Finite Group Databasebeta

← #49132775

Subgroups — #49132775

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×#22583639C211yesmaximal, characteristic, complemented
2C22×#17985380C211yesmaximal, characteristic, complemented
3#31827128C211yesmaximal, characteristic, complemented
4#31827109C211yesmaximal, characteristic
5#31826856C211yesmaximal, characteristic, complemented
6#31826710C211yesmaximal, characteristic, complemented
7#31826704C211yesmaximal, characteristic, complemented
8#31826686C211yesmaximal, characteristic, complemented
9#31826668C211yesmaximal, characteristic, complemented
10#31826628C211yesmaximal, characteristic, complemented
11#31826623C211yesmaximal, characteristic, complemented
12#31826602C211yesmaximal, characteristic, complemented
13#31826562C211yesmaximal, characteristic, complemented
14#31826533C211yesmaximal, characteristic, complemented
15#450×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C22×Dic566C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic, complemented
19#22583639—24no—
20C2×Dic1132C2211yescharacteristic, complemented
21C2×Dic1132C2211yescharacteristic, complemented
22C2×#17985382C2211yescharacteristic, complemented
23C2×#17985380—24no—
24C2×#17985380C2211yescharacteristic
25C2×#17985380C2211yescharacteristic
26C2×#17985380C2211yescharacteristic
27C2×#17985380C2211yescharacteristic
28C2×#17985375C2211yescharacteristic, complemented
29C22×C283⋊C8C2211yescharacteristic
30#22583609—21no—
31#22583608C2211yescharacteristic, complemented
32#22583597C2211yescharacteristic
33#22583591—21no—
34#22583590C2211yescharacteristic, complemented
35#22583589C2211yescharacteristic
36#22583588—21no—
37#22583587—21no—
38#22583582—21no—
39#22583581C2211yescharacteristic
40#22583579—21no—
41#22583575C2211yescharacteristic, complemented
42D4×Dic283C2211yescharacteristic
43C4×C22⋊D283C2211yescharacteristic
44(C22⋊D283)⋊C4—21no—
45C2×C22⋊Dic283—21no—
46#22583543C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583535C2211yescharacteristic, complemented
49#22583535C2211yescharacteristic, complemented
50#22583534C2211yescharacteristic, complemented
51#22583533C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583520C2211yescharacteristic, complemented
54C22×D4×C283C2211yescharacteristic
55C2×SD16×C283C2211yescharacteristic
56C2×SD16×C283C2211yescharacteristic
57C22⋊Q8×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59#102×C283C2211yescharacteristic
60C22⋊C8×C283C2211yescharacteristic
61#17985391—22no—
62#17985391—41no—
63C2×Dic566—21no—
64C2×Dic566—22no—
65C2×Dic566—21no—
66C2×Dic566—41no—
67C2×Dic566C2311yescharacteristic
68C2×Dic566C2311yescharacteristic
69C2×Dic566C2311yescharacteristic
70Q8×D283—22no—
71Q8×D283—41no—
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283D411yescharacteristic
75C22×Dic283—21no—
76C22×Dic283—21no—
77C22×Dic283—21no—
78C2×C4×D283—21no—
79C2×C4×D283C2311yescharacteristic
80Dic1132—22no—
81Dic1132—41no—
82#17985382—22no—
83#17985380—22no—
84#17985380—22no—
85#17985380—22no—
86#17985380D414yescomplemented
87#17985375—22no—
88C2×C283⋊C8—22no—
89C2×C283⋊C8C2311yescharacteristic
90C2×C283⋊C8C2311yescharacteristic
91C4⋊Dic283—21no—
92C4⋊Dic283C2311yescharacteristic
93Dic283⋊C4—21no—
94Dic283⋊C4—21no—
95Dic283⋊C4—21no—
96Dic283⋊C4—21no—
97Dic283⋊C4D411yescharacteristic
98C22⋊Dic283—21no—
99C22⋊Dic283—21no—
100C22⋊Dic283—21no—
101C22⋊Dic283—21no—
102C22⋊Dic283—21no—
103C22⋊Dic283D411yescharacteristic
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4D411yescharacteristic
106C4×Dic283—21no—
107C4×Dic283C2311yescharacteristic
108C24×C283—21no—
109C2×Q8×C283C2311yescharacteristic
110C2×D4×C283—22no—
111C2×D4×C283—21no—
112C2×D4×C283—41no—
113C2×D4×C283C2311yescharacteristic
114C2×D4×C283C2311yescharacteristic
115C22×C4×C283C2311yescharacteristic
116SD16×C283—22no—
117SD16×C283—41no—
118C2×C8×C283C2311yescharacteristic
119C2×C8×C283C2311yescharacteristic
120C4⋊C4×C283—21no—
121C4⋊C4×C283C2311yescharacteristic
122C22⋊C4×C283—21no—
123C22×D283C2×D411yescharacteristic
124Dic566—21no—
125Dic566—21no—
126Dic566—21no—
127Dic566—21no—
128Dic566—21no—
129Dic566—42no—
130Dic566—42no—
131Dic566C2×D412yes—
132C22⋊D283—22no—
133C22⋊D283—42no—
134C22⋊D283—81no—
135C2×Dic283—21no—
136C2×Dic283—21no—
137C2×Dic283—21no—
138C2×Dic283—21no—
139C2×Dic283—42no—
140C2×Dic283—42no—
141C2×Dic283—81no—
142C2×Dic283C2×D411yescharacteristic
143C2×Dic283C2×D411yescharacteristic
144C2×Dic283C2×D411yescharacteristic
145C4×D283—21no—
146C4×D283—21no—
147C4×D283—42no—
148C283⋊C8—21no—
149C283⋊C8C2×D412yes—
150C23×C283—21no—
151C23×C283—21no—
152C23×C283—42no—
153C23×C283—42no—
154C23×C283—81no—
155C23×C283C2×D411yescharacteristic
156Q8×C283—21no—
157Q8×C283—21no—
158D4×C283—21no—
159D4×C283—21no—
160D4×C283—21no—
161D4×C283—42no—
162D4×C283C2×D412yes—
163C2×C4×C283—21no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—21no—
167C2×C4×C283C2×D411yescharacteristic
168C2×C4×C283C2411yescharacteristic
169C8×C283—21no—
170C8×C283—21no—
171C22×C283—21no—
172C22×C283—21no—
173C22×C283—22no—
174C22×C283—41no—
175C22×C283—41no—
176C22×C283—41no—
177C22×C283—41no—
178C22×C283—41no—
179C22×C283—84no—
180C22×C283#13712yescomplemented
181C22×C283C22×D411yescharacteristic
182C2×D283—21no—
183C2×D283—21no—
184C2×D283—21no—
185C4×C283—21no—
186C4×C283—41no—
187C4×C283—41no—
188C4×C283C22×D411yescharacteristic
189C4×C283H(5,𝔽2)11yescharacteristic
190Dic283—21no—
191Dic283—22no—
192Dic283—21no—
193Dic283—21no—
194Dic283—41no—
195Dic283—41no—
196Dic283—41no—
197C2×C283—22no—
198C2×C283—42no—
199C2×C283—81no—
200C2×C283D4211yescharacteristic
201C2×C283C2×#13711yescharacteristic
202C2×C283#57711yescharacteristic
203D283—41no—
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#452—2831no—
216#451—2831no—
217#450D28311yescharacteristic, complemented
218#448—2831no—
219#440—2831no—
220#435—2831no—
221C2×#41—2831no—
222C22×Q8—2831no—
223C22×Q8—2831no—
224C22×D4C2×D28311yescharacteristic
225#137—5664no—
226C2×Q16—2831no—
227C2×Q16—2831no—
228C2×Q16—2831no—
229C2×SD16—2831no—
230C2×SD16—2831no—
231C2×SD16—2831no—
232C2×SD16—2831no—
233C2×SD16—5664no—
234C2×SD16C2×D28311yescharacteristic
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
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#103—2831no—
261#102C2×D28311yescharacteristic
262#102C2×D28311yescharacteristic
263C22⋊C8—2831no—
264C22⋊C8C2×D28311yescharacteristic
265#97—2831no—
266C24—21no—
267#41—5662no—
268#41—11321no—
269C2×Q8—2831no—
270C2×Q8—2831no—
271C2×Q8—2831no—
272C2×Q8—5661no—
273C2×Q8—5662no—
274C2×Q8—5661no—
275C2×Q8—5662no—
276C2×Q8—11321no—
277C2×Q8—11321no—
278C2×Q8C22×D28311yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—2831no—
283C2×D4—5661no—
284C2×D4—5661no—
285C2×D4C22×D28311yescharacteristic
286C2×D4C22×D28311yescharacteristic
287C22×C4—2831no—
288C22×C4—5661no—
289C22×C4—5661no—
290C22×C4—5661no—
291C22×C4—5661no—
292C22×C4C22×D28311yescharacteristic
293Q16—5662no—
294Q16—5662no—
295Q16—11321no—
296SD16—22no—
297SD16—41no—
298SD16—2834no—
299SD16—5662no—
300SD16—5662no—
301SD16—5662no—
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—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23—2831no—
334C23D4×D28311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—2832no—
338Q8—5661no—
339Q8—5661no—
340Q8—5661no—
341Q8—5661no—
342Q8—5661no—
343Q8—11322no—
344Q8—11322no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—5662no—
350D4—11322no—
351D4—22641no—
352D4D4×D28312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—2831no—
358C2×C4—2831no—
359C2×C4—2831no—
360C2×C4—5661no—
361C2×C4—5661no—
362C2×C4—5661no—
363C2×C4—5661no—
364C2×C4—5661no—
365C2×C4—5661no—
366C2×C4—11322no—
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—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—84no—
385C22—5661no—
386C22—5661no—
387C22—5661no—
388C22#2258363912yescomplemented
389C22C2×D4×D28311yescharacteristic
390C4—21no—
391C4—41no—
392C4—41no—
393C4—5661no—
394C4—5661no—
395C4—5661no—
396C4—5662no—
397C4—11321no—
398C4—11321no—
399C4—11321no—
400C4C2×D4×D28311yescharacteristic
401C4#2258365511yescharacteristic
402C2—22no—
403C2—42no—
404C2—81no—
405C2—11321no—
406C2#3182709711yescharacteristic
407C2C2×#2258363911yescharacteristic
408C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#450×C283C2
socleC22×C283C22×D4
prime radical#31827109C2
hypercenter#450D283
hypocenterC283#3266
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22583639C2semidirect
2. C22×#17985380C2semidirect
3. #31827128C2semidirect
5. #31826856C2semidirect
6. #31826710C2semidirect
7. #31826704C2semidirect
8. #31826686C2semidirect
9. #31826668C2semidirect
10. #31826628C2semidirect
11. #31826623C2semidirect
12. #31826602C2semidirect
13. #31826562C2semidirect
14. #31826533C2semidirect
15. #450×C283C2semidirect
18. C2×Q8×D283C22semidirect
20. C2×Dic1132C22semidirect
21. C2×Dic1132C22semidirect
22. C2×#17985382C22semidirect
28. C2×#17985375C22semidirect
31. #22583608C22semidirect
34. #22583590C22semidirect
41. #22583575C22semidirect
46. #22583543C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583535C22semidirect
49. #22583535C22semidirect
50. #22583534C22semidirect
51. #22583533C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583520C22semidirect
86. #17985380D4semidirect
180. C22×C283#137semidirect
204. C283#3266semidirect
217. #450D283semidirect
388. C22#22583639semidirect

Maximal subgroups

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

subgroupindex
C2×#225836392
C22×#179853802
#318271282
#318271092
#318268562
#318267102
#318267042
#318266862
#318266682
#318266282
#318266232
#318266022
#318265622
#318265332
#450×C2832
#3266283