Finite Group Databasebeta

← #34080823

Subgroups — #34080823

Order 19328.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#18012758C211yesmaximal, characteristic, complemented
2C2×SD16×D151C211yesmaximal, characteristic, complemented
3#23752870C211yesmaximal, characteristic, complemented
4#23752857C211yesmaximal, characteristic, complemented
5#23752594C211yesmaximal, characteristic, complemented
6#23752444C211yesmaximal, characteristic, complemented
7#23752440C211yesmaximal, characteristic, complemented
8#23752435C211yesmaximal, characteristic, complemented
9#23752371C211yesmaximal, characteristic, complemented
10#23752369C211yesmaximal, characteristic, complemented
11#23752358C211yesmaximal, characteristic, complemented
12#23752344C211yesmaximal, characteristic, complemented
13C4×#14558992C211yesmaximal, characteristic, complemented
14#23752277C211yesmaximal, characteristic, complemented
15#463×C151C211yesmaximal, characteristic, complemented
16C2×#14559003C2211yescharacteristic
17C2×#14559003C2211yescharacteristic, complemented
18C2×Q8×D151C2211yescharacteristic
19C2×D4×D151C2211yescharacteristic
20#18012758—24no—
21C2×Dic604C2211yescharacteristic, complemented
22C2×#14558994C2211yescharacteristic, complemented
23C2×#14558993C2211yescharacteristic
24C2×#14558992C2211yescharacteristic
25C2×#14558992C2211yescharacteristic, complemented
26C2×#14558991C2211yescharacteristic
27SD16×D151—24no—
28C2×#14558989C2211yescharacteristic
29C2×C8×D151C2211yescharacteristic
30C2×C8×D151C2211yescharacteristic, complemented
31#18012727C2211yescharacteristic
32#18012725—21no—
33#18012723—21no—
34#18012719C2211yescharacteristic
35#18012716—21no—
36#18012715C2211yescharacteristic, complemented
37#18012714C2211yescharacteristic
38#18012713—21no—
39#18012712—21no—
40#18012710C2211yescharacteristic
41#18012709—21no—
42C4×Dic302C2211yescharacteristic, complemented
43C4×D604C2211yescharacteristic
44(C22⋊D151)⋊C4—21no—
45C2×(C2×D151)⋊C4—21no—
46#18012670C2211yescharacteristic, complemented
47#18012664C2211yescharacteristic, complemented
48#18012662C2211yescharacteristic, complemented
49#18012661C2211yescharacteristic
50#18012660C2211yescharacteristic
51(C2×D151)⋊C8C2211yescharacteristic
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53C4×C151⋊C8C2211yescharacteristic, complemented
54C2×SD16×C151C2211yescharacteristic
55C2×D8×C151C2211yescharacteristic
56#124×C151C2211yescharacteristic, complemented
57C4×D4×C151C2211yescharacteristic
58C4⋊C8×C151C2211yescharacteristic, complemented
59#103×C151C2211yescharacteristic, complemented
60#102×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62#14559003—22no—
63#14559003—22no—
64#14559003—41no—
65#14559003—41no—
66C2×Dic302—21no—
67C2×Dic302C2311yescharacteristic
68C2×Dic302C2311yescharacteristic
69Q8×D151—22no—
70Q8×D151—41no—
71C2×D604—21no—
72C2×D604D411yescharacteristic
73C2×D604C2311yescharacteristic
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76C2×C22⋊D151—21no—
77C2×C22⋊D151—21no—
78D4×D151—22no—
79D4×D151—41no—
80C22×Dic151—21no—
81C22×Dic151—21no—
82C2×C4×D151—21no—
83C2×C4×D151C2311yescharacteristic
84C2×C4×D151C2311yescharacteristic
85Dic604—22no—
86#14558994—41no—
87#14558993—22no—
88#14558992—22no—
89#14558992D414yescomplemented
90#14558991—22no—
91#14558989—41no—
92C2×C151⋊C8C2311yescharacteristic
93C2×C151⋊C8C2311yescharacteristic
94C8×D151—22no—
95C8×D151—22no—
96C4⋊Dic151—21no—
97C4⋊Dic151D411yescharacteristic
98C4⋊Dic151C2311yescharacteristic
99Dic151⋊C4—21no—
100Dic151⋊C4—21no—
101C22⋊Dic151—21no—
102(C2×D151)⋊C4—21no—
103(C2×D151)⋊C4—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4D411yescharacteristic
108(C2×D151)⋊C4D411yescharacteristic
109C4×Dic151—21no—
110C2×Q8×C151C2311yescharacteristic
111C2×D4×C151C2311yescharacteristic
112C2×D4×C151C2311yescharacteristic
113C22×C4×C151—21no—
114SD16×C151—22no—
115D8×C151—22no—
116C2×C8×C151C2311yescharacteristic
117C2×C8×C151C2311yescharacteristic
118C4⋊C4×C151C2311yescharacteristic
119C22⋊C4×C151—21no—
120C22⋊C4×C151—21no—
121C22⋊C4×C151—21no—
122C42×C151C2311yescharacteristic
123C22×D151—21no—
124C22×D151—42no—
125C22×D151—42no—
126C22×D151—81no—
127C22×D151C2×D411yescharacteristic
128C22×D151C2×D411yescharacteristic
129Dic302—21no—
130Dic302—21no—
131Dic302—21no—
132Dic302—42no—
133Dic302C2×D412yes—
134D604—22no—
135D604—21no—
136D604—21no—
137D604—81no—
138C22⋊D151—42no—
139C22⋊D151—42no—
140C22⋊D151—42no—
141C22⋊D151—42no—
142C2×Dic151—21no—
143C2×Dic151—21no—
144C2×Dic151—21no—
145C2×Dic151—42no—
146C2×Dic151—42no—
147C2×Dic151C2×D411yescharacteristic
148C2×Dic151C2×D411yescharacteristic
149C4×D151—21no—
150C4×D151—21no—
151C4×D151—21no—
152C4×D151—21no—
153C4×D151—42no—
154C151⋊C8—21no—
155C151⋊C8C2×D412yes—
156C23×C151—21no—
157C23×C151—21no—
158Q8×C151—21no—
159Q8×C151—21no—
160D4×C151—21no—
161D4×C151—21no—
162D4×C151—21no—
163D4×C151C2×D412yes—
164C2×C4×C151—21no—
165C2×C4×C151—21no—
166C2×C4×C151—81no—
167C2×C4×C151C2×D411yescharacteristic
168C2×C4×C151C2×D411yescharacteristic
169C2×C4×C151C2411yescharacteristic
170C8×C151—21no—
171C8×C151—21no—
172C22×C151—22no—
173C22×C151—41no—
174C22×C151—41no—
175C22×C151—41no—
176C22×C151—41no—
177C22×C151—41no—
178C22×C151C22×D411yescharacteristic
179C2×D151—21no—
180C2×D151—21no—
181C2×D151—21no—
182C2×D151—21no—
183C2×D151—21no—
184C2×D151—41no—
185C2×D151—41no—
186C2×D151—41no—
187C2×D151—84no—
188C2×D151#13512yescomplemented
189C4×C151—21no—
190C4×C151—21no—
191C4×C151—41no—
192C4×C151—41no—
193C4×C151C22×D411yescharacteristic
194C4×C151H(5,𝔽2)11yescharacteristic
195Dic151—21no—
196Dic151—21no—
197Dic151—22no—
198Dic151—41no—
199Dic151—41no—
200C2×C151—42no—
201C2×C151—81no—
202C2×C151D4211yescharacteristic
203C2×C151C2×#13511yescharacteristic
204C2×C151#57711yescharacteristic
205D151—22no—
206D151—41no—
207D151—81no—
208C151#328211yescharacteristic, complemented, Sylow
209#3282—1511nomaximal, Sylow
210C2×#135—1511no—
211C22×SD16—1511no—
212#548—1511no—
213#546—1511no—
214#495—1511no—
215#465—1511no—
216#465—1511no—
217#464—1511no—
218#463D15111yescharacteristic, complemented
219#452—1511no—
220#451—1511no—
221#449—1511no—
222#447—1511no—
223C4×SD16—1511no—
224C8×D4—1511no—
225C2×#41—1511no—
226C2×#41—1511no—
227C22×Q8—1511no—
228C22×D4—1511no—
229#135—3024no—
230C2×Q16—1511no—
231C2×Q16—1511no—
232C2×SD16—3024no—
233C2×SD16—1511no—
234C2×SD16—1511no—
235C2×SD16—1511no—
236C2×SD16—1511no—
237C2×SD16C2×D15111yescharacteristic
238C2×D8—1511no—
239C2×D8C2×D15111yescharacteristic
240C22×C8—1511no—
241C22×C8—1511no—
242#124—3021no—
243#124—1511no—
244#124C2×D15111yescharacteristic, complemented
245#123—3021no—
246C22⋊Q8—3021no—
247C22⋊Q8—1511no—
248C22⋊Q8—1511no—
249#121—3021no—
250#121—3021no—
251#121—1511no—
252#121—1511no—
253#120—3021no—
254C4×Q8—1511no—
255C4×D4—3021no—
256C4×D4—1511no—
257C4×D4C2×D15111yescharacteristic
258C2×C22⋊C4—3021no—
259#106—1511no—
260C4⋊C8C2×D15111yescharacteristic, complemented
261#103—1511no—
262#103—1511no—
263#103C2×D15111yescharacteristic, complemented
264#102—1511no—
265#102—1511no—
266#102C2×D15111yescharacteristic, complemented
267C22⋊C8—1511no—
268C22⋊C8—1511no—
269C4×C8—1511no—
270C24—3021no—
271#41—3022no—
272#41—3022no—
273#41—6041no—
274#41—6041no—
275C2×Q8—3021no—
276C2×Q8—3022no—
277C2×Q8—6041no—
278C2×Q8—1511no—
279C2×Q8—1511no—
280C2×Q8C22×D15111yescharacteristic
281C2×D4—3021no—
282C2×D4—3022no—
283C2×D4—3021no—
284C2×D4—3021no—
285C2×D4—3021no—
286C2×D4—3021no—
287C2×D4—6041no—
288C2×D4—1511no—
289C2×D4—1511no—
290C2×D4C22×D15111yescharacteristic
291C2×D4C22×D15111yescharacteristic
292C22×C4—21no—
293C22×C4—3021no—
294C22×C4—3021no—
295C22×C4—3021no—
296C22×C4—1511no—
297C22×C4—1511no—
298Q16—3022no—
299Q16—6041no—
300SD16—22no—
301SD16—3022no—
302SD16—3022no—
303SD16—3022no—
304SD16—1514no—
305D8—22no—
306D8—6041no—
307C2×C8—3022no—
308C2×C8—3022no—
309C2×C8—1511no—
310C2×C8—1511no—
311C2×C8C22×D15111yescharacteristic
312C2×C8C22×D15111yescharacteristic
313C4⋊C4—3021no—
314C4⋊C4—3021no—
315C4⋊C4—3021no—
316C4⋊C4—1511no—
317C4⋊C4—1511no—
318C4⋊C4C22×D15111yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3021no—
323C22⋊C4—3021no—
324C22⋊C4—3021no—
325C22⋊C4—3021no—
326C22⋊C4—3021no—
327C22⋊C4—3021no—
328C22⋊C4—1511no—
329C22⋊C4—1511no—
330C42—3021no—
331C42C22×D15111yescharacteristic
332C23—21no—
333C23—21no—
334C23—3021no—
335C23—6042no—
336C23—6042no—
337C23—1511no—
338C23—1511no—
339C23—12081no—
340Q8—21no—
341Q8—21no—
342Q8—3021no—
343Q8—3021no—
344Q8—3021no—
345Q8—6042no—
346Q8—1512no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—3022no—
351D4—3021no—
352D4—3021no—
353D4—6042no—
354D4—6042no—
355D4—6042no—
356D4—6042no—
357D4—12081no—
358D4D4×D15112yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—81no—
362C2×C4—3021no—
363C2×C4—3021no—
364C2×C4—3021no—
365C2×C4—3021no—
366C2×C4—3021no—
367C2×C4—3021no—
368C2×C4—3021no—
369C2×C4—6042no—
370C2×C4—6042no—
371C2×C4—6042no—
372C2×C4—1511no—
373C2×C4—1511no—
374C2×C4D4×D15111yescharacteristic
375C2×C4D4×D15111yescharacteristic
376C2×C4C23×D15111yescharacteristic
377C8—21no—
378C8—21no—
379C8—3021no—
380C8—1512no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3021no—
388C22—3021no—
389C22—3021no—
390C22—3021no—
391C22—3021no—
392C22—6041no—
393C22—6041no—
394C22—6041no—
395C22—1512no—
396C22—12084no—
397C22C2×D4×D15111yescharacteristic
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3021no—
403C4—3021no—
404C4—3022no—
405C4—6041no—
406C4—6041no—
407C4C2×D4×D15111yescharacteristic
408C4#1801278511yescharacteristic
409C2—42no—
410C2—81no—
411C2—3022no—
412C2—6041no—
413C2—12081no—
414C2#2375284611yescharacteristic
415C2#135×D15111yescharacteristic
416C2#2375302211yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#463×C151C2
socleC22×C151C22×D4
hypercenter#463D151
hypocenterC151#3282
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#18012758C2semidirect
2. C2×SD16×D151C2semidirect
3. #23752870C2semidirect
4. #23752857C2semidirect
5. #23752594C2semidirect
6. #23752444C2semidirect
7. #23752440C2semidirect
8. #23752435C2semidirect
9. #23752371C2semidirect
10. #23752369C2semidirect
11. #23752358C2semidirect
12. #23752344C2semidirect
13. C4×#14558992C2semidirect
14. #23752277C2semidirect
15. #463×C151C2semidirect
17. C2×#14559003C22semidirect
21. C2×Dic604C22semidirect
22. C2×#14558994C22semidirect
25. C2×#14558992C22semidirect
30. C2×C8×D151C22semidirect
36. #18012715C22semidirect
42. C4×Dic302C22semidirect
46. #18012670C22semidirect
47. #18012664C22semidirect
48. #18012662C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. C4×C151⋊C8C22semidirect
56. #124×C151C22semidirect
58. C4⋊C8×C151C22semidirect
59. #103×C151C22semidirect
60. #102×C151C22semidirect
89. #14558992D4semidirect
188. C2×D151#135semidirect
208. C151#3282semidirect
218. #463D151semidirect
244. #124C2×D151semidirect
260. C4⋊C8C2×D151semidirect
263. #103C2×D151semidirect
266. #102C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127582
C2×SD16×D1512
#237528702
#237528572
#237525942
#237524442
#237524402
#237524352
#237523712
#237523692
#237523582
#237523442
C4×#145589922
#237522772
#463×C1512
#3282151