Finite Group Databasebeta

← #49133366

Subgroups — #49133366

Order 36224.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827124C211yesmaximal, characteristic
2#31826827C211yesmaximal, characteristic, complemented
3#31826795C211yesmaximal, characteristic, complemented
4#31826783C211yesmaximal, characteristic, complemented
5#31826747C211yesmaximal, characteristic, complemented
6#31826721C211yesmaximal, characteristic, complemented
7#31826696C211yesmaximal, characteristic, complemented
8#31826687C211yesmaximal, characteristic, complemented
9#31826625C211yesmaximal, characteristic, complemented
10#31826620C211yesmaximal, characteristic, complemented
11#31826570C211yesmaximal, characteristic, complemented
12#31826537C211yesmaximal, characteristic, complemented
13#31826464C211yesmaximal, characteristic, complemented
14C2×#22583533C211yesmaximal, characteristic, complemented
15#554×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×Dic566C2211yescharacteristic
18C2×#17985382C2211yescharacteristic
19C2×#17985382C2211yescharacteristic
20C2×#17985379C2211yescharacteristic
21C2×#17985376C2211yescharacteristic
22C22×C283⋊C8C2211yescharacteristic
23#22583598C2211yescharacteristic
24#22583596—21no—
25#22583589C2211yescharacteristic
26#22583588—21no—
27#22583584C2211yescharacteristic
28#22583582—21no—
29C4×Dic566C2211yescharacteristic
30(C22⋊D283)⋊C4—21no—
31#22583541C2211yescharacteristic
32#22583541C2211yescharacteristic
33#22583541C2211yescharacteristic
34C4⋊(C283⋊C8)C2211yescharacteristic
35#22583534C2211yescharacteristic
36#22583533C2211yescharacteristic
37#22583533C2211yescharacteristic
38#22583533C2211yescharacteristic
39#22583533C2211yescharacteristic
40#22583533C2211yescharacteristic
41#22583533C2211yescharacteristic
42#22583530C2211yescharacteristic
43#22583530C2211yescharacteristic
44#22583530C2211yescharacteristic
45C22⋊(C283⋊C8)C2211yescharacteristic
46C22⋊(C283⋊C8)C2211yescharacteristic
47#22583521C2211yescharacteristic
48C4⋊Q8×C283C2211yescharacteristic
49#125×C283—21no—
50C22⋊Q8×C283—21no—
51C22⋊Q8×C283—21no—
52C22⋊Q8×C283C2211yescharacteristic
53C22⋊Q8×C283C2211yescharacteristic
54C4×Q8×C283C2211yescharacteristic
55C4×D4×C283—21no—
56C4×D4×C283C2211yescharacteristic
57C2×C4⋊C4×C283C2211yescharacteristic
58C2×C4⋊C4×C283C2211yescharacteristic
59#17985392—22no—
60#17985392—41no—
61C2×Dic566—22no—
62C2×Dic566—21no—
63C2×Dic566—41no—
64C2×Dic566C2311yescharacteristic
65C2×Dic566C2311yescharacteristic
66C2×D1132C2311yescharacteristic
67C2×C22⋊D283—21no—
68C2×C22⋊D283—21no—
69C22×Dic283—21no—
70C2×C4×D283—21no—
71#17985382—22no—
72#17985382—41no—
73#17985379—41no—
74#17985376—41no—
75C2×C283⋊C8—41no—
76C2×C283⋊C8C2311yescharacteristic
77C2×C283⋊C8C2311yescharacteristic
78C2×C283⋊C8C2311yescharacteristic
79C2×C283⋊C8C2311yescharacteristic
80C4⋊Dic283C2311yescharacteristic
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83C22⋊Dic283—21no—
84(C2×D283)⋊C4—21no—
85(C2×D283)⋊C4—21no—
86(C2×D283)⋊C4—21no—
87C4×Dic283—21no—
88C2×Q8×C283—21no—
89C2×Q8×C283C2311yescharacteristic
90C2×D4×C283D411yescharacteristic
91C22×C4×C283—21no—
92C22×C4×C283—21no—
93C22×C4×C283C2311yescharacteristic
94C22×C4×C283C2311yescharacteristic
95C4⋊C4×C283—21no—
96C4⋊C4×C283—21no—
97C4⋊C4×C283—21no—
98C4⋊C4×C283—21no—
99C4⋊C4×C283—21no—
100C4⋊C4×C283—21no—
101C4⋊C4×C283D411yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C4⋊C4×C283C2311yescharacteristic
104C4⋊C4×C283C2311yescharacteristic
105C22⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C22⋊C4×C283D411yescharacteristic
108C22⋊C4×C283D411yescharacteristic
109C42×C283—21no—
110C42×C283C2311yescharacteristic
111C22×D283—21no—
112Dic566—21no—
113Dic566—21no—
114Dic566—21no—
115Dic566—42no—
116Dic566#4112yes—
117D1132—21no—
118D1132—21no—
119C22⋊D283—42no—
120C22⋊D283—42no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C2×Dic283—21no—
124C2×Dic283—42no—
125C4×D283—42no—
126C283⋊C8—21no—
127C283⋊C8—21no—
128C283⋊C8—21no—
129C283⋊C8—21no—
130C23×C283C2×D411yescharacteristic
131C23×C283C2×D411yescharacteristic
132Q8×C283—21no—
133Q8×C283—21no—
134Q8×C283—81no—
135D4×C283—22no—
136C2×C4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283—21no—
139C2×C4×C283—21no—
140C2×C4×C283—21no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—81no—
145C2×C4×C283—81no—
146C2×C4×C283C2×D411yescharacteristic
147C2×C4×C283C2×D411yescharacteristic
148C2×C4×C283C2×D411yescharacteristic
149C2×C4×C283C2×D411yescharacteristic
150C2×C4×C283C2411yescharacteristic
151C22×C283—21no—
152C22×C283—21no—
153C22×C283—21no—
154C22×C283—21no—
155C22×C283—21no—
156C22×C283#13712yes—
157C22×C283C22×D411yescharacteristic
158C2×D283—41no—
159C2×D283—41no—
160C2×D283—41no—
161C4×C283—21no—
162C4×C283—21no—
163C4×C283—21no—
164C4×C283—21no—
165C4×C283—41no—
166C4×C283—41no—
167C4×C283—41no—
168C4×C283—41no—
169C4×C283C2×#4111yescharacteristic
170C4×C283H(5,𝔽2)11yescharacteristic
171Dic283—22no—
172Dic283—41no—
173Dic283—41no—
174C2×C283—22no—
175C2×C283—41no—
176C2×C283#54611yescharacteristic
177C2×C283C2×#13711yescharacteristic
178C2×C283#57711yescharacteristic
179D283—81no—
180C283#331611yescharacteristic, complemented, Sylow
181#3316—2831nomaximal, Sylow
182#554D28311yescharacteristic, complemented
183#548—2831no—
184#489—2831no—
185#483—2831no—
186#481—2831no—
187#475—2831no—
188#470—2831no—
189#465—2831no—
190#464—2831no—
191#452—2831no—
192#451—2831no—
193#441—2831no—
194#435—2831no—
195#417—2831no—
196C2×#103—2831no—
197C2×#41—2831no—
198C22×Q8—2831no—
199C2×Q16—2831no—
200C2×Q16—2831no—
201C2×SD16—2831no—
202C2×M16—2831no—
203C22×C8—2831no—
204C4⋊Q8C2×D28311yescharacteristic
205#125—21no—
206#124—2831no—
207#124—5661no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—2831no—
211C22⋊Q8—5661no—
212C22⋊Q8C2×D28311yescharacteristic
213C22⋊Q8C2×D28311yescharacteristic
214#121—2831no—
215#121—5661no—
216C4×Q8—2831no—
217C4×Q8C2×D28311yescharacteristic
218C4×D4—21no—
219C4×D4—5661no—
220C4×D4C2×D28311yescharacteristic
221C2×C4⋊C4C2×D28311yescharacteristic
222C2×C4⋊C4C2×D28311yescharacteristic
223#106—2831no—
224#106—2831no—
225#106—2831no—
226C4⋊C8—2831no—
227#103—2831no—
228#103—2831no—
229#103—2831no—
230#103—2831no—
231#103—2831no—
232#103—2831no—
233#103—2831no—
234#102—2831no—
235#102—2831no—
236#102—2831no—
237C22⋊C8—2831no—
238C22⋊C8—2831no—
239#97—2831no—
240#41—5662no—
241#41—11321no—
242C2×Q8—21no—
243C2×Q8—2831no—
244C2×Q8—2831no—
245C2×Q8—5662no—
246C2×Q8—5661no—
247C2×Q8—11321no—
248C2×Q8C22×D28311yescharacteristic
249C2×D4—2831no—
250C2×D4—5661no—
251C2×D4—5661no—
252C2×D4C22⋊D28311yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—5661no—
256C22×C4—5661no—
257C22×C4C22×D28311yescharacteristic
258C22×C4C22×D28311yescharacteristic
259Q16—5662no—
260Q16—11321no—
261SD16—11321no—
262M16—11321no—
263C2×C8—2831no—
264C2×C8—2831no—
265C2×C8—2831no—
266C2×C8—2831no—
267C2×C8—11321no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—2831no—
275C4⋊C4—5661no—
276C4⋊C4—5661no—
277C4⋊C4C22⋊D28311yescharacteristic
278C4⋊C4C22×D28311yescharacteristic
279C4⋊C4C22×D28311yescharacteristic
280C4⋊C4C22×D28311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—5661no—
284C22⋊C4—5661no—
285C22⋊C4—5661no—
286C22⋊C4—5661no—
287C22⋊C4C22⋊D28311yescharacteristic
288C22⋊C4C22⋊D28311yescharacteristic
289C42—21no—
290C42—5661no—
291C42C22×D28311yescharacteristic
292C23—5661no—
293C23C2×C22⋊D28311yescharacteristic
294C23C2×C22⋊D28311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—81no—
298Q8—2832no—
299Q8—5661no—
300Q8—5661no—
301Q8—5661no—
302Q8—11322no—
303D4—22no—
304D4—5661no—
305D4—5661no—
306D4—11322no—
307D4—11322no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—5661no—
319C2×C4—5661no—
320C2×C4—5661no—
321C2×C4—11322no—
322C2×C4—11322no—
323C2×C4C2×C22⋊D28311yescharacteristic
324C2×C4C2×C22⋊D28311yescharacteristic
325C2×C4C2×C22⋊D28311yescharacteristic
326C2×C4C2×C22⋊D28311yescharacteristic
327C2×C4C23×D28311yescharacteristic
328C8—5661no—
329C8—5661no—
330C8—5661no—
331C8—5661no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—11321no—
338C22—11321no—
339C22—11321no—
340C22#2258364112yes—
341C22C22×C22⋊D28311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—5662no—
351C4—11321no—
352C4—11321no—
353C4#41×D28311yescharacteristic
354C4#2258365611yescharacteristic
355C2—22no—
356C2—41no—
357C2—22641no—
358C2#3182710311yescharacteristic
359C2C2×#2258364111yescharacteristic
360C2#3182727611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#554×C283C2
socleC22×C283C22×D4
prime radical#31827124C2
hypercenter#554D283
hypocenterC283#3316
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #31826827C2semidirect
3. #31826795C2semidirect
4. #31826783C2semidirect
5. #31826747C2semidirect
6. #31826721C2semidirect
7. #31826696C2semidirect
8. #31826687C2semidirect
9. #31826625C2semidirect
10. #31826620C2semidirect
11. #31826570C2semidirect
12. #31826537C2semidirect
13. #31826464C2semidirect
14. C2×#22583533C2semidirect
15. #554×C283C2semidirect
180. C283#3316semidirect
182. #554D283semidirect

Maximal subgroups

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

subgroupindex
#318271242
#318268272
#318267952
#318267832
#318267472
#318267212
#318266962
#318266872
#318266252
#318266202
#318265702
#318265372
#318264642
C2×#225835332
#554×C2832
#3316283