#41—41no—
215#41—41no—
216#41—41no—
217#41—41no—
218C2×Q8—21no—
219C2×D4—21no—
220C2×D4—41no—
221C22×C4—41no—
222C22×C4—1661no—
223C22×C4D4×D8311yescharacteristic
224Q16—21no—
225Q16—21no—
226SD16—41no—
227SD16—41no—
228D8—21no—
229D8—21no—
230M16—3321no—
231M16—3321no—
232M16—3321no—
233M16—1662no—
234C2×C8—3321no—
235C2×C8—3321no—
236C2×C8—831no—
237C2×C8—831no—
238C2×C8—1661no—
239C2×C8—1661no—
240C2×C8C22×Dic8311yescharacteristic
241C2×C8D4×D8311yescharacteristic
242C2×C8#1305058911yescharacteristic
243C2×C8#1305058911yescharacteristic
244C2×C8#1305059011yescharacteristic
245C2×C8#1305059011yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—21no—
248C22⋊C4—41no—
249C42—41no—
250C42—1661no—
251C42—1661no—
252C16—21no—
253C16—21no—
254C16—3321no—
255C23—41no—
256C23#1458181111yescharacteristic
257Q8—41no—
258Q8—41no—
259D4—41no—
260D4—41no—
261D4—81no—
262D4—81no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—3321no—
269C2×C4—3321no—
270C2×C4—1662no—
271C2×C4D4×Dic8311yescharacteristic
272C2×C4#1458176511yescharacteristic
273C2×C4#1458177411yescharacteristic
274C2×C4D8×D8311yescharacteristic
275C2×C4#1458181911yescharacteristic
276C2×C4#1458182111yescharacteristic
277C8—3322no—
278C8—6641no—
279C8—1661no—
280C8—1661no—
281C8#1458175111yescharacteristic
282C8D4×Dic8311yescharacteristic
283C8#1458177811yescharacteristic
284C8#1458178511yescharacteristic
285C22—21no—
286C22—21no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22D8×Dic8311yescharacteristic
291C22#1912792211yescharacteristic
292C22#1912802311yescharacteristic
293C4—81no—
294C4—81no—
295C4—81no—
296C4—3322no—
297C4#1912759411yescharacteristic
298C4#1912780211yescharacteristic
299C4#1912789511yescharacteristic
300C4#1912801311yescharacteristic
301C2—21no—
302C2—21no—
303C2—161no—
304C2#2490808511yescharacteristic
305C2#2490934411yescharacteristic
306C2#2490934911yescharacteristic
subgroupG/H
centerC2×C4D8×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#2138C2
socleC22×C83C4×D8
prime radicalC2×#135×C83C22
cosocleC22×C8×C83C23
hypercenter#2138D83
hypocenterC83#13134
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. C2×#19127531C2semidirect
4. #24907344C2semidirect
5. #24905558C2semidirect
6. #24905540C2semidirect
21. #358×C83C4semidirect
22. #358×C83C4semidirect
23. #357×C83C4semidirect
24. #357×C83C4semidirect
161. C83#13134semidirect
174. #358Dic83semidirect
175. #358Dic83semidirect
176. #357Dic83semidirect
177. #357Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275312
C2×#191275082
#249083522
#249073442
#249055582
#249055402
C83×#21382
#1313483