Finite Group Databasebeta

← #34080804

Subgroups — #34080804

Order 19328.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14558992C211yesmaximal, characteristic, complemented
2C2×SD16×D151C211yesmaximal, characteristic, complemented
3#23752875C211yesmaximal, characteristic, complemented
4D4×C22⋊D151C211yesmaximal, characteristic
5#23752583C211yesmaximal, characteristic, complemented
6#23752443C211yesmaximal, characteristic, complemented
7#23752440C211yesmaximal, characteristic, complemented
8#23752415C211yesmaximal, characteristic, complemented
9#23752411C211yesmaximal, characteristic, complemented
10#23752364C211yesmaximal, characteristic, complemented
11#23752353C211yesmaximal, characteristic, complemented
12#23752349C211yesmaximal, characteristic, complemented
13#23752303C211yesmaximal, characteristic, complemented
14(C22⋊D151)⋊C8C211yesmaximal, characteristic, complemented
15#450×C151C211yesmaximal, characteristic, complemented
16C22×Dic302C2211yescharacteristic, complemented
17C2×Q8×D151C2211yescharacteristic, complemented
18C22×C22⋊D151—21no—
19C2×D4×D151C2211yescharacteristic
20C2×#14558993C2211yescharacteristic, complemented
21C2×#14558993C2211yescharacteristic, complemented
22C2×#14558992—24no—
23C2×#14558992C2211yescharacteristic
24C2×#14558992C2211yescharacteristic
25C2×#14558992C2211yescharacteristic, complemented
26C2×#14558992C2211yescharacteristic, complemented
27C2×#14558991C2211yescharacteristic, complemented
28SD16×D151—24no—
29C22×C151⋊C8C2211yescharacteristic, complemented
30C2×C8×D151C2211yescharacteristic, complemented
31#18012738—21no—
32#18012737C2211yescharacteristic, complemented
33#18012735C2211yescharacteristic
34#18012719C2211yescharacteristic, complemented
35#18012717—21no—
36#18012716—21no—
37#18012713C2211yescharacteristic
38#18012711—21no—
39#18012710C2211yescharacteristic
40#18012708—21no—
41#18012707—21no—
42#18012704C2211yescharacteristic, complemented
43D4×Dic151C2211yescharacteristic
44C4×C22⋊D151C2211yescharacteristic
45(C22⋊D151)⋊C4—21no—
46#18012670C2211yescharacteristic, complemented
47Dic151⋊C8C2211yescharacteristic, complemented
48#18012664C2211yescharacteristic, complemented
49#18012664C2211yescharacteristic, complemented
50#18012663C2211yescharacteristic, complemented
51#18012659C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53C8×Dic151C2211yescharacteristic, complemented
54C22×D4×C151C2211yescharacteristic
55C2×SD16×C151C2211yescharacteristic
56C2×SD16×C151C2211yescharacteristic, complemented
57C22⋊Q8×C151C2211yescharacteristic, complemented
58#102×C151C2211yescharacteristic
59#102×C151C2211yescharacteristic, complemented
60C22⋊C8×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C2×Dic302—21no—
63C2×Dic302—22no—
64C2×Dic302—21no—
65C2×Dic302—41no—
66C2×Dic302C2311yescharacteristic
67C2×Dic302C2311yescharacteristic
68Q8×D151—22no—
69Q8×D151—41no—
70C2×D604C2311yescharacteristic
71C2×C22⋊D151—21no—
72C2×C22⋊D151—21no—
73C2×C22⋊D151—21no—
74C2×C22⋊D151—21no—
75C2×C22⋊D151—21no—
76C2×C22⋊D151—44no—
77C2×C22⋊D151D411yescharacteristic
78D4×D151—22no—
79D4×D151—41no—
80C22×Dic151—21no—
81C22×Dic151—21no—
82C2×C4×D151—21no—
83C2×C4×D151C2311yescharacteristic
84#14558993—22no—
85#14558993—41no—
86#14558992—22no—
87#14558992—22no—
88#14558992—22no—
89#14558992D414yescomplemented
90#14558991—22no—
91C2×C151⋊C8—22no—
92C2×C151⋊C8C2311yescharacteristic
93C2×C151⋊C8C2311yescharacteristic
94C8×D151—22no—
95C4⋊Dic151—21no—
96C4⋊Dic151C2311yescharacteristic
97Dic151⋊C4—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)⋊C4D411yescharacteristic
107C4×Dic151—21no—
108C4×Dic151C2311yescharacteristic
109C24×C151—21no—
110C2×Q8×C151C2311yescharacteristic
111C2×D4×C151—22no—
112C2×D4×C151—21no—
113C2×D4×C151—41no—
114C2×D4×C151C2311yescharacteristic
115C2×D4×C151C2311yescharacteristic
116C22×C4×C151C2311yescharacteristic
117SD16×C151—22no—
118SD16×C151—41no—
119C2×C8×C151C2311yescharacteristic
120C2×C8×C151C2311yescharacteristic
121C4⋊C4×C151—21no—
122C4⋊C4×C151C2311yescharacteristic
123C22⋊C4×C151—21no—
124C22×D151—21no—
125C22×D151—42no—
126C22×D151—42no—
127C22×D151—81no—
128C22×D151C2×D411yescharacteristic
129Dic302—21no—
130Dic302—21no—
131Dic302—21no—
132Dic302—42no—
133Dic302—42no—
134Dic302C2×D412yescomplemented
135D604—21no—
136D604—21no—
137C22⋊D151—42no—
138C22⋊D151—42no—
139C22⋊D151—42no—
140C22⋊D151—42no—
141C22⋊D151—81no—
142C22⋊D151SD1614yescomplemented
143C2×Dic151—21no—
144C2×Dic151—21no—
145C2×Dic151—21no—
146C2×Dic151—42no—
147C2×Dic151—42no—
148C2×Dic151C2×D411yescharacteristic
149C2×Dic151C2×D411yescharacteristic
150C2×Dic151C2×D411yescharacteristic
151C4×D151—21no—
152C4×D151—21no—
153C4×D151—42no—
154C151⋊C8—21no—
155C151⋊C8C2×D412yescomplemented
156C23×C151—21no—
157C23×C151—21no—
158C23×C151—42no—
159C23×C151—42no—
160C23×C151—81no—
161C23×C151C2×D411yescharacteristic
162Q8×C151—21no—
163Q8×C151—21no—
164D4×C151—21no—
165D4×C151—21no—
166D4×C151—21no—
167D4×C151—42no—
168D4×C151C2×D412yes—
169C2×C4×C151—21no—
170C2×C4×C151—21no—
171C2×C4×C151—21no—
172C2×C4×C151—21no—
173C2×C4×C151C2×D411yescharacteristic
174C2×C4×C151C2411yescharacteristic
175C8×C151—21no—
176C8×C151—21no—
177C22×C151—21no—
178C22×C151—21no—
179C22×C151—22no—
180C22×C151—41no—
181C22×C151—41no—
182C22×C151—41no—
183C22×C151—41no—
184C22×C151—41no—
185C22×C151—84no—
186C22×C151C2×SD1612yescomplemented
187C22×C151C22×D411yescharacteristic
188C2×D151—21no—
189C2×D151—21no—
190C2×D151—41no—
191C2×D151—41no—
192C2×D151—41no—
193C2×D151—84no—
194C2×D151C2×SD1612yescomplemented
195C4×C151—21no—
196C4×C151—41no—
197C4×C151—41no—
198C4×C151C22×D411yescharacteristic
199C4×C151H(5,𝔽2)11yescharacteristic
200Dic151—21no—
201Dic151—22no—
202Dic151—21no—
203Dic151—41no—
204Dic151—41no—
205Dic151C2×SD1612yescomplemented
206C2×C151—22no—
207C2×C151—42no—
208C2×C151—81no—
209C2×C151D4211yescharacteristic
210C2×C151C22×SD1611yescharacteristic
211C2×C151#57711yescharacteristic
212D151—22no—
213D151—81no—
214C151D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—1511nomaximal, Sylow
216C22×SD16—1511no—
217C22×SD16—1511no—
218D4×Q8—1511no—
219D42—1511no—
220#492—1511no—
221#465—1511no—
222#465—1511no—
223#461—1511no—
224#460—1511no—
225#450—1511no—
226#450D15111yescharacteristic, complemented
227#448—1511no—
228#448—1511no—
229C4×SD16—1511no—
230C8×D4—1511no—
231C22×Q8—1511no—
232C22×Q8—1511no—
233C22×D4—3021no—
234C22×D4—1511no—
235C22×D4C2×D15111yescharacteristic
236C2×SD16—3024no—
237C2×SD16—3024no—
238C2×SD16—1511no—
239C2×SD16—1511no—
240C2×SD16—1511no—
241C2×SD16—1511no—
242C2×SD16—1511no—
243C2×SD16—1511no—
244C2×SD16—1511no—
245C2×SD16C2×D15111yescharacteristic
246C2×SD16C2×D15111yescharacteristic, complemented
247C22×C8—1511no—
248C22×C8—1511no—
249C4⋊Q8—3021no—
250C4⋊Q8—1511no—
251#127—1511no—
252C22⋊Q8—3021no—
253C22⋊Q8—3021no—
254C22⋊Q8—1511no—
255C22⋊Q8C2×D15111yescharacteristic, complemented
256#121—3021no—
257#121—1511no—
258#121—1511no—
259#120—3021no—
260#120—3021no—
261C4×Q8—1511no—
262C4×D4—3021no—
263C4×D4—1511no—
264C4×D4—1511no—
265#106—1511no—
266C4⋊C8—1511no—
267#103—1511no—
268#103—1511no—
269#103—1511no—
270#102—1511no—
271#102C2×D15111yescharacteristic
272#102C2×D15111yescharacteristic, complemented
273C22⋊C8—1511no—
274C22⋊C8C2×D15111yescharacteristic, complemented
275C4×C8—1511no—
276C24—21no—
277C24—3021no—
278C2×Q8—3021no—
279C2×Q8—3022no—
280C2×Q8—3021no—
281C2×Q8—3022no—
282C2×Q8—6041no—
283C2×Q8—6041no—
284C2×Q8—1511no—
285C2×Q8—1511no—
286C2×Q8C22×D15111yescharacteristic
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—3022no—
291C2×D4—3021no—
292C2×D4—3021no—
293C2×D4—3021no—
294C2×D4—3021no—
295C2×D4—3021no—
296C2×D4—6041no—
297C2×D4—6044no—
298C2×D4—1511no—
299C2×D4—1511no—
300C2×D4C22×D15111yescharacteristic
301C2×D4C22×D15111yescharacteristic
302C22×C4—3021no—
303C22×C4—3021no—
304C22×C4—3021no—
305C22×C4—1511no—
306C22×C4C22×D15111yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—3022no—
310SD16—3022no—
311SD16—3022no—
312SD16—3022no—
313SD16—3022no—
314SD16—6041no—
315SD16—1514no—
316C2×C8—3022no—
317C2×C8—3022no—
318C2×C8—1511no—
319C2×C8—1511no—
320C2×C8C22×D15111yescharacteristic
321C2×C8C22×D15111yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—3021no—
324C4⋊C4—3021no—
325C4⋊C4—3021no—
326C4⋊C4—3021no—
327C4⋊C4—1511no—
328C4⋊C4—1511no—
329C4⋊C4C22×D15111yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—3021no—
332C22⋊C4—3021no—
333C22⋊C4—3021no—
334C22⋊C4—3021no—
335C22⋊C4—1511no—
336C22⋊C4—1511no—
337C42—3021no—
338C42—1511no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—3021no—
345C23—6042no—
346C23—6042no—
347C23—1511no—
348C23—12081no—
349C23D4×D15111yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—3021no—
353Q8—3021no—
354Q8—3021no—
355Q8—6042no—
356Q8—6042no—
357Q8—1512no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—3021no—
363D4—3021no—
364D4—6042no—
365D4—6042no—
366D4—6042no—
367D4—6042no—
368D4—1514no—
369D4—12081no—
370D4D4×D15112yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—3021no—
376C2×C4—3021no—
377C2×C4—3021no—
378C2×C4—3021no—
379C2×C4—3021no—
380C2×C4—6042no—
381C2×C4—6042no—
382C2×C4—6042no—
383C2×C4—1511no—
384C2×C4—1511no—
385C2×C4—1511no—
386C2×C4D4×D15111yescharacteristic
387C2×C4C23×D15111yescharacteristic
388C8—21no—
389C8—21no—
390C8—3021no—
391C8—1512no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—3021no—
402C22—3021no—
403C22—6041no—
404C22—6041no—
405C22—6041no—
406C22—1512no—
407C22—12084no—
408C22SD16×D15112yescomplemented
409C22C2×D4×D15111yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—3021no—
414C4—3021no—
415C4—3022no—
416C4—6041no—
417C4—6041no—
418C4—1512no—
419C4C2×D4×D15111yescharacteristic
420C4#1801278411yescharacteristic
421C2—22no—
422C2—42no—
423C2—81no—
424C2—3022no—
425C2—12081no—
426C2#2375284411yescharacteristic
427C2C2×SD16×D15111yescharacteristic
428C2#2375302211yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#450×C151C2
socleC22×C151C22×D4
prime radicalD4×C22⋊D151C2
hypercenter#450D151
hypocenterC151D4×SD16
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#14558992C2semidirect
2. C2×SD16×D151C2semidirect
3. #23752875C2semidirect
5. #23752583C2semidirect
6. #23752443C2semidirect
7. #23752440C2semidirect
8. #23752415C2semidirect
9. #23752411C2semidirect
10. #23752364C2semidirect
11. #23752353C2semidirect
12. #23752349C2semidirect
13. #23752303C2semidirect
14. (C22⋊D151)⋊C8C2semidirect
15. #450×C151C2semidirect
16. C22×Dic302C22semidirect
17. C2×Q8×D151C22semidirect
20. C2×#14558993C22semidirect
21. C2×#14558993C22semidirect
25. C2×#14558992C22semidirect
26. C2×#14558992C22semidirect
27. C2×#14558991C22semidirect
29. C22×C151⋊C8C22semidirect
30. C2×C8×D151C22semidirect
32. #18012737C22semidirect
34. #18012719C22semidirect
42. #18012704C22semidirect
46. #18012670C22semidirect
47. Dic151⋊C8C22semidirect
48. #18012664C22semidirect
49. #18012664C22semidirect
50. #18012663C22semidirect
51. #18012659C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. C8×Dic151C22semidirect
56. C2×SD16×C151C22semidirect
57. C22⋊Q8×C151C22semidirect
59. #102×C151C22semidirect
60. C22⋊C8×C151C22semidirect
89. #14558992D4semidirect
134. Dic302C2×D4semidirect
142. C22⋊D151SD16semidirect
155. C151⋊C8C2×D4semidirect
186. C22×C151C2×SD16semidirect
194. C2×D151C2×SD16semidirect
205. Dic151C2×SD16semidirect
214. C151D4×SD16semidirect
226. #450D151semidirect
246. C2×SD16C2×D151semidirect
255. C22⋊Q8C2×D151semidirect
272. #102C2×D151semidirect
274. C22⋊C8C2×D151semidirect
408. C22SD16×D151semidirect

Maximal subgroups

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

subgroupindex
C22×#145589922
C2×SD16×D1512
#237528752
D4×C22⋊D1512
#237525832
#237524432
#237524402
#237524152
#237524112
#237523642
#237523532
#237523492
#237523032
(C22⋊D151)⋊C82
#450×C1512
D4×SD16151