Finite Group Databasebeta

← #46822692

Subgroups — #46822692

Order 34432.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D1076C211yesmaximal, characteristic, complemented
2#31770953C211yesmaximal, characteristic, complemented
3#31770918C212yesmaximal, complemented
4#31770882C211yesmaximal, characteristic, complemented
5#31770839C212yesmaximal, complemented
6#31770800C211yesmaximal, characteristic, complemented
7#31770738C212yesmaximal, complemented
8C4×#16864387C211yesmaximal, characteristic, complemented
9D4×#14524130C211yesmaximal, characteristic, complemented
10C2×#22575373C212yesmaximal, complemented
11#547×C269C211yesmaximal, characteristic, complemented
12C22×D1076C2212yes—
13C2×D4×D269—21no—
14C2×#16864387C2211yescharacteristic
15C2×#16864387C2212yescomplemented
16C22×#14524130C2212yescomplemented
17#22575450C2211yescharacteristic
18#22575428—21no—
19#22575427C2212yes—
20#22575423—22no—
21C4×D1076C2211yescharacteristic
22#22575386C2212yescomplemented
23#22575386C2211yescharacteristic, complemented
24C4⋊#14524130C2211yescharacteristic, complemented
25#22575374C2211yescharacteristic, complemented
26#22575373C2212yes—
27#22575373C2211yescharacteristic
28#22575373C2212yescomplemented
29#22575373C2212yescomplemented
30#22575373C2212yescomplemented
31C22⋊#14524130C2212yescomplemented
32C4×#14524130C2211yescharacteristic, complemented
33C2×#41×C269—21no—
34C4⋊Q8×C269C2211yescharacteristic, complemented
35#123×C269—22no—
36C22⋊Q8×C269—21no—
37#121×C269C2212yescomplemented
38C4×D4×C269C2211yescharacteristic
39C4×D4×C269C2211yescharacteristic, complemented
40C2×C4⋊C4×C269C2212yescomplemented
41C23×D269—22no—
42C2×D1076—24no—
43C2×D1076—22no—
44C2×D1076—21no—
45C2×D1076—21no—
46C2×D1076—42no—
47C2×D1076C2312yes—
48C2×D1076C2311yescharacteristic
49C2×C22⋊D269—22no—
50D4×D269—44no—
51C2×C4×D269—21no—
52#16864387—22no—
53#16864387—42no—
54C2×#14524130—42no—
55C2×#14524130C2311yescharacteristic
56C2×#14524130C2311yescharacteristic
57C2×#14524130C2312yes—
58C4⋊Dic269C2311yescharacteristic
59#16864379—21no—
60#16864379—22no—
61#41×C269—44no—
62C2×Q8×C269—21no—
63C2×D4×C269—22no—
64C2×D4×C269D411yescharacteristic
65C2×D4×C269C2311yescharacteristic
66C22×C4×C269—22no—
67C22×C4×C269—21no—
68C22×C4×C269C2312yes—
69C4⋊C4×C269—22no—
70C4⋊C4×C269—21no—
71C4⋊C4×C269D411yescharacteristic
72C4⋊C4×C269C2311yescharacteristic
73C4⋊C4×C269C2312yes—
74C22⋊C4×C269—21no—
75C22⋊C4×C269—22no—
76C22⋊C4×C269D412yes—
77C42×C269C2311yescharacteristic
78C22×D269—21no—
79C22×D269—22no—
80C22×D269—44no—
81C22×D269—44no—
82C22×D269—82no—
83D1076—22no—
84D1076—21no—
85D1076—22no—
86D1076—44no—
87D1076—42no—
88D1076—81no—
89D1076#4112yescomplemented
90C22⋊D269—44no—
91C2×Dic269—21no—
92C4×D269—42no—
93#14524130—21no—
94#14524130—21no—
95#14524130—22no—
96C23×C269—21no—
97C23×C269C2×D412yes—
98Q8×C269—42no—
99D4×C269—21no—
100D4×C269—21no—
101D4×C269—44no—
102D4×C269D814yescomplemented
103C2×C4×C269—22no—
104C2×C4×C269—22no—
105C2×C4×C269—21no—
106C2×C4×C269—22no—
107C2×C4×C269—44no—
108C2×C4×C269—42no—
109C2×C4×C269C2×D411yescharacteristic
110C2×C4×C269C2×D412yes—
111C2×C4×C269C2×D411yescharacteristic
112C2×C4×C269C2411yescharacteristic
113C22×C269—22no—
114C22×C269—22no—
115C22×C269—41no—
116C22×C269—41no—
117C22×C269—41no—
118C22×C269C2×D814yes—
119C22×C269C22×D411yescharacteristic
120C2×D269—22no—
121C2×D269—41no—
122C2×D269—41no—
123C2×D269—42no—
124C2×D269—42no—
125C2×D269—42no—
126C2×D269—88no—
127C4×C269—22no—
128C4×C269—21no—
129C4×C269—41no—
130C4×C269—42no—
131C4×C269C2×D812yes—
132C4×C269C2×#4111yescharacteristic
133C4×C269H(5,𝔽2)11yescharacteristic
134Dic269—41no—
135C2×C269—24no—
136C2×C269—81no—
137C2×C269#54611yescharacteristic
138C2×C269C22×D811yescharacteristic
139C2×C269#57711yescharacteristic
140D269—42no—
141D269—82no—
142C269#329411yescharacteristic, complemented, Sylow
143#3294—2691nomaximal, Sylow
144#547D26911yescharacteristic, complemented
145D42—2691no—
146#486—2691no—
147#480—2692no—
148#474—2691no—
149#466—2692no—
150#459—2691no—
151#447—2692no—
152C4×D8—2691no—
153C8×D4—2691no—
154C2×#102—2692no—
155C2×#41—21no—
156C22×D4—2692no—
157C22×D4—5381no—
158C2×D8—2691no—
159C2×D8—2692no—
160C22×C8—2692no—
161C4⋊Q8C2×D26911yescharacteristic, complemented
162#127—2691no—
163#123—22no—
164C22⋊Q8—21no—
165#121—2692no—
166#121—5381no—
167#121C2×D26912yescomplemented
168#120—5382no—
169C4×D4—2691no—
170C4×D4C2×D26911yescharacteristic
171C4×D4C2×D26911yescharacteristic, complemented
172C2×C4⋊C4C2×D26912yescomplemented
173#107—2691no—
174#107—2692no—
175C4⋊C8—2691no—
176#102—2691no—
177#102—2691no—
178#102—2692no—
179#102—2692no—
180#102—2692no—
181#102—2692no—
182C22⋊C8—2692no—
183C4×C8—2691no—
184C24—5382no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—2691no—
189C2×D4—2692no—
190C2×D4—5384no—
191C2×D4—5382no—
192C2×D4—5382no—
193C2×D4—5381no—
194C2×D4—5381no—
195C2×D4—10762no—
196C2×D4—10764no—
197C2×D4C22⋊D26911yescharacteristic
198C2×D4C22×D26911yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—5381no—
202C22×C4C22×D26912yes—
203D8—5382no—
204D8—10762no—
205C2×C8—2691no—
206C2×C8—2691no—
207C2×C8—2692no—
208C2×C8—10762no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—2691no—
212C4⋊C4C22⋊D26911yescharacteristic
213C4⋊C4C22×D26911yescharacteristic
214C4⋊C4C22×D26912yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—5381no—
218C22⋊C4—5382no—
219C22⋊C4C22⋊D26912yes—
220C42C22×D26911yescharacteristic
221C23—21no—
222C23—5381no—
223C23—5382no—
224C23—10764no—
225C23—10764no—
226C23—21522no—
227C23C2×C22⋊D26912yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—2692no—
233D4—5382no—
234D4—5381no—
235D4—5382no—
236D4—10764no—
237D4—10764no—
238D4—10762no—
239D4—21521no—
240D4#1686438714yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—5381no—
248C2×C4—10762no—
249C2×C4C2×C22⋊D26911yescharacteristic
250C2×C4C2×C22⋊D26912yes—
251C2×C4C2×C22⋊D26911yescharacteristic
252C2×C4C23×D26911yescharacteristic
253C8—5381no—
254C8—5381no—
255C8—5382no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—5382no—
262C22—10761no—
263C22—10761no—
264C22—10762no—
265C22—10762no—
266C22—10762no—
267C22—21528no—
268C22C2×#1686438714yes—
269C22C22×C22⋊D26911yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—10761no—
275C4C2×#1686438712yes—
276C4#41×D26911yescharacteristic
277C4#2257549911yescharacteristic
278C2—24no—
279C2—81no—
280C2—10762no—
281C2—21522no—
282C2#3177124311yescharacteristic
283C2C22×#1686438711yescharacteristic
284C2#3177141311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#547×C269C2
socleC22×C269C22×D4
hypercenter#547D269
hypocenterC269#3294
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D1076C2semidirect
2. #31770953C2semidirect
3. #31770918C2semidirect
4. #31770882C2semidirect
5. #31770839C2semidirect
6. #31770800C2semidirect
7. #31770738C2semidirect
8. C4×#16864387C2semidirect
9. D4×#14524130C2semidirect
10. C2×#22575373C2semidirect
11. #547×C269C2semidirect
15. C2×#16864387C22semidirect
16. C22×#14524130C22semidirect
22. #22575386C22semidirect
23. #22575386C22semidirect
24. C4⋊#14524130C22semidirect
25. #22575374C22semidirect
28. #22575373C22semidirect
29. #22575373C22semidirect
30. #22575373C22semidirect
31. C22⋊#14524130C22semidirect
32. C4×#14524130C22semidirect
34. C4⋊Q8×C269C22semidirect
37. #121×C269C22semidirect
39. C4×D4×C269C22semidirect
40. C2×C4⋊C4×C269C22semidirect
89. D1076#41semidirect
102. D4×C269D8semidirect
142. C269#3294semidirect
144. #547D269semidirect
161. C4⋊Q8C2×D269semidirect
167. #121C2×D269semidirect
171. C4×D4C2×D269semidirect
172. C2×C4⋊C4C2×D269semidirect
240. D4#16864387semidirect

Maximal subgroups

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

subgroupindex
D4×D10762
#317709532
#317709182
#317708822
#317708392
#317708002
#317707382
C4×#168643872
D4×#145241302
C2×#225753732
#547×C2692
#3294269