Finite Group Databasebeta

← #49131555

Subgroups — #49131555

Order 36224.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827075C211yesmaximal, characteristic
2#31826943C211yesmaximal, characteristic
3#31826807C211yesmaximal, characteristic
4#31826803C211yesmaximal, characteristic
5#31826774C211yesmaximal, characteristic, complemented
6#31826770C211yesmaximal, characteristic, complemented
7#31826673C211yesmaximal, characteristic, complemented
8#31826673C211yesmaximal, characteristic, complemented
9#31826621C211yesmaximal, characteristic
10#31826572C211yesmaximal, characteristic, complemented
11#31826569C211yesmaximal, characteristic, complemented
12Q16×Dic283C211yesmaximal, characteristic, complemented
13#31826557C211yesmaximal, characteristic, complemented
14#31826523C211yesmaximal, characteristic
15#451×C283C211yesmaximal, characteristic
16C2×#17985382C2211yescharacteristic
17C2×#17985382C2211yescharacteristic
18#22583608C2211yescharacteristic
19#22583603—21no—
20#22583603—21no—
21#22583601—21no—
22#22583601C2211yescharacteristic
23#22583600—21no—
24#22583589C2211yescharacteristic
25Q8×Dic283—21no—
26Q8×Dic283—21no—
27Q8×Dic283C2211yescharacteristic
28Q8×Dic283C2211yescharacteristic
29Q8×Dic283C2211yescharacteristic
30Q8×Dic283C2211yescharacteristic
31Q8×Dic283C2211yescharacteristic
32#22583575C2211yescharacteristic
33#22583569—21no—
34#22583569—21no—
35#22583569C2211yescharacteristic
36#22583569C2211yescharacteristic
37#22583544C2211yescharacteristic
38#22583543C2211yescharacteristic
39#22583542C2211yescharacteristic
40#22583541C2211yescharacteristic
41Dic283⋊C8C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583534C2211yescharacteristic
47#22583534C2211yescharacteristic
48#22583534C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50#22583520C2211yescharacteristic
51C8×Dic283C2211yescharacteristic
52C22×Q8×C283C2211yescharacteristic
53C2×Q16×C283C2211yescharacteristic
54C2×Q16×C283C2211yescharacteristic
55C22⋊Q8×C283C2211yescharacteristic
56#103×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58C22⋊C8×C283C2211yescharacteristic
59C2×Dic566C2311yescharacteristic
60#17985382—22no—
61#17985382—22no—
62C2×C283⋊C8C2311yescharacteristic
63C2×C283⋊C8C2311yescharacteristic
64C4⋊Dic283—21no—
65C4⋊Dic283—21no—
66C4⋊Dic283—21no—
67C4⋊Dic283—21no—
68C4⋊Dic283—21no—
69C4⋊Dic283—21no—
70C4⋊Dic283D411yescharacteristic
71C4⋊Dic283D411yescharacteristic
72C4⋊Dic283C2311yescharacteristic
73C4⋊Dic283C2311yescharacteristic
74C4⋊Dic283C2311yescharacteristic
75Dic283⋊C4—21no—
76Dic283⋊C4—21no—
77Dic283⋊C4—21no—
78Dic283⋊C4—21no—
79Dic283⋊C4—21no—
80C22⋊Dic283—21no—
81C22⋊Dic283—21no—
82C22⋊Dic283D411yescharacteristic
83C22⋊Dic283D411yescharacteristic
84C4×Dic283—21no—
85C4×Dic283—21no—
86C4×Dic283—21no—
87C4×Dic283—21no—
88C4×Dic283—21no—
89C4×Dic283—21no—
90C4×Dic283—21no—
91C4×Dic283C2311yescharacteristic
92C4×Dic283C2311yescharacteristic
93C2×Q8×C283—22no—
94C2×Q8×C283—22no—
95C2×Q8×C283—21no—
96C2×Q8×C283C2311yescharacteristic
97C2×Q8×C283C2311yescharacteristic
98C2×Q8×C283C2311yescharacteristic
99C22×C4×C283—21no—
100C22×C4×C283C2311yescharacteristic
101Q16×C283—22no—
102Q16×C283—22no—
103C2×C8×C283C2311yescharacteristic
104C2×C8×C283C2311yescharacteristic
105C4⋊C4×C283—21no—
106C4⋊C4×C283C2311yescharacteristic
107C22⋊C4×C283—21no—
108Dic566—21no—
109Dic566—21no—
110C2×Dic283—21no—
111C2×Dic283—21no—
112C2×Dic283—21no—
113C2×Dic283—21no—
114C2×Dic283C2×D411yescharacteristic
115C2×Dic283C2×D411yescharacteristic
116C2×Dic283C2×D411yescharacteristic
117C2×Dic283C2×D411yescharacteristic
118C283⋊C8—21no—
119C283⋊C8—21no—
120C23×C283C2×D411yescharacteristic
121Q8×C283—24no—
122Q8×C283—21no—
123Q8×C283—21no—
124Q8×C283—21no—
125Q8×C283—21no—
126Q8×C283#4112yes—
127Q8×C283#4112yes—
128C2×C4×C283—24no—
129C2×C4×C283—21no—
130C2×C4×C283—21no—
131C2×C4×C283—21no—
132C2×C4×C283—21no—
133C2×C4×C283—21no—
134C2×C4×C283—21no—
135C2×C4×C283C2×D411yescharacteristic
136C2×C4×C283C2411yescharacteristic
137C8×C283—21no—
138C8×C283—21no—
139C22×C283—21no—
140C22×C283—21no—
141C22×C283—21no—
142C22×C283C22×D411yescharacteristic
143C4×C283—22no—
144C4×C283—22no—
145C4×C283—21no—
146C4×C283—41no—
147C4×C283—41no—
148C4×C283C2×#4111yescharacteristic
149C4×C283C2×#4111yescharacteristic
150Dic283—21no—
151Dic283—21no—
152Dic283—21no—
153Dic283—21no—
154Dic283—41no—
155Dic283—41no—
156Dic283—41no—
157Dic283—41no—
158C2×C283—41no—
159C2×C283#52511yescharacteristic
160C2×C283#57811yescharacteristic
161C2×C283#57811yescharacteristic
162C283#312711yescharacteristic, complemented, Sylow
163#3127—2831nomaximal, Sylow
164#541—2831no—
165#519—2831no—
166#485—2831no—
167#485—2831no—
168#479—2831no—
169#479—2831no—
170#462—2831no—
171#462—2831no—
172#451—2831no—
173#451D28311yescharacteristic
174#441—2831no—
175#441—2831no—
176C4×Q16—2831no—
177C4×Q16—2831no—
178#433—2831no—
179C22×Q8C2×D28311yescharacteristic
180C2×Q16—2831no—
181C2×Q16—2831no—
182C2×Q16C2×D28311yescharacteristic
183C2×Q16C2×D28311yescharacteristic
184C4⋊Q8—2831no—
185#126—5661no—
186#126—5661no—
187#125—2831no—
188#125—5661no—
189#125—5661no—
190C22⋊Q8—2831no—
191C22⋊Q8C2×D28311yescharacteristic
192C4×Q8—2831no—
193C4×Q8—2831no—
194C4×Q8—2831no—
195C4×Q8—2831no—
196C4×Q8—2831no—
197C4×Q8—2831no—
198C4×Q8—5661no—
199C4×Q8—5661no—
200#117—2831no—
201#117—2831no—
202#117—5661no—
203#117—5661no—
204#107—2831no—
205#107—2831no—
206#106—2831no—
207#106—2831no—
208C4⋊C8—2831no—
209C4⋊C8—2831no—
210#103—2831no—
211#103—2831no—
212#103—2831no—
213#103—2831no—
214#103—2831no—
215#103—2831no—
216#103C2×D28311yescharacteristic
217#103C2×D28311yescharacteristic
218C22⋊C8—2831no—
219C22⋊C8C2×D28311yescharacteristic
220#97—2831no—
221C4×C8—2831no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—2831no—
226C2×Q8C22×D28311yescharacteristic
227C2×Q8C22×D28311yescharacteristic
228C2×Q8C22×D28311yescharacteristic
229C22×C4—21no—
230C22×C4C22×D28311yescharacteristic
231Q16—22no—
232Q16—22no—
233Q16—5662no—
234Q16—5662no—
235C2×C8—2831no—
236C2×C8—2831no—
237C2×C8C22×D28311yescharacteristic
238C2×C8C22×D28311yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—2831no—
241C4⋊C4—2831no—
242C4⋊C4—2831no—
243C4⋊C4—2831no—
244C4⋊C4—2831no—
245C4⋊C4—5661no—
246C4⋊C4—5661no—
247C4⋊C4—5661no—
248C4⋊C4—5661no—
249C4⋊C4—5661no—
250C4⋊C4—5661no—
251C4⋊C4—5661no—
252C4⋊C4—5661no—
253C4⋊C4—5661no—
254C4⋊C4—5661no—
255C4⋊C4—5661no—
256C4⋊C4C22×D28311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—2831no—
259C22⋊C4—2831no—
260C22⋊C4—5661no—
261C22⋊C4—5661no—
262C42—2831no—
263C42—2831no—
264C42—5661no—
265C42—5661no—
266C42—5661no—
267C42—5661no—
268C42—5661no—
269C42—5661no—
270C42—5661no—
271C23D4×D28311yescharacteristic
272Q8—24no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—5661no—
278Q8—5661no—
279Q8#1798539112yes—
280Q8#1798539112yes—
281C2×C4—24no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—2831no—
289C2×C4—2831no—
290C2×C4—2831no—
291C2×C4—2831no—
292C2×C4—5661no—
293C2×C4—5661no—
294C2×C4—5661no—
295C2×C4—5661no—
296C2×C4D4×D28311yescharacteristic
297C2×C4C23×D28311yescharacteristic
298C8—21no—
299C8—21no—
300C8—5661no—
301C8—5661no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22C2×D4×D28311yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—5661no—
312C4—5661no—
313C4—5661no—
314C4—5661no—
315C4—11321no—
316C4—11321no—
317C4—11321no—
318C4—11321no—
319C4C2×#1798539111yescharacteristic
320C4C2×#1798539111yescharacteristic
321C2—41no—
322C2#3182697111yescharacteristic
323C2#3182727811yescharacteristic
324C2#3182728111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#451×C283C2
socleC22×C283C22×D4
prime radicalC23×C283C2×D4
hypercenter#451D283
hypocenterC283#3127
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
5. #31826774C2semidirect
6. #31826770C2semidirect
7. #31826673C2semidirect
8. #31826673C2semidirect
10. #31826572C2semidirect
11. #31826569C2semidirect
12. Q16×Dic283C2semidirect
13. #31826557C2semidirect
162. C283#3127semidirect

Maximal subgroups

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

subgroupindex
#318270752
#318269432
#318268072
#318268032
#318267742
#318267702
#318266732
#318266212
#318265722
#318265692
Q16×Dic2832
#318265572
#318265232
#451×C2832
#3127283