Finite Group Databasebeta

← #34080628

Subgroups — #34080628

Order 19328.

500 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×#18012763C211yesmaximal, characteristic, complemented
2C2×#18012759C211yesmaximal, characteristic, complemented
3#23752861C211yesmaximal, characteristic, complemented
4#23752859C211yesmaximal, characteristic, complemented
5#23752603C211yesmaximal, characteristic, complemented
6#23752469C211yesmaximal, characteristic, complemented
7#23752439C211yesmaximal, characteristic, complemented
8#23752425C211yesmaximal, characteristic, complemented
9#23752416C211yesmaximal, characteristic, complemented
10#23752376C211yesmaximal, characteristic, complemented
11#23752375C211yesmaximal, characteristic, complemented
12#23752361C211yesmaximal, characteristic, complemented
13#23752321C211yesmaximal, characteristic, complemented
14#23752280C211yesmaximal, characteristic, complemented
15#449×C151C211yesmaximal, characteristic, complemented
16C2×#14559004C2211yescharacteristic
17C2×#14559003C2211yescharacteristic
18C2×D4×D151C2211yescharacteristic
19#18012763—24no—
20#18012759—24no—
21C2×Dic604C2211yescharacteristic, complemented
22C2×#14558994C2211yescharacteristic, complemented
23C2×#14558993C2211yescharacteristic
24C2×#14558992C2211yescharacteristic
25C2×#14558992C2211yescharacteristic, complemented
26C2×#14558991C2211yescharacteristic
27C2×#14558989C2211yescharacteristic
28C2×#14558987C2211yescharacteristic
29C22×C151⋊C8C2211yescharacteristic, complemented
30#18012737C2211yescharacteristic, complemented
31#18012726C2211yescharacteristic
32#18012722—21no—
33#18012721—21no—
34#18012720—21no—
35#18012719C2211yescharacteristic
36#18012718C2211yescharacteristic, complemented
37#18012715C2211yescharacteristic
38#18012711—21no—
39#18012710—21no—
40#18012707—21no—
41D4×Dic151C2211yescharacteristic, complemented
42#18012701C2211yescharacteristic
43C4×C22⋊D151C2211yescharacteristic
44C2×Dic151⋊C4—21no—
45C22⋊C4×D151—21no—
46#18012670C2211yescharacteristic, complemented
47Dic151⋊C8C2211yescharacteristic, complemented
48#18012664C2211yescharacteristic, complemented
49#18012662C2211yescharacteristic, complemented
50#18012661C2211yescharacteristic
51#18012660C2211yescharacteristic
52(C2×D151)⋊C8C2211yescharacteristic
53#18012649C2211yescharacteristic, complemented
54C2×#41×C151C2211yescharacteristic, complemented
55C2×SD16×C151C2211yescharacteristic
56C2×D8×C151C2211yescharacteristic
57#121×C151C2211yescharacteristic
58#103×C151C2211yescharacteristic, complemented
59#102×C151C2211yescharacteristic, complemented
60C22⋊C8×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62#14559004—22no—
63#14559004—41no—
64#14559003—22no—
65#14559003—41no—
66C2×Dic302C2311yescharacteristic
67C2×Dic302C2311yescharacteristic
68C2×D604C2311yescharacteristic
69C2×C22⋊D151—21no—
70C2×C22⋊D151—21no—
71C2×C22⋊D151—21no—
72C2×C22⋊D151D411yescharacteristic
73D4×D151—22no—
74D4×D151—41no—
75C22×Dic151—21no—
76C22×Dic151—21no—
77C2×C4×D151—21no—
78C2×C4×D151—21no—
79C2×C4×D151C2311yescharacteristic
80Dic604—41no—
81#14558994—22no—
82#14558993—22no—
83#14558992—22no—
84#14558992—22no—
85#14558991—22no—
86#14558989D414yescomplemented
87#14558987—22no—
88C2×C151⋊C8—22no—
89C2×C151⋊C8C2311yescharacteristic
90C2×C151⋊C8C2311yescharacteristic
91C4⋊Dic151—21no—
92C4⋊Dic151C2311yescharacteristic
93Dic151⋊C4—21no—
94Dic151⋊C4—21no—
95Dic151⋊C4—21no—
96Dic151⋊C4—21no—
97Dic151⋊C4D411yescharacteristic
98C22⋊Dic151—21no—
99C22⋊Dic151—21no—
100C22⋊Dic151—21no—
101C22⋊Dic151—21no—
102C22⋊Dic151D411yescharacteristic
103(C2×D151)⋊C4—21no—
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4D411yescharacteristic
107C4×Dic151C2311yescharacteristic
108#41×C151—22no—
109#41×C151—41no—
110C2×Q8×C151C2311yescharacteristic
111C2×D4×C151—21no—
112C2×D4×C151—21no—
113C2×D4×C151C2311yescharacteristic
114C2×D4×C151C2311yescharacteristic
115C22×C4×C151—21no—
116C22×C4×C151C2311yescharacteristic
117SD16×C151—41no—
118D8×C151—22no—
119C2×C8×C151C2311yescharacteristic
120C2×C8×C151C2311yescharacteristic
121C4⋊C4×C151C2311yescharacteristic
122C22⋊C4×C151—21no—
123C22×D151—21no—
124C22×D151—42no—
125C22×D151—42no—
126C22×D151—81no—
127C22×D151C2×D411yescharacteristic
128Dic302—21no—
129Dic302—21no—
130Dic302—21no—
131Dic302—21no—
132D604—21no—
133D604C2×D412yes—
134C22⋊D151—41no—
135C22⋊D151—42no—
136C22⋊D151—42no—
137C22⋊D151—42no—
138C2×Dic151—21no—
139C2×Dic151—21no—
140C2×Dic151—21no—
141C2×Dic151—42no—
142C2×Dic151—42no—
143C2×Dic151C2×D411yescharacteristic
144C2×Dic151C2×D411yescharacteristic
145C2×Dic151C2×D411yescharacteristic
146C4×D151—21no—
147C4×D151—21no—
148C4×D151—42no—
149C4×D151—81no—
150C151⋊C8—21no—
151C151⋊C8C2×D412yes—
152C23×C151—21no—
153C23×C151—21no—
154C23×C151C2×D411yescharacteristic
155Q8×C151—21no—
156Q8×C151—21no—
157D4×C151—21no—
158D4×C151—21no—
159D4×C151—21no—
160D4×C151—42no—
161D4×C151—81no—
162D4×C151C2×D412yes—
163C2×C4×C151—21no—
164C2×C4×C151—21no—
165C2×C4×C151—21no—
166C2×C4×C151—21no—
167C2×C4×C151—42no—
168C2×C4×C151C2×D411yescharacteristic
169C2×C4×C151C2411yescharacteristic
170C8×C151—21no—
171C8×C151—21no—
172C22×C151—21no—
173C22×C151—22no—
174C22×C151—21no—
175C22×C151—21no—
176C22×C151—41no—
177C22×C151—41no—
178C22×C151—41no—
179C22×C151—41no—
180C22×C151—41no—
181C22×C151C22×D411yescharacteristic
182C2×D151—21no—
183C2×D151—22no—
184C2×D151—21no—
185C2×D151—21no—
186C2×D151—41no—
187C2×D151—41no—
188C2×D151—84no—
189C4×C151—21no—
190C4×C151—41no—
191C4×C151—41no—
192C4×C151C22×D411yescharacteristic
193C4×C151H(5,𝔽2)11yescharacteristic
194Dic151—21no—
195Dic151—21no—
196Dic151—21no—
197Dic151—41no—
198Dic151—41no—
199Dic151—41no—
200C2×C151—41no—
201C2×C151—42no—
202C2×C151—81no—
203C2×C151D4211yescharacteristic
204C2×C151#57511yescharacteristic
205C2×C151#57711yescharacteristic
206D151—41no—
207D151—42no—
208C151#326711yescharacteristic, complemented, Sylow
209#3267—1511nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1511no—
211C2×#135—1511no—
212#547—1511no—
213#546—1511no—
214#497—1511no—
215#470—1511no—
216#465—1511no—
217#463—1511no—
218#461—1511no—
219#452—1511no—
220#452—1511no—
221#450—1511no—
222#449D15111yescharacteristic, complemented
223#442—1511no—
224#435—1511no—
225C2×#41—1511no—
226C2×#41—1511no—
227C2×#41C2×D15111yescharacteristic, complemented
228C22×D4—1511no—
229AGL(1,ℤ/8)—3024no—
230#135—3024no—
231C2×Q16—1511no—
232C2×Q16—1511no—
233C2×SD16—1511no—
234C2×SD16—1511no—
235C2×SD16—1511no—
236C2×SD16—1511no—
237C2×SD16C2×D15111yescharacteristic
238C2×D8—1511no—
239C2×D8C2×D15111yescharacteristic
240C2×M16—1511no—
241C22×C8—1511no—
242C4⋊Q8—1511no—
243#124—1511no—
244#123—3021no—
245#123—3021no—
246#123—3021no—
247C22⋊Q8—1511no—
248C22⋊Q8—1511no—
249C22⋊Q8—1511no—
250#121—3021no—
251#121—3021no—
252#121C2×D15111yescharacteristic
253#120—3021no—
254C4×D4—1511no—
255C4×D4—1511no—
256C4×D4—1511no—
257C2×C4⋊C4—3021no—
258C2×C22⋊C4—3021no—
259#106—1511no—
260C4⋊C8—1511no—
261#103—1511no—
262#103—1511no—
263#103C2×D15111yescharacteristic, complemented
264#102—1511no—
265#102—1511no—
266#102C2×D15111yescharacteristic, complemented
267C22⋊C8—1511no—
268C22⋊C8C2×D15111yescharacteristic, complemented
269#97—1511no—
270C24—3021no—
271#41—22no—
272#41—41no—
273#41—3022no—
274#41—3022no—
275#41—6041no—
276#41—6041no—
277C2×Q8—1511no—
278C2×Q8—1511no—
279C2×Q8C22×D15111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—3022no—
283C2×D4—3021no—
284C2×D4—3021no—
285C2×D4—3021no—
286C2×D4—6041no—
287C2×D4—1511no—
288C2×D4—1511no—
289C2×D4C22×D15111yescharacteristic
290C2×D4C22×D15111yescharacteristic
291C22×C4—21no—
292C22×C4—3021no—
293C22×C4—3021no—
294C22×C4—3021no—
295C22×C4—3021no—
296C22×C4—1511no—
297C22×C4C22×D15111yescharacteristic
298Q16—3022no—
299Q16—6041no—
300SD16—41no—
301SD16—3022no—
302SD16—3022no—
303SD16—3022no—
304SD16—3022no—
305D8—22no—
306D8—1514no—
307M16—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—3021no—
317C4⋊C4—3021no—
318C4⋊C4—1511no—
319C4⋊C4—1511no—
320C4⋊C4C22×D15111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—3021no—
323C22⋊C4—3021no—
324C22⋊C4—3021no—
325C22⋊C4—3021no—
326C22⋊C4—3021no—
327C22⋊C4—3021no—
328C22⋊C4—3021no—
329C22⋊C4—1511no—
330C22⋊C4—1511no—
331C42—1511no—
332C23—21no—
333C23—21no—
334C23—3021no—
335C23—6042no—
336C23—6042no—
337C23—1511no—
338C23—12081no—
339C23D4×D15111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—3021no—
343Q8—3021no—
344Q8—3021no—
345Q8—3021no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—81no—
351D4—3021no—
352D4—6041no—
353D4—6042no—
354D4—6042no—
355D4—6042no—
356D4—1512no—
357D4D4×D15112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—3021no—
364C2×C4—3021no—
365C2×C4—3021no—
366C2×C4—3021no—
367C2×C4—3021no—
368C2×C4—6042no—
369C2×C4—6042no—
370C2×C4—6042no—
371C2×C4—1511no—
372C2×C4—1511no—
373C2×C4—1511no—
374C2×C4—12081no—
375C2×C4D4×D15111yescharacteristic
376C2×C4C23×D15111yescharacteristic
377C8—21no—
378C8—21no—
379C8—3021no—
380C8—1512no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—3021no—
391C22—3021no—
392C22—3021no—
393C22—3022no—
394C22—6041no—
395C22—6041no—
396C22—12084no—
397C22C2×D4×D15111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—3021no—
402C4—3021no—
403C4—3021no—
404C4—6041no—
405C4—6041no—
406C4—6041no—
407C4C2×D4×D15111yescharacteristic
408C4#1801278411yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—6041no—
413C2—6042no—
414C2#2375284411yescharacteristic
415C2#2375300911yescharacteristic
416C2#2375302211yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#449×C151C2
socleC22×C151C22×D4
hypercenter#449D151
hypocenterC151#3267
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×#18012763C2semidirect
2. C2×#18012759C2semidirect
3. #23752861C2semidirect
4. #23752859C2semidirect
5. #23752603C2semidirect
6. #23752469C2semidirect
7. #23752439C2semidirect
8. #23752425C2semidirect
9. #23752416C2semidirect
10. #23752376C2semidirect
11. #23752375C2semidirect
12. #23752361C2semidirect
13. #23752321C2semidirect
14. #23752280C2semidirect
15. #449×C151C2semidirect
21. C2×Dic604C22semidirect
22. C2×#14558994C22semidirect
25. C2×#14558992C22semidirect
29. C22×C151⋊C8C22semidirect
30. #18012737C22semidirect
36. #18012718C22semidirect
41. D4×Dic151C22semidirect
46. #18012670C22semidirect
47. Dic151⋊C8C22semidirect
48. #18012664C22semidirect
49. #18012662C22semidirect
53. #18012649C22semidirect
54. C2×#41×C151C22semidirect
58. #103×C151C22semidirect
59. #102×C151C22semidirect
60. C22⋊C8×C151C22semidirect
86. #14558989D4semidirect
208. C151#3267semidirect
222. #449D151semidirect
227. C2×#41C2×D151semidirect
263. #103C2×D151semidirect
266. #102C2×D151semidirect
268. C22⋊C8C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127632
C2×#180127592
#237528612
#237528592
#237526032
#237524692
#237524392
#237524252
#237524162
#237523762
#237523752
#237523612
#237523212
#237522802
#449×C1512
#3267151