Finite Group Databasebeta

← #46820136

Subgroups — #46820136

Order 34432.

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

normal subgroups only

This group has 806 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×#16864403C211yesmaximal, characteristic, complemented
2C2×#22575480C212yesmaximal, complemented
3#121×D269C211yesmaximal, characteristic, complemented
4#31770753C212yesmaximal, complemented
5#31770749C212yesmaximal, complemented
6#31770744C211yesmaximal, characteristic, complemented
7#31770741C211yesmaximal, characteristic, complemented
8#31770738C211yesmaximal, characteristic, complemented
9#31770605C212yesmaximal, complemented
10#31770567C211yesmaximal, characteristic, complemented
11#448×C269C211yesmaximal, characteristic, complemented
12C2×#16864403—28no—
13C2×#16864403—21no—
14C2×#16864403—21no—
15C2×#16864403C2212yes—
16C2×#16864403C2212yescomplemented
17C22×D1076—21no—
18C22×D1076C2211yescharacteristic, complemented
19C2×D4×D269—21no—
20C2×D4×D269C2211yescharacteristic, complemented
21C22×C4×D269—21no—
22C22×C4×D269C2211yescharacteristic
23#22575480—28no—
24C2×#16864389C2212yescomplemented
25C2×D2152C2212yescomplemented
26C2×#16864387C2212yescomplemented
27C2×#16864385C2212yescomplemented
28#22575428C2211yescharacteristic, complemented
29#22575427C2211yescharacteristic
30#22575426—21no—
31#22575425—21no—
32#22575424C2211yescharacteristic, complemented
33C4⋊C4×D269C2211yescharacteristic, complemented
34C22⋊C4×D269—21no—
35#22575377C2212yescomplemented
36#22575375C2212yescomplemented
37#22575373C2212yescomplemented
38C22⋊#14524130C2211yescharacteristic, complemented
39#22575365C2212yescomplemented
40C22×Q8×C269C2211yescharacteristic, complemented
41C2×SD16×C269C2212yescomplemented
42#121×C269C2211yescharacteristic
43#103×C269C2212yescomplemented
44C22⋊C8×C269C2211yescharacteristic, complemented
45C23×D269—21no—
46C23×D269—21no—
47C23×D269D411yescharacteristic
48#16864403—28no—
49#16864403—28no—
50#16864403—24no—
51#16864403—24no—
52#16864403—24no—
53#16864403D418yescomplemented
54C2×D1076—28no—
55C2×D1076—22no—
56C2×D1076—22no—
57C2×D1076—24no—
58C2×D1076—21no—
59C2×D1076—21no—
60C2×D1076C2311yescharacteristic
61C2×D1076C2312yes—
62C2×C22⋊D269—21no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269—21no—
65D4×D269—24no—
66D4×D269—44no—
67C22×Dic269D411yescharacteristic
68C2×C4×D269—28no—
69C2×C4×D269—22no—
70C2×C4×D269—22no—
71C2×C4×D269—24no—
72C2×C4×D269—21no—
73C2×C4×D269D411yescharacteristic
74C2×C4×D269D411yescharacteristic
75C2×C4×D269C2311yescharacteristic
76C2×C4×D269C2311yescharacteristic
77#16864389—24no—
78D2152—24no—
79#16864387—24no—
80#16864385—24no—
81C2×#14524130C2312yes—
82C4⋊Dic269C2311yescharacteristic
83#16864381—21no—
84C22⋊Dic269—21no—
85#16864379—21no—
86#16864379—21no—
87C2×Q8×C269—24no—
88C2×Q8×C269—21no—
89C2×Q8×C269C2312yes—
90C2×D4×C269—21no—
91C2×D4×C269C2311yescharacteristic
92C22×C4×C269—21no—
93C22×C4×C269C2311yescharacteristic
94SD16×C269—24no—
95C2×C8×C269C2312yes—
96C4⋊C4×C269C2311yescharacteristic
97C22⋊C4×C269—21no—
98C22×D269—24no—
99C22×D269—22no—
100C22×D269—22no—
101C22×D269—22no—
102C22×D269—21no—
103C22×D269—41no—
104C22×D269—42no—
105C22×D269—42no—
106C22×D269—42no—
107C22×D269—42no—
108C22×D269—42no—
109C22×D269C2×D411yescharacteristic
110C22×D269C2×D411yescharacteristic
111D1076—28no—
112D1076—24no—
113D1076—28no—
114D1076—22no—
115D1076—22no—
116D1076—42no—
117D1076C2×D414yescomplemented
118C22⋊D269—42no—
119C22⋊D269—42no—
120C22⋊D269—42no—
121C2×Dic269—22no—
122C2×Dic269—21no—
123C2×Dic269C2×D411yescharacteristic
124C2×Dic269C2×D411yescharacteristic
125C4×D269—28no—
126C4×D269—22no—
127C4×D269—22no—
128C4×D269—28no—
129C4×D269—22no—
130C4×D269—42no—
131C4×D269C2×D414yes—
132#14524130—22no—
133C23×C269—21no—
134C23×C269C2×D411yescharacteristic
135Q8×C269—24no—
136Q8×C269—22no—
137Q8×C269C2×D414yescomplemented
138D4×C269—22no—
139D4×C269—42no—
140C2×C4×C269—24no—
141C2×C4×C269—22no—
142C2×C4×C269—22no—
143C2×C4×C269—21no—
144C2×C4×C269C2×D411yescharacteristic
145C2×C4×C269C2411yescharacteristic
146C8×C269—22no—
147C22×C269—21no—
148C22×C269—21no—
149C22×C269—41no—
150C22×C269—41no—
151C22×C269—41no—
152C22×C269AGL(1,ℤ/8)12yescomplemented
153C22×C269C22×D411yescharacteristic
154C2×D269—21no—
155C2×D269—21no—
156C2×D269—21no—
157C2×D269—24no—
158C2×D269—21no—
159C2×D269—21no—
160C2×D269—44no—
161C2×D269—48no—
162C2×D269—42no—
163C2×D269—42no—
164C2×D269—41no—
165C2×D269—41no—
166C2×D269—41no—
167C2×D269—84no—
168C2×D269#12012yes—
169C4×C269—24no—
170C4×C269—21no—
171C4×C269—41no—
172C4×C269C22×D412yes—
173Dic269—21no—
174Dic269—41no—
175Dic269#12012yes—
176C2×C269—22no—
177C2×C269—81no—
178C2×C269C2×#12011yescharacteristic
179C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C269#57511yescharacteristic
181D269—22no—
182D269—41no—
183D269—44no—
184D269—81no—
185C269#300611yescharacteristic, complemented, Sylow
186#3006—2691nomaximal, Sylow
187C22×#41—2691no—
188C2×AGL(1,ℤ/8)—2692no—
189C2×#121—2691no—
190#449—2692no—
191#449—2692no—
192#448—2691no—
193#448D26911yescharacteristic, complemented
194#447—2691no—
195#447—2691no—
196#417—2692no—
197#407—2691no—
198C2×#41—2692no—
199C2×#41—2692no—
200C2×#41—5381no—
201C2×#41—5381no—
202C2×#41—5388no—
203C22×Q8C2×D26911yescharacteristic, complemented
204C22×D4—2691no—
205C22×D4—2691no—
206C22×D4—5381no—
207C22×D4—5381no—
208C23×C4—2691no—
209C23×C4—5381no—
210AGL(1,ℤ/8)—5388no—
211C2×SD16—2692no—
212C2×SD16C2×D26912yescomplemented
213C2×D8—2692no—
214C2×D8—2692no—
215C2×M16—2692no—
216#121—2691no—
217#121—2691no—
218#121—2691no—
219#121—5381no—
220#121—5381no—
221#121C2×D26911yescharacteristic
222C2×C4⋊C4—2691no—
223C2×C22⋊C4—5381no—
224#103—2692no—
225#103C2×D26912yescomplemented
226#102—2692no—
227#102—2692no—
228C22⋊C8—2691no—
229C22⋊C8—2692no—
230C22⋊C8C2×D26911yescharacteristic, complemented
231C24—2691no—
232C24—5381no—
233C24—5381no—
234#41—2698no—
235#41—5384no—
236#41—5384no—
237#41—5388no—
238#41—5388no—
239#41—5384no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D26912yes—
243C2×D4—21no—
244C2×D4—2692no—
245C2×D4—2691no—
246C2×D4—5384no—
247C2×D4—5382no—
248C2×D4—5382no—
249C2×D4—5388no—
250C2×D4—5381no—
251C2×D4—5384no—
252C2×D4—5381no—
253C2×D4—5381no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4—10764no—
257C2×D4C22×D26911yescharacteristic
258C22×C4—21no—
259C22×C4—2691no—
260C22×C4—2691no—
261C22×C4—2691no—
262C22×C4—2691no—
263C22×C4—2691no—
264C22×C4—5384no—
265C22×C4—5382no—
266C22×C4—5382no—
267C22×C4—5388no—
268C22×C4—5381no—
269C22×C4C22×D26911yescharacteristic
270SD16—24no—
271SD16—5384no—
272D8—5384no—
273D8—5384no—
274M16—5384no—
275C2×C8—2692no—
276C2×C8C22×D26912yes—
277C4⋊C4—2691no—
278C4⋊C4—5381no—
279C4⋊C4C22×D26911yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—5381no—
282C22⋊C4—5381no—
283C22⋊C4—5381no—
284C23—21no—
285C23—2691no—
286C23—2691no—
287C23—5382no—
288C23—5384no—
289C23—5382no—
290C23—5382no—
291C23—5381no—
292C23—10761no—
293C23—10762no—
294C23—10762no—
295C23—10762no—
296C23—10762no—
297C23—10762no—
298C23D4×D26911yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D26914yescomplemented
302D4—22no—
303D4—42no—
304D4—2694no—
305D4—5388no—
306D4—5388no—
307D4—5384no—
308D4—5382no—
309D4—5382no—
310D4—10762no—
311D4—10762no—
312D4—10762no—
313D4—10762no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—2691no—
319C2×C4—2694no—
320C2×C4—2691no—
321C2×C4—5382no—
322C2×C4—5388no—
323C2×C4—5382no—
324C2×C4—5388no—
325C2×C4—5382no—
326C2×C4—5382no—
327C2×C4—5381no—
328C2×C4—10762no—
329C2×C4D4×D26911yescharacteristic
330C2×C4C23×D26911yescharacteristic
331C8—22no—
332C8—5382no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—2692no—
339C22—5381no—
340C22—5381no—
341C22—5381no—
342C22—5381no—
343C22—5384no—
344C22—5381no—
345C22—10764no—
346C22—10768no—
347C22—10762no—
348C22—10762no—
349C22—10761no—
350C22—10761no—
351C22—10761no—
352C22—21524no—
353C22#2257548012yescomplemented
354C22C2×D4×D26911yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—2692no—
359C4—5381no—
360C4—10761no—
361C4C2×D4×D26912yes—
362C2—22no—
363C2—81no—
364C2—5382no—
365C2—10761no—
366C2—10764no—
367C2—21521no—
368C2#120×D26911yescharacteristic
369C2C2×#2257548011yescharacteristic
370C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#448×C269C2
socleC22×C269C22×D4
hypercenter#448D269
hypocenterC269#3006
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#16864403C2semidirect
2. C2×#22575480C2semidirect
3. #121×D269C2semidirect
4. #31770753C2semidirect
5. #31770749C2semidirect
6. #31770744C2semidirect
7. #31770741C2semidirect
8. #31770738C2semidirect
9. #31770605C2semidirect
10. #31770567C2semidirect
11. #448×C269C2semidirect
16. C2×#16864403C22semidirect
18. C22×D1076C22semidirect
20. C2×D4×D269C22semidirect
24. C2×#16864389C22semidirect
25. C2×D2152C22semidirect
26. C2×#16864387C22semidirect
27. C2×#16864385C22semidirect
28. #22575428C22semidirect
32. #22575424C22semidirect
33. C4⋊C4×D269C22semidirect
35. #22575377C22semidirect
36. #22575375C22semidirect
37. #22575373C22semidirect
38. C22⋊#14524130C22semidirect
39. #22575365C22semidirect
40. C22×Q8×C269C22semidirect
41. C2×SD16×C269C22semidirect
43. #103×C269C22semidirect
44. C22⋊C8×C269C22semidirect
53. #16864403D4semidirect
117. D1076C2×D4semidirect
137. Q8×C269C2×D4semidirect
152. C22×C269AGL(1,ℤ/8)semidirect
185. C269#3006semidirect
193. #448D269semidirect
203. C22×Q8C2×D269semidirect
212. C2×SD16C2×D269semidirect
225. #103C2×D269semidirect
230. C22⋊C8C2×D269semidirect
301. Q8D4×D269semidirect
353. C22#22575480semidirect

Maximal subgroups

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

subgroupindex
C22×#168644032
C2×#225754802
#121×D2692
#317707532
#317707492
#317707442
#317707412
#317707382
#317706052
#317705672
#448×C2692
#3006269