Finite Group Databasebeta

← #34080632

Subgroups — #34080632

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×#18012766C211yesmaximal, characteristic, complemented
2C2×#18012761C211yesmaximal, characteristic, complemented
3#23752864C211yesmaximal, characteristic, complemented
4#23752859C211yesmaximal, characteristic, complemented
5#23752603C211yesmaximal, characteristic, complemented
6#23752471C211yesmaximal, characteristic, complemented
7#23752426C211yesmaximal, characteristic, complemented
8#23752416C211yesmaximal, characteristic, complemented
9#23752377C211yesmaximal, characteristic, complemented
10#23752372C211yesmaximal, characteristic, complemented
11#23752364C211yesmaximal, characteristic, complemented
12#23752355C211yesmaximal, characteristic, complemented
13#23752323C211yesmaximal, characteristic, complemented
14#23752283C211yesmaximal, characteristic, complemented
15#465×C151C211yesmaximal, characteristic, complemented
16C2×#14559005C2211yescharacteristic
17C2×#14559004C2211yescharacteristic
18C2×#14559004C2211yescharacteristic, complemented
19C2×D4×D151C2211yescharacteristic
20#18012766—24no—
21#18012761—24no—
22C2×Dic604C2211yescharacteristic, complemented
23C2×#14558994C2211yescharacteristic, complemented
24C2×#14558993C2211yescharacteristic
25C2×#14558993C2211yescharacteristic, complemented
26C2×#14558992C2211yescharacteristic
27C2×#14558991C2211yescharacteristic
28C2×D1208C2211yescharacteristic
29C2×#14558989C2211yescharacteristic
30C2×#14558987C2211yescharacteristic
31#18012737C2211yescharacteristic, complemented
32#18012726C2211yescharacteristic
33#18012724—21no—
34#18012721—21no—
35#18012721—21no—
36#18012719C2211yescharacteristic
37#18012719C2211yescharacteristic
38#18012712—21no—
39#18012710—21no—
40#18012707—21no—
41#18012701C2211yescharacteristic
42#18012701C2211yescharacteristic, complemented
43C4×C22⋊D151C2211yescharacteristic
44C4⋊C4×D151—21no—
45C22⋊C4×D151—21no—
46Dic151⋊C8C2211yescharacteristic, complemented
47#18012662C2211yescharacteristic, complemented
48#18012662C2211yescharacteristic, complemented
49#18012659C2211yescharacteristic
50#18012659C2211yescharacteristic, complemented
51C22⋊(C151⋊C8)C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic
53#18012649C2211yescharacteristic, complemented
54C2×SD16×C151C2211yescharacteristic
55C22×C8×C151C2211yescharacteristic, complemented
56C22⋊Q8×C151C2211yescharacteristic, complemented
57#121×C151C2211yescharacteristic
58#106×C151C2211yescharacteristic, complemented
59#103×C151C2211yescharacteristic, complemented
60#102×C151C2211yescharacteristic
61C23×D151—21no—
62#14559005—22no—
63#14559005—41no—
64#14559004—22no—
65#14559004—22no—
66#14559004—41no—
67#14559004—41no—
68C2×Dic302C2311yescharacteristic
69C2×Dic302C2311yescharacteristic
70C2×D604—21no—
71C2×D604C2311yescharacteristic
72C2×D604C2311yescharacteristic
73C2×C22⋊D151—21no—
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76C2×C22⋊D151D411yescharacteristic
77D4×D151—22no—
78D4×D151—41no—
79C2×C4×D151—21no—
80C2×C4×D151—21no—
81C2×C4×D151—21no—
82C2×C4×D151—21no—
83C2×C4×D151—21no—
84C2×C4×D151C2311yescharacteristic
85Dic604—22no—
86#14558994—41no—
87#14558993—22no—
88#14558993—22no—
89#14558992—41no—
90#14558991—22no—
91D1208D414yescomplemented
92#14558989—22no—
93#14558987—22no—
94C2×C151⋊C8C2311yescharacteristic
95C2×C151⋊C8C2311yescharacteristic
96C4⋊Dic151—21no—
97C4⋊Dic151—21no—
98Dic151⋊C4—21no—
99Dic151⋊C4—21no—
100Dic151⋊C4D411yescharacteristic
101C22⋊Dic151—21no—
102C22⋊Dic151—21no—
103C22⋊Dic151D411yescharacteristic
104(C2×D151)⋊C4—21no—
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4—21no—
108(C2×D151)⋊C4D411yescharacteristic
109C4×Dic151C2311yescharacteristic
110C2×Q8×C151C2311yescharacteristic
111C2×D4×C151—21no—
112C2×D4×C151C2311yescharacteristic
113C22×C4×C151C2311yescharacteristic
114SD16×C151—22no—
115C2×C8×C151—22no—
116C2×C8×C151C2311yescharacteristic
117C2×C8×C151C2311yescharacteristic
118C4⋊C4×C151—21no—
119C4⋊C4×C151C2311yescharacteristic
120C4⋊C4×C151C2311yescharacteristic
121C22⋊C4×C151—21no—
122C22⋊C4×C151—21no—
123C22×D151—21no—
124C22×D151—21no—
125C22×D151—42no—
126C22×D151—42no—
127C22×D151—81no—
128C22×D151C2×D411yescharacteristic
129Dic302—21no—
130Dic302—21no—
131Dic302—21no—
132Dic302—21no—
133D604—21no—
134D604—21no—
135D604—42no—
136D604C2×D412yes—
137D604C2×D412yes—
138C22⋊D151—41no—
139C22⋊D151—42no—
140C22⋊D151—42no—
141C22⋊D151—42no—
142C2×Dic151—21no—
143C2×Dic151—21no—
144C2×Dic151C2×D411yescharacteristic
145C2×Dic151C2×D411yescharacteristic
146C2×Dic151C2×D411yescharacteristic
147C4×D151—21no—
148C4×D151—21no—
149C4×D151—42no—
150C4×D151—42no—
151C4×D151—42no—
152C4×D151—42no—
153C4×D151—81no—
154C151⋊C8—21no—
155C151⋊C8—21no—
156C23×C151—21no—
157C23×C151C2×D411yescharacteristic
158Q8×C151—21no—
159Q8×C151—21no—
160D4×C151—21no—
161D4×C151—21no—
162D4×C151—81no—
163C2×C4×C151—21no—
164C2×C4×C151—21no—
165C2×C4×C151—21no—
166C2×C4×C151—21no—
167C2×C4×C151—21no—
168C2×C4×C151C2×D411yescharacteristic
169C2×C4×C151C2411yescharacteristic
170C8×C151—21no—
171C8×C151C2×D412yes—
172C22×C151—21no—
173C22×C151—21no—
174C22×C151—21no—
175C22×C151—41no—
176C22×C151—41no—
177C22×C151—41no—
178C22×C151C22×D411yescharacteristic
179C2×D151—21no—
180C2×D151—22no—
181C2×D151—22no—
182C2×D151—21no—
183C2×D151—21no—
184C2×D151—41no—
185C2×D151—41no—
186C2×D151—41no—
187C2×D151—41no—
188C2×D151—84no—
189C4×C151—21no—
190C4×C151—41no—
191C4×C151—41no—
192C4×C151—41no—
193C4×C151C22×D411yescharacteristic
194C4×C151H(5,𝔽2)11yescharacteristic
195Dic151—21no—
196Dic151—21no—
197Dic151—21no—
198Dic151—41no—
199Dic151—41no—
200C2×C151—41no—
201C2×C151—81no—
202C2×C151D4211yescharacteristic
203C2×C151#57511yescharacteristic
204C2×C151#57711yescharacteristic
205D151—41no—
206D151—42no—
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#465D15111yescharacteristic, complemented
217#463—1511no—
218#461—1511no—
219#452—1511no—
220#452—1511no—
221#450—1511no—
222#449—1511no—
223#442—1511no—
224#435—1511no—
225C2×#41—1511no—
226C2×#41—1511no—
227C2×#41—1511no—
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×D8—1511no—
240C2×M16—1511no—
241C22×C8C2×D15111yescharacteristic, complemented
242C4⋊Q8—1511no—
243#124—1511no—
244#123—3021no—
245#123—3021no—
246#123—3021no—
247C22⋊Q8—1511no—
248C22⋊Q8—1511no—
249C22⋊Q8C2×D15111yescharacteristic, complemented
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#106C2×D15111yescharacteristic, complemented
260C4⋊C8—1511no—
261#103—1511no—
262#103—1511no—
263#103C2×D15111yescharacteristic, complemented
264#102—1511no—
265#102—1511no—
266#102C2×D15111yescharacteristic
267C22⋊C8—1511no—
268C22⋊C8—1511no—
269#97—1511no—
270C24—3021no—
271#41—3022no—
272#41—3022no—
273#41—3022no—
274#41—6041no—
275#41—6041no—
276#41—6041no—
277C2×Q8—1511no—
278C2×Q8—1511no—
279C2×Q8C22×D15111yescharacteristic
280C2×D4—21no—
281C2×D4—3021no—
282C2×D4—3022no—
283C2×D4—3021no—
284C2×D4—3021no—
285C2×D4—3021no—
286C2×D4—6041no—
287C2×D4—1511no—
288C2×D4—1511no—
289C2×D4—1511no—
290C2×D4C22×D15111yescharacteristic
291C22×C4—3021no—
292C22×C4—3021no—
293C22×C4—3021no—
294C22×C4—3021no—
295C22×C4—3021no—
296C22×C4—1511no—
297C22×C4C22×D15111yescharacteristic
298Q16—3022no—
299Q16—6041no—
300SD16—22no—
301SD16—3022no—
302SD16—3022no—
303SD16—3022no—
304SD16—6041no—
305D8—3022no—
306D8—1514no—
307M16—3022no—
308C2×C8—22no—
309C2×C8—1511no—
310C2×C8—1511no—
311C2×C8C22×D15111yescharacteristic
312C2×C8C22×D15111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3021no—
315C4⋊C4—3021no—
316C4⋊C4—3021no—
317C4⋊C4—3021no—
318C4⋊C4—1511no—
319C4⋊C4C22×D15111yescharacteristic
320C4⋊C4C22×D15111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
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—3021no—
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—81no—
349D4—3021no—
350D4—3021no—
351D4—6042no—
352D4—6041no—
353D4—6042no—
354D4—6042no—
355D4—6042no—
356D4—1512no—
357D4—1512no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—3021no—
364C2×C4—3021no—
365C2×C4—3021no—
366C2×C4—3021no—
367C2×C4—6042no—
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—3021no—
379C8—3021no—
380C8D4×D15112yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—3021no—
388C22—3021no—
389C22—3021no—
390C22—3022no—
391C22—3022no—
392C22—6041no—
393C22—6041no—
394C22—6041no—
395C22—6041no—
396C22—12084no—
397C22C2×D4×D15111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—3021no—
403C4—3021no—
404C4—3021no—
405C4—6041no—
406C4—6041no—
407C4C2×D4×D15111yescharacteristic
408C4#1801278511yescharacteristic
409C2—41no—
410C2—81no—
411C2—6041no—
412C2—6042no—
413C2—6042no—
414C2#2375284611yescharacteristic
415C2#2375300911yescharacteristic
416C2#2375301811yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#465×C151C2
socleC22×C151C22×D4
hypercenter#465D151
hypocenterC151#3267
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#18012766C2semidirect
2. C2×#18012761C2semidirect
3. #23752864C2semidirect
4. #23752859C2semidirect
5. #23752603C2semidirect
6. #23752471C2semidirect
7. #23752426C2semidirect
8. #23752416C2semidirect
9. #23752377C2semidirect
10. #23752372C2semidirect
11. #23752364C2semidirect
12. #23752355C2semidirect
13. #23752323C2semidirect
14. #23752283C2semidirect
15. #465×C151C2semidirect
18. C2×#14559004C22semidirect
22. C2×Dic604C22semidirect
23. C2×#14558994C22semidirect
25. C2×#14558993C22semidirect
31. #18012737C22semidirect
42. #18012701C22semidirect
46. Dic151⋊C8C22semidirect
47. #18012662C22semidirect
48. #18012662C22semidirect
50. #18012659C22semidirect
51. C22⋊(C151⋊C8)C22semidirect
53. #18012649C22semidirect
55. C22×C8×C151C22semidirect
56. C22⋊Q8×C151C22semidirect
58. #106×C151C22semidirect
59. #103×C151C22semidirect
91. D1208D4semidirect
208. C151#3267semidirect
216. #465D151semidirect
241. C22×C8C2×D151semidirect
249. C22⋊Q8C2×D151semidirect
259. #106C2×D151semidirect
263. #103C2×D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127662
C2×#180127612
#237528642
#237528592
#237526032
#237524712
#237524262
#237524162
#237523772
#237523722
#237523642
#237523552
#237523232
#237522832
#465×C1512
#3267151