Finite Group Databasebeta

← #49133297

Subgroups — #49133297

Order 36224.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827185C211yesmaximal, characteristic, complemented
2#31826824C211yesmaximal, characteristic, complemented
3#31826803C211yesmaximal, characteristic, complemented
4#31826786C211yesmaximal, characteristic, complemented
5#31826755C211yesmaximal, characteristic, complemented
6#31826721C211yesmaximal, characteristic, complemented
7#31826692C211yesmaximal, characteristic, complemented
8#31826684C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic, complemented
10#31826616C211yesmaximal, characteristic, complemented
11#31826565C211yesmaximal, characteristic, complemented
12#31826537C211yesmaximal, characteristic, complemented
13#31826470C211yesmaximal, characteristic, complemented
14#31826461C211yesmaximal, characteristic, complemented
15#546×C283C211yesmaximal, characteristic
16C2×#17985382C2211yescharacteristic, complemented
17C2×#17985380C2211yescharacteristic
18C2×#17985380C2211yescharacteristic
19C2×#17985376C2211yescharacteristic
20C22×C283⋊C8C2211yescharacteristic, complemented
21#22583610C2211yescharacteristic, complemented
22#22583603—21no—
23#22583603—21no—
24#22583597—21no—
25#22583589C2211yescharacteristic
26#22583589C2211yescharacteristic, complemented
27C4×Dic566C2211yescharacteristic, complemented
28#22583575—21no—
29#22583569C2211yescharacteristic
30#22583569C2211yescharacteristic, complemented
31#22583543C2211yescharacteristic, complemented
32#22583541C2211yescharacteristic, complemented
33#22583541C2211yescharacteristic, complemented
34C4⋊(C283⋊C8)C2211yescharacteristic, complemented
35#22583534C2211yescharacteristic, complemented
36#22583534C2211yescharacteristic, complemented
37#22583534C2211yescharacteristic, complemented
38#22583533C2211yescharacteristic, complemented
39#22583531C2211yescharacteristic
40#22583531C2211yescharacteristic
41#22583531C2211yescharacteristic
42#22583531C2211yescharacteristic
43#22583531C2211yescharacteristic
44#22583531C2211yescharacteristic
45C22⋊(C283⋊C8)C2211yescharacteristic
46C22⋊(C283⋊C8)C2211yescharacteristic, complemented
47#22583521C2211yescharacteristic, complemented
48C2×#41×C283C2211yescharacteristic
49C22×D4×C283C2211yescharacteristic
50#124×C283C2211yescharacteristic
51#123×C283—21no—
52C22⋊Q8×C283C2211yescharacteristic
53#121×C283—21no—
54#121×C283C2211yescharacteristic
55#120×C283—21no—
56C4×D4×C283C2211yescharacteristic
57C4×D4×C283C2211yescharacteristic
58C2×C22⋊C4×C283—21no—
59C2×Dic566—21no—
60C2×Dic566C2311yescharacteristic
61#17985382—41no—
62#17985380—22no—
63#17985380—41no—
64#17985376—41no—
65C2×C283⋊C8—41no—
66C2×C283⋊C8C2311yescharacteristic
67C2×C283⋊C8C2311yescharacteristic
68C2×C283⋊C8C2311yescharacteristic
69C2×C283⋊C8C2311yescharacteristic
70C4⋊Dic283—21no—
71C4⋊Dic283C2311yescharacteristic
72C4⋊Dic283C2311yescharacteristic
73C4⋊Dic283C2311yescharacteristic
74Dic283⋊C4—21no—
75Dic283⋊C4—21no—
76Dic283⋊C4—21no—
77C22⋊Dic283—21no—
78C22⋊Dic283—21no—
79C22⋊Dic283—21no—
80C22⋊Dic283—21no—
81C4×Dic283—21no—
82C4×Dic283—21no—
83C4×Dic283—21no—
84C24×C283—21no—
85#41×C283—22no—
86#41×C283—41no—
87C2×Q8×C283C2311yescharacteristic
88C2×D4×C283—22no—
89C2×D4×C283—21no—
90C2×D4×C283—21no—
91C2×D4×C283—21no—
92C2×D4×C283—41no—
93C2×D4×C283D411yescharacteristic
94C2×D4×C283C2311yescharacteristic
95C2×D4×C283C2311yescharacteristic
96C22×C4×C283—21no—
97C22×C4×C283—21no—
98C22×C4×C283C2311yescharacteristic
99C22×C4×C283C2311yescharacteristic
100C4⋊C4×C283—21no—
101C4⋊C4×C283D411yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C22⋊C4×C283—21no—
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C22⋊C4×C283—21no—
108C22⋊C4×C283D411yescharacteristic
109C22⋊C4×C283D411yescharacteristic
110C42×C283C2311yescharacteristic
111Dic566—21no—
112Dic566—21no—
113Dic566—42no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—21no—
118C283⋊C8—21no—
119C283⋊C8—21no—
120C283⋊C8—21no—
121C283⋊C8—21no—
122C23×C283—21no—
123C23×C283—21no—
124C23×C283—42no—
125C23×C283—42no—
126C23×C283—81no—
127C23×C283C2×D411yescharacteristic
128C23×C283C2×D411yescharacteristic
129Q8×C283—21no—
130Q8×C283—21no—
131D4×C283—21no—
132D4×C283—21no—
133D4×C283—21no—
134D4×C283—41no—
135D4×C283—42no—
136D4×C283—42no—
137D4×C283—81no—
138D4×C283#4112yes—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—42no—
146C2×C4×C283—81no—
147C2×C4×C283C2×D411yescharacteristic
148C2×C4×C283C2×D411yescharacteristic
149C2×C4×C283C2×D411yescharacteristic
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2411yescharacteristic
152C22×C283—21no—
153C22×C283—21no—
154C22×C283—22no—
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—21no—
158C22×C283—21no—
159C22×C283—41no—
160C22×C283—41no—
161C22×C283—41no—
162C22×C283—41no—
163C22×C283—41no—
164C22×C283—84no—
165C22×C283C22×D411yescharacteristic
166C4×C283—21no—
167C4×C283—21no—
168C4×C283—21no—
169C4×C283—21no—
170C4×C283—41no—
171C4×C283—41no—
172C4×C283C2×#4111yescharacteristic
173C4×C283H(5,𝔽2)11yescharacteristic
174Dic283—41no—
175Dic283—41no—
176Dic283—41no—
177Dic283—41no—
178C2×C283—41no—
179C2×C283—41no—
180C2×C283—42no—
181C2×C283—81no—
182C2×C283#54611yescharacteristic
183C2×C283#57511yescharacteristic
184C2×C283#57711yescharacteristic
185C283#331111yescharacteristic, complemented, Sylow
186#3311—2831nomaximal, Sylow
187#556—2831no—
188#546D28311yescharacteristic
189#489—2831no—
190#485—2831no—
191#482—2831no—
192#476—2831no—
193#470—2831no—
194#465—2831no—
195#464—2831no—
196#452—2831no—
197#450—2831no—
198#440—2831no—
199#435—2831no—
200#418—2831no—
201#416—2831no—
202C2×#41C2×D28311yescharacteristic
203C22×D4C2×D28311yescharacteristic
204C2×Q16—2831no—
205C2×SD16—2831no—
206C2×SD16—2831no—
207C2×M16—2831no—
208C22×C8—2831no—
209C4⋊Q8—2831no—
210#126—5661no—
211#126—5661no—
212#124—5661no—
213#124C2×D28311yescharacteristic
214#123—21no—
215C22⋊Q8—2831no—
216C22⋊Q8—2831no—
217C22⋊Q8C2×D28311yescharacteristic
218#121—21no—
219#121C2×D28311yescharacteristic
220#120—21no—
221C4×Q8—2831no—
222C4×Q8—5661no—
223C4×D4C2×D28311yescharacteristic
224C4×D4C2×D28311yescharacteristic
225#117—2831no—
226#117—2831no—
227C2×C22⋊C4—21no—
228#107—2831no—
229#106—2831no—
230#106—2831no—
231C4⋊C8—2831no—
232#103—2831no—
233#103—2831no—
234#103—2831no—
235#103—2831no—
236#102—2831no—
237#102—2831no—
238#102—2831no—
239#102—2831no—
240#102—2831no—
241#102—2831no—
242C22⋊C8—2831no—
243C22⋊C8—2831no—
244#97—2831no—
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—2831no—
249C2×Q8—5661no—
250C2×Q8C22×D28311yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4C22⋊D28311yescharacteristic
257C2×D4C22×D28311yescharacteristic
258C2×D4C22×D28311yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C22×D28311yescharacteristic
262C22×C4C22×D28311yescharacteristic
263Q16—11321no—
264SD16—5662no—
265SD16—11321no—
266M16—11321no—
267C2×C8—2831no—
268C2×C8—2831no—
269C2×C8—2831no—
270C2×C8—2831no—
271C2×C8—11321no—
272C4⋊C4—21no—
273C4⋊C4—2831no—
274C4⋊C4—2831no—
275C4⋊C4—2831no—
276C4⋊C4—5661no—
277C4⋊C4—5661no—
278C4⋊C4—5661no—
279C4⋊C4—5661no—
280C4⋊C4C22⋊D28311yescharacteristic
281C4⋊C4C22×D28311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—5661no—
288C22⋊C4—5661no—
289C22⋊C4—5661no—
290C22⋊C4—5661no—
291C22⋊C4C22⋊D28311yescharacteristic
292C22⋊C4C22⋊D28311yescharacteristic
293C42—5661no—
294C42—5661no—
295C42—5661no—
296C42C22×D28311yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D28311yescharacteristic
303C23C2×C22⋊D28311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—5661no—
307Q8—5661no—
308Q8—11322no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—81no—
316D4#1798539112yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—5661no—
328C2×C4—5661no—
329C2×C4C2×C22⋊D28311yescharacteristic
330C2×C4C2×C22⋊D28311yescharacteristic
331C2×C4C2×C22⋊D28311yescharacteristic
332C2×C4C2×C22⋊D28311yescharacteristic
333C2×C4C23×D28311yescharacteristic
334C8—5661no—
335C8—5661no—
336C8—5661no—
337C8—5661no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22C22×C22⋊D28311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—11321no—
359C4—11321no—
360C4—11321no—
361C4—11321no—
362C4C2×#1798539111yescharacteristic
363C4#2258365511yescharacteristic
364C2—41no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#3182710911yescharacteristic
369C2#3182726311yescharacteristic
370C2#3182727611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting, prime radical#546×C283C2
socleC22×C283C22×D4
hypercenter#546D283
hypocenterC283#3311
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #31827185C2semidirect
2. #31826824C2semidirect
3. #31826803C2semidirect
4. #31826786C2semidirect
5. #31826755C2semidirect
6. #31826721C2semidirect
7. #31826692C2semidirect
8. #31826684C2semidirect
9. #31826627C2semidirect
10. #31826616C2semidirect
11. #31826565C2semidirect
12. #31826537C2semidirect
13. #31826470C2semidirect
14. #31826461C2semidirect
16. C2×#17985382C22semidirect
20. C22×C283⋊C8C22semidirect
21. #22583610C22semidirect
26. #22583589C22semidirect
27. C4×Dic566C22semidirect
30. #22583569C22semidirect
31. #22583543C22semidirect
32. #22583541C22semidirect
33. #22583541C22semidirect
34. C4⋊(C283⋊C8)C22semidirect
35. #22583534C22semidirect
36. #22583534C22semidirect
37. #22583534C22semidirect
38. #22583533C22semidirect
46. C22⋊(C283⋊C8)C22semidirect
47. #22583521C22semidirect
185. C283#3311semidirect

Maximal subgroups

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

subgroupindex
#318271852
#318268242
#318268032
#318267862
#318267552
#318267212
#318266922
#318266842
#318266272
#318266162
#318265652
#318265372
#318264702
#318264612
#546×C2832
#3311283