Finite Group Databasebeta

← #46824017

Subgroups — #46824017

Order 34432.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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×#22575472C211yesmaximal, characteristic, complemented
2#135×D269C211yesmaximal, characteristic, complemented
3#31771048C211yesmaximal, characteristic, complemented
4#31771045C211yesmaximal, characteristic, complemented
5#31771044C212yesmaximal, complemented
6#31771041C211yesmaximal, characteristic, complemented
7#31771039C212yesmaximal, complemented
8#31771036C212yesmaximal, complemented
9#31771034C212yesmaximal, complemented
10M32×D269C211yesmaximal, characteristic, complemented
11#509×C269C211yesmaximal, characteristic, complemented
12C2×#16864401—21no—
13#41×D269—21no—
14#22575476C2211yescharacteristic, complemented
15#22575475C2211yescharacteristic
16#22575474—21no—
17#22575473—21no—
18#22575472—21no—
19#22575472—21no—
20#22575472C2212yes—
21#22575472C2211yescharacteristic, complemented
22#22575472C2212yescomplemented
23C2×Dic1076C2211yescharacteristic, complemented
24Q16×D269C2211yescharacteristic, complemented
25C2×#16864390—21no—
26SD16×D269—21no—
27D8×D269C2211yescharacteristic, complemented
28C2×C8×D269C2211yescharacteristic
29Dic2152C2212yescomplemented
30#22575399C2212yescomplemented
31#22575398C2212yescomplemented
32#22575397C2212yescomplemented
33#22575397C2212yescomplemented
34#22575394C2212yescomplemented
35#22575393C2211yescharacteristic, complemented
36#22575392C2212yescomplemented
37C16×D269C2212yescomplemented
38#135×C269C2211yescharacteristic
39C2×D8×C269C2211yescharacteristic, complemented
40SD32×C269C2212yescomplemented
41D16×C269C2212yescomplemented
42M32×C269C2211yescharacteristic, complemented
43#16864403—21no—
44#16864402—21no—
45#16864402—41no—
46#16864401—22no—
47#16864401—22no—
48#16864401—21no—
49#16864401—41no—
50#16864401—41no—
51#16864401—41no—
52C2×Dic538—21no—
53Q8×D269—21no—
54C2×C22⋊D269—41no—
55D4×D269—21no—
56D4×D269—41no—
57C22×Dic269—41no—
58C2×C4×D269—41no—
59C2×C4×D269D411yescharacteristic
60Dic1076—21no—
61Dic1076C2311yescharacteristic
62Dic1076C2312yes—
63#16864392—21no—
64#16864391—21no—
65#16864390—21no—
66#16864390—22no—
67#16864390—22no—
68#16864389—21no—
69D2152C2311yescharacteristic
70#16864387—21no—
71C2×#14524130D411yescharacteristic
72C8×D269D411yescharacteristic
73C8×D269D411yescharacteristic
74C8×D269C2311yescharacteristic
75C8×D269C2311yescharacteristic
76#16864377C2312yes—
77#41×C269—21no—
78C2×D4×C269—21no—
79Q16×C269C2311yescharacteristic
80SD16×C269—21no—
81D8×C269—21no—
82D8×C269C2312yes—
83D8×C269C2311yescharacteristic
84C2×C8×C269C2311yescharacteristic
85C16×C269C2312yes—
86C22×D269—41no—
87C22×D269D811yescharacteristic
88Dic538—22no—
89Dic538—21no—
90Dic538—41no—
91Dic538—41no—
92D1076—21no—
93D1076—41no—
94C22⋊D269—41no—
95C22⋊D269—41no—
96C22⋊D269—42no—
97C22⋊D269—42no—
98C22⋊D269—41no—
99C2×Dic269—41no—
100C2×Dic269—41no—
101C2×Dic269—42no—
102C2×Dic269—42no—
103C2×Dic269D811yescharacteristic
104C4×D269—41no—
105C4×D269—41no—
106C4×D269—41no—
107C4×D269D811yescharacteristic
108C4×D269D811yescharacteristic
109C4×D269C2×D411yescharacteristic
110C4×D269C2×D411yescharacteristic
111#14524130C2×D411yescharacteristic
112#14524130C2×D411yescharacteristic
113C23×C269—41no—
114Q8×C269—21no—
115D4×C269—22no—
116D4×C269—21no—
117D4×C269—41no—
118D4×C269—41no—
119C2×C4×C269—41no—
120C2×C4×C269C2×D411yescharacteristic
121C8×C269C2×D411yescharacteristic
122C8×C269C2411yescharacteristic
123C22×C269—41no—
124C22×C269—42no—
125C22×C269—82no—
126C22×C269C2×D811yescharacteristic
127C2×D269—41no—
128C2×D269—41no—
129C2×D269—82no—
130C2×D269C2×D811yescharacteristic
131C2×D269C2×D811yescharacteristic
132C4×C269—41no—
133C4×C269C2×D811yescharacteristic
134C4×C269C22×D411yescharacteristic
135Dic269—41no—
136Dic269—42no—
137Dic269C2×D811yescharacteristic
138Dic269C2×D811yescharacteristic
139C2×C269—21no—
140C2×C269—81no—
141C2×C269—82no—
142C2×C269C22×D811yescharacteristic
143D269—21no—
144D269—21no—
145D269—81no—
146C269#341411yescharacteristic, complemented, Sylow
147#3414—2691nomaximal, Sylow
148C2×#135—2691no—
149C2×#135—2691no—
150#510—2691no—
151#510—2692no—
152#510—2691no—
153#509—2691no—
154#509—2692no—
155#509D26911yescharacteristic, complemented
156#508—2692no—
157#508—2692no—
158C2×M32—2691no—
159C2×#41—5381no—
160C2×#41—5381no—
161#135—2691no—
162#135—2692no—
163#135—2692no—
164#135—2691no—
165#135—2691no—
166#135—5381no—
167#135—5381no—
168#135—5381no—
169#135—5381no—
170#135C2×D26911yescharacteristic
171C2×Q16—2691no—
172C2×Q16—2691no—
173C2×SD16—5381no—
174C2×SD16—5381no—
175C2×D8—2691no—
176C2×D8C2×D26911yescharacteristic, complemented
177C22×C8—2691no—
178Q32—2692no—
179Q32—2692no—
180SD32—2692no—
181SD32—2692no—
182SD32—2692no—
183SD32C2×D26912yescomplemented
184D16—2692no—
185D16C2×D26912yescomplemented
186M32—2691no—
187M32—2692no—
188M32C2×D26911yescharacteristic, complemented
189C2×C16—2692no—
190#41—21no—
191#41—5382no—
192#41—5382no—
193#41—5381no—
194#41—5381no—
195#41—5381no—
196#41—10761no—
197#41—10761no—
198#41—10761no—
199#41—10761no—
200C2×Q8—5381no—
201C2×Q8—5381no—
202C2×D4—21no—
203C2×D4—5381no—
204C2×D4—10761no—
205C2×D4—10761no—
206C22×C4—2691no—
207C22×C4—10761no—
208C22×C4—10761no—
209Q16—2691no—
210Q16—2692no—
211Q16—5381no—
212Q16—5381no—
213Q16C22×D26911yescharacteristic
214SD16—21no—
215SD16—5382no—
216SD16—5382no—
217SD16—5381no—
218SD16—5381no—
219SD16—5381no—
220D8—21no—
221D8—2691no—
222D8—5381no—
223D8C22×D26912yes—
224D8C22×D26911yescharacteristic
225C2×C8—2691no—
226C2×C8—2691no—
227C2×C8—2691no—
228C2×C8—2691no—
229C2×C8—2691no—
230C2×C8C22×D26911yescharacteristic
231C16—2692no—
232C16C22×D26912yes—
233C23—41no—
234C23—2691no—
235C23—10761no—
236Q8—21no—
237Q8—5382no—
238Q8—5381no—
239Q8—10761no—
240Q8—10761no—
241D4—22no—
242D4—21no—
243D4—41no—
244D4—41no—
245D4—5381no—
246D4—10761no—
247D4—10761no—
248D4—10761no—
249D4—10762no—
250D4—10762no—
251D4—10761no—
252C2×C4—41no—
253C2×C4—2691no—
254C2×C4—2691no—
255C2×C4—2691no—
256C2×C4—2691no—
257C2×C4—2691no—
258C2×C4—10761no—
259C2×C4—10761no—
260C2×C4—10761no—
261C2×C4—10762no—
262C2×C4—10762no—
263C2×C4—10761no—
264C2×C4—10761no—
265C2×C4D4×D26911yescharacteristic
266C8—2691no—
267C8—2691no—
268C8D4×D26911yescharacteristic
269C8C23×D26911yescharacteristic
270C22—41no—
271C22—42no—
272C22—82no—
273C22—2691no—
274C22—2691no—
275C22—10761no—
276C22—10761no—
277C22—21522no—
278C22D8×D26911yescharacteristic
279C4—41no—
280C4—2691no—
281C4—2691no—
282C4—10761no—
283C4—10762no—
284C4D8×D26911yescharacteristic
285C4C2×D4×D26911yescharacteristic
286C2—21no—
287C2—82no—
288C2—81no—
289C2—5381no—
290C2—5381no—
291C2—21521no—
292C2C2×D8×D26911yescharacteristic
subgroupG/H
centerC2C2×D8×D269
commutator, cosocleC8×C269C24
FrattiniC8C23×D269
Fitting#509×C269C2
socleC2×C269C22×D8
hypercenter#509D269
hypocenterC269#3414
LCSG ▷ C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. C2×#22575472C2semidirect
2. #135×D269C2semidirect
3. #31771048C2semidirect
4. #31771045C2semidirect
5. #31771044C2semidirect
6. #31771041C2semidirect
7. #31771039C2semidirect
8. #31771036C2semidirect
9. #31771034C2semidirect
10. M32×D269C2semidirect
11. #509×C269C2semidirect
14. #22575476C22semidirect
21. #22575472C22semidirect
22. #22575472C22semidirect
23. C2×Dic1076C22semidirect
24. Q16×D269C22semidirect
27. D8×D269C22semidirect
29. Dic2152C22semidirect
30. #22575399C22semidirect
31. #22575398C22semidirect
32. #22575397C22semidirect
33. #22575397C22semidirect
34. #22575394C22semidirect
35. #22575393C22semidirect
36. #22575392C22semidirect
37. C16×D269C22semidirect
39. C2×D8×C269C22semidirect
40. SD32×C269C22semidirect
41. D16×C269C22semidirect
42. M32×C269C22semidirect
146. C269#3414semidirect
155. #509D269semidirect
176. C2×D8C2×D269semidirect
183. SD32C2×D269semidirect
185. D16C2×D269semidirect
188. M32C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754722
#135×D2692
#317710482
#317710452
#317710442
#317710412
#317710392
#317710362
#317710342
M32×D2692
#509×C2692
#3414269