Finite Group Databasebeta

← #34078393

Subgroups — #34078393

Order 19328.

732 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012786C211yesmaximal, characteristic, complemented
2C2×#18012763C211yesmaximal, characteristic, complemented
3C2×SD16×D151C211yesmaximal, characteristic, complemented
4#23752778C211yesmaximal, characteristic, complemented
5#23752358C211yesmaximal, characteristic, complemented
6#23752357C211yesmaximal, characteristic, complemented
7#23752354C211yesmaximal, characteristic, complemented
8#23752353C211yesmaximal, characteristic, complemented
9#23752352C211yesmaximal, characteristic, complemented
10#23752350C211yesmaximal, characteristic, complemented
11#23752349C211yesmaximal, characteristic, complemented
12#23752219C211yesmaximal, characteristic, complemented
13#23752205C211yesmaximal, characteristic, complemented
14#23752177C211yesmaximal, characteristic, complemented
15#449×C151C211yesmaximal, characteristic, complemented
16#18012786—24no—
17#18012786—24no—
18C2×#14559004—21no—
19C2×#14559004C2211yescharacteristic
20C2×#14559003—21no—
21C2×#14559003—21no—
22C2×#14559003C2211yescharacteristic
23C2×#14559003C2211yescharacteristic, complemented
24C22×Dic302—21no—
25C22×Dic302C2211yescharacteristic, complemented
26C2×Q8×D151C2211yescharacteristic
27C2×Q8×D151C2211yescharacteristic, complemented
28C2×D4×D151C2211yescharacteristic
29#18012763—24no—
30C2×#14558993C2211yescharacteristic
31C2×#14558993C2211yescharacteristic
32C2×#14558992C2211yescharacteristic
33C2×#14558992C2211yescharacteristic
34C2×#14558991C2211yescharacteristic
35SD16×D151—24no—
36C2×#14558989C2211yescharacteristic
37C2×#14558987C2211yescharacteristic
38C2×C8×D151C2211yescharacteristic
39#18012735—21no—
40#18012725—21no—
41#18012714C2211yescharacteristic
42#18012713C2211yescharacteristic
43#18012710C2211yescharacteristic
44#18012709—21no—
45#18012707—21no—
46#18012697C2211yescharacteristic
47#18012664C2211yescharacteristic, complemented
48#18012664C2211yescharacteristic, complemented
49#18012663C2211yescharacteristic, complemented
50#18012662C2211yescharacteristic, complemented
51#18012662C2211yescharacteristic, complemented
52#18012660C2211yescharacteristic, complemented
53C22⋊(C151⋊C8)C2211yescharacteristic, complemented
54(C2×D151)⋊C8C2211yescharacteristic, complemented
55(C2×D151)⋊C8C2211yescharacteristic, complemented
56C2×#41×C151C2211yescharacteristic, complemented
57C2×SD16×C151C2211yescharacteristic
58C2×D8×C151C2211yescharacteristic
59#121×C151C2211yescharacteristic
60#103×C151C2211yescharacteristic, complemented
61#102×C151C2211yescharacteristic, complemented
62C22⋊C8×C151C2211yescharacteristic, complemented
63C23×D151—21no—
64#14559004—28no—
65#14559004—22no—
66#14559004—22no—
67#14559003—28no—
68#14559003—28no—
69#14559003—22no—
70#14559003—22no—
71#14559003—22no—
72#14559003D414yescomplemented
73C2×Dic302—28no—
74C2×Dic302—22no—
75C2×Dic302—22no—
76C2×Dic302—21no—
77C2×Dic302—21no—
78C2×Dic302—21no—
79C2×Dic302D411yescharacteristic
80C2×Dic302C2311yescharacteristic
81C2×Dic302C2311yescharacteristic
82Q8×D151—22no—
83Q8×D151—22no—
84Q8×D151—22no—
85Q8×D151D414yescomplemented
86C2×D604—21no—
87C2×D604D411yescharacteristic
88C2×D604C2311yescharacteristic
89C2×C22⋊D151—21no—
90C2×C22⋊D151—21no—
91C2×C22⋊D151—21no—
92C2×C22⋊D151—21no—
93C2×C22⋊D151—21no—
94C2×C22⋊D151D411yescharacteristic
95C2×C22⋊D151D411yescharacteristic
96D4×D151—22no—
97D4×D151—22no—
98C22×Dic151—21no—
99C22×Dic151—21no—
100C22×Dic151—21no—
101C2×C4×D151—21no—
102C2×C4×D151—21no—
103C2×C4×D151C2311yescharacteristic
104C2×C4×D151C2311yescharacteristic
105#14558993—22no—
106#14558993—22no—
107#14558992—22no—
108#14558992—22no—
109#14558991—22no—
110#14558989—22no—
111#14558987—22no—
112C2×C151⋊C8C2311yescharacteristic
113C2×C151⋊C8C2311yescharacteristic
114C8×D151—22no—
115C4⋊Dic151C2311yescharacteristic
116C22⋊Dic151—21no—
117(C2×D151)⋊C4—21no—
118(C2×D151)⋊C4—21no—
119(C2×D151)⋊C4—21no—
120C4×Dic151—21no—
121#41×C151—22no—
122#41×C151—22no—
123C2×Q8×C151C2311yescharacteristic
124C2×D4×C151—21no—
125C2×D4×C151—21no—
126C2×D4×C151C2311yescharacteristic
127C2×D4×C151C2311yescharacteristic
128C22×C4×C151—21no—
129C22×C4×C151C2311yescharacteristic
130SD16×C151—22no—
131D8×C151—22no—
132C2×C8×C151C2311yescharacteristic
133C2×C8×C151C2311yescharacteristic
134C4⋊C4×C151C2311yescharacteristic
135C22⋊C4×C151—21no—
136C22×D151—21no—
137C22×D151—42no—
138C22×D151—81no—
139C22×D151—81no—
140C22×D151C2×D411yescharacteristic
141C22×D151C2×D411yescharacteristic
142Dic302—24no—
143Dic302—24no—
144Dic302—21no—
145Dic302—21no—
146Dic302—24no—
147Dic302—22no—
148Dic302C2×D412yes—
149Dic302C2×D412yes—
150D604—22no—
151D604—21no—
152D604—21no—
153D604—81no—
154C22⋊D151—24no—
155C22⋊D151—24no—
156C22⋊D151—22no—
157C22⋊D151—22no—
158C22⋊D151—42no—
159C22⋊D151—81no—
160C22⋊D151—81no—
161C2×Dic151—24no—
162C2×Dic151—24no—
163C2×Dic151—21no—
164C2×Dic151—21no—
165C2×Dic151—24no—
166C2×Dic151—21no—
167C2×Dic151C2×D411yescharacteristic
168C2×Dic151C2×D411yescharacteristic
169C4×D151—24no—
170C4×D151—24no—
171C4×D151—21no—
172C4×D151—21no—
173C4×D151C2×D412yes—
174C4×D151C2×D412yes—
175C151⋊C8—21no—
176C151⋊C8—21no—
177C23×C151—21no—
178C23×C151—21no—
179C23×C151C2×D411yescharacteristic
180Q8×C151—21no—
181Q8×C151C2×D412yes—
182D4×C151—24no—
183D4×C151—21no—
184D4×C151—21no—
185D4×C151—21no—
186D4×C151—81no—
187D4×C151C2×D412yes—
188C2×C4×C151—24no—
189C2×C4×C151—21no—
190C2×C4×C151—21no—
191C2×C4×C151—21no—
192C2×C4×C151—21no—
193C2×C4×C151C2×D411yescharacteristic
194C2×C4×C151C2411yescharacteristic
195C8×C151—21no—
196C8×C151—21no—
197C22×C151—21no—
198C22×C151—22no—
199C22×C151—21no—
200C22×C151—21no—
201C22×C151—41no—
202C22×C151—41no—
203C22×C151—41no—
204C22×C151—41no—
205C22×C151—41no—
206C22×C151C22×D411yescharacteristic
207C2×D151—21no—
208C2×D151—21no—
209C2×D151—21no—
210C2×D151—21no—
211C2×D151—21no—
212C2×D151—41no—
213C2×D151—41no—
214C2×D151—41no—
215C2×D151—84no—
216C2×D151#12012yes—
217C4×C151—21no—
218C4×C151—22no—
219C4×C151—41no—
220C4×C151C22×D411yescharacteristic
221C4×C151C22×D411yescharacteristic
222Dic151—22no—
223Dic151—22no—
224Dic151—21no—
225Dic151—41no—
226Dic151#12012yes—
227C2×C151—41no—
228C2×C151—42no—
229C2×C151—81no—
230C2×C151C2×#12011yescharacteristic
231C2×C151#57711yescharacteristic
232C2×C151#57711yescharacteristic
233D151—22no—
234D151—41no—
235D151—81no—
236C151#301011yescharacteristic, complemented, Sylow
237#3010—1511nomaximal, Sylow
238C2×2−1+4—1511no—
239C2×AGL(1,ℤ/8)—1511no—
240C22×SD16—1511no—
241#535—1511no—
242#449—1511no—
243#449—1511no—
244#449—1511no—
245#449D15111yescharacteristic, complemented
246#448—1511no—
247#448—1511no—
248#448—1511no—
249#448—1511no—
250#419—1511no—
251#416—1511no—
252#408—1511no—
2532−1+4—3024no—
2542−1+4—3024no—
255C2×#41—3021no—
256C2×#41—3021no—
257C2×#41—3021no—
258C2×#41—1511no—
259C2×#41—1511no—
260C2×#41—1511no—
261C2×#41C2×D15111yescharacteristic, complemented
262C22×Q8—3021no—
263C22×Q8—1511no—
264C22×Q8—1511no—
265C22×Q8—1511no—
266C22×D4—1511no—
267AGL(1,ℤ/8)—3024no—
268C2×SD16—3024no—
269C2×SD16—1511no—
270C2×SD16—1511no—
271C2×SD16—1511no—
272C2×SD16—1511no—
273C2×SD16—1511no—
274C2×SD16C2×D15111yescharacteristic
275C2×D8—1511no—
276C2×D8C2×D15111yescharacteristic
277C2×M16—1511no—
278C22×C8—1511no—
279#127—3021no—
280#124—3021no—
281#121—1511no—
282#121—1511no—
283#121—1511no—
284#121C2×D15111yescharacteristic
285#120—3021no—
286#120—3021no—
287#117—1511no—
288#103—1511no—
289#103—1511no—
290#103—1511no—
291#103—1511no—
292#103—1511no—
293#103C2×D15111yescharacteristic, complemented
294#102—1511no—
295#102C2×D15111yescharacteristic, complemented
296C22⋊C8—1511no—
297C22⋊C8—1511no—
298C22⋊C8—1511no—
299C22⋊C8C2×D15111yescharacteristic, complemented
300C24—3021no—
301#41—22no—
302#41—22no—
303#41—3028no—
304#41—3022no—
305#41—3022no—
306#41—3028no—
307#41—3028no—
308#41—3022no—
309#41—3022no—
310#41—3022no—
311#41—1514no—
312C2×Q8—3028no—
313C2×Q8—3022no—
314C2×Q8—3022no—
315C2×Q8—3021no—
316C2×Q8—3021no—
317C2×Q8—3022no—
318C2×Q8—3022no—
319C2×Q8—3022no—
320C2×Q8—3021no—
321C2×Q8—1514no—
322C2×Q8—1511no—
323C2×Q8—1511no—
324C2×Q8—1511no—
325C2×Q8C22×D15111yescharacteristic
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—3021no—
329C2×D4—3021no—
330C2×D4—3021no—
331C2×D4—3021no—
332C2×D4—3021no—
333C2×D4—3022no—
334C2×D4—3022no—
335C2×D4—3021no—
336C2×D4—1511no—
337C2×D4—1511no—
338C2×D4—1511no—
339C2×D4—1511no—
340C2×D4C22×D15111yescharacteristic
341C2×D4C22×D15111yescharacteristic
342C22×C4—21no—
343C22×C4—3021no—
344C22×C4—3021no—
345C22×C4—3021no—
346C22×C4—3021no—
347C22×C4—3021no—
348C22×C4—1511no—
349C22×C4—1511no—
350C22×C4C22×D15111yescharacteristic
351SD16—22no—
352SD16—3022no—
353SD16—3022no—
354SD16—3022no—
355SD16—3022no—
356SD16—3022no—
357D8—22no—
358D8—3022no—
359M16—3022no—
360C2×C8—3022no—
361C2×C8—1511no—
362C2×C8—1511no—
363C2×C8C22×D15111yescharacteristic
364C2×C8C22×D15111yescharacteristic
365C4⋊C4—1511no—
366C4⋊C4C22×D15111yescharacteristic
367C22⋊C4—21no—
368C22⋊C4—3021no—
369C22⋊C4—3021no—
370C22⋊C4—3021no—
371C22⋊C4—3021no—
372C42—3021no—
373C23—21no—
374C23—21no—
375C23—3021no—
376C23—6042no—
377C23—1511no—
378C23—1511no—
379C23—12081no—
380C23—12081no—
381C23D4×D15111yescharacteristic
382Q8—21no—
383Q8—3024no—
384Q8—3024no—
385Q8—3022no—
386Q8—3024no—
387Q8—3021no—
388Q8—3021no—
389Q8—1512no—
390Q8—1512no—
391Q8D4×D15112yes—
392D4—24no—
393D4—21no—
394D4—21no—
395D4—21no—
396D4—81no—
397D4—3022no—
398D4—3024no—
399D4—3024no—
400D4—3022no—
401D4—3022no—
402D4—3021no—
403D4—3021no—
404D4—6042no—
405D4—12081no—
406D4—12081no—
407D4—12081no—
408D4D4×D15112yes—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—3024no—
415C2×C4—3024no—
416C2×C4—3024no—
417C2×C4—3024no—
418C2×C4—3021no—
419C2×C4—3021no—
420C2×C4—3024no—
421C2×C4—3021no—
422C2×C4—3021no—
423C2×C4—3021no—
424C2×C4—1511no—
425C2×C4—1511no—
426C2×C4—1512no—
427C2×C4—1512no—
428C2×C4D4×D15111yescharacteristic
429C2×C4C23×D15111yescharacteristic
430C8—21no—
431C8—21no—
432C8—3021no—
433C8—3021no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—3021no—
444C22—3021no—
445C22—3021no—
446C22—3021no—
447C22—3021no—
448C22—6041no—
449C22—6041no—
450C22—6041no—
451C22—1512no—
452C22—12084no—
453C22C2×D4×D15111yescharacteristic
454C4—21no—
455C4—22no—
456C4—41no—
457C4—3022no—
458C4—3022no—
459C4—3021no—
460C4—6041no—
461C4—1512no—
462C4C2×D4×D15111yescharacteristic
463C4C2×D4×D15111yescharacteristic
464C2—41no—
465C2—42no—
466C2—81no—
467C2—3022no—
468C2—6041no—
469C2—12081no—
470C2#120×D15111yescharacteristic
471C2#2375301911yescharacteristic
472C2#2375302211yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#449×C151C2
socleC22×C151C22×D4
hypercenter#449D151
hypocenterC151#3010
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#18012786C2semidirect
2. C2×#18012763C2semidirect
3. C2×SD16×D151C2semidirect
4. #23752778C2semidirect
5. #23752358C2semidirect
6. #23752357C2semidirect
7. #23752354C2semidirect
8. #23752353C2semidirect
9. #23752352C2semidirect
10. #23752350C2semidirect
11. #23752349C2semidirect
12. #23752219C2semidirect
13. #23752205C2semidirect
14. #23752177C2semidirect
15. #449×C151C2semidirect
23. C2×#14559003C22semidirect
25. C22×Dic302C22semidirect
27. C2×Q8×D151C22semidirect
47. #18012664C22semidirect
48. #18012664C22semidirect
49. #18012663C22semidirect
50. #18012662C22semidirect
51. #18012662C22semidirect
52. #18012660C22semidirect
53. C22⋊(C151⋊C8)C22semidirect
54. (C2×D151)⋊C8C22semidirect
55. (C2×D151)⋊C8C22semidirect
56. C2×#41×C151C22semidirect
60. #103×C151C22semidirect
61. #102×C151C22semidirect
62. C22⋊C8×C151C22semidirect
72. #14559003D4semidirect
85. Q8×D151D4semidirect
236. C151#3010semidirect
245. #449D151semidirect
261. C2×#41C2×D151semidirect
293. #103C2×D151semidirect
295. #102C2×D151semidirect
299. C22⋊C8C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127862
C2×#180127632
C2×SD16×D1512
#237527782
#237523582
#237523572
#237523542
#237523532
#237523522
#237523502
#237523492
#237522192
#237522052
#237521772
#449×C1512
#3010151