Finite Group Databasebeta

← #34078479

Subgroups — #34078479

Order 19328.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23753048C211yesmaximal, characteristic, complemented
2C2×#18012765C211yesmaximal, characteristic, complemented
3#23752779C211yesmaximal, characteristic, complemented
4#23752383C214yesmaximal, complemented
5#23752378C214yesmaximal, complemented
6#23752229C211yesmaximal, characteristic, complemented
7#23752228C211yesmaximal, characteristic, complemented
8#23752194C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C151C211yesmaximal, characteristic, complemented
10#18012787—21no—
11#18012786—21no—
12#18012786C2212yescomplemented
13#18012785—21no—
14#18012785C2212yescomplemented
15#18012784—21no—
16#18012784—22no—
17C2×#14559004—22no—
18C2×#14559004C2211yescharacteristic
19C2×#14559004C2211yescharacteristic
20#41×D151—22no—
21#41×D151—22no—
22#18012765—22no—
23#18012765C2214yescomplemented
24C2×#14558992—21no—
25C2×#14558989—21no—
26C2×#14558988C2211yescharacteristic, complemented
27#18012735C2212yescomplemented
28#18012726C2212yescomplemented
29#18012710—22no—
30#18012708—22no—
31#18012698C2211yescharacteristic, complemented
32#18012666C2214yescomplemented
33#18012665C2214yescomplemented
34#18012657C2212yescomplemented
35#18012655C2212yescomplemented
36C2×#41×C151C2211yescharacteristic
37C22×D4×C151C2211yescharacteristic, complemented
38AGL(1,ℤ/8)×C151—22no—
39AGL(1,ℤ/8)×C151C2214yescomplemented
40C2×SD16×C151—21no—
41C2×D8×C151—21no—
42C2×M16×C151C2211yescharacteristic, complemented
43#14559005—22no—
44#14559005—22no—
45#14559004—24no—
46#14559004—24no—
47#14559004—22no—
48#14559004—22no—
49#14559004—21no—
50#14559004—21no—
51#14559004—21no—
52#14559004—21no—
53#14559004—21no—
54#14559004—21no—
55#14559004D412yescomplemented
56#14559004C2312yes—
57#14559003—24no—
58#14559003—22no—
59#14559003—24no—
60#14559003—22no—
61C2×Dic302—22no—
62C2×Dic302D411yescharacteristic, complemented
63C2×Dic302C2311yescharacteristic
64Q8×D151—22no—
65Q8×D151—22no—
66C2×D604—22no—
67C2×D604D411yescharacteristic, complemented
68C2×D604C2311yescharacteristic
69C2×C22⋊D151—24no—
70C2×C22⋊D151—24no—
71C2×C22⋊D151—21no—
72C2×C22⋊D151D412yes—
73D4×D151—24no—
74D4×D151—22no—
75D4×D151—24no—
76D4×D151—22no—
77C2×C4×D151—24no—
78C2×C4×D151—21no—
79C2×C4×D151D412yes—
80#14558992—24no—
81#14558989—24no—
82#14558988—21no—
83#14558988C2312yes—
84C2×C151⋊C8—21no—
85C4⋊Dic151—21no—
86C22⋊Dic151—24no—
87C22⋊Dic151—21no—
88C4×Dic151C2312yes—
89C24×C151—21no—
90#41×C151—21no—
91#41×C151—21no—
92#41×C151D412yescomplemented
93#41×C151C2312yes—
94C2×Q8×C151D411yescharacteristic, complemented
95C2×D4×C151—21no—
96C2×D4×C151—21no—
97C2×D4×C151—21no—
98C2×D4×C151—21no—
99C2×D4×C151—21no—
100C2×D4×C151—21no—
101C2×D4×C151D411yescharacteristic, complemented
102C2×D4×C151C2312yes—
103C22×C4×C151—21no—
104C22×C4×C151C2311yescharacteristic
105SD16×C151—24no—
106D8×C151—24no—
107M16×C151—21no—
108M16×C151C2312yes—
109C2×C8×C151—21no—
110C22×D151—24no—
111C22×D151—21no—
112C22×D151C2×D412yes—
113Dic302—24no—
114Dic302—21no—
115Dic302—21no—
116Dic302—21no—
117Dic302C2×D412yes—
118D604—24no—
119D604—21no—
120D604—21no—
121D604—21no—
122D604C2×D412yes—
123C22⋊D151—24no—
124C22⋊D151—24no—
125C22⋊D151—22no—
126C22⋊D151—22no—
127C22⋊D151—22no—
128C22⋊D151—22no—
129C22⋊D151—84no—
130C2×Dic151—24no—
131C2×Dic151—21no—
132C2×Dic151—22no—
133C2×Dic151C2×D412yes—
134C4×D151—24no—
135C4×D151—24no—
136C4×D151—22no—
137C4×D151—22no—
138C4×D151—22no—
139C4×D151—22no—
140C151⋊C8—22no—
141C23×C151—21no—
142C23×C151—22no—
143C23×C151—41no—
144C23×C151—41no—
145C23×C151—82no—
146C23×C151C2×D411yescharacteristic
147Q8×C151—21no—
148Q8×C151C2×D412yes—
149D4×C151—22no—
150D4×C151—22no—
151D4×C151—21no—
152D4×C151—22no—
153D4×C151—22no—
154D4×C151—41no—
155D4×C151—41no—
156D4×C151C2×D412yes—
157C2×C4×C151—21no—
158C2×C4×C151—22no—
159C2×C4×C151—22no—
160C2×C4×C151C2×D411yescharacteristic
161C2×C4×C151C2×D411yescharacteristic
162C2×C4×C151C2×D411yescharacteristic
163C2×C4×C151C2×D411yescharacteristic
164C2×C4×C151C2×D411yescharacteristic
165C2×C4×C151C2411yescharacteristic
166C8×C151—22no—
167C22×C151—22no—
168C22×C151—41no—
169C22×C151—44no—
170C22×C151—42no—
171C22×C151—81no—
172C22×C151—82no—
173C22×C151—82no—
174C22×C151#12011yescharacteristic
175C22×C151#12011yescharacteristic
176C22×C151C22×D411yescharacteristic
177C2×D151—22no—
178C2×D151—22no—
179C2×D151—42no—
180C2×D151—81no—
181C2×D151—84no—
182C4×C151—22no—
183C4×C151#12011yescharacteristic
184C4×C151#12011yescharacteristic
185C4×C151C22×D411yescharacteristic
186C4×C151C22×D411yescharacteristic
187Dic151—22no—
188Dic151—22no—
189Dic151—42no—
190C2×C151—21no—
191C2×C151—21no—
192C2×C151—21no—
193C2×C151—42no—
194C2×C151—82no—
195C2×C151C2×#12011yescharacteristic
196D151—42no—
197D151—42no—
198C151#302111yescharacteristic, complemented, Sylow
199#3021—1511nomaximal, Sylow
200#585—1511no—
201C2×AGL(1,ℤ/8)—1511no—
202C2×AGL(1,ℤ/8)D15111yescharacteristic, complemented
203#535—1511no—
204#454—1514no—
205#453—1514no—
206#421—1511no—
207#421—1511no—
208#413—1511no—
2092−1+4—3021no—
2102−1+4—3021no—
2112−1+4—1512no—
212H(5,𝔽2)—3022no—
213H(5,𝔽2)—3021no—
214H(5,𝔽2)—3021no—
215H(5,𝔽2)—1512no—
216C2×#41—3022no—
217C2×#41—3022no—
218C2×#41—3022no—
219C2×#41—1511no—
220C2×#41—1511no—
221C2×#41C2×D15111yescharacteristic
222C22×D4C2×D15111yescharacteristic, complemented
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—3022no—
225AGL(1,ℤ/8)—1514no—
226AGL(1,ℤ/8)C2×D15114yescomplemented
227C2×SD16—21no—
228C2×SD16—3021no—
229C2×D8—21no—
230C2×D8—3021no—
231C2×M16—1511no—
232C2×M16C2×D15111yescharacteristic, complemented
233#127—1512no—
234#124—1512no—
235#121—3022no—
236#120—3022no—
237#117—1511no—
238#104—1514no—
239#104—1514no—
240#101—1512no—
241#100—1512no—
242C24—21no—
243#41—21no—
244#41—21no—
245#41—3024no—
246#41—3024no—
247#41—3022no—
248#41—3022no—
249#41—3024no—
250#41—3022no—
251#41—3022no—
252#41—3022no—
253#41—3022no—
254#41—3024no—
255#41—3021no—
256#41—3021no—
257#41—3021no—
258#41—3021no—
259#41—3021no—
260#41—3021no—
261#41—1512no—
262#41—1512no—
263#41C22⋊D15112yescomplemented
264#41C22×D15112yes—
265C2×Q8—3022no—
266C2×Q8—3022no—
267C2×Q8—3022no—
268C2×Q8—1511no—
269C2×Q8—1511no—
270C2×Q8C22⋊D15111yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—3022no—
278C2×D4—3024no—
279C2×D4—3022no—
280C2×D4—3024no—
281C2×D4—3024no—
282C2×D4—3021no—
283C2×D4—3022no—
284C2×D4—3024no—
285C2×D4—1512no—
286C2×D4—1511no—
287C2×D4—1511no—
288C2×D4C22⋊D15111yescharacteristic, complemented
289C2×D4C22×D15112yes—
290C22×C4—21no—
291C22×C4—3024no—
292C22×C4—3021no—
293C22×C4—1512no—
294C22×C4C22×D15111yescharacteristic
295SD16—24no—
296SD16—3024no—
297D8—24no—
298D8—3024no—
299M16—21no—
300M16—3021no—
301M16—1512no—
302M16C22×D15112yes—
303C2×C8—21no—
304C2×C8—3021no—
305C4⋊C4—3021no—
306C22⋊C4—3021no—
307C22⋊C4—3024no—
308C42—1512no—
309C23—21no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—82no—
314C23—3024no—
315C23—3021no—
316C23—1512no—
317C23D4×D15111yescharacteristic
318Q8—21no—
319Q8—3024no—
320Q8—3021no—
321Q8—3021no—
322Q8—3021no—
323Q8—1512no—
324Q8C2×C22⋊D15112yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—22no—
329D4—22no—
330D4—41no—
331D4—41no—
332D4—3024no—
333D4—3024no—
334D4—3024no—
335D4—3022no—
336D4—3022no—
337D4—3022no—
338D4—3022no—
339D4—3021no—
340D4—3021no—
341D4—3021no—
342D4—1512no—
343D4—12084no—
344D4C2×C22⋊D15112yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—3024no—
349C2×C4—3024no—
350C2×C4—3024no—
351C2×C4—3021no—
352C2×C4—3022no—
353C2×C4—3022no—
354C2×C4—3022no—
355C2×C4—3022no—
356C2×C4—3022no—
357C2×C4—1512no—
358C2×C4D4×D15111yescharacteristic
359C2×C4D4×D15111yescharacteristic
360C2×C4D4×D15111yescharacteristic
361C2×C4C2×C22⋊D15111yescharacteristic
362C2×C4C2×C22⋊D15111yescharacteristic
363C2×C4C23×D15111yescharacteristic
364C8—22no—
365C8—3022no—
366C22—22no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—3022no—
374C22—3022no—
375C22—6042no—
376C22—12081no—
377C22—12084no—
378C22#1801270711yescharacteristic
379C22#1801270711yescharacteristic
380C22C2×D4×D15111yescharacteristic
381C4—22no—
382C4—3022no—
383C4—3022no—
384C4—6042no—
385C4#1801270711yescharacteristic
386C4#1801270711yescharacteristic
387C4C2×D4×D15111yescharacteristic
388C4C22×C22⋊D15111yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—82no—
394C2—6042no—
395C2—6042no—
396C2C2×#1801270711yescharacteristic
subgroupG/H
centerC2C2×#18012707
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
FittingC2×AGL(1,ℤ/8)×C151C2
socleC2×C151C2×#120
hypercenterC2×AGL(1,ℤ/8)D151
hypocenterC151#3021
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23753048C2semidirect
2. C2×#18012765C2semidirect
3. #23752779C2semidirect
4. #23752383C2semidirect
5. #23752378C2semidirect
6. #23752229C2semidirect
7. #23752228C2semidirect
8. #23752194C2semidirect
9. C2×AGL(1,ℤ/8)×C151C2semidirect
12. #18012786C22semidirect
14. #18012785C22semidirect
23. #18012765C22semidirect
26. C2×#14558988C22semidirect
27. #18012735C22semidirect
28. #18012726C22semidirect
31. #18012698C22semidirect
32. #18012666C22semidirect
33. #18012665C22semidirect
34. #18012657C22semidirect
35. #18012655C22semidirect
37. C22×D4×C151C22semidirect
39. AGL(1,ℤ/8)×C151C22semidirect
42. C2×M16×C151C22semidirect
55. #14559004D4semidirect
62. C2×Dic302D4semidirect
67. C2×D604D4semidirect
92. #41×C151D4semidirect
94. C2×Q8×C151D4semidirect
101. C2×D4×C151D4semidirect
198. C151#3021semidirect
202. C2×AGL(1,ℤ/8)D151semidirect
222. C22×D4C2×D151semidirect
226. AGL(1,ℤ/8)C2×D151semidirect
232. C2×M16C2×D151semidirect
263. #41C22⋊D151semidirect
270. C2×Q8C22⋊D151semidirect
288. C2×D4C22⋊D151semidirect

Maximal subgroups

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

subgroupindex
#237530482
C2×#180127652
#237527792
#237523832
#237523782
#237522292
#237522282
#237521942
C2×AGL(1,ℤ/8)×C1512
#3021151