Finite Group Databasebeta

← #35318282

Subgroups — #35318282

Order 21248.

288 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 75 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#24909690C211yesmaximal, characteristic
2#24909605C211yesmaximal, characteristic
3#24906857C211yesmaximal, characteristic
4#24905512C211yesmaximal, characteristic
5Q8⋊(C83⋊C16)C211yesmaximal, characteristic
6#24905427C211yesmaximal, characteristic
7C83×#3395C211yesmaximal, characteristic
8#19128106C2211yescharacteristic
9#19127911—21no—
10#19127528—21no—
11C4⋊(C83⋊C16)C2211yescharacteristic
12#19127505—21no—
13#19127505—22no—
14C4×C83⋊C16C2211yescharacteristic
15#19127454C2211yescharacteristic
16#19127446—21no—
17#557×C83—21no—
18#494×C83—21no—
19#494×C83C2211yescharacteristic
20#462×C83—22no—
21#462×C83—21no—
22C8×Q8×C83C2211yescharacteristic
23C4×Q16×C83—21no—
24C4×Q16×C83C2211yescharacteristic
25C2×#13050580—41no—
26#14581792—21no—
27C2×C83⋊C16—21no—
28C2×C83⋊C16—21no—
29#14581726—21no—
30#14581726—21no—
31C4⋊(C83⋊C8)—21no—
32#14581715—41no—
33C2×Q16×C83—22no—
34C2×Q16×C83—21no—
35C2×Q16×C83D411yescharacteristic
36C2×Q16×C83D412yes—
37C4⋊Q8×C83—22no—
38C4⋊Q8×C83—21no—
39#125×C83—42no—
40#125×C83—41no—
41C4×Q8×C83—22no—
42C4×Q8×C83—21no—
43C4×Q8×C83—41no—
44C4×Q8×C83D411yescharacteristic
45#107×C83D411yescharacteristic
46C4⋊C8×C83—21no—
47C4⋊C8×C83D411yescharacteristic
48#103×C83—22no—
49#103×C83—21no—
50C4×C8×C83—21no—
51C4×C8×C83C2311yescharacteristic
52C2×Dic166—41no—
53#13050580—82no—
54C2×C83⋊C8—41no—
55C4⋊Dic83—41no—
56C4⋊Dic83—41no—
57C83⋊C16—41no—
58C83⋊C16—41no—
59C2×Q8×C83—21no—
60C2×Q8×C83—22no—
61C2×Q8×C83D811yescharacteristic
62Q16×C83—21no—
63Q16×C83—21no—
64Q16×C83—42no—
65Q16×C83—41no—
66Q16×C83—41no—
67Q16×C83—82no—
68C2×C8×C83—21no—
69C2×C8×C83C2×D411yescharacteristic
70C2×C8×C83C2×D411yescharacteristic
71C4⋊C4×C83—21no—
72C4⋊C4×C83—21no—
73C4⋊C4×C83—42no—
74C4⋊C4×C83—42no—
75C4⋊C4×C83—42no—
76C4⋊C4×C83—41no—
77C4⋊C4×C83—41no—
78C4⋊C4×C83—41no—
79C4⋊C4×C83D811yescharacteristic
80C42×C83—21no—
81C42×C83—42no—
82C42×C83—41no—
83C42×C83C2×D411yescharacteristic
84Q8×C83—21no—
85Q8×C83—41no—
86Q8×C83—42no—
87Q8×C83—42no—
88Q8×C83—41no—
89Q8×C83SD3212yes—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—41no—
93C2×C4×C83—42no—
94C2×C4×C83—41no—
95C2×C4×C83#12011yescharacteristic
96C2×C4×C83C2×D811yescharacteristic
97C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
98C2×Dic83—81no—
99Dic166—81no—
100Dic166—81no—
101C8×C83—21no—
102C8×C83—41no—
103C8×C83#13511yescharacteristic
104C8×C83#13711yescharacteristic
105C83⋊C8—81no—
106C22×C83#44711yescharacteristic
107C4×C83—22no—
108C4×C83—21no—
109C4×C83—41no—
110C4×C83—81no—
111C4×C83—82no—
112C4×C83—81no—
113C4×C83#45211yescharacteristic
114C4×C83#45511yescharacteristic
115C4×C83C2×SD3211yescharacteristic
116C4×C83#51011yescharacteristic
117Dic83—161no—
118#12382—831nomaximal, Sylow
119C2×C83#164111yescharacteristic
120C2×C83#218511yescharacteristic
121C2×C83#219511yescharacteristic
122#3395D8311yescharacteristic
123#2241—831no—
124#2225—831no—
125#1674—831no—
126#1362—831no—
127Q8⋊C16—831no—
128#1332—831no—
129C83#1238211yescharacteristic, complemented, Sylow
130#557—21no—
131#500—831no—
132#494—21no—
133#494C2×D8311yescharacteristic
134#462—22no—
135#462—21no—
136#462—1661no—
137C8×Q8C2×D8311yescharacteristic
138C4×Q16—21no—
139C4×Q16C2×D8311yescharacteristic
140#367—1661no—
141C4⋊C16—831no—
142#358—1661no—
143#358—1662no—
144C4×C16—831no—
145#335—831no—
146#332—1661no—
147C2×Q16—22no—
148C2×Q16—21no—
149C2×Q16—3321no—
150C2×Q16C22⋊D8311yescharacteristic
151C2×Q16C22⋊D8312yes—
152C4⋊Q8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—1661no—
155#125—42no—
156#125—41no—
157C4×Q8—22no—
158C4×Q8—21no—
159C4×Q8—41no—
160C4×Q8C22⋊D8311yescharacteristic
161C2×C16—1661no—
162C2×C16—1661no—
163#107—1661no—
164#107—1661no—
165#107C22⋊D8311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—1661no—
168C4⋊C8C22⋊D8311yescharacteristic
169#103—22no—
170#103—21no—
171#103—3321no—
172C4×C8—21no—
173C4×C8C22×D8311yescharacteristic
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—3321no—
177C2×Q8#1305057511yescharacteristic
178Q16—21no—
179Q16—21no—
180Q16—42no—
181Q16—41no—
182Q16—41no—
183Q16—82no—
184Q16—6642no—
185C2×C8—21no—
186C2×C8—3321no—
187C2×C8C2×C22⋊D8311yescharacteristic
188C2×C8C2×C22⋊D8311yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—3321no—
198C4⋊C4—3321no—
199C4⋊C4#1305057511yescharacteristic
200C42—21no—
201C42—42no—
202C42—41no—
203C42C2×C22⋊D8311yescharacteristic
204C16—3321no—
205C16—3321no—
206Q8—21no—
207Q8—41no—
208Q8—42no—
209Q8—42no—
210Q8—41no—
211Q8—6641no—
212Q8—6641no—
213Q8#1458173512yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—6641no—
220C2×C4#1458176111yescharacteristic
221C2×C4C2×#1305057511yescharacteristic
222C2×C4#1458181811yescharacteristic
223C8—21no—
224C8—41no—
225C8—6641no—
226C8#1458181211yescharacteristic
227C8#1458182211yescharacteristic
228C22#1912783511yescharacteristic
229C4—22no—
230C4—21no—
231C4—41no—
232C4—81no—
233C4—81no—
234C4—82no—
235C4—13281no—
236C4#1912786311yescharacteristic
237C4#1912787911yescharacteristic
238C4C2×#1458173511yescharacteristic
239C4#1912814211yescharacteristic
240C2#2490665611yescharacteristic
241C2#2490938011yescharacteristic
242C2#2490944111yescharacteristic
subgroupG/H
centerC22#19127835
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#3395C2
socle, prime radicalC22×C83#447
hypercenter#3395D83
hypocenterC83#12382
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
129. C83#12382semidirect

Maximal subgroups

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

subgroupindex
#249096902
#249096052
#249068572
#249055122
Q8⋊(C83⋊C16)2
#249054272
C83×#33952
#1238283