Finite Group Databasebeta

← #46820566

Subgroups — #46820566

Order 34432.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22575481C211yesmaximal, characteristic, complemented
2C2×#22575461C211yesmaximal, characteristic, complemented
3#31771186C211yesmaximal, characteristic, complemented
4#31771174C211yesmaximal, characteristic, complemented
5#31770859C212yesmaximal, complemented
6#31770852C212yesmaximal, complemented
7#31770842C212yesmaximal, complemented
8#31770835C212yesmaximal, complemented
9#31770599C211yesmaximal, characteristic, complemented
10C2×#22575373C211yesmaximal, characteristic, complemented
11#428×C269C211yesmaximal, characteristic, complemented
12C2×#16864402C2211yescharacteristic
13C2×#16864402C2211yescharacteristic
14C22×D1076C2211yescharacteristic
15#22575481—24no—
16C2×#16864391C2212yes—
17C2×D2152C2212yes—
18#22575461—24no—
19C2×#16864385C2212yescomplemented
20C22×#14524130C2211yescharacteristic
21C2×C8×D269C2212yescomplemented
22#22575450—21no—
23#22575441—21no—
24#22575433—21no—
25#22575433C2212yescomplemented
26#22575428—21no—
27#22575428C2212yes—
28#22575428C2212yes—
29#22575428C2212yescomplemented
30#22575427—22no—
31#22575423—22no—
32#22575376C2211yescharacteristic, complemented
33#22575376C2211yescharacteristic, complemented
34#22575373C2212yes—
35#22575373C2212yes—
36#22575373C2211yescharacteristic, complemented
37#22575373C2211yescharacteristic, complemented
38C2×M16×C269C2211yescharacteristic
39#117×C269C2211yescharacteristic
40C2×C4⋊C4×C269C2211yescharacteristic
41#107×C269C2212yescomplemented
42#106×C269C2212yescomplemented
43C23×D269—21no—
44#16864402—22no—
45#16864402—22no—
46#16864402—22no—
47#16864402—22no—
48C2×Dic538D411yescharacteristic
49C2×Dic538C2311yescharacteristic
50C2×D1076—22no—
51C2×D1076—22no—
52C2×D1076—22no—
53C2×D1076—22no—
54C2×D1076—21no—
55C2×D1076—22no—
56C2×D1076D411yescharacteristic
57C2×D1076C2312yes—
58C2×D1076C2311yescharacteristic
59C2×C22⋊D269—21no—
60C2×C22⋊D269D412yes—
61C2×C4×D269—21no—
62C2×C4×D269C2312yes—
63#16864391—24no—
64D2152—24no—
65#16864385—24no—
66C2×#14524130—22no—
67C2×#14524130C2311yescharacteristic
68C2×#14524130C2311yescharacteristic
69C8×D269—24no—
70C4⋊Dic269—22no—
71#16864379—22no—
72#16864379—22no—
73#16864379—22no—
74#16864379—22no—
75C22×C4×C269—21no—
76C22×C4×C269C2311yescharacteristic
77M16×C269D414yescomplemented
78C2×C8×C269C2312yes—
79C4⋊C4×C269—21no—
80C4⋊C4×C269C2312yes—
81C4⋊C4×C269C2311yescharacteristic
82C4⋊C4×C269C2311yescharacteristic
83C22⋊C4×C269—21no—
84C42×C269—21no—
85C22×D269—21no—
86C22×D269—22no—
87C22×D269—42no—
88C22×D269—81no—
89C22×D269—81no—
90C22×D269C2×D412yes—
91Dic538—22no—
92Dic538—21no—
93Dic538—21no—
94D1076—22no—
95D1076—21no—
96D1076—22no—
97D1076—22no—
98D1076—21no—
99D1076—42no—
100D1076—82no—
101D1076—82no—
102D1076—82no—
103C22⋊D269—24no—
104C22⋊D269—42no—
105C2×Dic269—21no—
106C2×Dic269C2×D412yes—
107C4×D269—22no—
108C4×D269—22no—
109C4×D269—42no—
110#14524130—21no—
111#14524130—21no—
112C23×C269C2×D411yescharacteristic
113C2×C4×C269—21no—
114C2×C4×C269—21no—
115C2×C4×C269—22no—
116C2×C4×C269—81no—
117C2×C4×C269C2×D411yescharacteristic
118C2×C4×C269C2×D412yes—
119C2×C4×C269#4112yes—
120C2×C4×C269C2411yescharacteristic
121C8×C269C2×D414yes—
122C22×C269—21no—
123C22×C269—21no—
124C22×C269#12112yes—
125C22×C269C22×D411yescharacteristic
126C2×D269—22no—
127C2×D269—22no—
128C2×D269—22no—
129C2×D269—41no—
130C2×D269—42no—
131C2×D269—41no—
132C2×D269—42no—
133C2×D269—42no—
134C2×D269—41no—
135C2×D269—84no—
136C4×C269—42no—
137C4×C269—41no—
138C4×C269—41no—
139C4×C269#12112yes—
140C4×C269C22×D411yescharacteristic
141C4×C269C2×#4111yescharacteristic
142Dic269—22no—
143Dic269—41no—
144C2×C269—22no—
145C2×C269C2×#12111yescharacteristic
146C2×C269#57611yescharacteristic
147C2×C269#57711yescharacteristic
148D269—42no—
149D269—81no—
150D269—82no—
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#428D26911yescharacteristic, complemented
162#416—2691no—
163C2×#102—2691no—
164C2×#41—2691no—
165C2×#41—2691no—
166C22×D4—2691no—
167AGL(1,ℤ/8)—5384no—
168C2×SD16—2692no—
169C2×D8—2692no—
170#131—5384no—
171C2×M16—2692no—
172C2×M16C2×D26911yescharacteristic
173C22×C8—2691no—
174C22×C8—2692no—
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#117C2×D26911yescharacteristic
186C2×C4⋊C4C2×D26911yescharacteristic
187#107C2×D26912yescomplemented
188#106C2×D26912yescomplemented
189#103—2691no—
190#103—2691no—
191#102—2692no—
192#102—2692no—
193#102—2691no—
194#102—2691no—
195C24—5381no—
196#41—5382no—
197#41—5382no—
198#41—5382no—
199#41—5382no—
200C2×Q8—2691no—
201C2×Q8—2691no—
202C2×D4—2692no—
203C2×D4—2691no—
204C2×D4—2691no—
205C2×D4—2692no—
206C2×D4—5382no—
207C2×D4—5382no—
208C2×D4—5382no—
209C2×D4—5381no—
210C2×D4—5381no—
211C2×D4—5382no—
212C2×D4—5382no—
213C22×C4—21no—
214C22×C4—2692no—
215C22×C4—5381no—
216C22×C4C22×D26911yescharacteristic
217SD16—5384no—
218D8—5384no—
219M16—5384no—
220M16D107614yescomplemented
221C2×C8—2691no—
222C2×C8—2691no—
223C2×C8—5382no—
224C2×C8—5384no—
225C2×C8C22×D26912yes—
226C4⋊C4—21no—
227C4⋊C4—5382no—
228C4⋊C4C22×D26912yes—
229C4⋊C4C22×D26911yescharacteristic
230C4⋊C4C22×D26911yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—5382no—
233C22⋊C4—5382no—
234C22⋊C4—5382no—
235C22⋊C4—5382no—
236C42—21no—
237C23—2692no—
238C23—5381no—
239C23—5382no—
240C23—10762no—
241C23—21521no—
242C23—21521no—
243C23D4×D26911yescharacteristic
244Q8—5382no—
245Q8—5381no—
246Q8—5381no—
247D4—5384no—
248D4—5382no—
249D4—5381no—
250D4—5382no—
251D4—5382no—
252D4—5381no—
253D4—10762no—
254D4—10762no—
255D4—21522no—
256D4—21522no—
257D4—21522no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—2692no—
263C2×C4—5382no—
264C2×C4—5382no—
265C2×C4—5381no—
266C2×C4—10762no—
267C2×C4D4×D26911yescharacteristic
268C2×C4C2×D107612yes—
269C2×C4#1686440312yes—
270C2×C4C23×D26911yescharacteristic
271C8—5381no—
272C8—5381no—
273C8C2×D107614yes—
274C22—21no—
275C22—21no—
276C22—5382no—
277C22—5382no—
278C22—5382no—
279C22—10761no—
280C22—10762no—
281C22—10762no—
282C22—10761no—
283C22—10761no—
284C22—10762no—
285C22—21524no—
286C22#2257542812yes—
287C22C2×D4×D26911yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—5382no—
292C4—10761no—
293C4#2257542812yes—
294C4C22×D107611yescharacteristic
295C4C2×#1686440311yescharacteristic
296C2—22no—
297C2—10762no—
298C2—21522no—
299C2—21521no—
300C2C2×#2257542811yescharacteristic
301C2#3177140911yescharacteristic
302C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#428×C269C2
socleC22×C269C22×D4
hypercenter#428D269
hypocenterC269#3062
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#22575481C2semidirect
2. C2×#22575461C2semidirect
3. #31771186C2semidirect
4. #31771174C2semidirect
5. #31770859C2semidirect
6. #31770852C2semidirect
7. #31770842C2semidirect
8. #31770835C2semidirect
9. #31770599C2semidirect
10. C2×#22575373C2semidirect
11. #428×C269C2semidirect
19. C2×#16864385C22semidirect
21. C2×C8×D269C22semidirect
25. #22575433C22semidirect
29. #22575428C22semidirect
32. #22575376C22semidirect
33. #22575376C22semidirect
36. #22575373C22semidirect
37. #22575373C22semidirect
41. #107×C269C22semidirect
42. #106×C269C22semidirect
77. M16×C269D4semidirect
151. C269#3062semidirect
161. #428D269semidirect
187. #107C2×D269semidirect
188. #106C2×D269semidirect
220. M16D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754812
C2×#225754612
#317711862
#317711742
#317708592
#317708522
#317708422
#317708352
#317705992
C2×#225753732
#428×C2692
#3062269