Finite Group Databasebeta

← #49130752

Subgroups — #49130752

Order 36224.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583642C211yesmaximal, characteristic, complemented
2C2×#22583637C211yesmaximal, characteristic
3#124×D283C211yesmaximal, characteristic, complemented
4C4×#17985393C211yesmaximal, characteristic, complemented
5#31826689C211yesmaximal, characteristic, complemented
6#31826687C211yesmaximal, characteristic, complemented
7#31826685C211yesmaximal, characteristic, complemented
8#31826683C211yesmaximal, characteristic, complemented
9#31826681C211yesmaximal, characteristic, complemented
10#31826679C211yesmaximal, characteristic, complemented
11#31826677C211yesmaximal, characteristic, complemented
12#31826487C211yesmaximal, characteristic, complemented
13#31826472C211yesmaximal, characteristic, complemented
14#31826468C211yesmaximal, characteristic, complemented
15#464×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×Q8×D283C2211yescharacteristic
18C2×D4×D283C2211yescharacteristic
19#22583642—24no—
20#22583637—24no—
21C2×#17985382C2211yescharacteristic, complemented
22C2×#17985381C2211yescharacteristic, complemented
23C2×#17985379C2211yescharacteristic
24C2×#17985379C2211yescharacteristic, complemented
25C2×D2264C2211yescharacteristic
26C2×#17985377C2211yescharacteristic
27C2×#17985375C2211yescharacteristic
28C2×#17985375C2211yescharacteristic
29#22583599C2211yescharacteristic
30#22583598C2211yescharacteristic
31#22583597C2211yescharacteristic
32#22583596—21no—
33#22583596—21no—
34Q8×Dic283C2211yescharacteristic, complemented
35C4×D1132—21no—
36C4×D1132C2211yescharacteristic, complemented
37#22583572—21no—
38#22583572C2211yescharacteristic, complemented
39#22583568—21no—
40#22583568C2211yescharacteristic
41C22⋊C4×D283—21no—
42C22⋊C4×D283—21no—
43C42×D283—21no—
44C42×D283C2211yescharacteristic
45C4⋊(C283⋊C8)C2211yescharacteristic, complemented
46Dic283⋊C8C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583535C2211yescharacteristic, complemented
49#22583534C2211yescharacteristic, complemented
50#22583533C2211yescharacteristic, complemented
51#22583532C2211yescharacteristic
52#22583531C2211yescharacteristic
53#22583530C2211yescharacteristic
54C2×Q16×C283C2211yescharacteristic, complemented
55C2×SD16×C283C2211yescharacteristic
56#124×C283C2211yescharacteristic
57C4×Q8×C283C2211yescharacteristic, complemented
58C4⋊C8×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#102×C283C2211yescharacteristic
61C23×D283—21no—
62#17985393—22no—
63#17985393D414yes—
64C2×Dic566—21no—
65C2×Dic566C2311yescharacteristic
66Q8×D283—22no—
67Q8×D283—22no—
68C2×D1132—21no—
69C2×D1132C2311yescharacteristic
70C2×D1132C2311yescharacteristic
71C2×C22⋊D283—21no—
72D4×D283—22no—
73D4×D283—22no—
74C2×C4×D283—21no—
75C2×C4×D283—21no—
76C2×C4×D283—21no—
77C2×C4×D283—21no—
78C2×C4×D283D411yescharacteristic
79C2×C4×D283D411yescharacteristic
80C2×C4×D283C2311yescharacteristic
81#17985382—22no—
82#17985381—22no—
83#17985379—22no—
84#17985379—22no—
85D2264—22no—
86#17985377—22no—
87#17985375—22no—
88#17985375—22no—
89C2×C283⋊C8C2311yescharacteristic
90C2×C283⋊C8C2311yescharacteristic
91C4⋊Dic283—21no—
92C4⋊Dic283C2311yescharacteristic
93C22⋊Dic283—21no—
94C22⋊Dic283—21no—
95(C2×D283)⋊C4—21no—
96(C2×D283)⋊C4—21no—
97(C2×D283)⋊C4—21no—
98(C2×D283)⋊C4—21no—
99(C2×D283)⋊C4—21no—
100(C2×D283)⋊C4—21no—
101(C2×D283)⋊C4—21no—
102C4×Dic283—21no—
103C4×Dic283—21no—
104C4×Dic283D411yescharacteristic
105C4×Dic283D411yescharacteristic
106C4×Dic283C2311yescharacteristic
107C2×Q8×C283C2311yescharacteristic
108C2×Q8×C283C2311yescharacteristic
109C2×D4×C283C2311yescharacteristic
110Q16×C283—22no—
111SD16×C283—22no—
112C2×C8×C283C2311yescharacteristic
113C2×C8×C283C2311yescharacteristic
114C4⋊C4×C283—21no—
115C4⋊C4×C283C2311yescharacteristic
116C22⋊C4×C283—21no—
117C22⋊C4×C283—21no—
118C42×C283—21no—
119C42×C283C2311yescharacteristic
120C22×D283—21no—
121C22×D283—21no—
122C22×D283—42no—
123C22×D283—42no—
124C22×D283—42no—
125C22×D283C2×D411yescharacteristic
126Dic566—21no—
127Dic566—21no—
128Dic566—42no—
129D1132—24no—
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132C2×D412yes—
134C22⋊D283—42no—
135C2×Dic283—21no—
136C2×Dic283—21no—
137C2×Dic283C2×D411yescharacteristic
138C2×Dic283C2×D411yescharacteristic
139C2×Dic283C2×D411yescharacteristic
140C4×D283—24no—
141C4×D283—41no—
142C4×D283—41no—
143C4×D283—42no—
144C4×D283—42no—
145C4×D283—42no—
146C4×D283C2×D412yes—
147C4×D283#4112yes—
148C283⋊C8—21no—
149C283⋊C8—21no—
150C23×C283—21no—
151Q8×C283—21no—
152Q8×C283—21no—
153Q8×C283—21no—
154Q8×C283C2×D412yes—
155D4×C283—21no—
156D4×C283—21no—
157C2×C4×C283—21no—
158C2×C4×C283—21no—
159C2×C4×C283—21no—
160C2×C4×C283C2×D411yescharacteristic
161C2×C4×C283C2×D411yescharacteristic
162C2×C4×C283C2411yescharacteristic
163C8×C283—21no—
164C8×C283—21no—
165C22×C283—41no—
166C22×C283—41no—
167C22×C283—41no—
168C22×C283C22×D411yescharacteristic
169C2×D283—21no—
170C2×D283—21no—
171C2×D283—22no—
172C2×D283—41no—
173C2×D283—41no—
174C2×D283—41no—
175C2×D283—41no—
176C2×D283—41no—
177C2×D283—84no—
178C2×D283#12112yes—
179C4×C283—22no—
180C4×C283—21no—
181C4×C283—21no—
182C4×C283—41no—
183C4×C283—41no—
184C4×C283C22×D411yescharacteristic
185C4×C283C2×#4111yescharacteristic
186Dic283—21no—
187Dic283—21no—
188Dic283—41no—
189Dic283—41no—
190Dic283#12112yes—
191C2×C283—81no—
192C2×C283C2×#12111yescharacteristic
193C2×C283#57511yescharacteristic
194C2×C283#57511yescharacteristic
195D283—22no—
196D283—42no—
197D283—81no—
198C283#304011yescharacteristic, complemented, Sylow
199#3040—2831nomaximal, Sylow
200C2×#137—2831no—
201C2×AGL(1,ℤ/8)—2831no—
202C2×#124—2831no—
203C4×#41—2831no—
204#464—2831no—
205#464—2831no—
206#464—2831no—
207#464D28311yescharacteristic, complemented
208#463—2831no—
209#463—2831no—
210#463—2831no—
211#463—2831no—
212#423—2831no—
213#419—2831no—
214#418—2831no—
215C2×#41—2831no—
216C22×Q8—2831no—
217C22×D4—2831no—
218#137—5664no—
219AGL(1,ℤ/8)—5664no—
220C2×Q16—2831no—
221C2×Q16C2×D28311yescharacteristic, complemented
222C2×SD16—2831no—
223C2×SD16—2831no—
224C2×SD16—2831no—
225C2×SD16C2×D28311yescharacteristic
226C2×D8—2831no—
227C2×D8—2831no—
228C2×M16—2831no—
229C2×M16—2831no—
230#124—2831no—
231#124—2831no—
232#124—2831no—
233#124—5661no—
234#124—5661no—
235#124C2×D28311yescharacteristic
236C4×Q8—2831no—
237C4×Q8C2×D28311yescharacteristic, complemented
238C4×D4—2831no—
239C4×D4—2831no—
240C4×D4—5661no—
241C4×D4—5661no—
242#117—2831no—
243#117—5661no—
244C2×C22⋊C4—5661no—
245C2×C22⋊C4—5661no—
246C2×C42—2831no—
247C2×C42—5661no—
248C4⋊C8—2831no—
249C4⋊C8—2831no—
250C4⋊C8—2831no—
251C4⋊C8C2×D28311yescharacteristic, complemented
252#103—2831no—
253#103—2831no—
254#103—2831no—
255#103C2×D28311yescharacteristic, complemented
256#102—2831no—
257#102—2831no—
258#102—2831no—
259#102C2×D28311yescharacteristic
260C24—5661no—
261#41—2834no—
262#41—5662no—
263C2×Q8—2831no—
264C2×Q8—5661no—
265C2×Q8—5662no—
266C2×Q8—5662no—
267C2×Q8C22×D28311yescharacteristic
268C2×Q8C22×D28311yescharacteristic
269C2×D4—2831no—
270C2×D4—2831no—
271C2×D4—5661no—
272C2×D4—5662no—
273C2×D4—5662no—
274C2×D4—5661no—
275C2×D4C22×D28311yescharacteristic
276C22×C4—2831no—
277C22×C4—2831no—
278C22×C4—2831no—
279C22×C4—5661no—
280C22×C4—5661no—
281C22×C4—5661no—
282C22×C4—5661no—
283Q16—22no—
284Q16—5662no—
285SD16—22no—
286SD16—5662no—
287SD16—5662no—
288SD16—5662no—
289D8—5662no—
290D8—5662no—
291M16—5662no—
292M16—5662no—
293C2×C8—2831no—
294C2×C8—2831no—
295C2×C8C22×D28311yescharacteristic
296C2×C8C22×D28311yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—2831no—
299C4⋊C4—5661no—
300C4⋊C4C22×D28311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—5661no—
304C22⋊C4—5661no—
305C22⋊C4—5661no—
306C22⋊C4—5661no—
307C22⋊C4—5661no—
308C22⋊C4—5661no—
309C22⋊C4—5661no—
310C22⋊C4—5661no—
311C22⋊C4—5661no—
312C42—21no—
313C42—2831no—
314C42—2831no—
315C42—2831no—
316C42—5661no—
317C42—5661no—
318C42C22×D28311yescharacteristic
319C23—21no—
320C23—2831no—
321C23—5661no—
322C23—5661no—
323C23—11322no—
324C23—11322no—
325C23—11322no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—5661no—
330Q8—5661no—
331Q8—11322no—
332Q8D4×D28312yes—
333D4—21no—
334D4—21no—
335D4—2832no—
336D4—5664no—
337D4—5661no—
338D4—5661no—
339D4—5661no—
340D4—11322no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2831no—
345C2×C4—2832no—
346C2×C4—2832no—
347C2×C4—2831no—
348C2×C4—2831no—
349C2×C4—5664no—
350C2×C4—5661no—
351C2×C4—5661no—
352C2×C4—11321no—
353C2×C4—11321no—
354C2×C4—11322no—
355C2×C4—11322no—
356C2×C4—11322no—
357C2×C4D4×D28311yescharacteristic
358C2×C4D4×D28311yescharacteristic
359C2×C4C23×D28311yescharacteristic
360C8—21no—
361C8—21no—
362C8—5661no—
363C8—5661no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—2832no—
368C22—5661no—
369C22—5661no—
370C22—5662no—
371C22—11321no—
372C22—11321no—
373C22—11321no—
374C22—11321no—
375C22—11321no—
376C22—22644no—
377C22C2×D4×D28311yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—2832no—
384C4—5661no—
385C4—5661no—
386C4—11321no—
387C4—11321no—
388C4C2×D4×D28311yescharacteristic
389C4#41×D28311yescharacteristic
390C2—81no—
391C2—5662no—
392C2—11322no—
393C2—22641no—
394C2#121×D28311yescharacteristic
395C2#3182726211yescharacteristic
396C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#464×C283C2
socleC22×C283C22×D4
prime radicalC2×#22583637C2
hypercenter#464D283
hypocenterC283#3040
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. C2×#22583642C2semidirect
3. #124×D283C2semidirect
4. C4×#17985393C2semidirect
5. #31826689C2semidirect
6. #31826687C2semidirect
7. #31826685C2semidirect
8. #31826683C2semidirect
9. #31826681C2semidirect
10. #31826679C2semidirect
11. #31826677C2semidirect
12. #31826487C2semidirect
13. #31826472C2semidirect
14. #31826468C2semidirect
15. #464×C283C2semidirect
21. C2×#17985382C22semidirect
22. C2×#17985381C22semidirect
24. C2×#17985379C22semidirect
34. Q8×Dic283C22semidirect
36. C4×D1132C22semidirect
38. #22583572C22semidirect
45. C4⋊(C283⋊C8)C22semidirect
46. Dic283⋊C8C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583535C22semidirect
49. #22583534C22semidirect
50. #22583533C22semidirect
54. C2×Q16×C283C22semidirect
57. C4×Q8×C283C22semidirect
58. C4⋊C8×C283C22semidirect
59. #103×C283C22semidirect
198. C283#3040semidirect
207. #464D283semidirect
221. C2×Q16C2×D283semidirect
237. C4×Q8C2×D283semidirect
251. C4⋊C8C2×D283semidirect
255. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836422
C2×#225836372
#124×D2832
C4×#179853932
#318266892
#318266872
#318266852
#318266832
#318266812
#318266792
#318266772
#318264872
#318264722
#318264682
#464×C2832
#3040283