Finite Group Databasebeta

← #34079049

Subgroups — #34079049

Order 19328.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×D151C211yesmaximal, characteristic
2#117×D151C211yesmaximal, characteristic, complemented
3#23752537C211yesmaximal, characteristic, complemented
4#23752536C211yesmaximal, characteristic, complemented
5#23752533C211yesmaximal, characteristic, complemented
6#23752532C211yesmaximal, characteristic, complemented
7#23752529C211yesmaximal, characteristic, complemented
8#23752527C211yesmaximal, characteristic, complemented
9#23752524C211yesmaximal, characteristic, complemented
10#23752253C211yesmaximal, characteristic, complemented
11#23752251C211yesmaximal, characteristic, complemented
12#23752215C211yesmaximal, characteristic, complemented
13#23752215C211yesmaximal, characteristic, complemented
14#23752174C211yesmaximal, characteristic, complemented
15#482×C151C211yesmaximal, characteristic, complemented
16C2×D4×D151—21no—
17C2×D4×D151C2211yescharacteristic
18C22×C4×D151C2211yescharacteristic
19C2×#14558987C2211yescharacteristic
20C2×#14558987C2211yescharacteristic
21#18012714C2211yescharacteristic
22#18012713C2211yescharacteristic
23#18012712—21no—
24#18012711—21no—
25#18012710C2211yescharacteristic
26#18012698C2211yescharacteristic, complemented
27#18012697C2211yescharacteristic
28#18012697C2211yescharacteristic
29#18012696—21no—
30#18012695—21no—
31C4⋊C4×D151C2211yescharacteristic
32C4⋊C4×D151C2211yescharacteristic, complemented
33C4⋊C4×D151C2211yescharacteristic, complemented
34C22⋊C4×D151—21no—
35C22⋊C4×D151—21no—
36C42×D151—21no—
37#18012673C2211yescharacteristic, complemented
38#18012672C2211yescharacteristic, complemented
39#18012672C2211yescharacteristic, complemented
40#18012671C2211yescharacteristic, complemented
41#18012670C2211yescharacteristic, complemented
42#18012670C2211yescharacteristic, complemented
43#18012661C2211yescharacteristic, complemented
44#18012661C2211yescharacteristic, complemented
45#18012660C2211yescharacteristic, complemented
46#18012660C2211yescharacteristic, complemented
47#18012659C2211yescharacteristic, complemented
48#18012659C2211yescharacteristic, complemented
49C22⋊(C151⋊C8)C2211yescharacteristic, complemented
50(C2×D151)⋊C8C2211yescharacteristic, complemented
51(C2×D151)⋊C8C2211yescharacteristic, complemented
52#121×C151C2211yescharacteristic
53#117×C151C2211yescharacteristic, complemented
54#107×C151C2211yescharacteristic, complemented
55#106×C151C2211yescharacteristic, complemented
56#102×C151C2211yescharacteristic, complemented
57#102×C151C2211yescharacteristic, complemented
58C22⋊C8×C151C2211yescharacteristic, complemented
59C23×D151—21no—
60C23×D151D411yescharacteristic
61C2×D604—21no—
62C2×D604C2311yescharacteristic
63C2×C22⋊D151—21no—
64C2×C22⋊D151—21no—
65C2×C22⋊D151—21no—
66D4×D151—22no—
67D4×D151—22no—
68D4×D151—44no—
69C22×Dic151D411yescharacteristic
70C2×C4×D151—22no—
71C2×C4×D151—22no—
72C2×C4×D151—21no—
73C2×C4×D151—21no—
74C2×C4×D151—21no—
75C2×C4×D151D411yescharacteristic
76C2×C4×D151D411yescharacteristic
77C2×C4×D151C2311yescharacteristic
78C2×C4×D151C2311yescharacteristic
79#14558987—22no—
80#14558987—22no—
81C2×C151⋊C8C2311yescharacteristic
82C2×C151⋊C8C2311yescharacteristic
83C4⋊Dic151C2311yescharacteristic
84C4⋊Dic151C2311yescharacteristic
85C4⋊Dic151C2311yescharacteristic
86Dic151⋊C4—21no—
87Dic151⋊C4—21no—
88Dic151⋊C4—21no—
89C22⋊Dic151—21no—
90C22⋊Dic151—21no—
91(C2×D151)⋊C4—21no—
92(C2×D151)⋊C4—21no—
93(C2×D151)⋊C4—21no—
94(C2×D151)⋊C4—21no—
95C4×Dic151—21no—
96C4×Dic151—21no—
97C4×Dic151—21no—
98C2×D4×C151—21no—
99C2×D4×C151C2311yescharacteristic
100C22×C4×C151C2311yescharacteristic
101C2×C8×C151C2311yescharacteristic
102C2×C8×C151C2311yescharacteristic
103C4⋊C4×C151C2311yescharacteristic
104C4⋊C4×C151C2311yescharacteristic
105C4⋊C4×C151C2311yescharacteristic
106C22⋊C4×C151—21no—
107C22⋊C4×C151—21no—
108C42×C151—21no—
109C22×D151—22no—
110C22×D151—22no—
111C22×D151—21no—
112C22×D151—41no—
113C22×D151—42no—
114C22×D151—42no—
115C22×D151—42no—
116C22×D151C2×D411yescharacteristic
117C22×D151C2×D411yescharacteristic
118D604—21no—
119D604—21no—
120D604—42no—
121C22⋊D151—42no—
122C22⋊D151—42no—
123C22⋊D151—42no—
124C2×Dic151—22no—
125C2×Dic151—21no—
126C2×Dic151—21no—
127C2×Dic151—21no—
128C2×Dic151C2×D411yescharacteristic
129C2×Dic151C2×D411yescharacteristic
130C4×D151—22no—
131C4×D151—21no—
132C4×D151—21no—
133C4×D151—22no—
134C4×D151—42no—
135C4×D151—42no—
136C4×D151—42no—
137C4×D151#4112yes—
138C4×D151#4112yes—
139C151⋊C8—21no—
140C151⋊C8—21no—
141C23×C151—21no—
142C23×C151C2×D411yescharacteristic
143D4×C151—21no—
144D4×C151—21no—
145D4×C151—42no—
146C2×C4×C151—21no—
147C2×C4×C151—21no—
148C2×C4×C151—21no—
149C2×C4×C151—21no—
150C2×C4×C151—21no—
151C2×C4×C151C2×D411yescharacteristic
152C2×C4×C151C2411yescharacteristic
153C8×C151—21no—
154C8×C151—21no—
155C22×C151—21no—
156C22×C151—21no—
157C22×C151—21no—
158C22×C151—41no—
159C22×C151—41no—
160C22×C151—41no—
161C22×C151C22×D411yescharacteristic
162C2×D151—21no—
163C2×D151—21no—
164C2×D151—21no—
165C2×D151—21no—
166C2×D151—44no—
167C2×D151—41no—
168C2×D151—41no—
169C2×D151—41no—
170C2×D151—84no—
171C2×D151#12312yes—
172C2×D151AGL(1,ℤ/8)12yes—
173C4×C151—21no—
174C4×C151—41no—
175C4×C151—41no—
176C4×C151—41no—
177C4×C151C2×#4111yescharacteristic
178C4×C151C2×#4111yescharacteristic
179Dic151—21no—
180Dic151—41no—
181Dic151—41no—
182Dic151—41no—
183Dic151#12312yes—
184C2×C151—41no—
185C2×C151—81no—
186C2×C151C2×#12311yescharacteristic
187C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C151#57511yescharacteristic
189D151—22no—
190D151—22no—
191D151—81no—
192C151#309411yescharacteristic, complemented, Sylow
193#3094—1511nomaximal, Sylow
194C2×#121—1511no—
195C2×#117—1511no—
196#482—1511no—
197#482—1511no—
198#482—1511no—
199#482D15111yescharacteristic, complemented
200#481—1511no—
201#481—1511no—
202#480—1511no—
203#480—1511no—
204#428—1511no—
205#428—1511no—
206#418—1511no—
207#418—1511no—
208#407—1511no—
209C22×D4—3021no—
210C22×D4—1511no—
211C23×C4—1511no—
212C2×M16—1511no—
213C2×M16—1511no—
214#121—3021no—
215#121—3021no—
216#121—1511no—
217#121—1511no—
218#121—1511no—
219#121C2×D15111yescharacteristic
220#117—3021no—
221#117—3021no—
222#117—1511no—
223#117—1511no—
224#117—1511no—
225#117C2×D15111yescharacteristic, complemented
226C2×C4⋊C4—1511no—
227C2×C4⋊C4—1511no—
228C2×C4⋊C4—1511no—
229C2×C22⋊C4—3021no—
230C2×C22⋊C4—3021no—
231C2×C42—3021no—
232#107—1511no—
233#107—1511no—
234#107—1511no—
235#107C2×D15111yescharacteristic, complemented
236#106—1511no—
237#106—1511no—
238#106—1511no—
239#106C2×D15111yescharacteristic, complemented
240#102—1511no—
241#102—1511no—
242#102—1511no—
243#102—1511no—
244#102—1511no—
245#102—1511no—
246#102C2×D15111yescharacteristic, complemented
247#102C2×D15111yescharacteristic, complemented
248C22⋊C8—1511no—
249C22⋊C8—1511no—
250C22⋊C8—1511no—
251C22⋊C8C2×D15111yescharacteristic, complemented
252C24—3021no—
253C24—1511no—
254C2×D4—21no—
255C2×D4—3021no—
256C2×D4—3021no—
257C2×D4—3021no—
258C2×D4—3022no—
259C2×D4—3022no—
260C2×D4—3021no—
261C2×D4—6044no—
262C2×D4—1511no—
263C2×D4C22×D15111yescharacteristic
264C22×C4—3022no—
265C22×C4—3022no—
266C22×C4—3021no—
267C22×C4—3021no—
268C22×C4—3021no—
269C22×C4—1511no—
270C22×C4—1511no—
271C22×C4—1511no—
272C22×C4—1511no—
273C22×C4—1511no—
274C22×C4C22×D15111yescharacteristic
275M16—3022no—
276M16—3022no—
277C2×C8—1511no—
278C2×C8—1511no—
279C2×C8C22×D15111yescharacteristic
280C2×C8C22×D15111yescharacteristic
281C4⋊C4—3021no—
282C4⋊C4—3021no—
283C4⋊C4—3021no—
284C4⋊C4—1511no—
285C4⋊C4—1511no—
286C4⋊C4—1511no—
287C4⋊C4C22×D15111yescharacteristic
288C4⋊C4C22×D15111yescharacteristic
289C4⋊C4C22×D15111yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—3021no—
293C22⋊C4—3021no—
294C22⋊C4—3021no—
295C22⋊C4—3021no—
296C22⋊C4—3021no—
297C22⋊C4—3021no—
298C42—21no—
299C42—3021no—
300C42—3021no—
301C42—3021no—
302C23—21no—
303C23—3022no—
304C23—3022no—
305C23—3021no—
306C23—6041no—
307C23—6042no—
308C23—6042no—
309C23—6042no—
310C23—1511no—
311C23—1511no—
312C23D4×D15111yescharacteristic
313D4—21no—
314D4—21no—
315D4—42no—
316D4—3021no—
317D4—3021no—
318D4—6042no—
319D4—6042no—
320D4—6042no—
321D4—6042no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—3022no—
328C2×C4—3022no—
329C2×C4—3022no—
330C2×C4—3021no—
331C2×C4—3021no—
332C2×C4—3021no—
333C2×C4—3021no—
334C2×C4—3021no—
335C2×C4—6042no—
336C2×C4—6042no—
337C2×C4—6042no—
338C2×C4—1511no—
339C2×C4—1511no—
340C2×C4—1512no—
341C2×C4—1512no—
342C2×C4D4×D15111yescharacteristic
343C2×C4C23×D15111yescharacteristic
344C8—21no—
345C8—21no—
346C8—3021no—
347C8—3021no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—3021no—
355C22—3021no—
356C22—3021no—
357C22—3021no—
358C22—6044no—
359C22—6041no—
360C22—6041no—
361C22—6041no—
362C22—1512no—
363C22—1512no—
364C22—12084no—
365C22C2×D4×D15111yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—3021no—
371C4—6041no—
372C4—6041no—
373C4—6041no—
374C4—1512no—
375C4#41×D15111yescharacteristic
376C4#41×D15111yescharacteristic
377C2—41no—
378C2—81no—
379C2—3022no—
380C2—3022no—
381C2—12081no—
382C2#123×D15111yescharacteristic
383C2AGL(1,ℤ/8)×D15111yescharacteristic
384C2#2375300911yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#482×C151C2
socleC22×C151C22×D4
prime radical#121×D151C2
hypercenter#482D151
hypocenterC151#3094
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D151C2semidirect
3. #23752537C2semidirect
4. #23752536C2semidirect
5. #23752533C2semidirect
6. #23752532C2semidirect
7. #23752529C2semidirect
8. #23752527C2semidirect
9. #23752524C2semidirect
10. #23752253C2semidirect
11. #23752251C2semidirect
12. #23752215C2semidirect
13. #23752215C2semidirect
14. #23752174C2semidirect
15. #482×C151C2semidirect
26. #18012698C22semidirect
32. C4⋊C4×D151C22semidirect
33. C4⋊C4×D151C22semidirect
37. #18012673C22semidirect
38. #18012672C22semidirect
39. #18012672C22semidirect
40. #18012671C22semidirect
41. #18012670C22semidirect
42. #18012670C22semidirect
43. #18012661C22semidirect
44. #18012661C22semidirect
45. #18012660C22semidirect
46. #18012660C22semidirect
47. #18012659C22semidirect
48. #18012659C22semidirect
49. C22⋊(C151⋊C8)C22semidirect
50. (C2×D151)⋊C8C22semidirect
51. (C2×D151)⋊C8C22semidirect
53. #117×C151C22semidirect
54. #107×C151C22semidirect
55. #106×C151C22semidirect
56. #102×C151C22semidirect
57. #102×C151C22semidirect
58. C22⋊C8×C151C22semidirect
192. C151#3094semidirect
199. #482D151semidirect
225. #117C2×D151semidirect
235. #107C2×D151semidirect
239. #106C2×D151semidirect
246. #102C2×D151semidirect
247. #102C2×D151semidirect
251. C22⋊C8C2×D151semidirect

Maximal subgroups

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

subgroupindex
#121×D1512
#117×D1512
#237525372
#237525362
#237525332
#237525322
#237525292
#237525272
#237525242
#237522532
#237522512
#237522152
#237521742
#482×C1512
#3094151