Finite Group Databasebeta

← #35322729

Subgroups — #35322729

Order 21248.

472 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 103 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#24912448C211yesmaximal, characteristic
2C2×#19127499C214yesmaximal
3C4×#14581711C211yesmaximal, characteristic
4C83×#1801C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic
6C2×#14581790C2212yes—
7C2×#14581711C2212yes—
8C4×#13050574—21no—
9#19127631—21no—
10#19127631—21no—
11#19127606C412yes—
12#19127599—21no—
13#19127599—21no—
14C4×C4⋊Dic83C411yescharacteristic
15#19127499—28no—
16#19127470—21no—
17C2×C4×Q8×C83C2211yescharacteristic
18#423×C83—21no—
19C2×#101×C83C2212yes—
20#343×C83—21no—
21C22×Dic166C2×C412yes—
22C2×#13050574—22no—
23#14581792—41no—
24#14581792—41no—
25#14581790—24no—
26#14581790—24no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—21no—
30C2×Dic83⋊C4—22no—
31C2×Dic83⋊C4—22no—
32C2×C4×Dic83C2×C412yes—
33#14581711—22no—
34#14581711D414yes—
35#14581703—41no—
36C4×C83⋊C8—41no—
37C83⋊#95—21no—
38C83⋊#95—22no—
39C83⋊#95—22no—
40C22×Q8×C83C2311yescharacteristic
41C2×M16×C83—22no—
42C4×Q8×C83—41no—
43C4×Q8×C83—41no—
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C4⋊C4×C83D411yescharacteristic
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C42×C83C2×C411yescharacteristic
48C2×C42×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50C4⋊C8×C83—41no—
51C4⋊C8×C83—41no—
52#101×C83—24no—
53C2×Dic166—24no—
54C2×Dic166—42no—
55C2×Dic166—42no—
56C22×Dic83—22no—
57C22×Dic83—22no—
58#13050574—24no—
59#13050574—42no—
60C2×C83⋊C8—42no—
61C4⋊Dic83—24no—
62C4⋊Dic83—42no—
63C4⋊Dic83—41no—
64C4⋊Dic83—41no—
65Dic83⋊C4—44no—
66Dic83⋊C4—42no—
67Dic83⋊C4—42no—
68C4×Dic83—24no—
69C2×Q8×C83—41no—
70C2×Q8×C83—41no—
71C2×Q8×C83C2×D412yes—
72C2×Q8×C83#4112yes—
73C22×C4×C83C22⋊C411yescharacteristic
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80M16×C83—42no—
81M16×C83—42no—
82C2×C8×C83—42no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—41no—
85C4⋊C4×C83—41no—
86C42×C83—21no—
87C42×C83—21no—
88C42×C83—21no—
89C42×C83—21no—
90C42×C83—21no—
91C42×C83—21no—
92C23×C83C2×C22⋊C411yescharacteristic
93Q8×C83—42no—
94Q8×C83—42no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—22no—
102C2×C4×C83—21no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83—41no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83#9911yescharacteristic
109C2×C4×C83#9911yescharacteristic
110C2×C4×C83C4×D412yes—
111C2×C4×C83#12112yes—
112C2×C4×C83#12411yescharacteristic
113C2×C4×C83#12711yescharacteristic
114C2×Dic83—24no—
115C2×Dic83—42no—
116C2×Dic83—42no—
117C2×Dic83—42no—
118C2×Dic83—42no—
119C2×Dic83—42no—
120Dic166—84no—
121C8×C83—82no—
122C83⋊C8—44no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83#39011yescharacteristic
126C22×C83C2×#9911yescharacteristic
127C22×C83#41011yescharacteristic
128C4×C83—21no—
129C4×C83—22no—
130C4×C83—21no—
131C4×C83—22no—
132C4×C83—21no—
133C4×C83—41no—
134C4×C83—41no—
135C4×C83—41no—
136C4×C83#35612yescomplemented
137Dic83—44no—
138Dic83—82no—
139#13227—831nomaximal, Sylow
140C2×C83—21no—
141C2×C83—21no—
142C2×C83#196411yescharacteristic
143C2×C83C2×#35611yescharacteristic
144C2×C83#213511yescharacteristic
145#2546—831no—
146C2×#356—834no—
147#1801D8311yescharacteristic
148C4×#101—831no—
149C83#1322711yescharacteristic, complemented, Sylow
150C2×C4⋊Q8—832no—
151C2×C4⋊Q8—831no—
152C2×C4×Q8C2×D8311yescharacteristic
153#423—21no—
154C2×#101—832no—
155C2×#101C2×D8312yes—
156C4×M16—1661no—
157#398—1661no—
158#398—1661no—
159#391—832no—
160#389—1661no—
161#389—1661no—
162C4×C4⋊C4—831no—
163#356—1668no—
164#343—21no—
165#343—1661no—
166C22×Q8—832no—
167C22×Q8C22×D8311yescharacteristic
168C2×M16—22no—
169C2×M16—1662no—
170C4⋊Q8—3321no—
171C4⋊Q8—3321no—
172C4⋊Q8—1664no—
173C4⋊Q8—1664no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—1661no—
177C2×C4⋊C4—1661no—
178C2×C4⋊C4—1662no—
179C2×C4⋊C4—1662no—
180C2×C4⋊C4—1661no—
181C2×C4⋊C4C4×D8311yescharacteristic
182C2×C4⋊C4D33211yescharacteristic
183C2×C4⋊C4C22⋊D8311yescharacteristic
184C2×C42—832no—
185C2×C42C4×D8311yescharacteristic
186C2×C42D33211yescharacteristic
187C2×C42C22⋊D8311yescharacteristic
188C4⋊C8—41no—
189C4⋊C8—41no—
190#101—24no—
191#101—834no—
192#101—1662no—
193#97—3321no—
194C4×C8—3321no—
195#95—1661no—
196#95—1662no—
197#95—1662no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—3322no—
201C2×Q8—3322no—
202C2×Q8—1664no—
203C2×Q8D4×D8312yes—
204C2×Q8#1305059112yes—
205C22×C4—1662no—
206C22×C4—1662no—
207C22×C4(C2×D83)⋊C411yescharacteristic
208C22×C4(C2×D83)⋊C411yescharacteristic
209C22×C4(C2×D83)⋊C411yescharacteristic
210C22×C4(C2×D83)⋊C411yescharacteristic
211C22×C4C2×C4×D8311yescharacteristic
212C22×C4C2×C22⋊D8311yescharacteristic
213C22×C4C2×D33211yescharacteristic
214M16—42no—
215M16—42no—
216M16—3322no—
217M16—1664no—
218C2×C8—42no—
219C2×C8—3322no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—3322no—
224C4⋊C4—3324no—
225C4⋊C4—3322no—
226C4⋊C4—3322no—
227C4⋊C4—3321no—
228C4⋊C4—3321no—
229C4⋊C4—1664no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—1664no—
237C23C2×(C2×D83)⋊C411yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—6644no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—3322no—
255C2×C4—3322no—
256C2×C4—3322no—
257C2×C4—3322no—
258C2×C4—3322no—
259C2×C4—1664no—
260C2×C4#1458170611yescharacteristic
261C2×C4#1458170611yescharacteristic
262C2×C4#1458175412yes—
263C2×C4#1458176712yes—
264C2×C4#1458178111yescharacteristic
265C2×C4#1458178811yescharacteristic
266C8—82no—
267C8—3324no—
268C22—21no—
269C22—21no—
270C22#1912760211yescharacteristic
271C22C2×#1458170611yescharacteristic
272C22#1912767411yescharacteristic
273C4—21no—
274C4—22no—
275C4—21no—
276C4—22no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—3324no—
282C4—6642no—
283C4#1912749912yescomplemented
284C2—21no—
285C2—21no—
286C2#2490829811yescharacteristic
287C2C2×#1912749911yescharacteristic
288C2#2490915711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC22×Q8C22×D83
FittingC83×#1801C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC22×Q8×C83C23
hypercenter#1801D83
hypocenterC83#13227
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1801

Complements

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

NH ≅ G/Ntype
136. C4×C83#356semidirect
149. C83#13227semidirect
283. C4#19127499semidirect

Maximal subgroups

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

subgroupindex
#249124482
C2×#191274992
C4×#145817112
C83×#18012
#1322783