Finite Group Databasebeta

← #49132777

Subgroups — #49132777

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
2C2×SD16×D283C211yesmaximal, characteristic, complemented
3#31827130C211yesmaximal, characteristic, complemented
4#31827110C211yesmaximal, characteristic
5#31826857C211yesmaximal, characteristic, complemented
6#31826714C211yesmaximal, characteristic, complemented
7#31826705C211yesmaximal, characteristic, complemented
8#31826688C211yesmaximal, characteristic, complemented
9#31826624C211yesmaximal, characteristic, complemented
10#31826622C211yesmaximal, characteristic, complemented
11#31826614C211yesmaximal, characteristic, complemented
12#31826605C211yesmaximal, characteristic, complemented
13#31826565C211yesmaximal, characteristic, complemented
14#31826535C211yesmaximal, characteristic, complemented
15#461×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×Q8×D283C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic, complemented
19C2×D4×D283C2211yescharacteristic
20#22583639—24no—
21C2×Dic1132C2211yescharacteristic, complemented
22C2×#17985382C2211yescharacteristic, complemented
23C2×#17985382C2211yescharacteristic, complemented
24C2×#17985381C2211yescharacteristic
25C2×#17985380C2211yescharacteristic
26C2×#17985379C2211yescharacteristic
27C2×#17985379C2211yescharacteristic
28SD16×D283—24no—
29C2×#17985375C2211yescharacteristic, complemented
30C2×C8×D283C2211yescharacteristic
31#22583609—21no—
32#22583598C2211yescharacteristic
33#22583594—21no—
34#22583590C2211yescharacteristic
35#22583588—21no—
36#22583588—21no—
37#22583586C2211yescharacteristic
38#22583586C2211yescharacteristic, complemented
39#22583585C2211yescharacteristic
40#22583584—21no—
41#22583580—21no—
42C4×Dic566C2211yescharacteristic, complemented
43C4×D1132C2211yescharacteristic
44#22583572—21no—
45C2×(C2×D283)⋊C4—21no—
46#22583543C2211yescharacteristic, complemented
47#22583535C2211yescharacteristic, complemented
48#22583533C2211yescharacteristic, complemented
49#22583532C2211yescharacteristic
50#22583531C2211yescharacteristic
51(C2×D283)⋊C8C2211yescharacteristic
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583521C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic
55C2×SD16×C283C2211yescharacteristic
56C4⋊Q8×C283C2211yescharacteristic, complemented
57C4×D4×C283C2211yescharacteristic
58C4⋊C8×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#103×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985391—22no—
63#17985391—41no—
64C2×Dic566—21no—
65C2×Dic566—21no—
66C2×Dic566C2311yescharacteristic
67C2×Dic566C2311yescharacteristic
68Q8×D283—22no—
69Q8×D283—22no—
70Q8×D283—41no—
71Q8×D283—41no—
72C2×D1132—21no—
73C2×D1132D411yescharacteristic
74C2×D1132C2311yescharacteristic
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77D4×D283—22no—
78D4×D283—41no—
79C22×Dic283—21no—
80C2×C4×D283—21no—
81C2×C4×D283—21no—
82C2×C4×D283C2311yescharacteristic
83C2×C4×D283C2311yescharacteristic
84Dic1132—22no—
85#17985382—22no—
86#17985382—41no—
87#17985381—22no—
88#17985380D414yescomplemented
89#17985379—22no—
90#17985379—41no—
91#17985375—22no—
92C2×C283⋊C8C2311yescharacteristic
93C2×C283⋊C8C2311yescharacteristic
94C8×D283—22no—
95C4⋊Dic283—21no—
96C4⋊Dic283D411yescharacteristic
97C4⋊Dic283C2311yescharacteristic
98Dic283⋊C4—21no—
99Dic283⋊C4—21no—
100Dic283⋊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)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4D411yescharacteristic
108(C2×D283)⋊C4D411yescharacteristic
109C4×Dic283—21no—
110C2×Q8×C283C2311yescharacteristic
111C2×Q8×C283C2311yescharacteristic
112C2×D4×C283C2311yescharacteristic
113C22×C4×C283—21no—
114SD16×C283—22no—
115SD16×C283—22no—
116C2×C8×C283C2311yescharacteristic
117C2×C8×C283C2311yescharacteristic
118C4⋊C4×C283—21no—
119C4⋊C4×C283—21no—
120C4⋊C4×C283C2311yescharacteristic
121C22⋊C4×C283—21no—
122C42×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283—42no—
125C22×D283—42no—
126C22×D283—81no—
127C22×D283C2×D411yescharacteristic
128C22×D283C2×D411yescharacteristic
129Dic566—21no—
130Dic566—21no—
131Dic566—21no—
132Dic566—42no—
133Dic566—42no—
134Dic566C2×D412yes—
135D1132—22no—
136D1132—21no—
137D1132—21no—
138D1132—81no—
139C22⋊D283—42no—
140C22⋊D283—42no—
141C2×Dic283—21no—
142C2×Dic283—21no—
143C2×Dic283—21no—
144C2×Dic283—42no—
145C2×Dic283C2×D411yescharacteristic
146C2×Dic283C2×D411yescharacteristic
147C4×D283—21no—
148C4×D283—21no—
149C4×D283—21no—
150C4×D283—21no—
151C4×D283—42no—
152C4×D283—42no—
153C283⋊C8—21no—
154C283⋊C8C2×D412yes—
155C23×C283—21no—
156Q8×C283—21no—
157Q8×C283—21no—
158Q8×C283—21no—
159Q8×C283—21no—
160D4×C283—21no—
161D4×C283C2×D412yes—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
164C2×C4×C283—21no—
165C2×C4×C283—81no—
166C2×C4×C283C2×D411yescharacteristic
167C2×C4×C283C2×D411yescharacteristic
168C2×C4×C283C2411yescharacteristic
169C8×C283—21no—
170C8×C283—21no—
171C22×C283—22no—
172C22×C283—41no—
173C22×C283—41no—
174C22×C283C22×D411yescharacteristic
175C2×D283—21no—
176C2×D283—21no—
177C2×D283—21no—
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—21no—
187C4×C283—41no—
188C4×C283—41no—
189C4×C283—41no—
190C4×C283C22×D411yescharacteristic
191C4×C283H(5,𝔽2)11yescharacteristic
192Dic283—21no—
193Dic283—21no—
194Dic283—22no—
195Dic283—41no—
196Dic283—41no—
197C2×C283—42no—
198C2×C283D4211yescharacteristic
199C2×C283C2×#13711yescharacteristic
200C2×C283#57711yescharacteristic
201D283—22no—
202D283—41no—
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#461D28311yescharacteristic, complemented
215#452—2831no—
216#451—2831no—
217#450—2831no—
218#448—2831no—
219#440—2831no—
220#435—2831no—
221C2×#41—2831no—
222C22×Q8—2831no—
223C22×Q8—2831no—
224C22×D4—2831no—
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—5661no—
239C4⋊Q8C2×D28311yescharacteristic, complemented
240#124—2831no—
241#123—5661no—
242C22⋊Q8—2831no—
243C22⋊Q8—2831no—
244C22⋊Q8—2831no—
245C22⋊Q8—5661no—
246C22⋊Q8—5661no—
247#121—2831no—
248#121—5661no—
249#120—5661no—
250C4×Q8—2831no—
251C4×D4—2831no—
252C4×D4—5661no—
253C4×D4C2×D28311yescharacteristic
254C2×C22⋊C4—5661no—
255#107—2831no—
256C4⋊C8C2×D28311yescharacteristic, complemented
257#103—2831no—
258#103—2831no—
259#103C2×D28311yescharacteristic, complemented
260#103C2×D28311yescharacteristic, complemented
261#102—2831no—
262#102—2831no—
263C22⋊C8—2831no—
264C22⋊C8—2831no—
265#97—2831no—
266C24—5661no—
267#41—5662no—
268#41—11321no—
269C2×Q8—2831no—
270C2×Q8—2831no—
271C2×Q8—5661no—
272C2×Q8—5661no—
273C2×Q8—5662no—
274C2×Q8—5662no—
275C2×Q8—11321no—
276C2×Q8—11321no—
277C2×Q8C22×D28311yescharacteristic
278C2×Q8C22×D28311yescharacteristic
279C2×D4—2831no—
280C2×D4—2831no—
281C2×D4—5661no—
282C2×D4—5661no—
283C2×D4—5662no—
284C2×D4—5661no—
285C2×D4—11321no—
286C2×D4C22×D28311yescharacteristic
287C22×C4—21no—
288C22×C4—2831no—
289C22×C4—2831no—
290C22×C4—5661no—
291C22×C4—5661no—
292C22×C4—5661no—
293Q16—5662no—
294Q16—5662no—
295Q16—11321no—
296SD16—22no—
297SD16—22no—
298SD16—2834no—
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—21no—
310C4⋊C4—2831no—
311C4⋊C4—2831no—
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—5661no—
327C42C22×D28311yescharacteristic
328C23—21no—
329C23—2831no—
330C23—2831no—
331C23—5661no—
332C23—11322no—
333C23—11322no—
334C23—22641no—
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—2832no—
340Q8—5661no—
341Q8—5661no—
342Q8—5661no—
343Q8—11322no—
344Q8—11322no—
345D4—21no—
346D4—5662no—
347D4—5661no—
348D4—5661no—
349D4—11322no—
350D4—11322no—
351D4—22641no—
352D4D4×D28312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—2831no—
358C2×C4—2831no—
359C2×C4—5661no—
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×C4D4×D28311yescharacteristic
370C2×C4D4×D28311yescharacteristic
371C2×C4C23×D28311yescharacteristic
372C8—21no—
373C8—21no—
374C8—2832no—
375C8—5661no—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—2832no—
380C22—5661no—
381C22—5661no—
382C22—5661no—
383C22—5661no—
384C22—5661no—
385C22—11321no—
386C22—11321no—
387C22—11321no—
388C22—22644no—
389C22C2×D4×D28311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—5661no—
396C4—5661no—
397C4—5662no—
398C4—11321no—
399C4—11321no—
400C4C2×D4×D28311yescharacteristic
401C4#2258365611yescharacteristic
402C2—42no—
403C2—5662no—
404C2—11321no—
405C2—22641no—
406C2#3182709911yescharacteristic
407C2#137×D28311yescharacteristic
408C2#3182727111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#461×C283C2
socleC22×C283C22×D4
prime radical#31827110C2
hypercenter#461D283
hypocenterC283#3266
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#22583639C2semidirect
2. C2×SD16×D283C2semidirect
3. #31827130C2semidirect
5. #31826857C2semidirect
6. #31826714C2semidirect
7. #31826705C2semidirect
8. #31826688C2semidirect
9. #31826624C2semidirect
10. #31826622C2semidirect
11. #31826614C2semidirect
12. #31826605C2semidirect
13. #31826565C2semidirect
14. #31826535C2semidirect
15. #461×C283C2semidirect
18. C2×Q8×D283C22semidirect
21. C2×Dic1132C22semidirect
22. C2×#17985382C22semidirect
23. C2×#17985382C22semidirect
29. C2×#17985375C22semidirect
38. #22583586C22semidirect
42. C4×Dic566C22semidirect
46. #22583543C22semidirect
47. #22583535C22semidirect
48. #22583533C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583521C22semidirect
56. C4⋊Q8×C283C22semidirect
58. C4⋊C8×C283C22semidirect
59. #103×C283C22semidirect
60. #103×C283C22semidirect
88. #17985380D4semidirect
184. C2×D283#137semidirect
204. C283#3266semidirect
214. #461D283semidirect
239. C4⋊Q8C2×D283semidirect
256. C4⋊C8C2×D283semidirect
259. #103C2×D283semidirect
260. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836392
C2×SD16×D2832
#318271302
#318271102
#318268572
#318267142
#318267052
#318266882
#318266242
#318266222
#318266142
#318266052
#318265652
#318265352
#461×C2832
#3266283