Finite Group Databasebeta

← #46820140

Subgroups — #46820140

Order 34432.

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

normal subgroups only

This group has 754 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
1C22×#16864401C211yesmaximal, characteristic
2C2×#22575482C212yesmaximal, complemented
3C22⋊Q8×D269C211yesmaximal, characteristic, complemented
4#31770768C212yesmaximal, complemented
5#31770764C212yesmaximal, complemented
6#31770763C211yesmaximal, characteristic, complemented
7#31770760C211yesmaximal, characteristic, complemented
8#31770756C211yesmaximal, characteristic, complemented
9#31770603C212yesmaximal, complemented
10#31770567C211yesmaximal, characteristic, complemented
11#450×C269C211yesmaximal, characteristic, complemented
12C2×#16864401—28no—
13C2×#16864401—21no—
14C2×#16864401—21no—
15C2×#16864401C2212yes—
16C2×#16864401C2212yes—
17C22×Dic538C2211yescharacteristic
18C2×Q8×D269C2211yescharacteristic, complemented
19C22×C22⋊D269—21no—
20C23×Dic269—21no—
21C22×C4×D269C2211yescharacteristic
22#22575482—28no—
23C2×Dic1076C2212yescomplemented
24C2×#16864392C2212yescomplemented
25C2×#16864390C2212yescomplemented
26C2×#16864385C2212yescomplemented
27#22575433C2211yescharacteristic, complemented
28#22575432C2211yescharacteristic
29#22575431—21no—
30#22575430—21no—
31#22575429C2211yescharacteristic, complemented
32C4⋊C4×D269—21no—
33C4⋊C4×D269C2211yescharacteristic, complemented
34C22⋊C4×D269—21no—
35#22575378C2212yescomplemented
36#22575376C2212yescomplemented
37#22575374C2212yescomplemented
38C22⋊#14524130C2211yescharacteristic
39#22575365C2212yescomplemented
40C22×D4×C269C2211yescharacteristic
41C2×SD16×C269C2212yescomplemented
42C22⋊Q8×C269C2211yescharacteristic
43#102×C269C2212yescomplemented
44C22⋊C8×C269C2211yescharacteristic
45C23×D269D411yescharacteristic
46#16864401—28no—
47#16864401—28no—
48#16864401—24no—
49#16864401—24no—
50#16864401—24no—
51#16864401D418yes—
52C2×Dic538—24no—
53C2×Dic538—21no—
54C2×Dic538—21no—
55C2×Dic538C2311yescharacteristic
56C2×Dic538C2312yes—
57Q8×D269—24no—
58C2×C22⋊D269—28no—
59C2×C22⋊D269—22no—
60C2×C22⋊D269—22no—
61C22×Dic269—28no—
62C22×Dic269—22no—
63C22×Dic269—22no—
64C22×Dic269—21no—
65C22×Dic269D411yescharacteristic
66C2×C4×D269—24no—
67C2×C4×D269—21no—
68C2×C4×D269—21no—
69C2×C4×D269D411yescharacteristic
70C2×C4×D269D411yescharacteristic
71C2×C4×D269C2311yescharacteristic
72C2×C4×D269C2311yescharacteristic
73Dic1076—24no—
74#16864392—24no—
75#16864390—24no—
76#16864385—24no—
77C2×#14524130C2312yes—
78C4⋊Dic269—21no—
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83C22⋊Dic269—21no—
84#16864379—21no—
85#16864379—21no—
86C24×C269—21no—
87C2×Q8×C269C2311yescharacteristic
88C2×D4×C269—24no—
89C2×D4×C269—21no—
90C2×D4×C269C2312yes—
91C22×C4×C269C2311yescharacteristic
92SD16×C269—24no—
93C2×C8×C269C2312yes—
94C4⋊C4×C269—21no—
95C4⋊C4×C269C2311yescharacteristic
96C22⋊C4×C269—21no—
97C22×D269—22no—
98C22×D269—22no—
99C22×D269—41no—
100C22×D269C2×D411yescharacteristic
101C22×D269C2×D411yescharacteristic
102Dic538—24no—
103Dic538—22no—
104Dic538—22no—
105Dic538—42no—
106Dic538C2×D414yes—
107C22⋊D269—28no—
108C22⋊D269—28no—
109C2×Dic269—24no—
110C2×Dic269—28no—
111C2×Dic269—22no—
112C2×Dic269—22no—
113C2×Dic269—28no—
114C2×Dic269—21no—
115C2×Dic269—21no—
116C2×Dic269C2×D411yescharacteristic
117C2×Dic269C2×D411yescharacteristic
118C4×D269—22no—
119C4×D269—22no—
120C4×D269—22no—
121C4×D269—42no—
122C4×D269—42no—
123C4×D269C2×D414yes—
124#14524130—22no—
125C23×C269—24no—
126C23×C269—22no—
127C23×C269—42no—
128C23×C269—42no—
129C23×C269C2×D411yescharacteristic
130Q8×C269—22no—
131D4×C269—24no—
132D4×C269—22no—
133D4×C269C2×D414yes—
134C2×C4×C269—22no—
135C2×C4×C269—21no—
136C2×C4×C269—21no—
137C2×C4×C269C2×D411yescharacteristic
138C2×C4×C269C2411yescharacteristic
139C8×C269—22no—
140C22×C269—21no—
141C22×C269—21no—
142C22×C269—24no—
143C22×C269—48no—
144C22×C269—42no—
145C22×C269—42no—
146C22×C269#13712yescomplemented
147C22×C269C22×D411yescharacteristic
148C2×D269—21no—
149C2×D269—21no—
150C2×D269—21no—
151C2×D269—21no—
152C2×D269—21no—
153C2×D269—44no—
154C2×D269#12012yes—
155C4×C269—21no—
156C4×C269—41no—
157C4×C269—41no—
158C4×C269C22×D412yes—
159Dic269—21no—
160Dic269—24no—
161Dic269—41no—
162Dic269—41no—
163Dic269#12012yes—
164C2×C269—22no—
165C2×C269—44no—
166C2×C269C2×#12011yescharacteristic
167C2×C269C2×#13711yescharacteristic
168C2×C269#57511yescharacteristic
169D269—22no—
170D269—41no—
171C269#300711yescharacteristic, complemented, Sylow
172#3007—2691nomaximal, Sylow
173C22×#41—2691no—
174C2×#137—2692no—
175C2×C22⋊Q8—2691no—
176#452—2692no—
177#452—2692no—
178#451—2691no—
179#451—2691no—
180#450—2691no—
181#450D26911yescharacteristic, complemented
182#417—2692no—
183#407—2691no—
184C2×#41—2692no—
185C2×#41—2692no—
186C2×#41—5381no—
187C2×#41—5381no—
188C2×#41—5388no—
189C22×Q8—2691no—
190C22×Q8—2691no—
191C22×D4—5381no—
192C22×D4C2×D26911yescharacteristic
193C23×C4—2691no—
194C23×C4—5381no—
195#137—5388no—
196C2×Q16—2692no—
197C2×Q16—2692no—
198C2×SD16—2692no—
199C2×SD16C2×D26912yescomplemented
200C2×M16—2692no—
201C22⋊Q8—2691no—
202C22⋊Q8—2691no—
203C22⋊Q8—2691no—
204C22⋊Q8—5381no—
205C22⋊Q8—5381no—
206C22⋊Q8C2×D26911yescharacteristic
207C2×C4⋊C4—2691no—
208C2×C4⋊C4—5381no—
209C2×C22⋊C4—5381no—
210#103—2692no—
211#103—2692no—
212#102—2692no—
213#102C2×D26912yescomplemented
214C22⋊C8—2691no—
215C22⋊C8—2692no—
216C22⋊C8C2×D26911yescharacteristic
217C24—21no—
218C24—2691no—
219#41—2698no—
220#41—5384no—
221#41—5384no—
222#41—5388no—
223#41—5388no—
224#41—5384no—
225C2×Q8—2692no—
226C2×Q8—2691no—
227C2×Q8—5381no—
228C2×Q8—5384no—
229C2×Q8—5384no—
230C2×Q8—5381no—
231C2×Q8C22×D26911yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—5382no—
235C2×D4—5382no—
236C2×D4—5388no—
237C2×D4C22×D26912yes—
238C22×C4—2691no—
239C22×C4—2691no—
240C22×C4—2691no—
241C22×C4—2691no—
242C22×C4—2691no—
243C22×C4—5384no—
244C22×C4—5381no—
245C22×C4—5381no—
246C22×C4—5382no—
247C22×C4—5382no—
248C22×C4—5388no—
249C22×C4—5381no—
250C22×C4C22×D26911yescharacteristic
251Q16—5384no—
252Q16—5384no—
253SD16—24no—
254SD16—5384no—
255M16—5384no—
256C2×C8—2692no—
257C2×C8C22×D26912yes—
258C4⋊C4—21no—
259C4⋊C4—2691no—
260C4⋊C4—5381no—
261C4⋊C4—5381no—
262C4⋊C4—5381no—
263C4⋊C4—5381no—
264C4⋊C4C22×D26911yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—5381no—
267C22⋊C4—5381no—
268C22⋊C4—5381no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—2691no—
274C23—2691no—
275C23—5382no—
276C23—5382no—
277C23—10761no—
278C23D4×D26911yescharacteristic
279Q8—22no—
280Q8—2694no—
281Q8—5384no—
282Q8—5382no—
283Q8—5382no—
284Q8—10762no—
285D4—24no—
286D4—22no—
287D4—5388no—
288D4—5388no—
289D4D4×D26914yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—2691no—
294C2×C4—2694no—
295C2×C4—2691no—
296C2×C4—5382no—
297C2×C4—5382no—
298C2×C4—5382no—
299C2×C4—5388no—
300C2×C4—5384no—
301C2×C4—5388no—
302C2×C4—5382no—
303C2×C4—5382no—
304C2×C4—5381no—
305C2×C4—5381no—
306C2×C4—10762no—
307C2×C4—10762no—
308C2×C4D4×D26911yescharacteristic
309C2×C4C23×D26911yescharacteristic
310C8—22no—
311C8—5382no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—2692no—
319C22—5381no—
320C22—5381no—
321C22—5381no—
322C22—5381no—
323C22—5381no—
324C22—10764no—
325C22#2257548212yescomplemented
326C22C2×D4×D26911yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—2692no—
331C4—5381no—
332C4—5384no—
333C4—10761no—
334C4—10761no—
335C4C2×D4×D26912yes—
336C2—22no—
337C2—44no—
338C2—5382no—
339C2—10761no—
340C2#120×D26911yescharacteristic
341C2C2×#2257548211yescharacteristic
342C2#3177140111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#450×C269C2
socleC22×C269C22×D4
prime radicalC22×#16864401C2
hypercenter#450D269
hypocenterC269#3007
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#22575482C2semidirect
3. C22⋊Q8×D269C2semidirect
4. #31770768C2semidirect
5. #31770764C2semidirect
6. #31770763C2semidirect
7. #31770760C2semidirect
8. #31770756C2semidirect
9. #31770603C2semidirect
10. #31770567C2semidirect
11. #450×C269C2semidirect
18. C2×Q8×D269C22semidirect
23. C2×Dic1076C22semidirect
24. C2×#16864392C22semidirect
25. C2×#16864390C22semidirect
26. C2×#16864385C22semidirect
27. #22575433C22semidirect
31. #22575429C22semidirect
33. C4⋊C4×D269C22semidirect
35. #22575378C22semidirect
36. #22575376C22semidirect
37. #22575374C22semidirect
39. #22575365C22semidirect
41. C2×SD16×C269C22semidirect
43. #102×C269C22semidirect
146. C22×C269#137semidirect
171. C269#3007semidirect
181. #450D269semidirect
199. C2×SD16C2×D269semidirect
213. #102C2×D269semidirect
325. C22#22575482semidirect

Maximal subgroups

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

subgroupindex
C22×#168644012
C2×#225754822
C22⋊Q8×D2692
#317707682
#317707642
#317707632
#317707602
#317707562
#317706032
#317705672
#450×C2692
#3007269