Finite Group Databasebeta

← #34080610

Subgroups — #34080610

Order 19328.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012767C211yesmaximal, characteristic, complemented
2C22×#14558993C211yesmaximal, characteristic, complemented
3#23752871C211yesmaximal, characteristic, complemented
4D4×D604C211yesmaximal, characteristic, complemented
5#23752601C211yesmaximal, characteristic, complemented
6#23752458C211yesmaximal, characteristic, complemented
7#23752448C211yesmaximal, characteristic, complemented
8#23752428C211yesmaximal, characteristic, complemented
9#23752413C211yesmaximal, characteristic, complemented
10#23752365C211yesmaximal, characteristic, complemented
11#23752359C211yesmaximal, characteristic, complemented
12#23752353C211yesmaximal, characteristic, complemented
13#23752348C211yesmaximal, characteristic, complemented
14#23752314C211yesmaximal, characteristic, complemented
15#435×C151C211yesmaximal, characteristic, complemented
16C2×#14559004C2211yescharacteristic, complemented
17C22×Dic302C2211yescharacteristic, complemented
18C22×D604C2211yescharacteristic
19C22×D604C2211yescharacteristic
20C2×D4×D151—21no—
21#18012767—24no—
22C2×#14558993—24no—
23C2×#14558993C2211yescharacteristic
24C2×#14558993C2211yescharacteristic
25C2×#14558993C2211yescharacteristic, complemented
26C2×#14558993C2211yescharacteristic, complemented
27C2×#14558993C2211yescharacteristic, complemented
28C2×#14558993C2211yescharacteristic, complemented
29C2×D1208C2211yescharacteristic
30C2×D1208C2211yescharacteristic, complemented
31C2×D1208C2211yescharacteristic, complemented
32#18012736C2211yescharacteristic
33#18012727C2211yescharacteristic, complemented
34#18012725—21no—
35#18012718C2211yescharacteristic, complemented
36#18012717—21no—
37#18012714—21no—
38#18012713C2211yescharacteristic
39#18012713C2211yescharacteristic
40#18012713C2211yescharacteristic, complemented
41#18012711—21no—
42#18012709—21no—
43#18012709—21no—
44C4×Dic302C2211yescharacteristic, complemented
45C4×D604C2211yescharacteristic
46(C22⋊D151)⋊C4—21no—
47#18012673C2211yescharacteristic, complemented
48#18012664C2211yescharacteristic, complemented
49#18012664C2211yescharacteristic, complemented
50#18012661C2211yescharacteristic
51#18012661C2211yescharacteristic, complemented
52#18012661C2211yescharacteristic, complemented
53#18012661C2211yescharacteristic, complemented
54C2×M16×C151C2211yescharacteristic, complemented
55C22×C8×C151C2211yescharacteristic, complemented
56C4×D4×C151C2211yescharacteristic
57C4⋊C8×C151C2211yescharacteristic, complemented
58C22⋊C8×C151C2211yescharacteristic, complemented
59C22⋊C8×C151C2211yescharacteristic, complemented
60#97×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63#14559004—22no—
64#14559004—41no—
65C2×Dic302—22no—
66C2×Dic302—21no—
67C2×Dic302—41no—
68C2×Dic302C2311yescharacteristic
69C2×Dic302C2311yescharacteristic
70C2×D604—22no—
71C2×D604—22no—
72C2×D604—21no—
73C2×D604—21no—
74C2×D604—21no—
75C2×D604—21no—
76C2×D604—41no—
77C2×D604—41no—
78C2×D604C2311yescharacteristic
79C2×D604C2311yescharacteristic
80C2×D604C2311yescharacteristic
81C2×D604C2311yescharacteristic
82C2×C22⋊D151—21no—
83C2×C22⋊D151—21no—
84C2×C22⋊D151—21no—
85C2×C22⋊D151—21no—
86D4×D151—44no—
87C22×Dic151—21no—
88C2×C4×D151—21no—
89C2×C4×D151—21no—
90#14558993—22no—
91#14558993—22no—
92#14558993—22no—
93#14558993—22no—
94#14558993—41no—
95#14558993D414yescomplemented
96D1208—22no—
97D1208—22no—
98D1208—41no—
99C4⋊Dic151C2311yescharacteristic
100C4⋊Dic151C2311yescharacteristic
101Dic151⋊C4—21no—
102Dic151⋊C4—21no—
103C22⋊Dic151—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4—21no—
108(C2×D151)⋊C4—21no—
109(C2×D151)⋊C4—21no—
110C4×Dic151—21no—
111C2×D4×C151D411yescharacteristic
112C22×C4×C151C2311yescharacteristic
113C22×C4×C151C2311yescharacteristic
114M16×C151—22no—
115C2×C8×C151—22no—
116C2×C8×C151C2311yescharacteristic
117C2×C8×C151C2311yescharacteristic
118C2×C8×C151C2311yescharacteristic
119C2×C8×C151C2311yescharacteristic
120C4⋊C4×C151D411yescharacteristic
121C22⋊C4×C151D411yescharacteristic
122C22⋊C4×C151D411yescharacteristic
123C42×C151C2311yescharacteristic
124C22×D151—21no—
125C22×D151—21no—
126C22×D151—21no—
127C22×D151—21no—
128C22×D151—42no—
129C22×D151—42no—
130C22×D151—42no—
131C22×D151—42no—
132C22×D151—81no—
133C22×D151—81no—
134Dic302—21no—
135Dic302—21no—
136Dic302—21no—
137Dic302—42no—
138Dic302C2×D412yescomplemented
139D604—21no—
140D604—21no—
141D604—21no—
142D604—21no—
143D604—21no—
144D604—21no—
145D604—21no—
146D604—42no—
147D604—42no—
148D604—42no—
149D604—81no—
150D604C2×D412yes—
151C22⋊D151—42no—
152C22⋊D151—42no—
153C22⋊D151—42no—
154C22⋊D151—42no—
155C2×Dic151—21no—
156C2×Dic151—21no—
157C2×Dic151—21no—
158C2×Dic151—21no—
159C2×Dic151—42no—
160C4×D151—42no—
161C4×D151—42no—
162C23×C151C2×D411yescharacteristic
163C23×C151C2×D411yescharacteristic
164D4×C151—22no—
165C2×C4×C151—21no—
166C2×C4×C151—21no—
167C2×C4×C151—21no—
168C2×C4×C151—21no—
169C2×C4×C151C2×D411yescharacteristic
170C2×C4×C151C2×D411yescharacteristic
171C2×C4×C151C2×D411yescharacteristic
172C2×C4×C151C2×D411yescharacteristic
173C2×C4×C151C2411yescharacteristic
174C8×C151—21no—
175C8×C151—21no—
176C8×C151—21no—
177C8×C151C2×D412yescomplemented
178C22×C151—21no—
179C22×C151—21no—
180C22×C151—21no—
181C22×C151—21no—
182C22×C151—21no—
183C22×C151AGL(1,ℤ/8)12yescomplemented
184C22×C151C22×D411yescharacteristic
185C2×D151—22no—
186C2×D151—41no—
187C2×D151—41no—
188C2×D151—41no—
189C2×D151—41no—
190C2×D151—41no—
191C2×D151—41no—
192C2×D151—41no—
193C2×D151—41no—
194C2×D151—41no—
195C2×D151—41no—
196C2×D151—41no—
197C2×D151—84no—
198C2×D151—84no—
199C4×C151—21no—
200C4×C151—21no—
201C4×C151—21no—
202C4×C151—21no—
203C4×C151C22×D411yescharacteristic
204C4×C151H(5,𝔽2)11yescharacteristic
205Dic151—22no—
206Dic151—41no—
207Dic151—41no—
208Dic151—41no—
209C2×C151—22no—
210C2×C151—41no—
211C2×C151D4211yescharacteristic
212C2×C151C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C151#57711yescharacteristic
214D151—42no—
215D151—81no—
216D151—81no—
217D151—81no—
218C151#326511yescharacteristic, complemented, Sylow
219#3265—1511nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1511no—
221C22×SD16—1511no—
222#548—1511no—
223D42—1511no—
224#496—1511no—
225#468—1511no—
226#466—1511no—
227#463—1511no—
228#460—1511no—
229#450—1511no—
230#449—1511no—
231#448—1511no—
232#447—1511no—
233#440—1511no—
234#435D15111yescharacteristic, complemented
235C2×#41—1511no—
236C22×Q8—1511no—
237C22×D4—3021no—
238C22×D4—1511no—
239C22×D4—1511no—
240AGL(1,ℤ/8)—3024no—
241C2×SD16—3024no—
242C2×SD16—1511no—
243C2×SD16—1511no—
244C2×SD16—1511no—
245C2×SD16—1511no—
246C2×SD16—1511no—
247C2×SD16—1511no—
248C2×D8—1511no—
249C2×D8—1511no—
250C2×D8—1511no—
251C2×M16C2×D15111yescharacteristic, complemented
252C22×C8C2×D15111yescharacteristic, complemented
253#127—1511no—
254#124—3021no—
255#124—1511no—
256C22⋊Q8—3021no—
257C22⋊Q8—1511no—
258#121—3021no—
259#121—3021no—
260#121—1511no—
261#121—1511no—
262#121—1511no—
263#120—3021no—
264#120—3021no—
265C4×Q8—1511no—
266C4×D4—3021no—
267C4×D4—1511no—
268C4×D4C2×D15111yescharacteristic
269#107—1511no—
270C4⋊C8C2×D15111yescharacteristic, complemented
271#103—1511no—
272#103—1511no—
273#102—1511no—
274#102—1511no—
275#102—1511no—
276#102—1511no—
277C22⋊C8C2×D15111yescharacteristic, complemented
278C22⋊C8C2×D15111yescharacteristic, complemented
279#97C2×D15111yescharacteristic, complemented
280C24—3021no—
281C24—3021no—
282#41—3022no—
283#41—6041no—
284C2×Q8—3022no—
285C2×Q8—3021no—
286C2×Q8—6041no—
287C2×Q8—1511no—
288C2×Q8—1511no—
289C2×D4—3022no—
290C2×D4—3022no—
291C2×D4—3021no—
292C2×D4—3021no—
293C2×D4—3021no—
294C2×D4—3021no—
295C2×D4—3021no—
296C2×D4—3021no—
297C2×D4—3021no—
298C2×D4—3021no—
299C2×D4—6041no—
300C2×D4—6041no—
301C2×D4—6044no—
302C2×D4—1511no—
303C2×D4—1511no—
304C2×D4—1511no—
305C2×D4—1511no—
306C2×D4D60411yescharacteristic
307C22×C4—3021no—
308C22×C4—3021no—
309C22×C4—3021no—
310C22×C4C22×D15111yescharacteristic
311C22×C4C22×D15111yescharacteristic
312SD16—3022no—
313SD16—3022no—
314SD16—3022no—
315SD16—3022no—
316SD16—6041no—
317SD16—1514no—
318D8—3022no—
319D8—3022no—
320D8—6041no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D15111yescharacteristic
324C2×C8C22×D15111yescharacteristic
325C2×C8C22×D15111yescharacteristic
326C2×C8C22×D15111yescharacteristic
327C4⋊C4—3021no—
328C4⋊C4—3021no—
329C4⋊C4—1511no—
330C4⋊C4—1511no—
331C4⋊C4D60411yescharacteristic
332C22⋊C4—3021no—
333C22⋊C4—3021no—
334C22⋊C4—3021no—
335C22⋊C4—3021no—
336C22⋊C4—3021no—
337C22⋊C4—3021no—
338C22⋊C4—3021no—
339C22⋊C4D60411yescharacteristic
340C22⋊C4D60411yescharacteristic
341C42—3021no—
342C42C22×D15111yescharacteristic
343C23—3021no—
344C23—3021no—
345C23—3021no—
346C23—3021no—
347C23—6042no—
348C23—6042no—
349C23—6042no—
350C23—6042no—
351C23—12081no—
352C23—12081no—
353C23C2×D60411yescharacteristic
354C23C2×D60411yescharacteristic
355Q8—3021no—
356Q8—3021no—
357Q8—3021no—
358Q8—6042no—
359Q8—1512no—
360D4—22no—
361D4—3021no—
362D4—3021no—
363D4—3021no—
364D4—3021no—
365D4—3021no—
366D4—3021no—
367D4—3021no—
368D4—6042no—
369D4—6042no—
370D4—6042no—
371D4—6042no—
372D4—6042no—
373D4—6042no—
374D4—6042no—
375D4—1512no—
376D4—12081no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—3021no—
382C2×C4—3021no—
383C2×C4—3021no—
384C2×C4—3021no—
385C2×C4—6042no—
386C2×C4—6042no—
387C2×C4—6042no—
388C2×C4C2×D60411yescharacteristic
389C2×C4C2×D60411yescharacteristic
390C2×C4C2×D60411yescharacteristic
391C2×C4C2×D60411yescharacteristic
392C2×C4C23×D15111yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D15112yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—3022no—
403C22—6041no—
404C22—6041no—
405C22—6041no—
406C22—6041no—
407C22—6041no—
408C22—6041no—
409C22—6041no—
410C22—6041no—
411C22—6041no—
412C22—6041no—
413C22—6041no—
414C22—12084no—
415C22—12084no—
416C22#1801276712yescomplemented
417C22C22×D60411yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—3022no—
423C4—6041no—
424C4—6041no—
425C4—6041no—
426C4C2×D4×D15111yescharacteristic
427C4#1801278511yescharacteristic
428C2—22no—
429C2—41no—
430C2—6042no—
431C2—12081no—
432C2—12081no—
433C2—12081no—
434C2D4×D60411yescharacteristic
435C2C2×#1801276711yescharacteristic
436C2#2375302411yescharacteristic
subgroupG/H
centerC22C22×D604
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#435×C151C2
socleC22×C151C22×D4
hypercenter#435D151
hypocenterC151#3265
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#18012767C2semidirect
2. C22×#14558993C2semidirect
3. #23752871C2semidirect
4. D4×D604C2semidirect
5. #23752601C2semidirect
6. #23752458C2semidirect
7. #23752448C2semidirect
8. #23752428C2semidirect
9. #23752413C2semidirect
10. #23752365C2semidirect
11. #23752359C2semidirect
12. #23752353C2semidirect
13. #23752348C2semidirect
14. #23752314C2semidirect
15. #435×C151C2semidirect
16. C2×#14559004C22semidirect
17. C22×Dic302C22semidirect
25. C2×#14558993C22semidirect
26. C2×#14558993C22semidirect
27. C2×#14558993C22semidirect
28. C2×#14558993C22semidirect
30. C2×D1208C22semidirect
31. C2×D1208C22semidirect
33. #18012727C22semidirect
35. #18012718C22semidirect
40. #18012713C22semidirect
44. C4×Dic302C22semidirect
47. #18012673C22semidirect
48. #18012664C22semidirect
49. #18012664C22semidirect
51. #18012661C22semidirect
52. #18012661C22semidirect
53. #18012661C22semidirect
54. C2×M16×C151C22semidirect
55. C22×C8×C151C22semidirect
57. C4⋊C8×C151C22semidirect
58. C22⋊C8×C151C22semidirect
59. C22⋊C8×C151C22semidirect
60. #97×C151C22semidirect
95. #14558993D4semidirect
138. Dic302C2×D4semidirect
177. C8×C151C2×D4semidirect
183. C22×C151AGL(1,ℤ/8)semidirect
218. C151#3265semidirect
234. #435D151semidirect
251. C2×M16C2×D151semidirect
252. C22×C8C2×D151semidirect
270. C4⋊C8C2×D151semidirect
277. C22⋊C8C2×D151semidirect
278. C22⋊C8C2×D151semidirect
279. #97C2×D151semidirect
396. C8D4×D151semidirect
416. C22#18012767semidirect

Maximal subgroups

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

subgroupindex
C2×#180127672
C22×#145589932
#237528712
D4×D6042
#237526012
#237524582
#237524482
#237524282
#237524132
#237523652
#237523592
#237523532
#237523482
#237523142
#435×C1512
#3265151