Finite Group Databasebeta

← #46820254

Subgroups — #46820254

Order 34432.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#31771441C211yesmaximal, characteristic, complemented
2C2×#22575479C211yesmaximal, characteristic, complemented
3#31771172C211yesmaximal, characteristic, complemented
4#31770776C214yesmaximal, complemented
5#31770771C214yesmaximal, complemented
6#31770622C211yesmaximal, characteristic, complemented
7#31770621C211yesmaximal, characteristic, complemented
8#31770587C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C269C211yesmaximal, characteristic, complemented
10#22575501—21no—
11#22575500—21no—
12#22575500C2212yescomplemented
13#22575499—21no—
14#22575499C2212yescomplemented
15#22575498—22no—
16#22575498—21no—
17C2×#16864402—22no—
18C2×#16864402C2211yescharacteristic
19C2×#16864402C2211yescharacteristic
20#41×D269—22no—
21#41×D269—22no—
22#22575479—22no—
23#22575479C2214yescomplemented
24C2×#16864390—21no—
25C2×#16864387—21no—
26C2×#16864386C2211yescharacteristic, complemented
27#22575449C2212yescomplemented
28#22575440C2212yescomplemented
29#22575424—22no—
30#22575422—22no—
31#22575412C2211yescharacteristic, complemented
32#22575380C2214yescomplemented
33#22575379C2214yescomplemented
34#22575371C2212yescomplemented
35#22575369C2212yescomplemented
36C2×#41×C269C2211yescharacteristic
37C22×D4×C269C2211yescharacteristic, complemented
38AGL(1,ℤ/8)×C269—22no—
39AGL(1,ℤ/8)×C269C2214yescomplemented
40C2×SD16×C269—21no—
41C2×D8×C269—21no—
42C2×M16×C269C2211yescharacteristic, complemented
43#16864403—22no—
44#16864403—22no—
45#16864402—24no—
46#16864402—24no—
47#16864402—22no—
48#16864402—22no—
49#16864402—21no—
50#16864402—21no—
51#16864402—21no—
52#16864402—21no—
53#16864402—21no—
54#16864402—21no—
55#16864402D412yescomplemented
56#16864402C2312yes—
57#16864401—24no—
58#16864401—22no—
59#16864401—24no—
60#16864401—22no—
61C2×Dic538—22no—
62C2×Dic538D411yescharacteristic, complemented
63C2×Dic538C2311yescharacteristic
64Q8×D269—22no—
65Q8×D269—22no—
66C2×D1076—22no—
67C2×D1076D411yescharacteristic, complemented
68C2×D1076C2311yescharacteristic
69C2×C22⋊D269—24no—
70C2×C22⋊D269—24no—
71C2×C22⋊D269—21no—
72C2×C22⋊D269D412yes—
73D4×D269—24no—
74D4×D269—22no—
75D4×D269—24no—
76D4×D269—22no—
77C2×C4×D269—24no—
78C2×C4×D269—21no—
79C2×C4×D269D412yes—
80#16864390—24no—
81#16864387—24no—
82#16864386—21no—
83#16864386C2312yes—
84C2×#14524130—21no—
85C4⋊Dic269—21no—
86C22⋊Dic269—24no—
87C22⋊Dic269—21no—
88C4×Dic269C2312yes—
89C24×C269—21no—
90#41×C269—21no—
91#41×C269—21no—
92#41×C269D412yescomplemented
93#41×C269C2312yes—
94C2×Q8×C269D411yescharacteristic, complemented
95C2×D4×C269—21no—
96C2×D4×C269—21no—
97C2×D4×C269—21no—
98C2×D4×C269—21no—
99C2×D4×C269—21no—
100C2×D4×C269—21no—
101C2×D4×C269D411yescharacteristic, complemented
102C2×D4×C269C2312yes—
103C22×C4×C269—21no—
104C22×C4×C269C2311yescharacteristic
105SD16×C269—24no—
106D8×C269—24no—
107M16×C269—21no—
108M16×C269C2312yes—
109C2×C8×C269—21no—
110C22×D269—24no—
111C22×D269—21no—
112C22×D269C2×D412yes—
113Dic538—24no—
114Dic538—21no—
115Dic538—21no—
116Dic538—21no—
117Dic538C2×D412yes—
118D1076—24no—
119D1076—21no—
120D1076—21no—
121D1076—21no—
122D1076C2×D412yes—
123C22⋊D269—24no—
124C22⋊D269—24no—
125C22⋊D269—22no—
126C22⋊D269—22no—
127C22⋊D269—22no—
128C22⋊D269—22no—
129C22⋊D269—84no—
130C2×Dic269—24no—
131C2×Dic269—21no—
132C2×Dic269—22no—
133C2×Dic269C2×D412yes—
134C4×D269—24no—
135C4×D269—24no—
136C4×D269—22no—
137C4×D269—22no—
138C4×D269—22no—
139C4×D269—22no—
140#14524130—22no—
141C23×C269—21no—
142C23×C269—22no—
143C23×C269—41no—
144C23×C269—41no—
145C23×C269—82no—
146C23×C269C2×D411yescharacteristic
147Q8×C269—21no—
148Q8×C269C2×D412yes—
149D4×C269—22no—
150D4×C269—22no—
151D4×C269—21no—
152D4×C269—22no—
153D4×C269—22no—
154D4×C269—41no—
155D4×C269—41no—
156D4×C269C2×D412yes—
157C2×C4×C269—21no—
158C2×C4×C269—22no—
159C2×C4×C269—22no—
160C2×C4×C269C2×D411yescharacteristic
161C2×C4×C269C2×D411yescharacteristic
162C2×C4×C269C2×D411yescharacteristic
163C2×C4×C269C2×D411yescharacteristic
164C2×C4×C269C2×D411yescharacteristic
165C2×C4×C269C2411yescharacteristic
166C8×C269—22no—
167C22×C269—22no—
168C22×C269—41no—
169C22×C269—44no—
170C22×C269—42no—
171C22×C269—81no—
172C22×C269—82no—
173C22×C269—82no—
174C22×C269#12011yescharacteristic
175C22×C269#12011yescharacteristic
176C22×C269C22×D411yescharacteristic
177C2×D269—22no—
178C2×D269—22no—
179C2×D269—42no—
180C2×D269—81no—
181C2×D269—84no—
182C4×C269—22no—
183C4×C269#12011yescharacteristic
184C4×C269#12011yescharacteristic
185C4×C269C22×D411yescharacteristic
186C4×C269C22×D411yescharacteristic
187Dic269—22no—
188Dic269—22no—
189Dic269—42no—
190C2×C269—21no—
191C2×C269—21no—
192C2×C269—21no—
193C2×C269—42no—
194C2×C269—82no—
195C2×C269C2×#12011yescharacteristic
196D269—42no—
197D269—42no—
198C269#302111yescharacteristic, complemented, Sylow
199#3021—2691nomaximal, Sylow
200#585—2691no—
201C2×AGL(1,ℤ/8)—2691no—
202C2×AGL(1,ℤ/8)D26911yescharacteristic, complemented
203#535—2691no—
204#454—2694no—
205#453—2694no—
206#421—2691no—
207#421—2691no—
208#413—2691no—
2092−1+4—2692no—
2102−1+4—5381no—
2112−1+4—5381no—
212H(5,𝔽2)—2692no—
213H(5,𝔽2)—5382no—
214H(5,𝔽2)—5381no—
215H(5,𝔽2)—5381no—
216C2×#41—2691no—
217C2×#41—2691no—
218C2×#41—5382no—
219C2×#41—5382no—
220C2×#41—5382no—
221C2×#41C2×D26911yescharacteristic
222C22×D4C2×D26911yescharacteristic, complemented
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—2694no—
225AGL(1,ℤ/8)—5382no—
226AGL(1,ℤ/8)C2×D26914yescomplemented
227C2×SD16—21no—
228C2×SD16—5381no—
229C2×D8—21no—
230C2×D8—5381no—
231C2×M16—2691no—
232C2×M16C2×D26911yescharacteristic, complemented
233#127—2692no—
234#124—2692no—
235#121—5382no—
236#120—5382no—
237#117—2691no—
238#104—2694no—
239#104—2694no—
240#101—2692no—
241#100—2692no—
242C24—21no—
243#41—21no—
244#41—21no—
245#41—2692no—
246#41—2692no—
247#41—5382no—
248#41—5382no—
249#41—5384no—
250#41—5384no—
251#41—5384no—
252#41—5384no—
253#41—5382no—
254#41—5382no—
255#41—5382no—
256#41—5382no—
257#41—5381no—
258#41—5381no—
259#41—5381no—
260#41—5381no—
261#41—5381no—
262#41—5381no—
263#41C22⋊D26912yescomplemented
264#41C22×D26912yes—
265C2×Q8—2691no—
266C2×Q8—2691no—
267C2×Q8—5382no—
268C2×Q8—5382no—
269C2×Q8—5382no—
270C2×Q8C22⋊D26911yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—2692no—
278C2×D4—2691no—
279C2×D4—2691no—
280C2×D4—5382no—
281C2×D4—5384no—
282C2×D4—5384no—
283C2×D4—5382no—
284C2×D4—5384no—
285C2×D4—5381no—
286C2×D4—5382no—
287C2×D4—5384no—
288C2×D4C22⋊D26911yescharacteristic, complemented
289C2×D4C22×D26912yes—
290C22×C4—21no—
291C22×C4—2692no—
292C22×C4—5384no—
293C22×C4—5381no—
294C22×C4C22×D26911yescharacteristic
295SD16—24no—
296SD16—5384no—
297D8—24no—
298D8—5384no—
299M16—21no—
300M16—2692no—
301M16—5381no—
302M16C22×D26912yes—
303C2×C8—21no—
304C2×C8—5381no—
305C4⋊C4—5381no—
306C22⋊C4—5384no—
307C22⋊C4—5381no—
308C42—2692no—
309C23—21no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—82no—
314C23—2692no—
315C23—5384no—
316C23—5381no—
317C23D4×D26911yescharacteristic
318Q8—21no—
319Q8—2692no—
320Q8—5384no—
321Q8—5381no—
322Q8—5381no—
323Q8—5381no—
324Q8C2×C22⋊D26912yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—22no—
329D4—22no—
330D4—41no—
331D4—41no—
332D4—2692no—
333D4—5384no—
334D4—5384no—
335D4—5384no—
336D4—5382no—
337D4—5382no—
338D4—5382no—
339D4—5382no—
340D4—5381no—
341D4—5381no—
342D4—5381no—
343D4—21524no—
344D4C2×C22⋊D26912yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—2692no—
349C2×C4—5384no—
350C2×C4—5384no—
351C2×C4—5384no—
352C2×C4—5381no—
353C2×C4—5382no—
354C2×C4—5382no—
355C2×C4—5382no—
356C2×C4—5382no—
357C2×C4—5382no—
358C2×C4D4×D26911yescharacteristic
359C2×C4D4×D26911yescharacteristic
360C2×C4D4×D26911yescharacteristic
361C2×C4C2×C22⋊D26911yescharacteristic
362C2×C4C2×C22⋊D26911yescharacteristic
363C2×C4C23×D26911yescharacteristic
364C8—22no—
365C8—5382no—
366C22—22no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—5382no—
374C22—5382no—
375C22—10762no—
376C22—21521no—
377C22—21524no—
378C22#2257542111yescharacteristic
379C22#2257542111yescharacteristic
380C22C2×D4×D26911yescharacteristic
381C4—22no—
382C4—5382no—
383C4—5382no—
384C4—10762no—
385C4#2257542111yescharacteristic
386C4#2257542111yescharacteristic
387C4C2×D4×D26911yescharacteristic
388C4C22×C22⋊D26911yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—82no—
394C2—10762no—
395C2—10762no—
396C2C2×#2257542111yescharacteristic
subgroupG/H
centerC2C2×#22575421
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×AGL(1,ℤ/8)×C269C2
socleC2×C269C2×#120
hypercenterC2×AGL(1,ℤ/8)D269
hypocenterC269#3021
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. C2×#22575479C2semidirect
3. #31771172C2semidirect
4. #31770776C2semidirect
5. #31770771C2semidirect
6. #31770622C2semidirect
7. #31770621C2semidirect
8. #31770587C2semidirect
9. C2×AGL(1,ℤ/8)×C269C2semidirect
12. #22575500C22semidirect
14. #22575499C22semidirect
23. #22575479C22semidirect
26. C2×#16864386C22semidirect
27. #22575449C22semidirect
28. #22575440C22semidirect
31. #22575412C22semidirect
32. #22575380C22semidirect
33. #22575379C22semidirect
34. #22575371C22semidirect
35. #22575369C22semidirect
37. C22×D4×C269C22semidirect
39. AGL(1,ℤ/8)×C269C22semidirect
42. C2×M16×C269C22semidirect
55. #16864402D4semidirect
62. C2×Dic538D4semidirect
67. C2×D1076D4semidirect
92. #41×C269D4semidirect
94. C2×Q8×C269D4semidirect
101. C2×D4×C269D4semidirect
198. C269#3021semidirect
202. C2×AGL(1,ℤ/8)D269semidirect
222. C22×D4C2×D269semidirect
226. AGL(1,ℤ/8)C2×D269semidirect
232. C2×M16C2×D269semidirect
263. #41C22⋊D269semidirect
270. C2×Q8C22⋊D269semidirect
288. C2×D4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
C2×#225754792
#317711722
#317707762
#317707712
#317706222
#317706212
#317705872
C2×AGL(1,ℤ/8)×C2692
#3021269