Finite Group Databasebeta

← #49131860

Subgroups — #49131860

Order 36224.

364 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827017C211yesmaximal, characteristic, complemented
2#31826996C211yesmaximal, characteristic
3#31826873C211yesmaximal, characteristic, complemented
4#31826856C211yesmaximal, characteristic, complemented
5#31826832C211yesmaximal, characteristic, complemented
6#31826823C211yesmaximal, characteristic, complemented
7#31826723C211yesmaximal, characteristic, complemented
8#31826721C211yesmaximal, characteristic, complemented
9#31826709C211yesmaximal, characteristic
10#31826559C211yesmaximal, characteristic, complemented
11#31826557C211yesmaximal, characteristic, complemented
12#31826556C211yesmaximal, characteristic, complemented
13SD16×Dic283C211yesmaximal, characteristic, complemented
14M16×Dic283C211yesmaximal, characteristic, complemented
15#468×C283C211yesmaximal, characteristic
16C2×Dic1132C2211yescharacteristic
17C2×#17985382C2211yescharacteristic
18C2×#17985380C2211yescharacteristic
19C2×#17985376C2211yescharacteristic
20#22583609C2211yescharacteristic
21#22583608C2211yescharacteristic
22#22583603—21no—
23#22583601C2211yescharacteristic
24#22583600—21no—
25#22583597C2211yescharacteristic
26#22583591—21no—
27#22583589C2211yescharacteristic
28#22583589C2211yescharacteristic
29#22583587—21no—
30Q8×Dic283C2211yescharacteristic
31#22583575—21no—
32#22583575C2211yescharacteristic
33#22583575C2211yescharacteristic
34D4×Dic283—21no—
35D4×Dic283C2211yescharacteristic
36#22583566—21no—
37#22583566—21no—
38C2×C4×Dic283C2211yescharacteristic
39#22583543C2211yescharacteristic
40#22583542C2211yescharacteristic
41#22583541C2211yescharacteristic
42#22583535C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583534C2211yescharacteristic
45#22583533C2211yescharacteristic
46#22583533C2211yescharacteristic
47#22583531C2211yescharacteristic
48#22583520C2211yescharacteristic
49#22583520C2211yescharacteristic
50C8×Dic283C2211yescharacteristic
51C8×Dic283C2211yescharacteristic
52C2×SD16×C283C2211yescharacteristic
53C2×M16×C283C2211yescharacteristic
54C22⋊Q8×C283C2211yescharacteristic
55#121×C283C2211yescharacteristic
56#107×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59C2×Dic566—21no—
60C2×Dic566C2311yescharacteristic
61C2×Dic566C2311yescharacteristic
62C22×Dic283—21no—
63C22×Dic283D411yescharacteristic
64C22×Dic283D411yescharacteristic
65Dic1132—22no—
66#17985382—22no—
67#17985380—22no—
68#17985376—22no—
69C2×C283⋊C8C2311yescharacteristic
70C2×C283⋊C8C2311yescharacteristic
71C4⋊Dic283—21no—
72C4⋊Dic283—21no—
73C4⋊Dic283C2311yescharacteristic
74C4⋊Dic283C2311yescharacteristic
75Dic283⋊C4—21no—
76Dic283⋊C4—21no—
77Dic283⋊C4—21no—
78Dic283⋊C4—21no—
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83C22⋊Dic283—21no—
84C22⋊Dic283—21no—
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88C4×Dic283—21no—
89C4×Dic283—21no—
90C4×Dic283—21no—
91C4×Dic283D411yescharacteristic
92C4×Dic283D411yescharacteristic
93C4×Dic283C2311yescharacteristic
94C4×Dic283C2311yescharacteristic
95C2×Q8×C283C2311yescharacteristic
96C2×D4×C283—21no—
97C2×D4×C283C2311yescharacteristic
98C22×C4×C283C2311yescharacteristic
99SD16×C283—22no—
100M16×C283—22no—
101C2×C8×C283C2311yescharacteristic
102C2×C8×C283C2311yescharacteristic
103C4⋊C4×C283—21no—
104C4⋊C4×C283C2311yescharacteristic
105C4⋊C4×C283C2311yescharacteristic
106C22⋊C4×C283—21no—
107C22⋊C4×C283—21no—
108Dic566—21no—
109Dic566—21no—
110Dic566—21no—
111Dic566—21no—
112Dic566—42no—
113C2×Dic283—22no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283—21no—
117C2×Dic283—21no—
118C2×Dic283—41no—
119C2×Dic283—42no—
120C2×Dic283C2×D411yescharacteristic
121C2×Dic283C2×D411yescharacteristic
122C2×Dic283C2×D411yescharacteristic
123C2×Dic283C2×D411yescharacteristic
124C283⋊C8—21no—
125C283⋊C8#4112yes—
126C23×C283—21no—
127C23×C283C2×D411yescharacteristic
128Q8×C283—21no—
129Q8×C283—21no—
130D4×C283—21no—
131D4×C283—21no—
132D4×C283—42no—
133C2×C4×C283—21no—
134C2×C4×C283—21no—
135C2×C4×C283—21no—
136C2×C4×C283—21no—
137C2×C4×C283—21no—
138C2×C4×C283C2×D411yescharacteristic
139C2×C4×C283C2411yescharacteristic
140C8×C283—21no—
141C8×C283#4112yes—
142C22×C283—21no—
143C22×C283—21no—
144C22×C283—21no—
145C22×C283—41no—
146C22×C283—41no—
147C22×C283—41no—
148C22×C283C22×D411yescharacteristic
149C4×C283—21no—
150C4×C283—41no—
151C4×C283—41no—
152C4×C283—41no—
153C4×C283C2×#4111yescharacteristic
154C4×C283C2×#4111yescharacteristic
155Dic283—21no—
156Dic283—21no—
157Dic283—21no—
158Dic283—41no—
159Dic283—41no—
160Dic283—41no—
161Dic283—41no—
162Dic283#13712yes—
163C2×C283—41no—
164C2×C283—81no—
165C2×C283#53211yescharacteristic
166C2×C283C2×#13711yescharacteristic
167C2×C283#57511yescharacteristic
168C283#317211yescharacteristic, complemented, Sylow
169#3172—2831nomaximal, Sylow
170#533—2831no—
171#529—2831no—
172#501—2831no—
173#497—2831no—
174#491—2831no—
175#489—2831no—
176#470—2831no—
177#470—2831no—
178#468—2831no—
179#468D28311yescharacteristic
180C4×Q16—2831no—
181C4×Q16—2831no—
182C4×SD16—2831no—
183C4×SD16—2831no—
184C4×M16—2831no—
185C2×Q16—2831no—
186C2×Q16—2831no—
187C2×SD16—2831no—
188C2×SD16C2×D28311yescharacteristic
189C2×M16—2831no—
190C2×M16C2×D28311yescharacteristic
191C4⋊Q8—2831no—
192C4⋊Q8—2831no—
193#126—5661no—
194#125—2831no—
195#125—5661no—
196#124—2831no—
197#123—5661no—
198C22⋊Q8—2831no—
199C22⋊Q8—2831no—
200C22⋊Q8—5661no—
201C22⋊Q8C2×D28311yescharacteristic
202#121C2×D28311yescharacteristic
203C4×Q8—2831no—
204C4×Q8—2831no—
205C4×Q8—2831no—
206C4×Q8—5661no—
207C4×D4—2831no—
208C4×D4—5661no—
209#117—5661no—
210#117—5661no—
211C2×C42—2831no—
212#107—2831no—
213#107C2×D28311yescharacteristic
214#106—2831no—
215#106—2831no—
216#103—2831no—
217#103—2831no—
218#103—2831no—
219#103—2831no—
220#103—2831no—
221#103C2×D28311yescharacteristic
222#102—2831no—
223#102C2×D28311yescharacteristic
224#97—2831no—
225#97—2831no—
226C4×C8—2831no—
227C4×C8—2831no—
228C2×Q8—2831no—
229C2×Q8—2831no—
230C2×Q8—5661no—
231C2×Q8C22×D28311yescharacteristic
232C2×D4—21no—
233C2×D4C22×D28311yescharacteristic
234C22×C4—2831no—
235C22×C4—2831no—
236C22×C4—5661no—
237C22×C4C22×D28311yescharacteristic
238Q16—5662no—
239Q16—5662no—
240SD16—22no—
241SD16—5662no—
242M16—22no—
243M16—5662no—
244C2×C8—2831no—
245C2×C8—2831no—
246C2×C8C22×D28311yescharacteristic
247C2×C8C22×D28311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—2831no—
250C4⋊C4—2831no—
251C4⋊C4—5661no—
252C4⋊C4—5661no—
253C4⋊C4—5661no—
254C4⋊C4—5661no—
255C4⋊C4—5661no—
256C4⋊C4—5661no—
257C4⋊C4—5661no—
258C4⋊C4—5661no—
259C4⋊C4—5661no—
260C4⋊C4—5661no—
261C4⋊C4C22×D28311yescharacteristic
262C4⋊C4C22×D28311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—5661no—
266C22⋊C4—5661no—
267C22⋊C4—5661no—
268C22⋊C4—5661no—
269C22⋊C4—5661no—
270C42—2831no—
271C42—2831no—
272C42—2831no—
273C42—2831no—
274C42—5661no—
275C42—5661no—
276C42—5661no—
277C23—21no—
278C23D4×D28311yescharacteristic
279Q8—21no—
280Q8—21no—
281Q8—5661no—
282Q8—5661no—
283Q8—5661no—
284Q8—5661no—
285Q8—11322no—
286D4—21no—
287D4—21no—
288D4—42no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—2831no—
295C2×C4—2831no—
296C2×C4—2831no—
297C2×C4—2831no—
298C2×C4—5662no—
299C2×C4—5661no—
300C2×C4—5661no—
301C2×C4—5661no—
302C2×C4—5661no—
303C2×C4—11322no—
304C2×C4—11321no—
305C2×C4D4×D28311yescharacteristic
306C2×C4C23×D28311yescharacteristic
307C8—21no—
308C8—2832no—
309C8—5661no—
310C8#1798539112yes—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C2×D4×D28311yescharacteristic
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—2832no—
323C4—5661no—
324C4—5661no—
325C4—5661no—
326C4—11321no—
327C4—11321no—
328C4—11321no—
329C4—11321no—
330C4#41×D28311yescharacteristic
331C4C2×#1798539111yescharacteristic
332C2—41no—
333C2—81no—
334C2#3182701011yescharacteristic
335C2#137×D28311yescharacteristic
336C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#468×C283C2
socleC22×C283C22×D4
prime radical#121×C283C22
hypercenter#468D283
hypocenterC283#3172
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #31827017C2semidirect
3. #31826873C2semidirect
4. #31826856C2semidirect
5. #31826832C2semidirect
6. #31826823C2semidirect
7. #31826723C2semidirect
8. #31826721C2semidirect
10. #31826559C2semidirect
11. #31826557C2semidirect
12. #31826556C2semidirect
13. SD16×Dic283C2semidirect
14. M16×Dic283C2semidirect
168. C283#3172semidirect

Maximal subgroups

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

subgroupindex
#318270172
#318269962
#318268732
#318268562
#318268322
#318268232
#318267232
#318267212
#318267092
#318265592
#318265572
#318265562
SD16×Dic2832
M16×Dic2832
#468×C2832
#3172283