Finite Group Databasebeta

← #46820191

Subgroups — #46820191

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×#22575499C211yesmaximal, characteristic, complemented
2C2×#22575481C211yesmaximal, characteristic, complemented
3C2×#22575473C211yesmaximal, characteristic, complemented
4#31771186C211yesmaximal, characteristic, complemented
5#31770770C212yesmaximal, complemented
6#31770757C212yesmaximal, complemented
7#31770753C212yesmaximal, complemented
8#31770740C212yesmaximal, complemented
9C2×#22575373C211yesmaximal, characteristic, complemented
10#31770573C211yesmaximal, characteristic, complemented
11#417×C269C211yesmaximal, characteristic, complemented
12#22575499—24no—
13#22575499—24no—
14C2×#16864403C2212yes—
15C2×#16864402—21no—
16C2×#16864402C2211yescharacteristic
17C2×#16864402C2211yescharacteristic, complemented
18C22×D1076—21no—
19C22×D1076C2211yescharacteristic
20C2×D4×D269—22no—
21C2×D4×D269C2212yes—
22#22575481—24no—
23#22575473—24no—
24C2×#16864392C2211yescharacteristic, complemented
25C2×#16864391C2212yescomplemented
26C2×#16864390C2211yescharacteristic, complemented
27C2×#16864389C2211yescharacteristic, complemented
28C2×D2152C2212yescomplemented
29C2×#16864387C2211yescharacteristic, complemented
30C22×#14524130C2211yescharacteristic, complemented
31#22575433—21no—
32#22575433C2212yescomplemented
33#22575428—21no—
34#22575428C2212yescomplemented
35#22575427—22no—
36#22575373C2212yescomplemented
37#22575373C2212yescomplemented
38#22575365C2212yescomplemented
39#22575365C2212yescomplemented
40C2×#41×C269C2211yescharacteristic
41C2×M16×C269C2211yescharacteristic, complemented
42C2×C4⋊C4×C269C2211yescharacteristic, complemented
43#103×C269C2212yescomplemented
44#102×C269C2212yescomplemented
45C23×D269—22no—
46C23×D269—21no—
47#16864403—24no—
48#16864403—24no—
49#16864402—28no—
50#16864402—22no—
51#16864402—22no—
52#16864402—22no—
53#16864402—22no—
54C2×Dic538D411yescharacteristic
55C2×Dic538C2311yescharacteristic
56C2×D1076—28no—
57C2×D1076—22no—
58C2×D1076—22no—
59C2×D1076—21no—
60C2×D1076—22no—
61C2×D1076D411yescharacteristic
62C2×D1076D414yescomplemented
63C2×D1076C2311yescharacteristic
64C2×D1076C2312yes—
65C2×C22⋊D269—22no—
66C2×C22⋊D269—21no—
67C2×C22⋊D269D412yes—
68D4×D269—216no—
69D4×D269—24no—
70D4×D269—24no—
71C2×C4×D269—22no—
72C2×C4×D269—21no—
73C2×C4×D269C2312yes—
74#16864392—22no—
75#16864391—24no—
76#16864390—22no—
77#16864389—22no—
78D2152—24no—
79#16864387—22no—
80C2×#14524130—22no—
81C2×#14524130C2311yescharacteristic
82C2×#14524130C2311yescharacteristic
83C4⋊Dic269—22no—
84#16864379—22no—
85#16864379—22no—
86#41×C269—22no—
87#41×C269D414yescomplemented
88C2×Q8×C269C2311yescharacteristic
89C2×D4×C269—21no—
90C2×D4×C269C2311yescharacteristic
91C22×C4×C269—21no—
92C22×C4×C269—21no—
93C22×C4×C269C2311yescharacteristic
94M16×C269—22no—
95C2×C8×C269C2312yes—
96C4⋊C4×C269—21no—
97C4⋊C4×C269C2312yes—
98C22×D269—22no—
99C22×D269—28no—
100C22×D269—24no—
101C22×D269—21no—
102C22×D269—82no—
103C22×D269—82no—
104C22×D269—81no—
105C22×D269—81no—
106C22×D269C2×D412yes—
107Dic538—22no—
108Dic538—21no—
109Dic538—21no—
110D1076—28no—
111D1076—22no—
112D1076—24no—
113D1076—22no—
114D1076—21no—
115D1076—21no—
116D1076—82no—
117D1076C2×D414yes—
118C22⋊D269—28no—
119C22⋊D269—24no—
120C22⋊D269—42no—
121C2×Dic269—21no—
122C2×Dic269C2×D412yes—
123C4×D269—28no—
124C4×D269—22no—
125C4×D269—22no—
126C4×D269—42no—
127#14524130—21no—
128#14524130—21no—
129C23×C269—21no—
130C23×C269C2×D411yescharacteristic
131Q8×C269—21no—
132Q8×C269C2×D412yes—
133D4×C269—24no—
134D4×C269—21no—
135D4×C269C2×D412yes—
136C2×C4×C269—21no—
137C2×C4×C269—24no—
138C2×C4×C269—22no—
139C2×C4×C269—81no—
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—22no—
148C22×C269—41no—
149C22×C269—41no—
150C22×C269#12012yes—
151C22×C269C22×D411yescharacteristic
152C2×D269—22no—
153C2×D269—24no—
154C2×D269—22no—
155C2×D269—22no—
156C2×D269—42no—
157C2×D269—42no—
158C2×D269—41no—
159C2×D269—41no—
160C2×D269—41no—
161C2×D269—84no—
162C2×D269—88no—
163C4×C269—22no—
164C4×C269—42no—
165C4×C269#12012yes—
166C4×C269C22×D411yescharacteristic
167C4×C269C22×D411yescharacteristic
168Dic269—22no—
169Dic269—41no—
170C2×C269—22no—
171C2×C269—42no—
172C2×C269C2×#12011yescharacteristic
173C2×C269#57611yescharacteristic
174C2×C269#57711yescharacteristic
175D269—42no—
176D269—44no—
177D269—81no—
178C269#301211yescharacteristic, complemented, Sylow
179#3012—2691nomaximal, Sylow
180C2×H(5,𝔽2)—2691no—
181C2×AGL(1,ℤ/8)—2691no—
182C2×#135—2691no—
183#537—2691no—
184#452—2692no—
185#450—2692no—
186#449—2692no—
187#447—2692no—
188#417D26911yescharacteristic, complemented
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
198C22×D4—2692no—
199C22×D4—2691no—
200C22×D4—5381no—
201C22×D4—5382no—
202AGL(1,ℤ/8)—5384no—
203#135—5384no—
204C2×Q16—2691no—
205C2×SD16—2691no—
206C2×SD16—2691no—
207C2×SD16—2692no—
208C2×D8—2691no—
209C2×D8—2692no—
210C2×M16C2×D26911yescharacteristic, complemented
211C22×C8—2691no—
212C22⋊Q8—2692no—
213C22⋊Q8—5381no—
214#121—2692no—
215#121—5381no—
216#121—5382no—
217C2×C4⋊C4C2×D26911yescharacteristic, complemented
218#103C2×D26912yescomplemented
219#102—2692no—
220#102—2692no—
221#102C2×D26912yescomplemented
222C22⋊C8—2692no—
223C22⋊C8—2692no—
224C24—5382no—
225C24—5381no—
226#41—22no—
227#41—5388no—
228#41—5384no—
229#41—5384no—
230#41—5382no—
231#41—5382no—
232#41—5382no—
233#41—5382no—
234#41D107614yescomplemented
235C2×Q8—2691no—
236C2×Q8—2691no—
237C2×Q8C22×D26911yescharacteristic
238C2×D4—21no—
239C2×D4—2692no—
240C2×D4—2694no—
241C2×D4—2691no—
242C2×D4—2692no—
243C2×D4—2691no—
244C2×D4—5382no—
245C2×D4—5388no—
246C2×D4—5382no—
247C2×D4—5382no—
248C2×D4—5384no—
249C2×D4—53816no—
250C2×D4—5384no—
251C2×D4—5381no—
252C2×D4—5381no—
253C2×D4—5382no—
254C2×D4C22×D26911yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—2692no—
258C22×C4—5382no—
259C22×C4—5381no—
260C22×C4C22×D26911yescharacteristic
261Q16—5382no—
262SD16—5384no—
263SD16—5382no—
264SD16—5382no—
265D8—5384no—
266D8—5382no—
267M16—22no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8—5382no—
271C2×C8C22×D26912yes—
272C4⋊C4—21no—
273C4⋊C4—5382no—
274C4⋊C4C22×D26912yes—
275C22⋊C4—5382no—
276C22⋊C4—5382no—
277C23—21no—
278C23—2692no—
279C23—5388no—
280C23—5382no—
281C23—5384no—
282C23—5381no—
283C23—21521no—
284C23—21521no—
285C23—21522no—
286C23—21522no—
287C23D4×D26911yescharacteristic
288Q8—21no—
289Q8—5382no—
290Q8—5381no—
291Q8—5381no—
292Q8C2×D107612yes—
293D4—24no—
294D4—21no—
295D4—2694no—
296D4—5388no—
297D4—5388no—
298D4—5384no—
299D4—5384no—
300D4—5382no—
301D4—5381no—
302D4—5382no—
303D4—5381no—
304D4—10762no—
305D4—21522no—
306D4C2×D107612yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—2692no—
312C2×C4—5388no—
313C2×C4—5382no—
314C2×C4—5382no—
315C2×C4—5381no—
316C2×C4—10762no—
317C2×C4D4×D26911yescharacteristic
318C2×C4D4×D26912yes—
319C2×C4C2×D107612yes—
320C2×C4C23×D26911yescharacteristic
321C8—22no—
322C8—5381no—
323C8—5381no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—5382no—
330C22—5382no—
331C22—5384no—
332C22—5382no—
333C22—10762no—
334C22—10762no—
335C22—10761no—
336C22—10761no—
337C22—10761no—
338C22—21524no—
339C22—21528no—
340C22#2257542312yes—
341C22C2×D4×D26911yescharacteristic
342C4—22no—
343C4—42no—
344C4—5382no—
345C4—10761no—
346C4#2257542312yes—
347C4C2×D4×D26911yescharacteristic
348C4C22×D107611yescharacteristic
349C2—22no—
350C2—42no—
351C2—10762no—
352C2—10764no—
353C2—21521no—
354C2C2×#2257542311yescharacteristic
355C2#3177140911yescharacteristic
356C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#417×C269C2
socleC22×C269C22×D4
hypercenter#417D269
hypocenterC269#3012
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22575499C2semidirect
2. C2×#22575481C2semidirect
3. C2×#22575473C2semidirect
4. #31771186C2semidirect
5. #31770770C2semidirect
6. #31770757C2semidirect
7. #31770753C2semidirect
8. #31770740C2semidirect
9. C2×#22575373C2semidirect
10. #31770573C2semidirect
11. #417×C269C2semidirect
17. C2×#16864402C22semidirect
24. C2×#16864392C22semidirect
25. C2×#16864391C22semidirect
26. C2×#16864390C22semidirect
27. C2×#16864389C22semidirect
28. C2×D2152C22semidirect
29. C2×#16864387C22semidirect
30. C22×#14524130C22semidirect
32. #22575433C22semidirect
34. #22575428C22semidirect
36. #22575373C22semidirect
37. #22575373C22semidirect
38. #22575365C22semidirect
39. #22575365C22semidirect
41. C2×M16×C269C22semidirect
42. C2×C4⋊C4×C269C22semidirect
43. #103×C269C22semidirect
44. #102×C269C22semidirect
62. C2×D1076D4semidirect
87. #41×C269D4semidirect
178. C269#3012semidirect
188. #417D269semidirect
210. C2×M16C2×D269semidirect
217. C2×C4⋊C4C2×D269semidirect
218. #103C2×D269semidirect
221. #102C2×D269semidirect
234. #41D1076semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754812
C2×#225754732
#317711862
#317707702
#317707572
#317707532
#317707402
C2×#225753732
#317705732
#417×C2692
#3012269