Finite Group Databasebeta

← #46820244

Subgroups — #46820244

Order 34432.

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

normal subgroups only

This group has 762 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
1C2×#22575499C211yesmaximal, characteristic, complemented
2#31771405C211yesmaximal, characteristic, complemented
3#31771403C211yesmaximal, characteristic, complemented
4#31771174C211yesmaximal, characteristic, complemented
5#31770784C214yesmaximal, complemented
6#31770774C214yesmaximal, complemented
7#31770623C211yesmaximal, characteristic, complemented
8C2×#22575369C211yesmaximal, characteristic, complemented
9#421×C269C211yesmaximal, characteristic, complemented
10#22575499—22no—
11#22575499—22no—
12#22575499—22no—
13#22575499C2214yes—
14C2×#16864403—21no—
15C2×#16864402—21no—
16C2×#16864402C2211yescharacteristic
17C2×#16864402C2211yescharacteristic, complemented
18C22×D1076—21no—
19C22×D1076C2211yescharacteristic
20C2×D4×D269—22no—
21C2×D4×D269—21no—
22#22575486—21no—
23#22575486C2212yescomplemented
24#22575484—21no—
25#22575483C2212yescomplemented
26#22575481—21no—
27#22575481C2212yescomplemented
28#22575479—21no—
29#22575478C2212yescomplemented
30#22575475—22no—
31#22575473—22no—
32C2×#16864386C2211yescharacteristic, complemented
33#22575450C2212yescomplemented
34#22575441C2212yescomplemented
35#22575428—22no—
36#22575423—22no—
37#22575381C2214yescomplemented
38#22575369C2214yescomplemented
39C2×#41×C269C2211yescharacteristic
40C2×M16×C269C2211yescharacteristic, complemented
41#117×C269C2211yescharacteristic, complemented
42#104×C269C2214yescomplemented
43C23×D269—22no—
44C23×D269D412yes—
45#16864403—24no—
46#16864403—24no—
47#16864402—24no—
48#16864402—24no—
49#16864402—21no—
50#16864402—21no—
51#16864402—21no—
52#16864402—21no—
53#16864402—21no—
54#16864402—21no—
55#16864402D412yescomplemented
56#16864402C2312yes—
57C2×Dic538D411yescharacteristic, complemented
58C2×Dic538C2311yescharacteristic, complemented
59C2×D1076—22no—
60C2×D1076—24no—
61C2×D1076—24no—
62C2×D1076—24no—
63C2×D1076—21no—
64C2×D1076—21no—
65C2×D1076—21no—
66C2×D1076—21no—
67C2×D1076D412yes—
68C2×D1076D411yescharacteristic, complemented
69C2×D1076C2312yes—
70C2×D1076C2311yescharacteristic, complemented
71C2×C22⋊D269—22no—
72C2×C22⋊D269—21no—
73C2×C22⋊D269—21no—
74D4×D269—28no—
75D4×D269—24no—
76D4×D269—28no—
77D4×D269—24no—
78C2×C4×D269—22no—
79C2×C4×D269—21no—
80C2×C4×D269—21no—
81Dic1076—22no—
82#16864392—22no—
83#16864391—22no—
84#16864391—22no—
85#16864390—22no—
86#16864389—22no—
87D2152—22no—
88#16864387—22no—
89#16864386—21no—
90#16864386C2312yescomplemented
91C2×#14524130—21no—
92#16864379—24no—
93#41×C269—21no—
94#41×C269—21no—
95#41×C269D412yescomplemented
96#41×C269C2312yes—
97C2×Q8×C269D411yescharacteristic, complemented
98C2×D4×C269—21no—
99C2×D4×C269D411yescharacteristic, complemented
100C22×C4×C269—21no—
101C22×C4×C269C2311yescharacteristic
102M16×C269—21no—
103M16×C269C2312yescomplemented
104C2×C8×C269—21no—
105C4⋊C4×C269—21no—
106C22⋊C4×C269—21no—
107C42×C269C2312yescomplemented
108C22×D269—22no—
109C22×D269—21no—
110C22×D269—28no—
111C22×D269—22no—
112C22×D269—21no—
113C22×D269—82no—
114C22×D269—84no—
115C22×D269C2×D414yes—
116Dic538—21no—
117Dic538—21no—
118Dic538—21no—
119Dic538C2×D412yes—
120D1076—28no—
121D1076—22no—
122D1076—22no—
123D1076—21no—
124D1076—22no—
125D1076—22no—
126D1076—21no—
127D1076—21no—
128D1076—84no—
129D1076C2×D412yes—
130C22⋊D269—28no—
131C22⋊D269—22no—
132C22⋊D269—22no—
133C22⋊D269—41no—
134C22⋊D269—41no—
135C2×Dic269—21no—
136C2×Dic269—21no—
137C4×D269—28no—
138C4×D269—22no—
139C4×D269—22no—
140C4×D269—41no—
141C4×D269—41no—
142#14524130—22no—
143C23×C269—21no—
144C23×C269C2×D411yescharacteristic
145Q8×C269—21no—
146Q8×C269C2×D412yes—
147D4×C269—22no—
148D4×C269—22no—
149D4×C269—21no—
150D4×C269C2×D412yes—
151C2×C4×C269—21no—
152C2×C4×C269—22no—
153C2×C4×C269—22no—
154C2×C4×C269—22no—
155C2×C4×C269C2×D411yescharacteristic
156C2×C4×C269C2×D411yescharacteristic
157C2×C4×C269C2×D411yescharacteristic
158C2×C4×C269C2×D411yescharacteristic
159C2×C4×C269C2×D411yescharacteristic
160C2×C4×C269C2411yescharacteristic
161C8×C269—22no—
162C22×C269—22no—
163C22×C269—41no—
164C22×C269—42no—
165C22×C269#12011yescharacteristic
166C22×C269#12011yescharacteristic
167C22×C269C22×D411yescharacteristic
168C2×D269—24no—
169C2×D269—22no—
170C2×D269—44no—
171C2×D269—42no—
172C2×D269—44no—
173C2×D269—42no—
174C2×D269—41no—
175C2×D269—82no—
176C2×D269—88no—
177C4×C269—22no—
178C4×C269—42no—
179C4×C269#12011yescharacteristic
180C4×C269#12011yescharacteristic
181C4×C269C22×D411yescharacteristic
182C4×C269C22×D411yescharacteristic
183Dic269—22no—
184Dic269—41no—
185C2×C269—21no—
186C2×C269—21no—
187C2×C269—21no—
188C2×C269—42no—
189C2×C269C2×#12011yescharacteristic
190D269—44no—
191D269—42no—
192D269—81no—
193C269#301911yescharacteristic, complemented, Sylow
194#3019—2691nomaximal, Sylow
195C2×H(5,𝔽2)—2691no—
196#575—2691no—
197#575—2691no—
198#535—2691no—
199#455—2694no—
200#453—2694no—
201#421—2691no—
202#421D26911yescharacteristic, complemented
203C2×#100—2691no—
204H(5,𝔽2)—2694no—
205H(5,𝔽2)—5382no—
206H(5,𝔽2)—5382no—
207H(5,𝔽2)—5382no—
208C2×#41—2691no—
209C2×#41—2691no—
210C2×#41—5381no—
211C2×#41—5381no—
212C2×#41C2×D26911yescharacteristic
213C22×D4—2691no—
214C22×D4—5381no—
215C22×D4—5382no—
216C22×D4—5381no—
217#137—2692no—
218#137—2692no—
219#137—5381no—
220#137—5381no—
221AGL(1,ℤ/8)—2692no—
222AGL(1,ℤ/8)—2692no—
223AGL(1,ℤ/8)—5381no—
224AGL(1,ℤ/8)—5381no—
225#135—5382no—
226#135—5382no—
227C2×M16—2691no—
228C2×M16C2×D26911yescharacteristic, complemented
229#127—2692no—
230#124—2692no—
231#121—5382no—
232#120—5382no—
233#117C2×D26911yescharacteristic, complemented
234#104—2694no—
235#104C2×D26914yescomplemented
236#100—2694no—
237C24—2692no—
238C24—5382no—
239#41—21no—
240#41—21no—
241#41—2692no—
242#41—2692no—
243#41—5384no—
244#41—5384no—
245#41—5384no—
246#41—5384no—
247#41—5381no—
248#41—5381no—
249#41—5381no—
250#41—5381no—
251#41—5381no—
252#41—5381no—
253#41D107612yescomplemented
254#41C22×D26912yes—
255C2×Q8—2691no—
256C2×Q8—2691no—
257C2×Q8D107611yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—2691no—
260C2×D4—2692no—
261C2×D4—2692no—
262C2×D4—2691no—
263C2×D4—5384no—
264C2×D4—5382no—
265C2×D4—5384no—
266C2×D4—5384no—
267C2×D4—5388no—
268C2×D4—5388no—
269C2×D4—5384no—
270C2×D4—5382no—
271C2×D4—5381no—
272C2×D4—5384no—
273C2×D4—5381no—
274C2×D4—5381no—
275C2×D4—5381no—
276C2×D4—5381no—
277C2×D4—5381no—
278C2×D4D107611yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—5382no—
281C22×C4—5381no—
282C22×C4—5381no—
283C22×C4C22×D26911yescharacteristic
284Q16—5382no—
285Q16—5382no—
286SD16—5382no—
287SD16—5382no—
288SD16—5382no—
289SD16—5382no—
290D8—5382no—
291D8—5382no—
292M16—21no—
293M16—2692no—
294M16—5381no—
295M16C22×D26912yescomplemented
296C2×C8—21no—
297C2×C8—5381no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—5384no—
301C42C22×D26912yescomplemented
302C23—21no—
303C23—2694no—
304C23—5388no—
305C23—5381no—
306C23—5382no—
307C23—5382no—
308C23—5381no—
309C23—21522no—
310C23—21524no—
311C23D4×D26911yescharacteristic
312Q8—21no—
313Q8—2692no—
314Q8—5381no—
315Q8—5381no—
316Q8—5381no—
317Q8C2×D107612yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—2692no—
322D4—5388no—
323D4—5388no—
324D4—5382no—
325D4—5382no—
326D4—5381no—
327D4—5382no—
328D4—5382no—
329D4—5382no—
330D4—5381no—
331D4—5382no—
332D4—5381no—
333D4—10761no—
334D4—10761no—
335D4—21524no—
336D4C2×D107612yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—5388no—
342C2×C4—5381no—
343C2×C4—5382no—
344C2×C4—5382no—
345C2×C4—5381no—
346C2×C4—10761no—
347C2×C4—10761no—
348C2×C4D4×D26911yescharacteristic
349C2×C4D4×D26911yescharacteristic
350C2×C4D4×D26911yescharacteristic
351C2×C4C2×D107611yescharacteristic
352C2×C4C2×D107611yescharacteristic
353C2×C4C23×D26911yescharacteristic
354C8—22no—
355C8—5382no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—5382no—
360C22—5384no—
361C22—10764no—
362C22—10764no—
363C22—10762no—
364C22—10762no—
365C22—10761no—
366C22—21522no—
367C22—21528no—
368C22#2257542311yescharacteristic
369C22#2257542311yescharacteristic
370C22C2×D4×D26911yescharacteristic
371C4—22no—
372C4—42no—
373C4—5382no—
374C4—10761no—
375C4#2257542311yescharacteristic
376C4#2257542311yescharacteristic
377C4C2×D4×D26911yescharacteristic
378C4C22×D107611yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—10764no—
384C2—10762no—
385C2—21521no—
386C2C2×#2257542311yescharacteristic
subgroupG/H
centerC2C2×#22575423
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#421×C269C2
socleC2×C269C2×#120
hypercenter#421D269
hypocenterC269#3019
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
1. C2×#22575499C2semidirect
2. #31771405C2semidirect
3. #31771403C2semidirect
4. #31771174C2semidirect
5. #31770784C2semidirect
6. #31770774C2semidirect
7. #31770623C2semidirect
8. C2×#22575369C2semidirect
9. #421×C269C2semidirect
17. C2×#16864402C22semidirect
23. #22575486C22semidirect
25. #22575483C22semidirect
27. #22575481C22semidirect
29. #22575478C22semidirect
32. C2×#16864386C22semidirect
33. #22575450C22semidirect
34. #22575441C22semidirect
37. #22575381C22semidirect
38. #22575369C22semidirect
40. C2×M16×C269C22semidirect
41. #117×C269C22semidirect
42. #104×C269C22semidirect
55. #16864402D4semidirect
57. C2×Dic538D4semidirect
58. C2×Dic538C23semidirect
68. C2×D1076D4semidirect
70. C2×D1076C23semidirect
90. #16864386C23semidirect
95. #41×C269D4semidirect
97. C2×Q8×C269D4semidirect
99. C2×D4×C269D4semidirect
103. M16×C269C23semidirect
107. C42×C269C23semidirect
193. C269#3019semidirect
202. #421D269semidirect
228. C2×M16C2×D269semidirect
233. #117C2×D269semidirect
235. #104C2×D269semidirect
253. #41D1076semidirect
257. C2×Q8D1076semidirect
278. C2×D4D1076semidirect
295. M16C22×D269semidirect
301. C42C22×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754992
#317714052
#317714032
#317711742
#317707842
#317707742
#317706232
C2×#225753692
#421×C2692
#3019269