Finite Group Databasebeta

← #34080803

Subgroups — #34080803

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×#14558993C211yesmaximal, characteristic, complemented
2C2×SD16×D151C211yesmaximal, characteristic, complemented
3#23752875C211yesmaximal, characteristic, complemented
4#23752846C211yesmaximal, characteristic
5#23752583C211yesmaximal, characteristic, complemented
6#23752442C211yesmaximal, characteristic, complemented
7#23752416C211yesmaximal, characteristic, complemented
8#23752411C211yesmaximal, characteristic, complemented
9#23752364C211yesmaximal, characteristic, complemented
10#23752362C211yesmaximal, characteristic, complemented
11#23752352C211yesmaximal, characteristic, complemented
12#23752350C211yesmaximal, characteristic, complemented
13#23752299C211yesmaximal, characteristic, complemented
14C8×C22⋊D151C211yesmaximal, characteristic, complemented
15#465×C151C211yesmaximal, characteristic, complemented
16C22×Dic302C2211yescharacteristic, complemented
17C2×Q8×D151C2211yescharacteristic, complemented
18C22×D604C2211yescharacteristic
19C2×D4×D151—21no—
20C2×D4×D151C2211yescharacteristic
21C2×#14558993—24no—
22C2×#14558993C2211yescharacteristic
23C2×#14558993C2211yescharacteristic
24C2×#14558993C2211yescharacteristic, complemented
25C2×#14558993C2211yescharacteristic, complemented
26C2×#14558992C2211yescharacteristic, complemented
27C2×#14558992C2211yescharacteristic, complemented
28C2×#14558991C2211yescharacteristic
29C2×#14558991C2211yescharacteristic, complemented
30SD16×D151—24no—
31C2×C8×D151C2211yescharacteristic, complemented
32#18012738—21no—
33#18012737C2211yescharacteristic, complemented
34#18012735C2211yescharacteristic
35#18012719C2211yescharacteristic, complemented
36#18012717—21no—
37#18012716—21no—
38#18012714C2211yescharacteristic
39#18012712—21no—
40#18012709—21no—
41#18012707—21no—
42#18012704C2211yescharacteristic, complemented
43#18012701C2211yescharacteristic
44C4×C22⋊D151C2211yescharacteristic
45(C22⋊D151)⋊C4—21no—
46Dic151⋊C8C2211yescharacteristic, complemented
47#18012662C2211yescharacteristic, complemented
48#18012662C2211yescharacteristic, complemented
49#18012659C2211yescharacteristic
50#18012659C2211yescharacteristic, complemented
51C22⋊(C151⋊C8)C2211yescharacteristic, complemented
52(C2×D151)⋊C8C2211yescharacteristic, complemented
53C8×Dic151C2211yescharacteristic, complemented
54C2×SD16×C151C2211yescharacteristic, complemented
55C22×C8×C151C2211yescharacteristic, complemented
56C22⋊Q8×C151C2211yescharacteristic, complemented
57#121×C151C2211yescharacteristic
58#106×C151C2211yescharacteristic, complemented
59#103×C151C2211yescharacteristic, complemented
60#102×C151C2211yescharacteristic, complemented
61C23×D151—21no—
62C23×D151—21no—
63C2×Dic302—21no—
64C2×Dic302—22no—
65C2×Dic302—21no—
66C2×Dic302—41no—
67C2×Dic302C2311yescharacteristic
68C2×Dic302C2311yescharacteristic
69Q8×D151—22no—
70Q8×D151—41no—
71C2×D604—22no—
72C2×D604—21no—
73C2×D604—21no—
74C2×D604—41no—
75C2×D604C2311yescharacteristic
76C2×D604C2311yescharacteristic
77C2×C22⋊D151—21no—
78C2×C22⋊D151—21no—
79C2×C22⋊D151—21no—
80C2×C22⋊D151D411yescharacteristic
81D4×D151—22no—
82D4×D151—41no—
83D4×D151—44no—
84C22×Dic151—21no—
85C2×C4×D151—21no—
86C2×C4×D151—21no—
87C2×C4×D151C2311yescharacteristic
88#14558993—22no—
89#14558993—22no—
90#14558993—22no—
91#14558993D414yescomplemented
92#14558992—22no—
93#14558992—41no—
94#14558991—22no—
95#14558991—41no—
96C2×C151⋊C8C2311yescharacteristic
97C2×C151⋊C8C2311yescharacteristic
98C8×D151—22no—
99C4⋊Dic151—21no—
100Dic151⋊C4—21no—
101Dic151⋊C4—21no—
102Dic151⋊C4—21no—
103Dic151⋊C4D411yescharacteristic
104C22⋊Dic151D411yescharacteristic
105(C2×D151)⋊C4—21no—
106(C2×D151)⋊C4—21no—
107(C2×D151)⋊C4—21no—
108(C2×D151)⋊C4D411yescharacteristic
109C4×Dic151—21no—
110C4×Dic151C2311yescharacteristic
111C2×Q8×C151C2311yescharacteristic
112C2×D4×C151—21no—
113C2×D4×C151C2311yescharacteristic
114C22×C4×C151C2311yescharacteristic
115SD16×C151—22no—
116C2×C8×C151—22no—
117C2×C8×C151C2311yescharacteristic
118C2×C8×C151C2311yescharacteristic
119C4⋊C4×C151—21no—
120C4⋊C4×C151C2311yescharacteristic
121C4⋊C4×C151C2311yescharacteristic
122C22⋊C4×C151—21no—
123C22⋊C4×C151—21no—
124C22×D151—21no—
125C22×D151—21no—
126C22×D151—42no—
127C22×D151—42no—
128C22×D151—42no—
129C22×D151—42no—
130C22×D151—81no—
131C22×D151—81no—
132C22×D151C2×D411yescharacteristic
133Dic302—21no—
134Dic302—21no—
135Dic302—21no—
136Dic302—42no—
137Dic302—42no—
138Dic302C2×D412yescomplemented
139D604—21no—
140D604—21no—
141D604—21no—
142D604—42no—
143D604—42no—
144D604C2×D412yes—
145C22⋊D151—42no—
146C22⋊D151—42no—
147C22⋊D151—81no—
148C22⋊D151SD1614yescomplemented
149C2×Dic151—21no—
150C2×Dic151—21no—
151C2×Dic151—42no—
152C2×Dic151C2×D411yescharacteristic
153C2×Dic151C2×D411yescharacteristic
154C2×Dic151C2×D411yescharacteristic
155C4×D151—21no—
156C4×D151—21no—
157C4×D151—42no—
158C4×D151—42no—
159C151⋊C8—21no—
160C151⋊C8—21no—
161C23×C151—21no—
162C23×C151C2×D411yescharacteristic
163Q8×C151—21no—
164Q8×C151—21no—
165D4×C151—21no—
166D4×C151—21no—
167D4×C151—42no—
168C2×C4×C151—21no—
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×C151C2×D412yescomplemented
177C22×C151—21no—
178C22×C151—21no—
179C22×C151—41no—
180C22×C151—41no—
181C22×C151—41no—
182C22×C151C2×SD1612yescomplemented
183C22×C151C22×D411yescharacteristic
184C2×D151—21no—
185C2×D151—21no—
186C2×D151—22no—
187C2×D151—41no—
188C2×D151—41no—
189C2×D151—41no—
190C2×D151—41no—
191C2×D151—41no—
192C2×D151—84no—
193C2×D151—84no—
194C2×D151C2×SD1612yescomplemented
195C4×C151—21no—
196C4×C151—41no—
197C4×C151—41no—
198C4×C151—41no—
199C4×C151C22×D411yescharacteristic
200C4×C151H(5,𝔽2)11yescharacteristic
201Dic151—21no—
202Dic151—22no—
203Dic151—21no—
204Dic151—41no—
205Dic151C2×SD1612yescomplemented
206C2×C151—22no—
207C2×C151—81no—
208C2×C151D4211yescharacteristic
209C2×C151C22×SD1611yescharacteristic
210C2×C151#57711yescharacteristic
211D151—22no—
212D151—42no—
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#465D15111yescharacteristic, complemented
223#461—1511no—
224#460—1511no—
225#450—1511no—
226#450—1511no—
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×D4—1511no—
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×SD16—1511no—
246C2×SD16C2×D15111yescharacteristic, complemented
247C22×C8—1511no—
248C22×C8C2×D15111yescharacteristic, complemented
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#121C2×D15111yescharacteristic
259#120—3021no—
260#120—3021no—
261C4×Q8—1511no—
262C4×D4—3021no—
263C4×D4—1511no—
264C4×D4—1511no—
265#106C2×D15111yescharacteristic, complemented
266C4⋊C8—1511no—
267#103—1511no—
268#103—1511no—
269#103C2×D15111yescharacteristic, complemented
270#102—1511no—
271#102—1511no—
272#102C2×D15111yescharacteristic, complemented
273C22⋊C8—1511no—
274C22⋊C8—1511no—
275C4×C8—1511no—
276C24—3021no—
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—21no—
288C2×D4—3022no—
289C2×D4—3021no—
290C2×D4—3022no—
291C2×D4—3021no—
292C2×D4—3021no—
293C2×D4—3021no—
294C2×D4—3021no—
295C2×D4—6041no—
296C2×D4—6041no—
297C2×D4—6044no—
298C2×D4—1511no—
299C2×D4—1511no—
300C2×D4—1511no—
301C2×D4C22×D15111yescharacteristic
302C22×C4—3021no—
303C22×C4—3021no—
304C22×C4—3021no—
305C22×C4—1511no—
306C22×C4C22×D15111yescharacteristic
307SD16—22no—
308SD16—3022no—
309SD16—3022no—
310SD16—3022no—
311SD16—3022no—
312SD16—3022no—
313SD16—6041no—
314SD16—6041no—
315SD16—1514no—
316C2×C8—22no—
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⋊C4C22×D15111yescharacteristic
329C4⋊C4C22×D15111yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—3021no—
333C22⋊C4—3021no—
334C22⋊C4—3021no—
335C22⋊C4—1511no—
336C22⋊C4—1511no—
337C42—3021no—
338C42—1511no—
339C23—21no—
340C23—3021no—
341C23—3021no—
342C23—6042no—
343C23—6042no—
344C23—6042no—
345C23—6042no—
346C23—1511no—
347C23—12081no—
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—42no—
361D4—3021no—
362D4—3021no—
363D4—3021no—
364D4—6042no—
365D4—6042no—
366D4—6042no—
367D4—6042no—
368D4—1514no—
369D4—1512no—
370D4—12081no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
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—3021no—
390C8—3021no—
391C8D4×D15112yescomplemented
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—3021no—
398C22—3021no—
399C22—3022no—
400C22—6041no—
401C22—6041no—
402C22—6041no—
403C22—6041no—
404C22—6041no—
405C22—1512no—
406C22—12084no—
407C22—12084no—
408C22SD16×D15112yescomplemented
409C22C2×D4×D15111yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—3021no—
415C4—3021no—
416C4—3022no—
417C4—6041no—
418C4—1512no—
419C4C2×D4×D15111yescharacteristic
420C4#1801278511yescharacteristic
421C2—22no—
422C2—81no—
423C2—3022no—
424C2—6042no—
425C2—12081no—
426C2#2375284611yescharacteristic
427C2C2×SD16×D15111yescharacteristic
428C2#2375301911yescharacteristic
subgroupG/H
centerC22C2×D4×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#465×C151C2
socleC22×C151C22×D4
prime radical#23752846C2
hypercenter#465D151
hypocenterC151D4×SD16
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. C22×#14558993C2semidirect
2. C2×SD16×D151C2semidirect
3. #23752875C2semidirect
5. #23752583C2semidirect
6. #23752442C2semidirect
7. #23752416C2semidirect
8. #23752411C2semidirect
9. #23752364C2semidirect
10. #23752362C2semidirect
11. #23752352C2semidirect
12. #23752350C2semidirect
13. #23752299C2semidirect
14. C8×C22⋊D151C2semidirect
15. #465×C151C2semidirect
16. C22×Dic302C22semidirect
17. C2×Q8×D151C22semidirect
24. C2×#14558993C22semidirect
25. C2×#14558993C22semidirect
26. C2×#14558992C22semidirect
27. C2×#14558992C22semidirect
29. C2×#14558991C22semidirect
31. C2×C8×D151C22semidirect
33. #18012737C22semidirect
35. #18012719C22semidirect
42. #18012704C22semidirect
46. Dic151⋊C8C22semidirect
47. #18012662C22semidirect
48. #18012662C22semidirect
50. #18012659C22semidirect
51. C22⋊(C151⋊C8)C22semidirect
52. (C2×D151)⋊C8C22semidirect
53. C8×Dic151C22semidirect
54. C2×SD16×C151C22semidirect
55. C22×C8×C151C22semidirect
56. C22⋊Q8×C151C22semidirect
58. #106×C151C22semidirect
59. #103×C151C22semidirect
60. #102×C151C22semidirect
91. #14558993D4semidirect
138. Dic302C2×D4semidirect
148. C22⋊D151SD16semidirect
176. C8×C151C2×D4semidirect
182. C22×C151C2×SD16semidirect
194. C2×D151C2×SD16semidirect
205. Dic151C2×SD16semidirect
214. C151D4×SD16semidirect
222. #465D151semidirect
246. C2×SD16C2×D151semidirect
248. C22×C8C2×D151semidirect
255. C22⋊Q8C2×D151semidirect
265. #106C2×D151semidirect
269. #103C2×D151semidirect
272. #102C2×D151semidirect
391. C8D4×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×#145589932
C2×SD16×D1512
#237528752
#237528462
#237525832
#237524422
#237524162
#237524112
#237523642
#237523622
#237523522
#237523502
#237522992
C8×C22⋊D1512
#465×C1512
D4×SD16151