Finite Group Databasebeta

← #46822399

Subgroups — #46822399

Order 34432.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575482C211yesmaximal, characteristic, complemented
2C22×#16864390C211yesmaximal, characteristic, complemented
3#31771268C211yesmaximal, characteristic, complemented
4#31771249C211yesmaximal, characteristic
5#31770996C211yesmaximal, characteristic, complemented
6#31770850C211yesmaximal, characteristic, complemented
7#31770844C211yesmaximal, characteristic, complemented
8#31770826C211yesmaximal, characteristic, complemented
9#31770808C211yesmaximal, characteristic, complemented
10#31770768C211yesmaximal, characteristic, complemented
11#31770763C211yesmaximal, characteristic, complemented
12#31770742C211yesmaximal, characteristic, complemented
13#31770702C211yesmaximal, characteristic, complemented
14#31770673C211yesmaximal, characteristic, complemented
15#450×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C22×Dic538C2211yescharacteristic
18C2×Q8×D269C2211yescharacteristic, complemented
19#22575482—24no—
20C2×Dic1076C2211yescharacteristic, complemented
21C2×Dic1076C2211yescharacteristic, complemented
22C2×#16864392C2211yescharacteristic, complemented
23C2×#16864390—24no—
24C2×#16864390C2211yescharacteristic
25C2×#16864390C2211yescharacteristic
26C2×#16864390C2211yescharacteristic
27C2×#16864390C2211yescharacteristic
28C2×#16864385C2211yescharacteristic, complemented
29C22×#14524130C2211yescharacteristic
30#22575452—21no—
31#22575451C2211yescharacteristic, complemented
32#22575440C2211yescharacteristic
33#22575434—21no—
34#22575433C2211yescharacteristic, complemented
35#22575432C2211yescharacteristic
36#22575431—21no—
37#22575430—21no—
38#22575425—21no—
39#22575424C2211yescharacteristic
40#22575422—21no—
41#22575418C2211yescharacteristic, complemented
42D4×Dic269C2211yescharacteristic
43C4×C22⋊D269C2211yescharacteristic
44#22575413—21no—
45C2×C22⋊Dic269—21no—
46#22575386C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575378C2211yescharacteristic, complemented
49#22575378C2211yescharacteristic, complemented
50#22575377C2211yescharacteristic, complemented
51#22575376C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic, complemented
53#22575363C2211yescharacteristic, complemented
54C22×D4×C269C2211yescharacteristic
55C2×SD16×C269C2211yescharacteristic
56C2×SD16×C269C2211yescharacteristic
57C22⋊Q8×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59#102×C269C2211yescharacteristic
60C22⋊C8×C269C2211yescharacteristic
61#16864401—22no—
62#16864401—41no—
63C2×Dic538—21no—
64C2×Dic538—22no—
65C2×Dic538—21no—
66C2×Dic538—41no—
67C2×Dic538C2311yescharacteristic
68C2×Dic538C2311yescharacteristic
69C2×Dic538C2311yescharacteristic
70Q8×D269—22no—
71Q8×D269—41no—
72C2×C22⋊D269—21no—
73C2×C22⋊D269—21no—
74C2×C22⋊D269D411yescharacteristic
75C22×Dic269—21no—
76C22×Dic269—21no—
77C22×Dic269—21no—
78C2×C4×D269—21no—
79C2×C4×D269C2311yescharacteristic
80Dic1076—22no—
81Dic1076—41no—
82#16864392—22no—
83#16864390—22no—
84#16864390—22no—
85#16864390—22no—
86#16864390D414yescomplemented
87#16864385—22no—
88C2×#14524130—22no—
89C2×#14524130C2311yescharacteristic
90C2×#14524130C2311yescharacteristic
91C4⋊Dic269—21no—
92C4⋊Dic269C2311yescharacteristic
93#16864381—21no—
94#16864381—21no—
95#16864381—21no—
96#16864381—21no—
97#16864381D411yescharacteristic
98C22⋊Dic269—21no—
99C22⋊Dic269—21no—
100C22⋊Dic269—21no—
101C22⋊Dic269—21no—
102C22⋊Dic269—21no—
103C22⋊Dic269D411yescharacteristic
104#16864379—21no—
105#16864379D411yescharacteristic
106C4×Dic269—21no—
107C4×Dic269C2311yescharacteristic
108C24×C269—21no—
109C2×Q8×C269C2311yescharacteristic
110C2×D4×C269—22no—
111C2×D4×C269—21no—
112C2×D4×C269—41no—
113C2×D4×C269C2311yescharacteristic
114C2×D4×C269C2311yescharacteristic
115C22×C4×C269C2311yescharacteristic
116SD16×C269—22no—
117SD16×C269—41no—
118C2×C8×C269C2311yescharacteristic
119C2×C8×C269C2311yescharacteristic
120C4⋊C4×C269—21no—
121C4⋊C4×C269C2311yescharacteristic
122C22⋊C4×C269—21no—
123C22×D269C2×D411yescharacteristic
124Dic538—21no—
125Dic538—21no—
126Dic538—21no—
127Dic538—21no—
128Dic538—21no—
129Dic538—42no—
130Dic538—42no—
131Dic538C2×D412yes—
132C22⋊D269—22no—
133C22⋊D269—42no—
134C22⋊D269—81no—
135C2×Dic269—21no—
136C2×Dic269—21no—
137C2×Dic269—21no—
138C2×Dic269—21no—
139C2×Dic269—42no—
140C2×Dic269—42no—
141C2×Dic269—81no—
142C2×Dic269C2×D411yescharacteristic
143C2×Dic269C2×D411yescharacteristic
144C2×Dic269C2×D411yescharacteristic
145C4×D269—21no—
146C4×D269—21no—
147C4×D269—42no—
148#14524130—21no—
149#14524130C2×D412yes—
150C23×C269—21no—
151C23×C269—21no—
152C23×C269—42no—
153C23×C269—42no—
154C23×C269—81no—
155C23×C269C2×D411yescharacteristic
156Q8×C269—21no—
157Q8×C269—21no—
158D4×C269—21no—
159D4×C269—21no—
160D4×C269—21no—
161D4×C269—42no—
162D4×C269C2×D412yes—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269C2×D411yescharacteristic
168C2×C4×C269C2411yescharacteristic
169C8×C269—21no—
170C8×C269—21no—
171C22×C269—21no—
172C22×C269—21no—
173C22×C269—22no—
174C22×C269—41no—
175C22×C269—41no—
176C22×C269—41no—
177C22×C269—41no—
178C22×C269—41no—
179C22×C269—84no—
180C22×C269#13712yescomplemented
181C22×C269C22×D411yescharacteristic
182C2×D269—21no—
183C2×D269—21no—
184C2×D269—21no—
185C4×C269—21no—
186C4×C269—41no—
187C4×C269—41no—
188C4×C269C22×D411yescharacteristic
189C4×C269H(5,𝔽2)11yescharacteristic
190Dic269—21no—
191Dic269—22no—
192Dic269—21no—
193Dic269—21no—
194Dic269—41no—
195Dic269—41no—
196Dic269—41no—
197C2×C269—22no—
198C2×C269—42no—
199C2×C269—81no—
200C2×C269D4211yescharacteristic
201C2×C269C2×#13711yescharacteristic
202C2×C269#57711yescharacteristic
203D269—41no—
204C269#326611yescharacteristic, complemented, Sylow
205#3266—2691nomaximal, Sylow
206C2×#137—2691no—
207C22×SD16—2691no—
208D4×Q8—2691no—
209#546—2691no—
210#497—2691no—
211#468—2691no—
212#467—2691no—
213#464—2691no—
214#461—2691no—
215#452—2691no—
216#451—2691no—
217#450D26911yescharacteristic, complemented
218#448—2691no—
219#440—2691no—
220#435—2691no—
221C2×#41—2691no—
222C22×Q8—2691no—
223C22×Q8—2691no—
224C22×D4C2×D26911yescharacteristic
225#137—5384no—
226C2×Q16—2691no—
227C2×Q16—2691no—
228C2×Q16—2691no—
229C2×SD16—2691no—
230C2×SD16—2691no—
231C2×SD16—2691no—
232C2×SD16—2691no—
233C2×SD16—5384no—
234C2×SD16C2×D26911yescharacteristic
235C2×SD16C2×D26911yescharacteristic
236C2×M16—2691no—
237C22×C8—2691no—
238C4⋊Q8—2691no—
239C4⋊Q8—5381no—
240#124—2691no—
241#123—5381no—
242C22⋊Q8—2691no—
243C22⋊Q8—2691no—
244C22⋊Q8—5381no—
245C22⋊Q8—5381no—
246C22⋊Q8C2×D26911yescharacteristic
247#121—2691no—
248#121—5381no—
249#120—5381no—
250C4×Q8—2691no—
251C4×D4—2691no—
252C4×D4—2691no—
253C4×D4—5381no—
254C2×C22⋊C4—5381no—
255#107—2691no—
256C4⋊C8—2691no—
257#103—2691no—
258#103—2691no—
259#103—2691no—
260#103—2691no—
261#102C2×D26911yescharacteristic
262#102C2×D26911yescharacteristic
263C22⋊C8—2691no—
264C22⋊C8C2×D26911yescharacteristic
265#97—2691no—
266C24—21no—
267#41—5382no—
268#41—10761no—
269C2×Q8—2691no—
270C2×Q8—2691no—
271C2×Q8—2691no—
272C2×Q8—5381no—
273C2×Q8—5382no—
274C2×Q8—5381no—
275C2×Q8—5382no—
276C2×Q8—10761no—
277C2×Q8—10761no—
278C2×Q8C22×D26911yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—2691no—
283C2×D4—5381no—
284C2×D4—5381no—
285C2×D4C22×D26911yescharacteristic
286C2×D4C22×D26911yescharacteristic
287C22×C4—2691no—
288C22×C4—5381no—
289C22×C4—5381no—
290C22×C4—5381no—
291C22×C4—5381no—
292C22×C4C22×D26911yescharacteristic
293Q16—5382no—
294Q16—5382no—
295Q16—10761no—
296SD16—22no—
297SD16—41no—
298SD16—2694no—
299SD16—5382no—
300SD16—5382no—
301SD16—5382no—
302M16—5382no—
303C2×C8—2691no—
304C2×C8—2691no—
305C2×C8—5382no—
306C2×C8C22×D26911yescharacteristic
307C2×C8C22×D26911yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2691no—
310C4⋊C4—2691no—
311C4⋊C4—5381no—
312C4⋊C4—5381no—
313C4⋊C4—5381no—
314C4⋊C4—5381no—
315C4⋊C4—5381no—
316C4⋊C4C22×D26911yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2691no—
319C22⋊C4—2691no—
320C22⋊C4—5381no—
321C22⋊C4—5381no—
322C22⋊C4—5381no—
323C22⋊C4—5381no—
324C22⋊C4—5381no—
325C22⋊C4—5381no—
326C42—2691no—
327C42—5381no—
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—2691no—
334C23D4×D26911yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—2692no—
338Q8—5381no—
339Q8—5381no—
340Q8—5381no—
341Q8—5381no—
342Q8—5381no—
343Q8—10762no—
344Q8—10762no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—5382no—
350D4—10762no—
351D4—21521no—
352D4D4×D26912yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—2691no—
358C2×C4—2691no—
359C2×C4—2691no—
360C2×C4—5381no—
361C2×C4—5381no—
362C2×C4—5381no—
363C2×C4—5381no—
364C2×C4—5381no—
365C2×C4—5381no—
366C2×C4—10762no—
367C2×C4—10762no—
368C2×C4—10762no—
369C2×C4—21521no—
370C2×C4D4×D26911yescharacteristic
371C2×C4C23×D26911yescharacteristic
372C8—21no—
373C8—21no—
374C8—2692no—
375C8—5381no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—5381no—
386C22—5381no—
387C22—5381no—
388C22#2257548212yescomplemented
389C22C2×D4×D26911yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—5381no—
394C4—5381no—
395C4—5381no—
396C4—5382no—
397C4—10761no—
398C4—10761no—
399C4—10761no—
400C4C2×D4×D26911yescharacteristic
401C4#2257549811yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—10761no—
406C2#3177123711yescharacteristic
407C2C2×#2257548211yescharacteristic
408C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#450×C269C2
socleC22×C269C22×D4
prime radical#31771249C2
hypercenter#450D269
hypocenterC269#3266
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22575482C2semidirect
2. C22×#16864390C2semidirect
3. #31771268C2semidirect
5. #31770996C2semidirect
6. #31770850C2semidirect
7. #31770844C2semidirect
8. #31770826C2semidirect
9. #31770808C2semidirect
10. #31770768C2semidirect
11. #31770763C2semidirect
12. #31770742C2semidirect
13. #31770702C2semidirect
14. #31770673C2semidirect
15. #450×C269C2semidirect
18. C2×Q8×D269C22semidirect
20. C2×Dic1076C22semidirect
21. C2×Dic1076C22semidirect
22. C2×#16864392C22semidirect
28. C2×#16864385C22semidirect
31. #22575451C22semidirect
34. #22575433C22semidirect
41. #22575418C22semidirect
46. #22575386C22semidirect
47. #22575382C22semidirect
48. #22575378C22semidirect
49. #22575378C22semidirect
50. #22575377C22semidirect
51. #22575376C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
86. #16864390D4semidirect
180. C22×C269#137semidirect
204. C269#3266semidirect
217. #450D269semidirect
388. C22#22575482semidirect

Maximal subgroups

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

subgroupindex
C2×#225754822
C22×#168643902
#317712682
#317712492
#317709962
#317708502
#317708442
#317708262
#317708082
#317707682
#317707632
#317707422
#317707022
#317706732
#450×C2692
#3266269