Finite Group Databasebeta

← #49122319

Subgroups — #49122319

Order 36224.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#31827263C211yesmaximal, characteristic
2#31827039C211yesmaximal, characteristic, complemented
3#31826481C211yesmaximal, characteristic, complemented
4#31826474C211yesmaximal, characteristic, complemented
5#31826439C211yesmaximal, characteristic, complemented
6#31826237C211yesmaximal, characteristic, complemented
7C2×#137×C283C211yesmaximal, characteristic, complemented
8C2×#17985392C2211yescharacteristic
9#22583641—21no—
10#22583640—21no—
11#22583636—21no—
12#22583635—21no—
13#22583630—22no—
14C2×#17985376C2211yescharacteristic
15#22583608—21no—
16#22583599—21no—
17#22583593—22no—
18#22583586—22no—
19#22583569C2211yescharacteristic
20#22583569C2211yescharacteristic
21#22583536—22no—
22#22583534—22no—
23#22583525—22no—
24C2×#41×C283C2211yescharacteristic
25C22×Q8×C283C2211yescharacteristic
26#137×C283—22no—
27#137×C283—22no—
28C2×Q16×C283—21no—
29C2×SD16×C283—21no—
30C2×M16×C283C2211yescharacteristic
31#17985392—21no—
32#17985392—21no—
33#17985392—21no—
34#17985392—21no—
35C2×Dic566D411yescharacteristic
36C2×D1132D411yescharacteristic
37C2×C22⋊D283—21no—
38C2×C4×D283—21no—
39#17985382—22no—
40#17985380—22no—
41#17985379—22no—
42#17985377—22no—
43#17985376—21no—
44#17985376—21no—
45C2×C283⋊C8D412yes—
46C4⋊Dic283—21no—
47C4⋊Dic283—21no—
48Dic283⋊C4—24no—
49C22⋊Dic283—21no—
50C22⋊Dic283D412yes—
51(C2×D283)⋊C4—24no—
52C4×Dic283—21no—
53C4×Dic283—21no—
54#41×C283—21no—
55#41×C283—21no—
56#41×C283—21no—
57#41×C283—21no—
58C2×Q8×C283—21no—
59C2×Q8×C283—21no—
60C2×Q8×C283—21no—
61C2×Q8×C283—21no—
62C2×Q8×C283—21no—
63C2×Q8×C283D412yes—
64C2×Q8×C283D411yescharacteristic
65C2×D4×C283—21no—
66C2×D4×C283D411yescharacteristic
67C22×C4×C283—21no—
68C22×C4×C283—21no—
69C22×C4×C283C2311yescharacteristic
70Q16×C283—24no—
71SD16×C283—24no—
72M16×C283—21no—
73M16×C283D412yes—
74C2×C8×C283—21no—
75C22×D283—21no—
76Dic566—21no—
77Dic566—21no—
78D1132—21no—
79D1132—21no—
80C22⋊D283—41no—
81C22⋊D283—41no—
82C2×Dic283—22no—
83C2×Dic283—21no—
84C2×Dic283—41no—
85C4×D283—41no—
86C4×D283—41no—
87C283⋊C8—22no—
88C23×C283—21no—
89C23×C283C2×D411yescharacteristic
90Q8×C283—22no—
91Q8×C283—21no—
92Q8×C283—21no—
93Q8×C283—22no—
94Q8×C283—41no—
95Q8×C283—41no—
96D4×C283—21no—
97D4×C283—21no—
98D4×C283—41no—
99D4×C283—41no—
100C2×C4×C283—22no—
101C2×C4×C283—21no—
102C2×C4×C283—41no—
103C2×C4×C283—41no—
104C2×C4×C283—41no—
105C2×C4×C283—41no—
106C2×C4×C283C2×D411yescharacteristic
107C2×C4×C283C2×D411yescharacteristic
108C2×C4×C283C2×D411yescharacteristic
109C2×C4×C283C2×D411yescharacteristic
110C2×C4×C283C2×D411yescharacteristic
111C2×C4×C283#4111yescharacteristic
112C8×C283—22no—
113C22×C283—41no—
114C22×C283—41no—
115C22×C283—81no—
116C22×C283#12011yescharacteristic
117C22×C283#12111yescharacteristic
118C22×C283#12711yescharacteristic
119C2×D283—42no—
120C2×D283—41no—
121C4×C283—41no—
122C4×C283—42no—
123C4×C283#12011yescharacteristic
124C4×C283#12011yescharacteristic
125C4×C283#12111yescharacteristic
126C4×C283#12111yescharacteristic
127Dic283—42no—
128Dic283—41no—
129Dic283—81no—
130C2×C283—21no—
131C2×C283—21no—
132C2×C283—21no—
133C2×C283—81no—
134C2×C283#39211yescharacteristic
135D283—81no—
136C283#200911yescharacteristic, complemented, Sylow
137#2009—2831nomaximal, Sylow
138#575—2831no—
139C2×#137D28311yescharacteristic, complemented
140#536—2831no—
141#421—2831no—
142#419—2831no—
143#410—2831no—
144#344—2831no—
145C2×#41—2831no—
146C2×#41C2×D28311yescharacteristic
147C22×Q8C2×D28311yescharacteristic
148#137—22no—
149#137—22no—
150#137—5661no—
151#137—5661no—
152AGL(1,ℤ/8)—5661no—
153AGL(1,ℤ/8)—5661no—
154#135—5662no—
155C2×Q16—21no—
156C2×SD16—21no—
157C2×M16—2831no—
158C2×M16C2×D28311yescharacteristic
159C4⋊Q8—5661no—
160#124—5661no—
161#123—5662no—
162C22⋊Q8—5662no—
163#117—2831no—
164#117—2831no—
165#104—5662no—
166#103—5662no—
167#99—5662no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—5661no—
173#41—5661no—
174#41—5661no—
175#41—5661no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—2831no—
182C2×Q8C22⋊D28312yes—
183C2×Q8C22⋊D28311yescharacteristic
184C2×D4—21no—
185C2×D4—2831no—
186C2×D4—5661no—
187C2×D4C22⋊D28311yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—5661no—
191C22×C4C22×D28311yescharacteristic
192Q16—24no—
193Q16—5662no—
194SD16—24no—
195SD16—5662no—
196SD16—5662no—
197D8—5662no—
198M16—21no—
199M16—5661no—
200M16—5661no—
201M16C22⋊D28312yes—
202C2×C8—21no—
203C2×C8—2832no—
204C4⋊C4—5664no—
205C4⋊C4—5661no—
206C4⋊C4—5661no—
207C22⋊C4—2832no—
208C22⋊C4—5664no—
209C22⋊C4—5661no—
210C42—5661no—
211C42—5661no—
212C23—21no—
213C23—5661no—
214C23D4×D28311yescharacteristic
215Q8—21no—
216Q8—22no—
217Q8—21no—
218Q8—22no—
219Q8—41no—
220Q8—41no—
221Q8—5661no—
222Q8—5661no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—5661no—
228D4—5661no—
229D4—11321no—
230D4—11321no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—5662no—
238C2×C4—5661no—
239C2×C4—11321no—
240C2×C4—11321no—
241C2×C4—11321no—
242C2×C4D4×D28311yescharacteristic
243C2×C4D4×D28311yescharacteristic
244C2×C4C2×C22⋊D28311yescharacteristic
245C2×C4C2×C22⋊D28311yescharacteristic
246C2×C4C2×C22⋊D28311yescharacteristic
247C2×C4#1798539111yescharacteristic
248C8—22no—
249C8—5662no—
250C22—41no—
251C22—41no—
252C22—81no—
253C22—11322no—
254C22—11321no—
255C22#2258357811yescharacteristic
256C22#2258358211yescharacteristic
257C22#2258360611yescharacteristic
258C4—41no—
259C4—42no—
260C4—11322no—
261C4—11321no—
262C4—22641no—
263C4#2258357811yescharacteristic
264C4#2258357911yescharacteristic
265C4#2258358111yescharacteristic
266C4#2258358211yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—22641no—
272C2#3182637311yescharacteristic
subgroupG/H
centerC2#31826373
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
FittingC2×#137×C283C2
socleC2×C283#392
prime radical#31827263C2
hypercenterC2×#137D283
hypocenterC283#2009
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #31827039C2semidirect
3. #31826481C2semidirect
4. #31826474C2semidirect
5. #31826439C2semidirect
6. #31826237C2semidirect
7. C2×#137×C283C2semidirect
136. C283#2009semidirect
139. C2×#137D283semidirect

Maximal subgroups

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

subgroupindex
#318272632
#318270392
#318264812
#318264742
#318264392
#318262372
C2×#137×C2832
#2009283