Finite Group Databasebeta

← #46820188

Subgroups — #46820188

Order 34432.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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×#22575498C211yesmaximal, characteristic, complemented
2C2×#22575473C211yesmaximal, characteristic, complemented
3#31771183C211yesmaximal, characteristic, complemented
4#31770764C212yesmaximal, complemented
5#31770754C212yesmaximal, complemented
6#31770747C212yesmaximal, complemented
7#31770737C212yesmaximal, complemented
8#31770607C211yesmaximal, characteristic, complemented
9C2×#22575374C211yesmaximal, characteristic, complemented
10#31770573C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
12#22575498—24no—
13#22575498—24no—
14C2×#16864402C2211yescharacteristic
15C2×#16864402C2211yescharacteristic
16C2×#16864401—21no—
17C2×#16864401C2212yes—
18C22×C22⋊D269—22no—
19C2×D4×D269—21no—
20C2×D4×D269C2212yes—
21#22575473—24no—
22C2×#16864392C2211yescharacteristic, complemented
23C2×#16864390C2211yescharacteristic, complemented
24C2×#16864389C2211yescharacteristic, complemented
25C2×#16864387C2211yescharacteristic, complemented
26C22×#14524130C2211yescharacteristic, complemented
27#22575432—21no—
28#22575429C2212yescomplemented
29#22575427—21no—
30#22575425—22no—
31#22575424C2212yescomplemented
32C2×C4⋊Dic269C2211yescharacteristic, complemented
33#22575378C2212yescomplemented
34#22575375C2212yescomplemented
35#22575374C2212yescomplemented
36#22575374C2212yescomplemented
37#22575365C2212yescomplemented
38#22575365C2212yescomplemented
39C2×#41×C269C2211yescharacteristic, complemented
40C22×D4×C269C2211yescharacteristic
41AGL(1,ℤ/8)×C269—24no—
42C2×SD16×C269C2212yescomplemented
43C2×D8×C269C2212yescomplemented
44C2×M16×C269C2211yescharacteristic, complemented
45C23×D269—22no—
46#16864402—22no—
47#16864402—22no—
48#16864402—22no—
49#16864402D414yescomplemented
50#16864401—28no—
51#16864401—24no—
52#16864401—24no—
53C2×Dic538D411yescharacteristic
54C2×Dic538C2311yescharacteristic
55C2×D1076D411yescharacteristic
56C2×D1076C2311yescharacteristic
57C2×C22⋊D269—216no—
58C2×C22⋊D269—22no—
59C2×C22⋊D269—22no—
60C2×C22⋊D269—22no—
61C2×C22⋊D269—21no—
62C2×C22⋊D269D412yes—
63D4×D269—28no—
64D4×D269—24no—
65D4×D269—24no—
66C22×Dic269—22no—
67C22×Dic269—21no—
68C2×C4×D269—21no—
69C2×C4×D269C2312yes—
70#16864392—22no—
71#16864390—22no—
72#16864389—22no—
73#16864387—22no—
74C2×#14524130—22no—
75C2×#14524130C2311yescharacteristic
76C2×#14524130C2311yescharacteristic
77C4⋊Dic269—21no—
78C4⋊Dic269C2312yes—
79#16864381—22no—
80C22⋊Dic269—22no—
81#16864379—22no—
82C24×C269—21no—
83#41×C269—22no—
84#41×C269—22no—
85C2×Q8×C269C2311yescharacteristic
86C2×D4×C269—21no—
87C2×D4×C269—21no—
88C2×D4×C269—22no—
89C2×D4×C269D414yescomplemented
90C2×D4×C269C2311yescharacteristic
91C2×D4×C269C2312yes—
92C22×C4×C269—21no—
93C22×C4×C269C2311yescharacteristic
94SD16×C269—24no—
95D8×C269—24no—
96M16×C269—22no—
97C2×C8×C269C2312yes—
98C22×D269—21no—
99C22×D269—28no—
100C22×D269—82no—
101C22×D269—82no—
102C22×D269C2×D412yes—
103Dic538—22no—
104Dic538—21no—
105Dic538C2×D412yes—
106D1076—22no—
107D1076—21no—
108D1076C2×D412yes—
109C22⋊D269—28no—
110C22⋊D269—28no—
111C22⋊D269—24no—
112C22⋊D269—24no—
113C22⋊D269—82no—
114C2×Dic269—28no—
115C2×Dic269—21no—
116C2×Dic269—22no—
117C2×Dic269—81no—
118C2×Dic269C2×D412yes—
119C4×D269—24no—
120C4×D269—22no—
121C4×D269—22no—
122#14524130—21no—
123#14524130—21no—
124C23×C269—22no—
125C23×C269—24no—
126C23×C269—21no—
127C23×C269—81no—
128C23×C269—81no—
129C23×C269C2×D411yescharacteristic
130Q8×C269—21no—
131Q8×C269—21no—
132D4×C269—24no—
133D4×C269—22no—
134D4×C269—21no—
135D4×C269—21no—
136D4×C269—42no—
137D4×C269C2×D414yes—
138C2×C4×C269—21no—
139C2×C4×C269—42no—
140C2×C4×C269C2×D411yescharacteristic
141C2×C4×C269C2×D412yes—
142C2×C4×C269C2×D412yes—
143C2×C4×C269C2411yescharacteristic
144C8×C269—22no—
145C22×C269—21no—
146C22×C269—21no—
147C22×C269—24no—
148C22×C269—42no—
149C22×C269—42no—
150C22×C269—41no—
151C22×C269—41no—
152C22×C269—41no—
153C22×C269—84no—
154C22×C269#12012yes—
155C22×C269C22×D411yescharacteristic
156C2×D269—22no—
157C2×D269—22no—
158C2×D269—22no—
159C2×D269—22no—
160C2×D269—41no—
161C2×D269—41no—
162C2×D269—88no—
163C4×C269—41no—
164C4×C269#12012yes—
165C4×C269C22×D411yescharacteristic
166C4×C269C22×D411yescharacteristic
167Dic269—22no—
168Dic269—22no—
169Dic269—42no—
170C2×C269—22no—
171C2×C269—44no—
172C2×C269—81no—
173C2×C269C2×#12011yescharacteristic
174C2×C269#57611yescharacteristic
175C2×C269#57711yescharacteristic
176D269—42no—
177D269—42no—
178C269#301211yescharacteristic, complemented, Sylow
179#3012—2691nomaximal, Sylow
180C2×H(5,𝔽2)—2691no—
181C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
182C2×#135—2691no—
183#537—2691no—
184#452—2692no—
185#450—2692no—
186#449—2692no—
187#447—2692no—
188#417—2691no—
189C2×#102—2691no—
190#408—2691no—
191H(5,𝔽2)—5384no—
192H(5,𝔽2)—5384no—
193C2×#41—2691no—
194C2×#41—2692no—
195C2×#41—2691no—
196C2×#41—5381no—
197C2×#41C2×D26911yescharacteristic, complemented
198C22×D4—2692no—
199C22×D4—5381no—
200C22×D4—5382no—
201C22×D4C2×D26911yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—5384no—
204C2×Q16—2691no—
205C2×SD16—2691no—
206C2×SD16—2691no—
207C2×SD16C2×D26912yescomplemented
208C2×D8—2691no—
209C2×D8C2×D26912yescomplemented
210C2×M16C2×D26911yescharacteristic, complemented
211C22×C8—2691no—
212C22⋊Q8—2692no—
213C22⋊Q8—5381no—
214#121—2692no—
215#121—5381no—
216#121—5382no—
217C2×C4⋊C4—2691no—
218#103—2692no—
219#102—2692no—
220#102—2692no—
221#102—2692no—
222C22⋊C8—2692no—
223C22⋊C8—2692no—
224C24—21no—
225C24—5382no—
226#41—22no—
227#41—22no—
228#41—2694no—
229#41—5384no—
230#41—5388no—
231#41—5384no—
232#41—5382no—
233#41—5382no—
234#41—5382no—
235C2×Q8—2691no—
236C2×Q8—2691no—
237C2×Q8C22×D26911yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—2692no—
242C2×D4—2691no—
243C2×D4—2691no—
244C2×D4—5382no—
245C2×D4—5384no—
246C2×D4—5388no—
247C2×D4—5382no—
248C2×D4—53816no—
249C2×D4—5381no—
250C2×D4—5384no—
251C2×D4—5382no—
252C2×D4C22⋊D26914yescomplemented
253C2×D4C22×D26911yescharacteristic
254C2×D4C22×D26912yes—
255C22×C4—21no—
256C22×C4—2692no—
257C22×C4—5382no—
258C22×C4—5381no—
259C22×C4—5381no—
260C22×C4C22×D26911yescharacteristic
261Q16—5382no—
262SD16—24no—
263SD16—5382no—
264SD16—5382no—
265D8—24no—
266D8—5382no—
267M16—22no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8—5382no—
271C2×C8C22×D26912yes—
272C4⋊C4—2692no—
273C4⋊C4—5382no—
274C4⋊C4—5381no—
275C22⋊C4—5382no—
276C22⋊C4—5382no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—2692no—
283C23—5388no—
284C23—5381no—
285C23—21522no—
286C23—21522no—
287C23D4×D26911yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2692no—
291Q8—5382no—
292Q8—5381no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—2692no—
299D4—5388no—
300D4—5388no—
301D4—5384no—
302D4—5384no—
303D4—5382no—
304D4—5381no—
305D4—21522no—
306D4C2×C22⋊D26914yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—2692no—
310C2×C4—5388no—
311C2×C4—5381no—
312C2×C4—5384no—
313C2×C4—5382no—
314C2×C4—5382no—
315C2×C4—5382no—
316C2×C4—21521no—
317C2×C4D4×D26911yescharacteristic
318C2×C4D4×D26912yes—
319C2×C4C2×C22⋊D26912yes—
320C2×C4C23×D26911yescharacteristic
321C8—22no—
322C8—5381no—
323C8—5381no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—5382no—
334C22—5382no—
335C22—5382no—
336C22—5382no—
337C22—10761no—
338C22—10761no—
339C22—21528no—
340C22#2257542112yes—
341C22C2×D4×D26911yescharacteristic
342C4—41no—
343C4—5382no—
344C4—5382no—
345C4—10762no—
346C4#2257542112yes—
347C4C2×D4×D26911yescharacteristic
348C4C22×C22⋊D26911yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—10762no—
353C2—10762no—
354C2C2×#2257542111yescharacteristic
355C2#3177140911yescharacteristic
356C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×AGL(1,ℤ/8)×C269C2
socleC22×C269C22×D4
hypercenterC2×AGL(1,ℤ/8)D269
hypocenterC269#3012
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22575498C2semidirect
2. C2×#22575473C2semidirect
3. #31771183C2semidirect
4. #31770764C2semidirect
5. #31770754C2semidirect
6. #31770747C2semidirect
7. #31770737C2semidirect
8. #31770607C2semidirect
9. C2×#22575374C2semidirect
10. #31770573C2semidirect
11. C2×AGL(1,ℤ/8)×C269C2semidirect
22. C2×#16864392C22semidirect
23. C2×#16864390C22semidirect
24. C2×#16864389C22semidirect
25. C2×#16864387C22semidirect
26. C22×#14524130C22semidirect
28. #22575429C22semidirect
31. #22575424C22semidirect
32. C2×C4⋊Dic269C22semidirect
33. #22575378C22semidirect
34. #22575375C22semidirect
35. #22575374C22semidirect
36. #22575374C22semidirect
37. #22575365C22semidirect
38. #22575365C22semidirect
39. C2×#41×C269C22semidirect
42. C2×SD16×C269C22semidirect
43. C2×D8×C269C22semidirect
44. C2×M16×C269C22semidirect
49. #16864402D4semidirect
89. C2×D4×C269D4semidirect
178. C269#3012semidirect
181. C2×AGL(1,ℤ/8)D269semidirect
197. C2×#41C2×D269semidirect
207. C2×SD16C2×D269semidirect
209. C2×D8C2×D269semidirect
210. C2×M16C2×D269semidirect
252. C2×D4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754982
C2×#225754732
#317711832
#317707642
#317707542
#317707472
#317707372
#317706072
C2×#225753742
#317705732
C2×AGL(1,ℤ/8)×C2692
#3012269