Finite Group Databasebeta

← #46820563

Subgroups — #46820563

Order 34432.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#22575479C211yesmaximal, characteristic, complemented
2C2×#22575459C211yesmaximal, characteristic, complemented
3#31771186C211yesmaximal, characteristic, complemented
4#31771172C211yesmaximal, characteristic, complemented
5#31770857C212yesmaximal, complemented
6#31770850C212yesmaximal, complemented
7#31770840C212yesmaximal, complemented
8#31770833C212yesmaximal, complemented
9#31770645C211yesmaximal, characteristic, complemented
10#31770602C211yesmaximal, characteristic, complemented
11C2×#102×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C2×#16864402C2211yescharacteristic
14#22575479—24no—
15C2×#16864390C2212yes—
16C2×#16864387C2212yes—
17#22575459—24no—
18C2×#16864386C2211yescharacteristic
19C2×#16864385C2212yescomplemented
20C2×C8×D269C2212yescomplemented
21#22575449—21no—
22#22575440—21no—
23#22575433—21no—
24#22575433C2212yescomplemented
25#22575428—21no—
26#22575428C2212yescomplemented
27#22575427—22no—
28#22575424C2212yes—
29#22575424C2212yes—
30#22575422—22no—
31#22575412C2211yescharacteristic
32#22575386C2212yescomplemented
33#22575384C2212yescomplemented
34#22575378C2211yescharacteristic, complemented
35#22575378C2211yescharacteristic, complemented
36#22575375C2211yescharacteristic, complemented
37#22575375C2211yescharacteristic, complemented
38C22×D4×C269C2211yescharacteristic
39C22×C8×C269C2211yescharacteristic
40C2×C4⋊C4×C269C2211yescharacteristic
41#102×C269C2212yes—
42#102×C269C2212yes—
43#16864402—22no—
44#16864402—22no—
45#16864402—22no—
46#16864402—22no—
47C2×Dic538D411yescharacteristic
48C2×Dic538C2311yescharacteristic
49C2×D1076—22no—
50C2×D1076D411yescharacteristic
51C2×D1076C2311yescharacteristic
52C2×C22⋊D269—22no—
53C2×C22⋊D269—22no—
54C2×C22⋊D269—21no—
55C2×C22⋊D269D412yes—
56C2×C4×D269—21no—
57C2×C4×D269C2312yes—
58#16864390—24no—
59#16864387—24no—
60#16864386D414yescomplemented
61#16864385—24no—
62C2×#14524130C2312yes—
63C8×D269—24no—
64C4⋊Dic269—22no—
65C4⋊Dic269C2311yescharacteristic
66C4⋊Dic269C2311yescharacteristic
67C22⋊Dic269—22no—
68C22⋊Dic269—22no—
69C22⋊Dic269—21no—
70#16864379—22no—
71#16864379—22no—
72C4×Dic269—21no—
73C24×C269—21no—
74C2×D4×C269—22no—
75C2×D4×C269—21no—
76C2×D4×C269—22no—
77C2×D4×C269C2312yes—
78C22×C4×C269—21no—
79C22×C4×C269C2311yescharacteristic
80C2×C8×C269—22no—
81C2×C8×C269C2311yescharacteristic
82C2×C8×C269C2311yescharacteristic
83C4⋊C4×C269—21no—
84C4⋊C4×C269C2312yes—
85C22×D269—21no—
86C22×D269C2×D412yes—
87Dic538—22no—
88Dic538—21no—
89Dic538—21no—
90D1076—22no—
91D1076—21no—
92D1076—21no—
93D1076—82no—
94C22⋊D269—24no—
95C22⋊D269—42no—
96C22⋊D269—82no—
97C22⋊D269—82no—
98C2×Dic269—21no—
99C2×Dic269—21no—
100C2×Dic269—21no—
101C2×Dic269C2×D412yes—
102C4×D269—22no—
103C4×D269—22no—
104C4×D269—42no—
105#14524130C2×D414yes—
106C23×C269—22no—
107C23×C269—42no—
108C23×C269—81no—
109C23×C269—81no—
110C23×C269C2×D411yescharacteristic
111D4×C269—22no—
112D4×C269—22no—
113D4×C269—42no—
114C2×C4×C269—22no—
115C2×C4×C269—81no—
116C2×C4×C269C2×D411yescharacteristic
117C2×C4×C269C2×D412yes—
118C2×C4×C269#4112yes—
119C2×C4×C269C2411yescharacteristic
120C8×C269—21no—
121C8×C269—21no—
122C22×C269—21no—
123C22×C269—21no—
124C22×C269—42no—
125C22×C269—42no—
126C22×C269—42no—
127C22×C269—84no—
128C22×C269#12112yes—
129C22×C269C22×D411yescharacteristic
130C2×D269—22no—
131C2×D269—22no—
132C2×D269—22no—
133C2×D269—41no—
134C2×D269—41no—
135C2×D269—41no—
136C4×C269—42no—
137C4×C269#12112yes—
138C4×C269C22×D411yescharacteristic
139C4×C269C2×#4111yescharacteristic
140Dic269—22no—
141Dic269—41no—
142Dic269—41no—
143Dic269—41no—
144C2×C269—22no—
145C2×C269—82no—
146C2×C269C2×#12111yescharacteristic
147C2×C269#57611yescharacteristic
148C2×C269#57711yescharacteristic
149D269—42no—
150D269—81no—
151C269#306211yescharacteristic, complemented, Sylow
152#3062—2691nomaximal, Sylow
153C2×AGL(1,ℤ/8)—2691no—
154C2×#131—2691no—
155#537—2691no—
156#535—2691no—
157#469—2692no—
158#468—2692no—
159#466—2692no—
160#465—2692no—
161#428—2691no—
162#416—2691no—
163C2×#102D26911yescharacteristic, complemented
164C2×#41—2691no—
165C2×#41—2691no—
166C22×D4C2×D26911yescharacteristic
167AGL(1,ℤ/8)—5384no—
168C2×SD16—2692no—
169C2×D8—2692no—
170#131—5384no—
171C2×M16—2691no—
172C2×M16—2692no—
173C22×C8—2692no—
174C22×C8C2×D26911yescharacteristic
175#127—5381no—
176#124—5381no—
177C22⋊Q8—2692no—
178C22⋊Q8—5381no—
179#121—2692no—
180#121—2692no—
181#121—2692no—
182#121—5381no—
183#121—5382no—
184#120—5382no—
185#117—2691no—
186C2×C4⋊C4C2×D26911yescharacteristic
187#107—2692no—
188#106—2692no—
189#103—2691no—
190#103—2691no—
191#102—2691no—
192#102—2691no—
193#102C2×D26912yes—
194#102C2×D26912yes—
195C24—21no—
196#41—5382no—
197#41—5382no—
198#41—5382no—
199#41—5382no—
200C2×Q8—2691no—
201C2×Q8—2691no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—2692no—
206C2×D4—2691no—
207C2×D4—2691no—
208C2×D4—5382no—
209C2×D4—5381no—
210C2×D4—5382no—
211C2×D4—5382no—
212C2×D4C22×D26912yes—
213C22×C4—21no—
214C22×C4—2692no—
215C22×C4—5381no—
216C22×C4C22×D26911yescharacteristic
217SD16—5384no—
218D8—5384no—
219M16—2694no—
220M16—5384no—
221C2×C8—22no—
222C2×C8—2692no—
223C2×C8—5384no—
224C2×C8C22×D26911yescharacteristic
225C2×C8C22×D26911yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—2691no—
228C4⋊C4—2691no—
229C4⋊C4—5382no—
230C4⋊C4C22×D26912yes—
231C22⋊C4—5382no—
232C22⋊C4—5382no—
233C22⋊C4—5382no—
234C22⋊C4—5382no—
235C22⋊C4—5381no—
236C42—5381no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—2692no—
242C23—5381no—
243C23D4×D26911yescharacteristic
244Q8—5382no—
245Q8—5381no—
246Q8—5381no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—5384no—
251D4—5382no—
252D4—5381no—
253D4—5381no—
254D4—10762no—
255D4—21522no—
256D4—21522no—
257D4—21522no—
258C2×C4—22no—
259C2×C4—81no—
260C2×C4—2692no—
261C2×C4—5382no—
262C2×C4—5382no—
263C2×C4—5381no—
264C2×C4—5381no—
265C2×C4—5381no—
266C2×C4—10762no—
267C2×C4D4×D26911yescharacteristic
268C2×C4D4×D26912yes—
269C2×C4#1686440212yes—
270C2×C4C23×D26911yescharacteristic
271C8—21no—
272C8—21no—
273C8—2694no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—84no—
280C22—5382no—
281C22—5382no—
282C22—5382no—
283C22—10761no—
284C22—10761no—
285C22—10761no—
286C22#2257542612yes—
287C22C2×D4×D26911yescharacteristic
288C4—42no—
289C4—5382no—
290C4—10761no—
291C4—10761no—
292C4—10761no—
293C4#2257542612yes—
294C4C2×D4×D26911yescharacteristic
295C4C2×#1686440211yescharacteristic
296C2—22no—
297C2—82no—
298C2—10762no—
299C2—21521no—
300C2C2×#2257542611yescharacteristic
301C2#3177140611yescharacteristic
302C2#3177141111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#102×C269C2
socleC22×C269C22×D4
hypercenterC2×#102D269
hypocenterC269#3062
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#22575479C2semidirect
2. C2×#22575459C2semidirect
3. #31771186C2semidirect
4. #31771172C2semidirect
5. #31770857C2semidirect
6. #31770850C2semidirect
7. #31770840C2semidirect
8. #31770833C2semidirect
9. #31770645C2semidirect
10. #31770602C2semidirect
11. C2×#102×C269C2semidirect
19. C2×#16864385C22semidirect
20. C2×C8×D269C22semidirect
24. #22575433C22semidirect
26. #22575428C22semidirect
32. #22575386C22semidirect
33. #22575384C22semidirect
34. #22575378C22semidirect
35. #22575378C22semidirect
36. #22575375C22semidirect
37. #22575375C22semidirect
60. #16864386D4semidirect
151. C269#3062semidirect
163. C2×#102D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754792
C2×#225754592
#317711862
#317711722
#317708572
#317708502
#317708402
#317708332
#317706452
#317706022
C2×#102×C2692
#3062269