Finite Group Databasebeta

← #49130997

Subgroups — #49130997

Order 36224.

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

normal subgroups only

This group has 446 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#31827265C211yesmaximal, characteristic
2C2×#22583641C211yesmaximal, characteristic, complemented
3C2×#22583641C211yesmaximal, characteristic, complemented
4#31827231C211yesmaximal, characteristic, complemented
5#31826739C214yesmaximal, complemented
6#31826731C214yesmaximal, complemented
7#31826513C211yesmaximal, characteristic, complemented
8#31826447C211yesmaximal, characteristic, complemented
9C2×#101×C283C211yesmaximal, characteristic, complemented
10C2×#17985392C2211yescharacteristic
11C2×#17985392C2211yescharacteristic
12#22583643—21no—
13#22583643C2212yes—
14#22583641—22no—
15#22583641—22no—
16#22583641—22no—
17#22583641C2214yes—
18#22583638—21no—
19#22583638C2212yes—
20#22583632—22no—
21C2×#17985382—21no—
22C2×#17985382C2212yescomplemented
23C2×#17985379—21no—
24C2×#17985379C2212yescomplemented
25#22583618C2214yes—
26#22583616—22no—
27C2×#17985376C2211yescharacteristic
28C2×#17985376C2211yescharacteristic
29M16×D283—22no—
30#22583545C2214yescomplemented
31#22583528C2212yescomplemented
32#22583526C2212yescomplemented
33C22×Q8×C283C2211yescharacteristic
34C2×M16×C283C2211yescharacteristic
35C2×M16×C283C2211yescharacteristic
36#101×C283C2214yescomplemented
37#17985392—21no—
38#17985392—21no—
39#17985392—21no—
40#17985392—21no—
41#17985392—21no—
42#17985392—21no—
43#17985392D412yes—
44#17985392C2312yes—
45C2×Dic566D411yescharacteristic
46C2×Dic566C2311yescharacteristic
47C2×D1132D411yescharacteristic
48C2×D1132C2311yescharacteristic
49C2×C22⋊D283—21no—
50C2×C22⋊D283—21no—
51C2×C4×D283—21no—
52C2×C4×D283—21no—
53Dic1132—22no—
54#17985382—24no—
55#17985382—24no—
56#17985381—22no—
57#17985381—22no—
58#17985379—24no—
59#17985379—24no—
60D2264—22no—
61#17985376—21no—
62#17985376D412yes—
63#17985376D412yes—
64#17985376C2312yes—
65#17985375—24no—
66C2×C283⋊C8—21no—
67C2×C283⋊C8C2312yes—
68C8×D283—24no—
69C2×Q8×C283—21no—
70C2×Q8×C283—21no—
71C2×Q8×C283—21no—
72C2×Q8×C283—21no—
73C2×Q8×C283—21no—
74C2×Q8×C283C2312yes—
75C22×C4×C283—21no—
76C22×C4×C283C2311yescharacteristic
77M16×C283—21no—
78M16×C283—21no—
79M16×C283C2312yes—
80M16×C283C2312yes—
81C2×C8×C283—21no—
82C2×C8×C283—21no—
83C22×D283—21no—
84C22×D283—21no—
85Dic566—21no—
86Dic566—21no—
87Dic566—21no—
88Dic566C2×D412yes—
89D1132—21no—
90D1132—21no—
91D1132—21no—
92D1132C2×D412yes—
93C22⋊D283—22no—
94C22⋊D283—22no—
95C22⋊D283—41no—
96C22⋊D283—41no—
97C2×Dic283—21no—
98C2×Dic283—21no—
99C4×D283—22no—
100C4×D283—22no—
101C4×D283—41no—
102C4×D283—41no—
103C283⋊C8—22no—
104C283⋊C8C2×D414yes—
105C23×C283#4111yescharacteristic
106Q8×C283—22no—
107Q8×C283—22no—
108Q8×C283—41no—
109Q8×C283—41no—
110C2×C4×C283—22no—
111C2×C4×C283—41no—
112C2×C4×C283—41no—
113C2×C4×C283C2×D411yescharacteristic
114C2×C4×C283C2×D411yescharacteristic
115C2×C4×C283C2×D411yescharacteristic
116C2×C4×C283C2×D411yescharacteristic
117C2×C4×C283#4111yescharacteristic
118C2×C4×C283C2411yescharacteristic
119C8×C283—22no—
120C8×C283—22no—
121C22×C283—41no—
122C22×C283#12111yescharacteristic
123C22×C283#12111yescharacteristic
124C22×C283C2×#4111yescharacteristic
125C2×D283—22no—
126C2×D283—41no—
127C2×D283—81no—
128C2×D283—81no—
129C4×C283—42no—
130C4×C283#12111yescharacteristic
131C4×C283#12111yescharacteristic
132C4×C283C22×D411yescharacteristic
133C4×C283C22×D411yescharacteristic
134Dic283—22no—
135Dic283—41no—
136C2×C283—21no—
137C2×C283—21no—
138C2×C283—21no—
139C2×C283C2×#12111yescharacteristic
140D283—42no—
141D283—81no—
142C283#306911yescharacteristic, complemented, Sylow
143#3069—2831nomaximal, Sylow
144#575—2831no—
145C2×#137—2831no—
146C2×#137—2831no—
147#568—2831no—
148#473—2834no—
149#471—2834no—
150#430—2831no—
151#413—2831no—
152C2×#101D28311yescharacteristic, complemented
153C2×#41—2831no—
154C2×#41—2831no—
155C22×Q8C2×D28311yescharacteristic
156#137—2834no—
157#137—2832no—
158#137—5661no—
159#137—5662no—
160#137—5662no—
161#137—5662no—
162AGL(1,ℤ/8)—2832no—
163AGL(1,ℤ/8)—5661no—
164#135—5662no—
165C2×Q16—2832no—
166C2×Q16—5661no—
167C2×SD16—2832no—
168C2×SD16—5661no—
169#131—2834no—
170#131—5662no—
171C2×M16—2831no—
172C2×M16—2831no—
173C2×M16—5662no—
174C2×M16C2×D28311yescharacteristic
175C2×M16C2×D28311yescharacteristic
176#108—2834no—
177#101—2832no—
178#101C2×D28314yescomplemented
179#100—2832no—
180#41—2832no—
181#41—2832no—
182#41—5661no—
183#41—5661no—
184#41—5661no—
185#41—5661no—
186#41—5661no—
187#41—5661no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—2831no—
194C2×Q8—2831no—
195C2×Q8C22×D28312yes—
196C2×D4—2831no—
197C2×D4—2831no—
198C2×D4—5661no—
199C2×D4—5661no—
200C22×C4—21no—
201C22×C4—5661no—
202C22×C4—5661no—
203C22×C4C22×D28311yescharacteristic
204Q16—5662no—
205Q16—5664no—
206Q16—5664no—
207SD16—5662no—
208SD16—5662no—
209SD16—5664no—
210SD16—5664no—
211D8—5662no—
212M16—21no—
213M16—21no—
214M16—2832no—
215M16—2832no—
216M16—2832no—
217M16—5664no—
218M16—5661no—
219M16C22×D28312yes—
220M16C22×D28312yes—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—2832no—
224C2×C8—5664no—
225C2×C8—5661no—
226C23—5661no—
227C23—5661no—
228C23#1798539211yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—41no—
232Q8—41no—
233Q8—2832no—
234Q8—5661no—
235Q8—5661no—
236Q8—5661no—
237D4—2832no—
238D4—5662no—
239D4—5662no—
240D4—5661no—
241D4—5661no—
242D4—5661no—
243D4—11321no—
244D4—11321no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—5661no—
249C2×C4—5662no—
250C2×C4—5662no—
251C2×C4—5661no—
252C2×C4—11321no—
253C2×C4—11321no—
254C2×C4D4×D28311yescharacteristic
255C2×C4D4×D28311yescharacteristic
256C2×C4D4×D28311yescharacteristic
257C2×C4D4×D28311yescharacteristic
258C2×C4#1798539211yescharacteristic
259C2×C4C23×D28311yescharacteristic
260C8—22no—
261C8—22no—
262C8—2834no—
263C8—5662no—
264C22—41no—
265C22—5662no—
266C22—11321no—
267C22—22641no—
268C22—22641no—
269C22#2258358311yescharacteristic
270C22#2258358311yescharacteristic
271C22C2×#1798539211yescharacteristic
272C4—42no—
273C4—5662no—
274C4—11321no—
275C4#2258358311yescharacteristic
276C4#2258358311yescharacteristic
277C4C2×D4×D28311yescharacteristic
278C4C2×D4×D28311yescharacteristic
279C2—21no—
280C2—21no—
281C2—21no—
282C2—11322no—
283C2—22641no—
284C2C2×#2258358311yescharacteristic
subgroupG/H
centerC2C2×#22583583
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×#101×C283C2
socleC2×C283C2×#121
prime radical#31827265C2
hypercenterC2×#101D283
hypocenterC283#3069
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#22583641C2semidirect
3. C2×#22583641C2semidirect
4. #31827231C2semidirect
5. #31826739C2semidirect
6. #31826731C2semidirect
7. #31826513C2semidirect
8. #31826447C2semidirect
9. C2×#101×C283C2semidirect
22. C2×#17985382C22semidirect
24. C2×#17985379C22semidirect
30. #22583545C22semidirect
31. #22583528C22semidirect
32. #22583526C22semidirect
36. #101×C283C22semidirect
142. C283#3069semidirect
152. C2×#101D283semidirect
178. #101C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272652
C2×#225836412
#318272312
#318267392
#318267312
#318265132
#318264472
C2×#101×C2832
#3069283