Finite Group Databasebeta

← #49133142

Subgroups — #49133142

Order 36224.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827124C211yesmaximal, characteristic
2#31826822C211yesmaximal, characteristic, complemented
3#31826806C211yesmaximal, characteristic, complemented
4#31826778C211yesmaximal, characteristic, complemented
5#31826769C211yesmaximal, characteristic, complemented
6#31826704C211yesmaximal, characteristic, complemented
7#31826696C211yesmaximal, characteristic, complemented
8#31826687C211yesmaximal, characteristic, complemented
9#31826625C211yesmaximal, characteristic, complemented
10#31826620C211yesmaximal, characteristic, complemented
11C4×#17985382C211yesmaximal, characteristic, complemented
12D4×C283⋊C8C211yesmaximal, characteristic, complemented
13#31826459C211yesmaximal, characteristic, complemented
14C2×#22583533C211yesmaximal, characteristic, complemented
15#552×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×Dic566C2211yescharacteristic
18C2×#17985382C2211yescharacteristic
19C2×#17985382C2211yescharacteristic
20C2×#17985379C2211yescharacteristic
21C22×C283⋊C8C2211yescharacteristic
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#22583543C2211yescharacteristic
32#22583543C2211yescharacteristic
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
47C4×C283⋊C8C2211yescharacteristic
48#126×C283—21no—
49#125×C283—21no—
50#125×C283C2211yescharacteristic
51#123×C283—21no—
52C22⋊Q8×C283C2211yescharacteristic
53C22⋊Q8×C283C2211yescharacteristic
54C4×Q8×C283C2211yescharacteristic
55C4×D4×C283C2211yescharacteristic
56#117×C283—21no—
57#117×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—
74C2×C283⋊C8—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×C283C2311yescharacteristic
89C2×D4×C283D411yescharacteristic
90C22×C4×C283—21no—
91C22×C4×C283C2311yescharacteristic
92C22×C4×C283C2311yescharacteristic
93C4⋊C4×C283—21no—
94C4⋊C4×C283—21no—
95C4⋊C4×C283—21no—
96C4⋊C4×C283—21no—
97C4⋊C4×C283—21no—
98C4⋊C4×C283—21no—
99C4⋊C4×C283D411yescharacteristic
100C4⋊C4×C283C2311yescharacteristic
101C4⋊C4×C283C2311yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C22⋊C4×C283—21no—
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C22⋊C4×C283D411yescharacteristic
107C22⋊C4×C283D411yescharacteristic
108C42×C283—21no—
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—
134D4×C283—22no—
135C2×C4×C283—21no—
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—42no—
144C2×C4×C283C2×D411yescharacteristic
145C2×C4×C283C2×D411yescharacteristic
146C2×C4×C283C2×D411yescharacteristic
147C2×C4×C283C2×D411yescharacteristic
148C2×C4×C283C2411yescharacteristic
149C22×C283—21no—
150C22×C283—21no—
151C22×C283—21no—
152C22×C283—21no—
153C22×C283—21no—
154C22×C283#13512yes—
155C22×C283C22×D411yescharacteristic
156C2×D283—41no—
157C2×D283—41no—
158C2×D283—41no—
159C4×C283—21no—
160C4×C283—21no—
161C4×C283—21no—
162C4×C283—21no—
163C4×C283—41no—
164C4×C283—41no—
165C4×C283—41no—
166C4×C283—41no—
167C4×C283C2×#4111yescharacteristic
168C4×C283H(5,𝔽2)11yescharacteristic
169Dic283—22no—
170Dic283—41no—
171Dic283—41no—
172C2×C283—22no—
173C2×C283—41no—
174C2×C283#54611yescharacteristic
175C2×C283C2×#13511yescharacteristic
176C2×C283#57811yescharacteristic
177D283—81no—
178C283#330011yescharacteristic, complemented, Sylow
179#3300—2831nomaximal, Sylow
180#552D28311yescharacteristic, complemented
181#548—2831no—
182#488—2831no—
183#485—2831no—
184#480—2831no—
185#479—2831no—
186#467—2831no—
187#465—2831no—
188#464—2831no—
189#452—2831no—
190#451—2831no—
191C4×Q16—2831no—
192C8×D4—2831no—
193#416—2831no—
194C2×#103—2831no—
195C2×#41—2831no—
196C22×Q8—2831no—
197C2×Q16—2831no—
198C2×Q16—2831no—
199C2×SD16—2831no—
200C22×C8—2831no—
201C22×C8—2831no—
202#126—21no—
203#125—21no—
204#125C2×D28311yescharacteristic
205#124—2831no—
206#124—5661no—
207#123—21no—
208C22⋊Q8—2831no—
209C22⋊Q8—5661no—
210C22⋊Q8C2×D28311yescharacteristic
211C22⋊Q8C2×D28311yescharacteristic
212#121—2831no—
213#121—5661no—
214C4×Q8—2831no—
215C4×Q8C2×D28311yescharacteristic
216C4×D4—5661no—
217C4×D4C2×D28311yescharacteristic
218#117—21no—
219#117C2×D28311yescharacteristic
220C2×C4⋊C4C2×D28311yescharacteristic
221#107—2831no—
222#107—2831no—
223#106—2831no—
224C4⋊C8—2831no—
225#103—2831no—
226#103—2831no—
227#103—2831no—
228#103—2831no—
229#103—2831no—
230#103—2831no—
231#103—2831no—
232#102—2831no—
233#102—2831no—
234#102—2831no—
235C22⋊C8—2831no—
236C22⋊C8—2831no—
237C4×C8—2831no—
238#41—5662no—
239#41—11321no—
240C2×Q8—2831no—
241C2×Q8—2831no—
242C2×Q8—5662no—
243C2×Q8—5661no—
244C2×Q8—11321no—
245C2×Q8C22×D28311yescharacteristic
246C2×D4—2831no—
247C2×D4—5661no—
248C2×D4—5661no—
249C2×D4C22⋊D28311yescharacteristic
250C22×C4—21no—
251C22×C4—5661no—
252C22×C4—5661no—
253C22×C4C22×D28311yescharacteristic
254C22×C4C22×D28311yescharacteristic
255Q16—5662no—
256Q16—11321no—
257SD16—11321no—
258C2×C8—2831no—
259C2×C8—2831no—
260C2×C8—2831no—
261C2×C8—2831no—
262C2×C8—11321no—
263C2×C8—11321no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—2831no—
271C4⋊C4—5661no—
272C4⋊C4—5661no—
273C4⋊C4C22⋊D28311yescharacteristic
274C4⋊C4C22×D28311yescharacteristic
275C4⋊C4C22×D28311yescharacteristic
276C4⋊C4C22×D28311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—5661no—
281C22⋊C4—5661no—
282C22⋊C4—5661no—
283C22⋊C4—5661no—
284C22⋊C4C22⋊D28311yescharacteristic
285C22⋊C4C22⋊D28311yescharacteristic
286C42—21no—
287C42—21no—
288C42—5661no—
289C42C22×D28311yescharacteristic
290C23—5661no—
291C23C2×C22⋊D28311yescharacteristic
292C23C2×C22⋊D28311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2832no—
296Q8—5661no—
297Q8—5661no—
298Q8—5661no—
299Q8—11322no—
300D4—22no—
301D4—5661no—
302D4—5661no—
303D4—11322no—
304D4—11322no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—5661no—
315C2×C4—5661no—
316C2×C4—5661no—
317C2×C4—11322no—
318C2×C4—11322no—
319C2×C4C2×C22⋊D28311yescharacteristic
320C2×C4C2×C22⋊D28311yescharacteristic
321C2×C4C2×C22⋊D28311yescharacteristic
322C2×C4C2×C22⋊D28311yescharacteristic
323C2×C4C23×D28311yescharacteristic
324C8—5661no—
325C8—5661no—
326C8—5661no—
327C8—5661no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—11321no—
334C22—11321no—
335C22—11321no—
336C22#2258363012yes—
337C22C22×C22⋊D28311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—5662no—
347C4—11321no—
348C4—11321no—
349C4#41×D28311yescharacteristic
350C4#2258365611yescharacteristic
351C2—22no—
352C2—41no—
353C2—22641no—
354C2#3182710311yescharacteristic
355C2C2×#2258363011yescharacteristic
356C2#3182728011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#552×C283C2
socleC22×C283C22×D4
prime radical#31827124C2
hypercenter#552D283
hypocenterC283#3300
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #31826822C2semidirect
3. #31826806C2semidirect
4. #31826778C2semidirect
5. #31826769C2semidirect
6. #31826704C2semidirect
7. #31826696C2semidirect
8. #31826687C2semidirect
9. #31826625C2semidirect
10. #31826620C2semidirect
11. C4×#17985382C2semidirect
12. D4×C283⋊C8C2semidirect
13. #31826459C2semidirect
14. C2×#22583533C2semidirect
15. #552×C283C2semidirect
178. C283#3300semidirect
180. #552D283semidirect

Maximal subgroups

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

subgroupindex
#318271242
#318268222
#318268062
#318267782
#318267692
#318267042
#318266962
#318266872
#318266252
#318266202
C4×#179853822
D4×C283⋊C82
#318264592
C2×#225835332
#552×C2832
#3300283