Finite Group Databasebeta

← #49132769

Subgroups — #49132769

Order 36224.

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×#22583640C211yesmaximal, characteristic, complemented
2C2×SD16×D283C211yesmaximal, characteristic, complemented
3#31827128C211yesmaximal, characteristic, complemented
4#31827103C211yesmaximal, characteristic
5#31826858C211yesmaximal, characteristic, complemented
6#31826713C211yesmaximal, characteristic, complemented
7#31826705C211yesmaximal, characteristic, complemented
8#31826690C211yesmaximal, characteristic, complemented
9#31826668C211yesmaximal, characteristic, complemented
10#31826619C211yesmaximal, characteristic, complemented
11#31826617C211yesmaximal, characteristic, complemented
12#31826606C211yesmaximal, characteristic, complemented
13#31826562C211yesmaximal, characteristic, complemented
14#31826539C211yesmaximal, characteristic, complemented
15#452×C283C211yesmaximal, characteristic, complemented
16C22×Dic566C2211yescharacteristic, complemented
17C2×Q8×D283C2211yescharacteristic
18C2×D4×D283C2211yescharacteristic
19#22583640—24no—
20C2×#17985382C2211yescharacteristic, complemented
21C2×#17985382C2211yescharacteristic, complemented
22C2×#17985381C2211yescharacteristic
23C2×#17985381C2211yescharacteristic
24C2×#17985380C2211yescharacteristic
25C2×#17985380C2211yescharacteristic
26C2×#17985379C2211yescharacteristic
27SD16×D283—24no—
28C2×#17985376C2211yescharacteristic, complemented
29C2×C8×D283C2211yescharacteristic
30#22583609—21no—
31#22583608C2211yescharacteristic, complemented
32#22583599C2211yescharacteristic
33#22583593—21no—
34#22583590C2211yescharacteristic
35#22583588—21no—
36#22583587—21no—
37#22583586C2211yescharacteristic
38#22583584C2211yescharacteristic
39#22583582—21no—
40#22583578—21no—
41#22583575C2211yescharacteristic, complemented
42D4×Dic283C2211yescharacteristic
43C4×C22⋊D283C2211yescharacteristic
44(C22⋊D283)⋊C4—21no—
45C2×(C2×D283)⋊C4—21no—
46#22583543C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583535C2211yescharacteristic, complemented
49#22583535C2211yescharacteristic, complemented
50#22583534C2211yescharacteristic, complemented
51#22583530C2211yescharacteristic
52(C2×D283)⋊C8C2211yescharacteristic
53#22583520C2211yescharacteristic, complemented
54C2×#41×C283C2211yescharacteristic
55C2×Q16×C283C2211yescharacteristic, complemented
56C2×SD16×C283C2211yescharacteristic
57C22⋊Q8×C283C2211yescharacteristic, complemented
58#103×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic
60C22⋊C8×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62C2×Dic566—21no—
63C2×Dic566—22no—
64C2×Dic566—21no—
65C2×Dic566—41no—
66C2×Dic566C2311yescharacteristic
67C2×Dic566C2311yescharacteristic
68Q8×D283—22no—
69Q8×D283—41no—
70C2×D1132C2311yescharacteristic
71C2×C22⋊D283—21no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283D411yescharacteristic
74D4×D283—22no—
75D4×D283—41no—
76C22×Dic283—21no—
77C22×Dic283—21no—
78C2×C4×D283—21no—
79C2×C4×D283C2311yescharacteristic
80#17985382—22no—
81#17985382—22no—
82#17985381—22no—
83#17985381—41no—
84#17985380—22no—
85#17985380D414yescomplemented
86#17985379—22no—
87#17985376—22no—
88C2×C283⋊C8C2311yescharacteristic
89C2×C283⋊C8C2311yescharacteristic
90C8×D283—22no—
91C4⋊Dic283—21no—
92C4⋊Dic283C2311yescharacteristic
93Dic283⋊C4—21no—
94Dic283⋊C4—21no—
95Dic283⋊C4—21no—
96Dic283⋊C4—21no—
97Dic283⋊C4D411yescharacteristic
98C22⋊Dic283—21no—
99C22⋊Dic283D411yescharacteristic
100(C2×D283)⋊C4—21no—
101(C2×D283)⋊C4—21no—
102(C2×D283)⋊C4—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4D411yescharacteristic
106C4×Dic283—21no—
107C4×Dic283C2311yescharacteristic
108#41×C283—22no—
109#41×C283—41no—
110C2×Q8×C283C2311yescharacteristic
111C2×Q8×C283C2311yescharacteristic
112C2×D4×C283—21no—
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283—21no—
115C22×C4×C283C2311yescharacteristic
116Q16×C283—41no—
117SD16×C283—22no—
118C2×C8×C283C2311yescharacteristic
119C2×C8×C283C2311yescharacteristic
120C4⋊C4×C283—21no—
121C4⋊C4×C283C2311yescharacteristic
122C22⋊C4×C283—21no—
123C22×D283—21no—
124C22×D283—42no—
125C22×D283—42no—
126C22×D283—81no—
127C22×D283C2×D411yescharacteristic
128Dic566—21no—
129Dic566—21no—
130Dic566—21no—
131Dic566—42no—
132Dic566—42no—
133Dic566C2×D412yes—
134D1132—21no—
135D1132—21no—
136C22⋊D283—22no—
137C22⋊D283—42no—
138C22⋊D283—81no—
139C2×Dic283—21no—
140C2×Dic283—21no—
141C2×Dic283—21no—
142C2×Dic283—42no—
143C2×Dic283—81no—
144C2×Dic283C2×D411yescharacteristic
145C2×Dic283C2×D411yescharacteristic
146C2×Dic283C2×D411yescharacteristic
147C4×D283—21no—
148C4×D283—21no—
149C4×D283—42no—
150C283⋊C8—21no—
151C283⋊C8C2×D412yes—
152C23×C283—21no—
153C23×C283C2×D411yescharacteristic
154Q8×C283—21no—
155Q8×C283—21no—
156Q8×C283—21no—
157Q8×C283—21no—
158D4×C283—21no—
159D4×C283—42no—
160D4×C283C2×D412yes—
161C2×C4×C283—21no—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—42no—
167C2×C4×C283C2×D411yescharacteristic
168C2×C4×C283C2411yescharacteristic
169C8×C283—21no—
170C8×C283—21no—
171C22×C283—21no—
172C22×C283—22no—
173C22×C283—21no—
174C22×C283—21no—
175C22×C283—41no—
176C22×C283—41no—
177C22×C283C22×D411yescharacteristic
178C2×D283—21no—
179C2×D283—21no—
180C2×D283—41no—
181C2×D283—41no—
182C2×D283—41no—
183C2×D283—84no—
184C2×D283#13712yescomplemented
185C4×C283—21no—
186C4×C283—41no—
187C4×C283—41no—
188C4×C283—41no—
189C4×C283C22×D411yescharacteristic
190C4×C283H(5,𝔽2)11yescharacteristic
191Dic283—21no—
192Dic283—22no—
193Dic283—21no—
194Dic283—21no—
195Dic283—41no—
196Dic283—41no—
197C2×C283—42no—
198C2×C283—41no—
199C2×C283D4211yescharacteristic
200C2×C283C2×#13711yescharacteristic
201C2×C283#57711yescharacteristic
202D283—22no—
203D283—81no—
204C283#326611yescharacteristic, complemented, Sylow
205#3266—2831nomaximal, Sylow
206C2×#137—2831no—
207C22×SD16—2831no—
208D4×Q8—2831no—
209#546—2831no—
210#497—2831no—
211#468—2831no—
212#467—2831no—
213#464—2831no—
214#461—2831no—
215#452D28311yescharacteristic, complemented
216#451—2831no—
217#450—2831no—
218#448—2831no—
219#440—2831no—
220#435—2831no—
221C2×#41C2×D28311yescharacteristic
222C22×Q8—2831no—
223C22×Q8—2831no—
224C22×D4—2831no—
225#137—5664no—
226C2×Q16—2831no—
227C2×Q16—2831no—
228C2×Q16C2×D28311yescharacteristic, complemented
229C2×SD16—2831no—
230C2×SD16—2831no—
231C2×SD16—2831no—
232C2×SD16—2831no—
233C2×SD16—2831no—
234C2×SD16—5664no—
235C2×SD16C2×D28311yescharacteristic
236C2×M16—2831no—
237C22×C8—2831no—
238C4⋊Q8—2831no—
239C4⋊Q8—5661no—
240#124—2831no—
241#123—5661no—
242C22⋊Q8—2831no—
243C22⋊Q8—2831no—
244C22⋊Q8—5661no—
245C22⋊Q8—5661no—
246C22⋊Q8C2×D28311yescharacteristic, complemented
247#121—2831no—
248#121—5661no—
249#120—5661no—
250C4×Q8—2831no—
251C4×D4—2831no—
252C4×D4—2831no—
253C4×D4—5661no—
254C2×C22⋊C4—5661no—
255#107—2831no—
256C4⋊C8—2831no—
257#103—2831no—
258#103—2831no—
259#103—2831no—
260#103C2×D28311yescharacteristic, complemented
261#102—2831no—
262#102C2×D28311yescharacteristic
263C22⋊C8—2831no—
264C22⋊C8C2×D28311yescharacteristic, complemented
265#97—2831no—
266C24—5661no—
267#41—22no—
268#41—41no—
269C2×Q8—2831no—
270C2×Q8—2831no—
271C2×Q8—5661no—
272C2×Q8—5662no—
273C2×Q8—5661no—
274C2×Q8—5662no—
275C2×Q8—11321no—
276C2×Q8—11321no—
277C2×Q8C22×D28311yescharacteristic
278C2×Q8C22×D28311yescharacteristic
279C2×D4—21no—
280C2×D4—2831no—
281C2×D4—2831no—
282C2×D4—5661no—
283C2×D4—5662no—
284C2×D4—5661no—
285C2×D4—11321no—
286C2×D4C22×D28311yescharacteristic
287C22×C4—21no—
288C22×C4—2831no—
289C22×C4—5661no—
290C22×C4—5661no—
291C22×C4—5661no—
292C22×C4C22×D28311yescharacteristic
293Q16—41no—
294Q16—5662no—
295Q16—5662no—
296SD16—22no—
297SD16—2834no—
298SD16—5662no—
299SD16—5662no—
300SD16—5662no—
301SD16—11321no—
302M16—5662no—
303C2×C8—2831no—
304C2×C8—2831no—
305C2×C8—5662no—
306C2×C8C22×D28311yescharacteristic
307C2×C8C22×D28311yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—2831no—
310C4⋊C4—2831no—
311C4⋊C4—5661no—
312C4⋊C4—5661no—
313C4⋊C4—5661no—
314C4⋊C4—5661no—
315C4⋊C4—5661no—
316C4⋊C4C22×D28311yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—2831no—
319C22⋊C4—2831no—
320C22⋊C4—5661no—
321C22⋊C4—5661no—
322C22⋊C4—5661no—
323C22⋊C4—5661no—
324C22⋊C4—5661no—
325C22⋊C4—5661no—
326C42—2831no—
327C42—5661no—
328C23—21no—
329C23—2831no—
330C23—5661no—
331C23—11322no—
332C23—11322no—
333C23—22641no—
334C23D4×D28311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—2832no—
340Q8—5661no—
341Q8—5661no—
342Q8—5661no—
343Q8—11322no—
344Q8—11322no—
345D4—21no—
346D4—42no—
347D4—5662no—
348D4—5661no—
349D4—5661no—
350D4—11322no—
351D4—22641no—
352D4D4×D28312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—2831no—
360C2×C4—2831no—
361C2×C4—2831no—
362C2×C4—5661no—
363C2×C4—5661no—
364C2×C4—5661no—
365C2×C4—5661no—
366C2×C4—5661no—
367C2×C4—11322no—
368C2×C4—11322no—
369C2×C4—22641no—
370C2×C4D4×D28311yescharacteristic
371C2×C4C23×D28311yescharacteristic
372C8—21no—
373C8—21no—
374C8—2832no—
375C8—5661no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—2832no—
383C22—5661no—
384C22—5661no—
385C22—11321no—
386C22—11321no—
387C22—11321no—
388C22—22644no—
389C22C2×D4×D28311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—5661no—
395C4—5661no—
396C4—5661no—
397C4—5662no—
398C4—11321no—
399C4—11321no—
400C4C2×D4×D28311yescharacteristic
401C4#2258365511yescharacteristic
402C2—41no—
403C2—42no—
404C2—5662no—
405C2—22641no—
406C2#3182709711yescharacteristic
407C2#137×D28311yescharacteristic
408C2#3182727211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#452×C283C2
socleC22×C283C22×D4
prime radical#31827103C2
hypercenter#452D283
hypocenterC283#3266
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22583640C2semidirect
2. C2×SD16×D283C2semidirect
3. #31827128C2semidirect
5. #31826858C2semidirect
6. #31826713C2semidirect
7. #31826705C2semidirect
8. #31826690C2semidirect
9. #31826668C2semidirect
10. #31826619C2semidirect
11. #31826617C2semidirect
12. #31826606C2semidirect
13. #31826562C2semidirect
14. #31826539C2semidirect
15. #452×C283C2semidirect
16. C22×Dic566C22semidirect
20. C2×#17985382C22semidirect
21. C2×#17985382C22semidirect
28. C2×#17985376C22semidirect
31. #22583608C22semidirect
41. #22583575C22semidirect
46. #22583543C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583535C22semidirect
49. #22583535C22semidirect
50. #22583534C22semidirect
53. #22583520C22semidirect
55. C2×Q16×C283C22semidirect
57. C22⋊Q8×C283C22semidirect
58. #103×C283C22semidirect
60. C22⋊C8×C283C22semidirect
85. #17985380D4semidirect
184. C2×D283#137semidirect
204. C283#3266semidirect
215. #452D283semidirect
228. C2×Q16C2×D283semidirect
246. C22⋊Q8C2×D283semidirect
260. #103C2×D283semidirect
264. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836402
C2×SD16×D2832
#318271282
#318271032
#318268582
#318267132
#318267052
#318266902
#318266682
#318266192
#318266172
#318266062
#318265622
#318265392
#452×C2832
#3266283