Finite Group Databasebeta

← #49131491

Subgroups — #49131491

Order 36224.

370 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#31827088C211yesmaximal, characteristic
2#31826943C211yesmaximal, characteristic, complemented
3#31826803C211yesmaximal, characteristic, complemented
4#31826790C211yesmaximal, characteristic
5#31826758C211yesmaximal, characteristic, complemented
6#31826748C211yesmaximal, characteristic, complemented
7#31826689C211yesmaximal, characteristic, complemented
8#31826689C211yesmaximal, characteristic, complemented
9#31826604C211yesmaximal, characteristic
10#31826572C211yesmaximal, characteristic, complemented
11#31826568C211yesmaximal, characteristic, complemented
12Q16×Dic283C211yesmaximal, characteristic, complemented
13#31826550C211yesmaximal, characteristic, complemented
14#31826523C211yesmaximal, characteristic, complemented
15#451×C283C211yesmaximal, characteristic, complemented
16C2×#17985379C2211yescharacteristic
17C2×#17985379C2211yescharacteristic
18#22583609C2211yescharacteristic
19#22583605—21no—
20#22583599C2211yescharacteristic
21#22583596—21no—
22#22583592—21no—
23#22583590—21no—
24#22583584C2211yescharacteristic
25Q8×Dic283—21no—
26Q8×Dic283—21no—
27Q8×Dic283C2211yescharacteristic
28Q8×Dic283C2211yescharacteristic
29Q8×Dic283C2211yescharacteristic
30Q8×Dic283C2211yescharacteristic
31Q8×Dic283C2211yescharacteristic
32#22583572C2211yescharacteristic
33#22583569—21no—
34#22583569—21no—
35#22583569C2211yescharacteristic
36#22583569C2211yescharacteristic
37#22583544C2211yescharacteristic
38#22583543C2211yescharacteristic
39#22583542C2211yescharacteristic
40#22583541C2211yescharacteristic
41Dic283⋊C8C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43#22583534C2211yescharacteristic
44#22583534C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583534C2211yescharacteristic
47#22583532C2211yescharacteristic
48#22583532C2211yescharacteristic
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×Dic566—21no—
60C2×D1132C2311yescharacteristic
61C2×C22⋊D283—21no—
62C2×C4×D283—21no—
63#17985379—22no—
64#17985379—22no—
65C2×C283⋊C8C2311yescharacteristic
66C2×C283⋊C8C2311yescharacteristic
67C4⋊Dic283—21no—
68C4⋊Dic283—21no—
69C4⋊Dic283—21no—
70C4⋊Dic283—21no—
71C4⋊Dic283—21no—
72C4⋊Dic283D411yescharacteristic
73C4⋊Dic283D411yescharacteristic
74C4⋊Dic283C2311yescharacteristic
75C4⋊Dic283C2311yescharacteristic
76C4⋊Dic283C2311yescharacteristic
77Dic283⋊C4—21no—
78C22⋊Dic283—21no—
79C22⋊Dic283D411yescharacteristic
80C22⋊Dic283D411yescharacteristic
81(C2×D283)⋊C4—21no—
82(C2×D283)⋊C4—21no—
83(C2×D283)⋊C4—21no—
84(C2×D283)⋊C4—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—
108C22×D283—21no—
109Dic566—81no—
110D1132—21no—
111D1132—21no—
112C22⋊D283—81no—
113C2×Dic283—21no—
114C2×Dic283—21no—
115C2×Dic283—21no—
116C2×Dic283C2×D411yescharacteristic
117C2×Dic283C2×D411yescharacteristic
118C2×Dic283C2×D411yescharacteristic
119C2×Dic283C2×D411yescharacteristic
120C4×D283—81no—
121C283⋊C8—21no—
122C283⋊C8—21no—
123C23×C283C2×D411yescharacteristic
124Q8×C283—24no—
125Q8×C283—21no—
126Q8×C283—21no—
127Q8×C283—21no—
128Q8×C283—21no—
129Q8×C283#4112yes—
130Q8×C283#4112yes—
131C2×C4×C283—24no—
132C2×C4×C283—21no—
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—21no—
142C22×C283—21no—
143C22×C283—21no—
144C22×C283—21no—
145C22×C283C22×D411yescharacteristic
146C2×D283—41no—
147C2×D283—41no—
148C2×D283—41no—
149C4×C283—22no—
150C4×C283—22no—
151C4×C283—21no—
152C4×C283—41no—
153C4×C283—41no—
154C4×C283C2×#4111yescharacteristic
155C4×C283C2×#4111yescharacteristic
156Dic283—21no—
157Dic283—21no—
158Dic283—21no—
159Dic283—21no—
160Dic283—41no—
161Dic283—41no—
162Dic283—41no—
163C2×C283—41no—
164C2×C283#52511yescharacteristic
165C2×C283#57711yescharacteristic
166C2×C283#57811yescharacteristic
167D283—81no—
168C283#312111yescharacteristic, complemented, Sylow
169#3121—2831nomaximal, Sylow
170#542—2831no—
171#519—2831no—
172#485—2831no—
173#482—2831no—
174#477—2831no—
175#475—2831no—
176#464—2831no—
177#464—2831no—
178#451D28311yescharacteristic, complemented
179#448—2831no—
180#441—2831no—
181#440—2831no—
182C4×Q16—2831no—
183C4×SD16—2831no—
184#433—2831no—
185C22×Q8C2×D28311yescharacteristic
186C2×Q16C2×D28311yescharacteristic
187C2×Q16C2×D28311yescharacteristic
188C2×SD16—2831no—
189C2×SD16—2831no—
190C4⋊Q8—2831no—
191#126—5661no—
192#124—2831no—
193#124—5661no—
194#123—5661no—
195C22⋊Q8—5661no—
196C22⋊Q8C2×D28311yescharacteristic
197#121—2831no—
198C4×Q8—2831no—
199C4×Q8—2831no—
200C4×Q8—2831no—
201C4×Q8—2831no—
202C4×Q8—2831no—
203C4×Q8—5661no—
204C4×Q8—5661no—
205C4×D4—2831no—
206#117—2831no—
207#117—2831no—
208#117—5661no—
209#117—5661no—
210#107—2831no—
211#107—2831no—
212#106—2831no—
213#106—2831no—
214C4⋊C8—2831no—
215C4⋊C8—2831no—
216#103—2831no—
217#103—2831no—
218#103—2831no—
219#103—2831no—
220#103C2×D28311yescharacteristic
221#103C2×D28311yescharacteristic
222#102—2831no—
223#102—2831no—
224C22⋊C8—2831no—
225C22⋊C8C2×D28311yescharacteristic
226#97—2831no—
227C4×C8—2831no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—5661no—
232C2×Q8C22×D28311yescharacteristic
233C2×Q8C22×D28311yescharacteristic
234C2×Q8C22×D28311yescharacteristic
235C2×D4—2831no—
236C2×D4—5661no—
237C22×C4—21no—
238C22×C4—5661no—
239C22×C4C22×D28311yescharacteristic
240Q16—22no—
241Q16—22no—
242SD16—5662no—
243SD16—5662no—
244C2×C8—2831no—
245C2×C8—2831no—
246C2×C8C22×D28311yescharacteristic
247C2×C8C22×D28311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—2831no—
250C4⋊C4—2831no—
251C4⋊C4—2831no—
252C4⋊C4—2831no—
253C4⋊C4—2831no—
254C4⋊C4—5661no—
255C4⋊C4—5661no—
256C4⋊C4—5661no—
257C4⋊C4—5661no—
258C4⋊C4—5661no—
259C4⋊C4—5661no—
260C4⋊C4C22×D28311yescharacteristic
261C22⋊C4—21no—
262C22⋊C4—2831no—
263C22⋊C4—2831no—
264C22⋊C4—5661no—
265C22⋊C4—5661no—
266C22⋊C4—5661no—
267C22⋊C4—5661no—
268C22⋊C4—5661no—
269C42—2831no—
270C42—2831no—
271C42—5661no—
272C42—5661no—
273C42—5661no—
274C42—5661no—
275C42—5661no—
276C42—5661no—
277C23—5661no—
278C23D4×D28311yescharacteristic
279Q8—24no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—22641no—
285Q8#1798539112yes—
286Q8#1798539112yes—
287D4—5661no—
288D4—5661no—
289D4—22641no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—2831no—
298C2×C4—2831no—
299C2×C4—2831no—
300C2×C4—2831no—
301C2×C4—5661no—
302C2×C4—5661no—
303C2×C4—5661no—
304C2×C4—22641no—
305C2×C4D4×D28311yescharacteristic
306C2×C4C23×D28311yescharacteristic
307C8—21no—
308C8—21no—
309C8—5661no—
310C8—5661no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—11321no—
315C22—11321no—
316C22—11321no—
317C22C2×D4×D28311yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—41no—
322C4—41no—
323C4—5661no—
324C4—5661no—
325C4—5661no—
326C4—5661no—
327C4—11321no—
328C4—11321no—
329C4—11321no—
330C4C2×#1798539111yescharacteristic
331C4C2×#1798539111yescharacteristic
332C2—41no—
333C2—22641no—
334C2#3182697111yescharacteristic
335C2#3182727411yescharacteristic
336C2#3182727811yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#451×C283C2
socleC22×C283C22×D4
prime radical#22583584C22
hypercenter#451D283
hypocenterC283#3121
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
2. #31826943C2semidirect
3. #31826803C2semidirect
5. #31826758C2semidirect
6. #31826748C2semidirect
7. #31826689C2semidirect
8. #31826689C2semidirect
10. #31826572C2semidirect
11. #31826568C2semidirect
12. Q16×Dic283C2semidirect
13. #31826550C2semidirect
14. #31826523C2semidirect
15. #451×C283C2semidirect
168. C283#3121semidirect
178. #451D283semidirect

Maximal subgroups

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

subgroupindex
#318270882
#318269432
#318268032
#318267902
#318267582
#318267482
#318266892
#318266042
#318265722
#318265682
Q16×Dic2832
#318265502
#318265232
#451×C2832
#3121283