Finite Group Databasebeta

← #46820143

Subgroups — #46820143

Order 34432.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#16864402C211yesmaximal, characteristic
2C2×#22575484C212yesmaximal, complemented
3#31770765C214yesmaximal, complemented
4#31770760C212yesmaximal, complemented
5#31770755C212yesmaximal, complemented
6#31770604C212yesmaximal, complemented
7#31770569C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C269C211yesmaximal, characteristic, complemented
9C2×#16864402—28no—
10C2×#16864402—21no—
11C2×#16864402—21no—
12C2×#16864402C2212yes—
13C2×#16864402C2212yes—
14C22×Dic538C2211yescharacteristic
15C22×D1076C2211yescharacteristic
16C22×C22⋊D269—21no—
17C22×C4×D269—21no—
18#22575484—28no—
19C2×#16864392C2214yescomplemented
20C2×#16864389C2214yescomplemented
21C2×#16864386C2212yescomplemented
22#22575376C2214yescomplemented
23#22575373C2214yescomplemented
24C22⋊#14524130C2212yes—
25C22⋊#14524130C2212yescomplemented
26C22×Q8×C269C2211yescharacteristic, complemented
27C23×C4×C269C2211yescharacteristic
28C22⋊Q8×C269—21no—
29C22⋊Q8×C269—21no—
30C22⋊Q8×C269C2212yes—
31C22⋊Q8×C269C2212yescomplemented
32C2×C4⋊C4×C269—21no—
33C2×C4⋊C4×C269C2211yescharacteristic, complemented
34C2×C22⋊C4×C269—21no—
35C23×D269—21no—
36#16864402—28no—
37#16864402—28no—
38#16864402—24no—
39#16864402—24no—
40#16864402—24no—
41#16864402D418yes—
42C2×Dic538—24no—
43C2×Dic538—21no—
44C2×Dic538C2312yes—
45C2×D1076—24no—
46C2×D1076—21no—
47C2×D1076C2312yes—
48C2×C22⋊D269—28no—
49C2×C22⋊D269—22no—
50C2×C22⋊D269—22no—
51C22×Dic269—21no—
52C2×C4×D269—28no—
53C2×C4×D269—22no—
54C2×C4×D269—22no—
55#16864392—28no—
56#16864389—28no—
57#16864386—24no—
58C2×#14524130C2314yes—
59C24×C269D411yescharacteristic
60C2×Q8×C269—21no—
61C2×Q8×C269—24no—
62C2×Q8×C269C2312yes—
63C22×C4×C269—24no—
64C22×C4×C269—21no—
65C22×C4×C269—21no—
66C22×C4×C269D411yescharacteristic
67C22×C4×C269D411yescharacteristic
68C22×C4×C269D411yescharacteristic
69C22×C4×C269C2311yescharacteristic
70C22×C4×C269C2311yescharacteristic
71C22×C4×C269C2311yescharacteristic
72C4⋊C4×C269—22no—
73C4⋊C4×C269—22no—
74C4⋊C4×C269—21no—
75C4⋊C4×C269C2312yes—
76C22⋊C4×C269—22no—
77C22⋊C4×C269—22no—
78C22×D269—24no—
79C22×D269—22no—
80C22×D269—42no—
81C22×D269—42no—
82Dic538—24no—
83Dic538—22no—
84Dic538C2×D414yes—
85D1076—24no—
86D1076—22no—
87D1076C2×D414yes—
88C22⋊D269—28no—
89C22⋊D269—28no—
90C2×Dic269—24no—
91C2×Dic269—22no—
92C4×D269—28no—
93C4×D269—28no—
94#14524130—24no—
95C23×C269—22no—
96C23×C269—22no—
97C23×C269—41no—
98C23×C269C2×D411yescharacteristic
99C23×C269C2×D411yescharacteristic
100C23×C269C2×D411yescharacteristic
101Q8×C269—24no—
102Q8×C269—42no—
103C2×C4×C269—22no—
104C2×C4×C269—22no—
105C2×C4×C269—22no—
106C2×C4×C269—22no—
107C2×C4×C269—22no—
108C2×C4×C269—22no—
109C2×C4×C269—22no—
110C2×C4×C269—42no—
111C2×C4×C269—42no—
112C2×C4×C269C2×D414yes—
113C2×C4×C269C2×D411yescharacteristic
114C2×C4×C269C2×D411yescharacteristic
115C2×C4×C269C2×D411yescharacteristic
116C2×C4×C269C2411yescharacteristic
117C22×C269—21no—
118C22×C269—21no—
119C22×C269—21no—
120C22×C269—21no—
121C22×C269—21no—
122C22×C269—21no—
123C22×C269—21no—
124C22×C269—44no—
125C22×C269#12012yes—
126C22×C269#13712yescomplemented
127C22×C269C22×D411yescharacteristic
128C2×D269—24no—
129C2×D269—48no—
130C2×D269—42no—
131C2×D269—42no—
132C4×C269—21no—
133C4×C269—21no—
134C4×C269—42no—
135C4×C269—42no—
136C4×C269#12012yes—
137C4×C269C22×D412yes—
138Dic269—24no—
139C2×C269—22no—
140C2×C269—22no—
141C2×C269—41no—
142C2×C269C2×#12011yescharacteristic
143C2×C269C2×#13711yescharacteristic
144C2×C269#57511yescharacteristic
145D269—44no—
146C269#300711yescharacteristic, complemented, Sylow
147#3007—2691nomaximal, Sylow
148C22×#41—2691no—
149C2×#137—2692no—
150C2×C22⋊Q8D26911yescharacteristic, complemented
151#452—2694no—
152#451—2692no—
153#450—2692no—
154#417—2692no—
155#407—2691no—
156C2×#41—2692no—
157C2×#41—2692no—
158C2×#41—5388no—
159C2×#41—5381no—
160C2×#41—5381no—
161C22×Q8—2691no—
162C22×Q8C2×D26911yescharacteristic, complemented
163C22×D4—2691no—
164C22×D4—5381no—
165C23×C4—5381no—
166C23×C4C2×D26911yescharacteristic
167#137—5388no—
168C2×Q16—2694no—
169C2×SD16—2694no—
170C2×M16—2692no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D26912yes—
174C22⋊Q8C2×D26912yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D26911yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—2694no—
179#102—2694no—
180C22⋊C8—2692no—
181C22⋊C8—2692no—
182C24—5381no—
183C24C22⋊D26911yescharacteristic
184#41—2698no—
185#41—5384no—
186#41—5384no—
187#41—5388no—
188#41—5388no—
189#41—5384no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—2692no—
193C2×Q8—5384no—
194C2×Q8—5381no—
195C2×Q8C22×D26912yes—
196C2×D4—2692no—
197C2×D4—5384no—
198C2×D4—5388no—
199C2×D4—5382no—
200C2×D4—5382no—
201C2×D4—5381no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—5388no—
206C22×C4—5381no—
207C22×C4—5382no—
208C22×C4—5382no—
209C22×C4C22⋊D26911yescharacteristic
210C22×C4C22⋊D26911yescharacteristic
211C22×C4C22⋊D26911yescharacteristic
212C22×C4C22×D26911yescharacteristic
213C22×C4C22×D26911yescharacteristic
214C22×C4C22×D26911yescharacteristic
215Q16—5388no—
216SD16—5388no—
217M16—5384no—
218C2×C8—2694no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D26912yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—5384no—
229C23—5382no—
230C23—10762no—
231C23—10762no—
232C23C2×C22⋊D26911yescharacteristic
233C23C2×C22⋊D26911yescharacteristic
234C23C2×C22⋊D26911yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—2694no—
238Q8—5384no—
239Q8—5382no—
240D4—2694no—
241D4—5384no—
242D4—5388no—
243D4—5388no—
244D4—5382no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—5388no—
255C2×C4—5384no—
256C2×C4—5382no—
257C2×C4—5388no—
258C2×C4D4×D26914yes—
259C2×C4C2×C22⋊D26911yescharacteristic
260C2×C4C2×C22⋊D26911yescharacteristic
261C2×C4C2×C22⋊D26911yescharacteristic
262C2×C4C23×D26911yescharacteristic
263C8—5384no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—5384no—
273C22—10768no—
274C22—10762no—
275C22—10762no—
276C22#2257542112yes—
277C22#2257548412yescomplemented
278C22C22×C22⋊D26911yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—5384no—
284C4#2257542112yes—
285C4C2×D4×D26912yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—10764no—
290C2C2×#2257542111yescharacteristic
291C2C2×#2257548411yescharacteristic
292C2#3177140311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×C22⋊Q8×C269C2
socleC22×C269C22×D4
prime radicalC22×#16864402C2
hypercenterC2×C22⋊Q8D269
hypocenterC269#3007
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#22575484C2semidirect
3. #31770765C2semidirect
4. #31770760C2semidirect
5. #31770755C2semidirect
6. #31770604C2semidirect
7. #31770569C2semidirect
8. C2×C22⋊Q8×C269C2semidirect
19. C2×#16864392C22semidirect
20. C2×#16864389C22semidirect
21. C2×#16864386C22semidirect
22. #22575376C22semidirect
23. #22575373C22semidirect
25. C22⋊#14524130C22semidirect
26. C22×Q8×C269C22semidirect
31. C22⋊Q8×C269C22semidirect
33. C2×C4⋊C4×C269C22semidirect
126. C22×C269#137semidirect
146. C269#3007semidirect
150. C2×C22⋊Q8D269semidirect
162. C22×Q8C2×D269semidirect
174. C22⋊Q8C2×D269semidirect
176. C2×C4⋊C4C2×D269semidirect
277. C22#22575484semidirect

Maximal subgroups

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

subgroupindex
C22×#168644022
C2×#225754842
#317707652
#317707602
#317707552
#317706042
#317705692
C2×C22⋊Q8×C2692
#3007269