Finite Group Databasebeta

← #35306269

Subgroups — #35306269

Order 21248.

488 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#24918408C211yesmaximal, characteristic
2C2×#19128053C211yesmaximal, characteristic
3#24908452C211yesmaximal, characteristic
4#24906144C211yesmaximal, characteristic
5#24905918C211yesmaximal, characteristic
6#24905301C211yesmaximal, characteristic
7C83×#3165C211yesmaximal, characteristic
8C2×#14581792C2211yescharacteristic
9#19128080—21no—
10#19128053—22no—
11#19128053—22no—
12#19128053—21no—
13#19127911—22no—
14#19127856—22no—
15#19127754C2211yescharacteristic
16#19127725C2211yescharacteristic
17C2×#14581715—22no—
18C2×C4×C83⋊C8C2211yescharacteristic
19#19127469—22no—
20#19127450—22no—
21Q8⋊(C83⋊C8)—22no—
22#533×C83C2211yescharacteristic
23C2×C4⋊Q8×C83C2211yescharacteristic
24#501×C83—22no—
25#500×C83—21no—
26#498×C83—21no—
27#428×C83—22no—
28C4×M16×C83C2211yescharacteristic
29C22×Dic166D412yes—
30C2×#13050580—24no—
31C2×#13050574—21no—
32C22×C83⋊C8—21no—
33#14581792—21no—
34#14581792—21no—
35#14581792—21no—
36#14581792—21no—
37C2×C4⋊Dic83—21no—
38C2×C4⋊Dic83—21no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)D412yes—
42#14581715—24no—
43#14581715—28no—
44C22⋊(C83⋊C8)—21no—
45C4×C83⋊C8—21no—
46C4×C83⋊C8—21no—
47C4×C83⋊C8D412yes—
48C22×Q8×C83—21no—
49C2×M16×C83D412yes—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83—21no—
52C4⋊Q8×C83—21no—
53C4⋊Q8×C83D411yescharacteristic
54C4⋊Q8×C83D411yescharacteristic
55C4⋊Q8×C83D412yes—
56#125×C83—21no—
57C22⋊Q8×C83—21no—
58C22⋊Q8×C83—21no—
59C4×Q8×C83—21no—
60C4×Q8×C83—21no—
61#117×C83—21no—
62C2×C4⋊C4×C83—21no—
63C2×C4⋊C4×C83—21no—
64C2×C42×C83C2311yescharacteristic
65#107×C83—24no—
66#106×C83—24no—
67#97×C83—21no—
68C4×C8×C83—21no—
69C2×Dic166—22no—
70C2×Dic166—22no—
71C2×Dic166—44no—
72C22×Dic83—22no—
73#13050580—84no—
74#13050574—42no—
75C2×C83⋊C8—22no—
76C2×C83⋊C8—22no—
77C2×C83⋊C8—42no—
78C2×C83⋊C8—41no—
79C4⋊Dic83—41no—
80C4⋊Dic83—41no—
81C4⋊Dic83—41no—
82C4⋊Dic83—41no—
83C2×Q8×C83—21no—
84C2×Q8×C83—21no—
85C2×Q8×C83—22no—
86C2×Q8×C83—41no—
87C2×Q8×C83—82no—
88C22×C4×C83—41no—
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83#4111yescharacteristic
92M16×C83—22no—
93M16×C83Q1614yescomplemented
94C2×C8×C83—22no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83—22no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—22no—
99C4⋊C4×C83—21no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83—41no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C22⋊C4×C83—41no—
106C42×C83—41no—
107C42×C83C2×D411yescharacteristic
108C42×C83C2×D411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83C2×D411yescharacteristic
111C23×C83#12111yescharacteristic
112Q8×C83—42no—
113Q8×C83—42no—
114Q8×C83—82no—
115Q8×C83—82no—
116C2×C4×C83—22no—
117C2×C4×C83—22no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—42no—
121C2×C4×C83—82no—
122C2×C4×C83#12011yescharacteristic
123C2×C4×C83#12011yescharacteristic
124C2×C4×C83#12011yescharacteristic
125C2×C4×C83#12111yescharacteristic
126C2×C4×C83#12111yescharacteristic
127C2×C4×C83#12711yescharacteristic
128C2×C4×C83C2×Q1612yes—
129C2×C4×C83#13712yes—
130C2×Dic83—42no—
131C2×Dic83—82no—
132Dic166—44no—
133Dic166—44no—
134C8×C83—24no—
135C83⋊C8—42no—
136C83⋊C8—42no—
137C83⋊C8—81no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83#39211yescharacteristic
141C22×C83#46212yes—
142C4×C83—21no—
143C4×C83—21no—
144C4×C83—81no—
145C4×C83—82no—
146C4×C83—81no—
147C4×C83#45112yes—
148C4×C83#45512yes—
149C4×C83#47112yes—
150C4×C83#49411yescharacteristic
151C4×C83#49711yescharacteristic
152Dic83—82no—
153#10163—831nomaximal, Sylow
154C2×C83—22no—
155C2×C83#200411yescharacteristic
156C2×C83#201611yescharacteristic
157C2×C83#201911yescharacteristic
158#3209—831no—
159#3165D8311yescharacteristic
160C2×#487—831no—
161#1997—831no—
162#1548—831no—
163#1493—831no—
164#1278—831no—
165C83#1016311yescharacteristic, complemented, Sylow
166#533C2×D8311yescharacteristic
167C2×C4⋊Q8—831no—
168C2×C4⋊Q8C2×D8311yescharacteristic
169#501—22no—
170#500—21no—
171#498—21no—
172#494—1661no—
173#487—1662no—
174#487—1662no—
175#487—1661no—
176#462—1662no—
177#451—1662no—
178#431—831no—
179#428—22no—
180#423—831no—
181C2×#103—1662no—
182C4×M16C2×D8311yescharacteristic
183C2×C4×C8—831no—
184#343—1662no—
185#333—1662no—
186Q8⋊C8—1662no—
187C22×Q8—21no—
188C22×Q8—832no—
189C2×Q16—1664no—
190C2×M16—1661no—
191C2×M16C22⋊D8312yes—
192C22×C8—1661no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—1661no—
197C4⋊Q8—1661no—
198C4⋊Q8—1661no—
199C4⋊Q8—1661no—
200C4⋊Q8C22⋊D8312yes—
201C4⋊Q8C22⋊D8311yescharacteristic
202C4⋊Q8C22⋊D8311yescharacteristic
203#125—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208#117—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—1661no—
212C2×C4⋊C4—1661no—
213C2×C42C22×D8311yescharacteristic
214#107—24no—
215#106—24no—
216C4⋊C8—832no—
217C4⋊C8—1661no—
218C4⋊C8—1661no—
219#103—1668no—
220#103—1664no—
221C22⋊C8—1661no—
222#97—21no—
223C4×C8—21no—
224C4×C8—832no—
225C4×C8—1661no—
226C4×C8—1661no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—82no—
232C2×Q8—3324no—
233C2×Q8—1662no—
234C2×Q8—1662no—
235C22×C4—41no—
236C22×C4—1662no—
237C22×C4D4×D8311yescharacteristic
238C22×C4C2×C22⋊D8311yescharacteristic
239C22×C4#1305058911yescharacteristic
240Q16—6644no—
241M16—22no—
242M16—3322no—
243M16#1305058014yescomplemented
244C2×C8—22no—
245C2×C8—3322no—
246C2×C8—3321no—
247C2×C8—1662no—
248C2×C8—1662no—
249C4⋊C4—22no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—3321no—
260C4⋊C4—3321no—
261C4⋊C4—3321no—
262C4⋊C4—3321no—
263C22⋊C4—41no—
264C42—41no—
265C42D4×D8311yescharacteristic
266C42D4×D8311yescharacteristic
267C42C2×C22⋊D8311yescharacteristic
268C42C2×C22⋊D8311yescharacteristic
269C23#1458176311yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—82no—
273Q8—82no—
274Q8—3324no—
275Q8—3324no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—3322no—
283C2×C4—6642no—
284C2×C4#1458176011yescharacteristic
285C2×C4#1458176011yescharacteristic
286C2×C4#1458176111yescharacteristic
287C2×C4#1458176411yescharacteristic
288C2×C4#1458176411yescharacteristic
289C2×C4#1458178811yescharacteristic
290C2×C4C2×#1305058012yes—
291C2×C4#1458182312yes—
292C8—24no—
293C8—3322no—
294C8—3322no—
295C8—6641no—
296C22—21no—
297C22—21no—
298C22#1912760911yescharacteristic
299C22#1912790812yes—
300C4—21no—
301C4—21no—
302C4—82no—
303C4—81no—
304C4—81no—
305C4—6642no—
306C4#1912785712yes—
307C4#1912787712yes—
308C4#1912796212yes—
309C4#1912808011yescharacteristic
310C4#1912809311yescharacteristic
311C2—22no—
312C2#2490848911yescharacteristic
313C2#2490856311yescharacteristic
314C2#2490857711yescharacteristic
subgroupG/H
centerC22#19127609
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#3165C2
socleC22×C83#392
prime radicalC23×C83#121
hypercenter#3165D83
hypocenterC83#10163
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
93. M16×C83Q16semidirect
165. C83#10163semidirect
243. M16#13050580semidirect

Maximal subgroups

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

subgroupindex
#249184082
C2×#191280532
#249084522
#249061442
#249059182
#249053012
C83×#31652
#1016383