Finite Group Databasebeta

← #49131405

Subgroups — #49131405

Order 36224.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827010C211yesmaximal, characteristic, complemented
2#41×Dic283C211yesmaximal, characteristic, complemented
3#31826790C211yesmaximal, characteristic, complemented
4#31826786C211yesmaximal, characteristic, complemented
5#31826748C211yesmaximal, characteristic, complemented
6#31826743C211yesmaximal, characteristic, complemented
7#31826663C211yesmaximal, characteristic, complemented
8#31826658C211yesmaximal, characteristic, complemented
9#31826610C211yesmaximal, characteristic, complemented
10#31826577C211yesmaximal, characteristic, complemented
11#31826574C211yesmaximal, characteristic, complemented
12#31826568C211yesmaximal, characteristic, complemented
13#31826566C211yesmaximal, characteristic, complemented
14#31826519C211yesmaximal, characteristic, complemented
15#449×C283C211yesmaximal, characteristic, complemented
16C2×#17985391—21no—
17C2×#17985379C2211yescharacteristic
18C2×#17985377C2211yescharacteristic
19#22583609C2211yescharacteristic
20#22583606C2211yescharacteristic
21#22583596—21no—
22#22583584C2211yescharacteristic
23#22583582—21no—
24Q8×Dic283C2211yescharacteristic
25Q8×Dic283C2211yescharacteristic
26D4×Dic283—21no—
27D4×Dic283—21no—
28D4×Dic283C2211yescharacteristic
29D4×Dic283C2211yescharacteristic
30D4×Dic283C2211yescharacteristic
31#22583572C2211yescharacteristic
32(C22⋊D283)⋊C4—21no—
33#22583569—21no—
34#22583569C2211yescharacteristic
35C2×C4×Dic283—21no—
36C2×C4×Dic283C2211yescharacteristic
37#22583544C2211yescharacteristic
38#22583543C2211yescharacteristic
39#22583542C2211yescharacteristic
40#22583541C2211yescharacteristic
41Dic283⋊C8C2211yescharacteristic
42Dic283⋊C8C2211yescharacteristic
43#22583534C2211yescharacteristic
44#22583532C2211yescharacteristic
45#22583532C2211yescharacteristic
46#22583531C2211yescharacteristic
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50#22583520C2211yescharacteristic
51#22583520C2211yescharacteristic
52C2×#41×C283C2211yescharacteristic
53C2×SD16×C283C2211yescharacteristic
54C2×D8×C283C2211yescharacteristic
55#121×C283C2211yescharacteristic
56#103×C283C2211yescharacteristic
57#102×C283C2211yescharacteristic
58C22⋊C8×C283C2211yescharacteristic
59#17985391—44no—
60C2×Dic566—21no—
61C2×D1132C2311yescharacteristic
62C2×C22⋊D283—21no—
63C2×C22⋊D283—21no—
64C2×C22⋊D283—21no—
65C22×Dic283—21no—
66C22×Dic283—21no—
67C22×Dic283—21no—
68C22×Dic283D411yescharacteristic
69C22×Dic283D411yescharacteristic
70C2×C4×D283—21no—
71#17985379—22no—
72#17985377—22no—
73C2×C283⋊C8C2311yescharacteristic
74C2×C283⋊C8C2311yescharacteristic
75C4⋊Dic283—21no—
76C4⋊Dic283—21no—
77C4⋊Dic283C2311yescharacteristic
78C4⋊Dic283C2311yescharacteristic
79C4⋊Dic283C2311yescharacteristic
80Dic283⋊C4—21no—
81C22⋊Dic283—21no—
82C22⋊Dic283—21no—
83C22⋊Dic283—21no—
84C22⋊Dic283—21no—
85(C2×D283)⋊C4—21no—
86(C2×D283)⋊C4—21no—
87C4×Dic283—21no—
88C4×Dic283—21no—
89C4×Dic283—21no—
90C4×Dic283D411yescharacteristic
91C4×Dic283D411yescharacteristic
92C4×Dic283C2311yescharacteristic
93C4×Dic283C2311yescharacteristic
94#41×C283—22no—
95#41×C283—22no—
96C2×Q8×C283C2311yescharacteristic
97C2×D4×C283—21no—
98C2×D4×C283—21no—
99C2×D4×C283C2311yescharacteristic
100C2×D4×C283C2311yescharacteristic
101C22×C4×C283—21no—
102C22×C4×C283C2311yescharacteristic
103SD16×C283—22no—
104D8×C283—22no—
105C2×C8×C283C2311yescharacteristic
106C2×C8×C283C2311yescharacteristic
107C4⋊C4×C283C2311yescharacteristic
108C22⋊C4×C283—21no—
109C22×D283—21no—
110Dic566—42no—
111D1132—21no—
112D1132—21no—
113C22⋊D283—42no—
114C22⋊D283—42no—
115C22⋊D283—42no—
116C2×Dic283—22no—
117C2×Dic283—21no—
118C2×Dic283—21no—
119C2×Dic283—21no—
120C2×Dic283—41no—
121C2×Dic283—42no—
122C2×Dic283—42no—
123C2×Dic283—42no—
124C2×Dic283C2×D411yescharacteristic
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C2×Dic283C2×D411yescharacteristic
128C4×D283—42no—
129C283⋊C8—21no—
130C283⋊C8—21no—
131C23×C283—21no—
132C23×C283—21no—
133C23×C283C2×D411yescharacteristic
134Q8×C283—21no—
135Q8×C283#4112yescomplemented
136D4×C283—24no—
137D4×C283—21no—
138D4×C283—21no—
139D4×C283—21no—
140D4×C283—42no—
141D4×C283#4112yescomplemented
142C2×C4×C283—24no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283C2×D411yescharacteristic
148C2×C4×C283C2411yescharacteristic
149C8×C283—21no—
150C8×C283—21no—
151C22×C283—21no—
152C22×C283—22no—
153C22×C283—21no—
154C22×C283—21no—
155C22×C283—41no—
156C22×C283—41no—
157C22×C283—41no—
158C22×C283—41no—
159C22×C283—41no—
160C22×C283C22×D411yescharacteristic
161C2×D283—41no—
162C2×D283—41no—
163C2×D283—41no—
164C4×C283—21no—
165C4×C283—22no—
166C4×C283—41no—
167C4×C283C2×#4111yescharacteristic
168C4×C283C2×#4111yescharacteristic
169Dic283—21no—
170Dic283—21no—
171Dic283—21no—
172Dic283—41no—
173Dic283—41no—
174Dic283—41no—
175Dic283AGL(1,ℤ/8)12yes—
176C2×C283—41no—
177C2×C283—42no—
178C2×C283—81no—
179C2×C283#52511yescharacteristic
180C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
181C2×C283#57511yescharacteristic
182D283—81no—
183C283#311511yescharacteristic, complemented, Sylow
184#3115—2831nomaximal, Sylow
185#532—2831no—
186C4×#41—2831no—
187#482—2831no—
188#482—2831no—
189#475—2831no—
190#474—2831no—
191#460—2831no—
192#459—2831no—
193#449—2831no—
194#449D28311yescharacteristic, complemented
195#442—2831no—
196#442—2831no—
197#440—2831no—
198#440—2831no—
199#432—2831no—
200C2×#41—5661no—
201C2×#41C2×D28311yescharacteristic
202C2×SD16—2831no—
203C2×SD16C2×D28311yescharacteristic
204C2×D8—2831no—
205C2×D8C2×D28311yescharacteristic
206C4⋊Q8—2831no—
207#127—2831no—
208#124—5661no—
209#121—2831no—
210#121—5661no—
211#121C2×D28311yescharacteristic
212C4×Q8—2831no—
213C4×Q8—2831no—
214C4×D4—2831no—
215C4×D4—2831no—
216C4×D4—2831no—
217C4×D4—2831no—
218C4×D4—5661no—
219C4×D4—5661no—
220C4×D4—5661no—
221#117—2831no—
222#117—5661no—
223C2×C42—2831no—
224C2×C42—5661no—
225#107—2831no—
226#107—2831no—
227#106—2831no—
228#106—2831no—
229C4⋊C8—2831no—
230C4⋊C8—2831no—
231#103—2831no—
232#103C2×D28311yescharacteristic
233#102—2831no—
234#102—2831no—
235#102—2831no—
236#102—2831no—
237#102—2831no—
238#102C2×D28311yescharacteristic
239C22⋊C8—2831no—
240C22⋊C8C2×D28311yescharacteristic
241#97—2831no—
242#97—2831no—
243#41—22no—
244#41—22no—
245#41—11324no—
246C2×Q8—5661no—
247C2×Q8C22×D28311yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—2831no—
251C2×D4—5661no—
252C2×D4—5661no—
253C2×D4—5661no—
254C2×D4C22×D28311yescharacteristic
255C2×D4C22×D28311yescharacteristic
256C22×C4—21no—
257C22×C4—2831no—
258C22×C4—2831no—
259C22×C4—5661no—
260C22×C4—5661no—
261C22×C4—5661no—
262C22×C4—5661no—
263C22×C4C22×D28311yescharacteristic
264SD16—22no—
265SD16—5662no—
266D8—22no—
267D8—5662no—
268C2×C8—2831no—
269C2×C8—2831no—
270C2×C8C22×D28311yescharacteristic
271C2×C8C22×D28311yescharacteristic
272C4⋊C4—2831no—
273C4⋊C4—2831no—
274C4⋊C4—2831no—
275C4⋊C4—5661no—
276C4⋊C4—5661no—
277C4⋊C4—5661no—
278C4⋊C4C22×D28311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—5661no—
281C22⋊C4—5661no—
282C22⋊C4—5661no—
283C22⋊C4—5661no—
284C22⋊C4—5661no—
285C22⋊C4—5661no—
286C42—2831no—
287C42—2831no—
288C42—2831no—
289C42—2831no—
290C42—5661no—
291C42—5661no—
292C42—5661no—
293C23—21no—
294C23—21no—
295C23—5661no—
296C23D4×D28311yescharacteristic
297Q8—21no—
298Q8—11322no—
299Q8#1798539112yescomplemented
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—5661no—
306D4—5661no—
307D4—11322no—
308D4—11322no—
309D4—11322no—
310D4#1798539112yescomplemented
311C2×C4—24no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—2831no—
317C2×C4—2831no—
318C2×C4—2831no—
319C2×C4—2831no—
320C2×C4—5662no—
321C2×C4—5661no—
322C2×C4—5661no—
323C2×C4—5661no—
324C2×C4—11322no—
325C2×C4—11322no—
326C2×C4—11321no—
327C2×C4—11322no—
328C2×C4—11322no—
329C2×C4D4×D28311yescharacteristic
330C2×C4C23×D28311yescharacteristic
331C8—21no—
332C8—21no—
333C8—5661no—
334C8—5661no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—11321no—
345C22—11321no—
346C22—11321no—
347C22C2×D4×D28311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—2832no—
352C4—5661no—
353C4—5661no—
354C4—5661no—
355C4—11321no—
356C4—11321no—
357C4—11321no—
358C4C2×#1798539111yescharacteristic
359C4C2×#1798539111yescharacteristic
360C2—41no—
361C2—42no—
362C2—81no—
363C2—22641no—
364C2#3182697111yescharacteristic
365C2AGL(1,ℤ/8)×D28311yescharacteristic
366C2#3182726211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#449×C283C2
socleC22×C283C22×D4
hypercenter#449D283
hypocenterC283#3115
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31827010C2semidirect
2. #41×Dic283C2semidirect
3. #31826790C2semidirect
4. #31826786C2semidirect
5. #31826748C2semidirect
6. #31826743C2semidirect
7. #31826663C2semidirect
8. #31826658C2semidirect
9. #31826610C2semidirect
10. #31826577C2semidirect
11. #31826574C2semidirect
12. #31826568C2semidirect
13. #31826566C2semidirect
14. #31826519C2semidirect
15. #449×C283C2semidirect
135. Q8×C283#41semidirect
141. D4×C283#41semidirect
183. C283#3115semidirect
194. #449D283semidirect
299. Q8#17985391semidirect
310. D4#17985391semidirect

Maximal subgroups

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

subgroupindex
#318270102
#41×Dic2832
#318267902
#318267862
#318267482
#318267432
#318266632
#318266582
#318266102
#318265772
#318265742
#318265682
#318265662
#318265192
#449×C2832
#3115283