Finite Group Databasebeta

← #49131652

Subgroups — #49131652

Order 36224.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#31827011C211yesmaximal, characteristic
2C2×#22583609C211yesmaximal, characteristic, complemented
3#31826826C212yesmaximal, complemented
4#31826823C212yesmaximal, complemented
5#31826814C212yesmaximal, complemented
6#31826810C212yesmaximal, complemented
7#31826467C211yesmaximal, characteristic, complemented
8#31826466C211yesmaximal, characteristic, complemented
9#31826464C211yesmaximal, characteristic, complemented
10M16×Dic283C211yesmaximal, characteristic, complemented
11#417×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×Dic566—21no—
14C2×#17985376C2211yescharacteristic
15#22583609—21no—
16#22583609—21no—
17#22583609C2212yes—
18#22583609C2212yes—
19#22583608C2211yescharacteristic
20#22583606C2211yescharacteristic
21#22583599C2211yescharacteristic
22#22583597C2211yescharacteristic
23#22583582—21no—
24#22583582—21no—
25C4×C22⋊D283—21no—
26C4×C22⋊D283—21no—
27C2×C4⋊Dic283C2211yescharacteristic
28C2×Dic283⋊C4—21no—
29C2×C4×Dic283C2211yescharacteristic
30#22583535C2212yes—
31#22583534C2212yes—
32#22583533C2212yes—
33#22583532C2212yes—
34#22583531C2212yes—
35#22583530C2212yes—
36#22583520C2212yes—
37C8×Dic283C2212yes—
38C2×#41×C283C2211yescharacteristic
39C2×M16×C283C2211yescharacteristic
40C2×C4⋊C4×C283C2211yescharacteristic
41#103×C283C2212yes—
42#102×C283C2212yes—
43#17985392—22no—
44#17985392—22no—
45C2×Dic566—22no—
46C2×Dic566—22no—
47C2×Dic566—44no—
48C2×Dic566C2311yescharacteristic
49C2×D1132C2311yescharacteristic
50C2×C22⋊D283—21no—
51C2×C22⋊D283—21no—
52C2×C22⋊D283—21no—
53C22×Dic283—21no—
54C22×Dic283D411yescharacteristic
55C22×Dic283D411yescharacteristic
56C2×C4×D283—21no—
57#17985376—22no—
58C2×C283⋊C8C2312yes—
59C4⋊Dic283—21no—
60C4⋊Dic283C2312yes—
61Dic283⋊C4—22no—
62Dic283⋊C4—22no—
63Dic283⋊C4—21no—
64Dic283⋊C4—21no—
65C22⋊Dic283—21no—
66C22⋊Dic283—21no—
67(C2×D283)⋊C4—21no—
68(C2×D283)⋊C4—21no—
69C4×Dic283D411yescharacteristic
70C4×Dic283D411yescharacteristic
71C4×Dic283C2311yescharacteristic
72C4×Dic283C2311yescharacteristic
73#41×C283—22no—
74#41×C283—22no—
75C2×Q8×C283C2311yescharacteristic
76C2×D4×C283—21no—
77C2×D4×C283C2311yescharacteristic
78C22×C4×C283—21no—
79C22×C4×C283—21no—
80C22×C4×C283C2311yescharacteristic
81M16×C283—22no—
82C2×C8×C283C2312yes—
83C4⋊C4×C283—21no—
84C4⋊C4×C283C2312yes—
85C22×D283—21no—
86Dic566—21no—
87Dic566—21no—
88Dic566—44no—
89Dic566—44no—
90D1132—21no—
91D1132—21no—
92C22⋊D283—42no—
93C22⋊D283—42no—
94C22⋊D283—42no—
95C2×Dic283—22no—
96C2×Dic283—22no—
97C2×Dic283—21no—
98C2×Dic283—22no—
99C2×Dic283—42no—
100C2×Dic283C2×D411yescharacteristic
101C2×Dic283C2×D411yescharacteristic
102C2×Dic283C2×D411yescharacteristic
103C2×Dic283C2×D411yescharacteristic
104C4×D283—42no—
105C283⋊C8—22no—
106C23×C283—21no—
107C23×C283C2×D411yescharacteristic
108Q8×C283—21no—
109Q8×C283—21no—
110D4×C283—21no—
111D4×C283—21no—
112D4×C283—42no—
113C2×C4×C283—22no—
114C2×C4×C283—21no—
115C2×C4×C283—42no—
116C2×C4×C283—42no—
117C2×C4×C283C2×D411yescharacteristic
118C2×C4×C283#4112yes—
119C2×C4×C283#4112yes—
120C2×C4×C283C2411yescharacteristic
121C8×C283—22no—
122C22×C283—21no—
123C22×C283—21no—
124C22×C283—41no—
125C22×C283—41no—
126C22×C283—41no—
127C22×C283#12412yes—
128C22×C283C22×D411yescharacteristic
129C2×D283—41no—
130C2×D283—41no—
131C2×D283—41no—
132C4×C283—41no—
133C4×C283—42no—
134C4×C283#12412yes—
135C4×C283C2×#4111yescharacteristic
136C4×C283C2×#4111yescharacteristic
137Dic283—21no—
138Dic283—21no—
139Dic283—41no—
140Dic283—42no—
141Dic283AGL(1,ℤ/8)12yes—
142Dic283#13712yes—
143C2×C283—22no—
144C2×C283—81no—
145C2×C283C2×#12411yescharacteristic
146C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C283C2×#13711yescharacteristic
148D283—81no—
149C283#314111yescharacteristic, complemented, Sylow
150#3141—2831nomaximal, Sylow
151#532—2831no—
152C2×C4⋊Q8—2831no—
153#489—2832no—
154#489—2832no—
155#487—2832no—
156#486—2832no—
157#417—2831no—
158#417—2831no—
159#417—2831no—
160#417D28311yescharacteristic, complemented
161C4×M16—2831no—
162C2×#41—2831no—
163C2×#41C2×D28311yescharacteristic
164C22×Q8—5661no—
165C2×M16—2831no—
166C2×M16C2×D28311yescharacteristic
167C4⋊Q8—2832no—
168C4⋊Q8—2832no—
169C4⋊Q8—2831no—
170C4⋊Q8—5661no—
171C4⋊Q8—5661no—
172#127—2831no—
173#124—2831no—
174#124—2831no—
175#121—5661no—
176#121—5661no—
177C4×D4—5661no—
178C4×D4—5661no—
179C2×C4⋊C4—2831no—
180C2×C4⋊C4—5661no—
181C2×C4⋊C4C2×D28311yescharacteristic
182C2×C42—2831no—
183#103—2832no—
184#103—2832no—
185#103—2832no—
186#103C2×D28312yes—
187#102—2832no—
188#102—2832no—
189#102—2832no—
190#102C2×D28312yes—
191#97—2832no—
192C4×C8—2832no—
193#41—22no—
194#41—22no—
195#41—5662no—
196#41—5662no—
197C2×Q8—2831no—
198C2×Q8—5662no—
199C2×Q8—5662no—
200C2×Q8—11324no—
201C2×Q8C22×D28311yescharacteristic
202C2×D4—21no—
203C2×D4—2831no—
204C2×D4—5661no—
205C2×D4—5661no—
206C2×D4—5661no—
207C2×D4C22×D28311yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—2831no—
211C22×C4—2831no—
212C22×C4—5661no—
213C22×C4—5661no—
214C22×C4C22×D28311yescharacteristic
215M16—22no—
216M16—5662no—
217C2×C8—2832no—
218C2×C8C22×D28312yes—
219C4⋊C4—21no—
220C4⋊C4—2832no—
221C4⋊C4—5662no—
222C4⋊C4—5662no—
223C4⋊C4—5661no—
224C4⋊C4—5661no—
225C4⋊C4—5661no—
226C4⋊C4C22×D28312yes—
227C22⋊C4—5661no—
228C22⋊C4—5661no—
229C22⋊C4—5661no—
230C22⋊C4—5661no—
231C42—2831no—
232C42—2831no—
233C42—2831no—
234C42—2831no—
235C23—21no—
236C23—5661no—
237C23D4×D28311yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—5661no—
241Q8—5661no—
242Q8—11324no—
243Q8—11324no—
244D4—21no—
245D4—21no—
246D4—42no—
247D4—5661no—
248D4—5661no—
249D4—11322no—
250D4—11322no—
251D4—11322no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—2831no—
257C2×C4—2831no—
258C2×C4—2831no—
259C2×C4—2831no—
260C2×C4—5662no—
261C2×C4—5662no—
262C2×C4—5661no—
263C2×C4—5662no—
264C2×C4—11322no—
265C2×C4—11322no—
266C2×C4D4×D28311yescharacteristic
267C2×C4#1798539112yes—
268C2×C4#1798539212yes—
269C2×C4C23×D28311yescharacteristic
270C8—22no—
271C8—5662no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—11321no—
278C22—11321no—
279C22—11321no—
280C22#2258359612yes—
281C22C2×D4×D28311yescharacteristic
282C4—41no—
283C4—42no—
284C4—2832no—
285C4—2832no—
286C4—5661no—
287C4—5661no—
288C4—11321no—
289C4—11322no—
290C4#2258359612yes—
291C4C2×#1798539111yescharacteristic
292C4C2×#1798539211yescharacteristic
293C2—22no—
294C2—81no—
295C2—22641no—
296C2C2×#2258359611yescharacteristic
297C2AGL(1,ℤ/8)×D28311yescharacteristic
298C2#137×D28311yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
prime radical#31827011C2
hypercenter#417D283
hypocenterC283#3141
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#22583609C2semidirect
3. #31826826C2semidirect
4. #31826823C2semidirect
5. #31826814C2semidirect
6. #31826810C2semidirect
7. #31826467C2semidirect
8. #31826466C2semidirect
9. #31826464C2semidirect
10. M16×Dic283C2semidirect
11. #417×C283C2semidirect
149. C283#3141semidirect
160. #417D283semidirect

Maximal subgroups

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

subgroupindex
#318270112
C2×#225836092
#318268262
#318268232
#318268142
#318268102
#318264672
#318264662
#318264642
M16×Dic2832
#417×C2832
#3141283