Finite Group Databasebeta

← #456×D151 #34078449

Subgroups — #456×D151 #34078449

Order 19328.

708 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 135 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
12−1+4×D151C211yesmaximal, characteristic
2#137×D151C212yesmaximal, complemented
3C4⋊Q8×D151C211yesmaximal, characteristic, complemented
4#23752397C211yesmaximal, characteristic, complemented
5#23752396C212yesmaximal, complemented
6#23752395C211yesmaximal, characteristic, complemented
7#23752394C211yesmaximal, characteristic, complemented
8#23752393C212yesmaximal, complemented
9#104×D151C212yesmaximal, complemented
10#101×D151C211yesmaximal, characteristic, complemented
11#456×C151C211yesmaximal, characteristic, complemented
12#18012787—22no—
13#18012787C2211yescharacteristic
14#18012786—22no—
15#18012786—22no—
16#18012786C2212yes—
17#41×D151—22no—
18#41×D151—22no—
19#41×D151C2212yes—
20C2×Q8×D151—22no—
21C2×Q8×D151C2211yescharacteristic
22C2×Q8×D151C2211yescharacteristic
23#18012772C2212yescomplemented
24#18012771—22no—
25#18012770C2212yescomplemented
26#18012769C2212yescomplemented
27#18012768—22no—
28Q16×D151—22no—
29SD16×D151—22no—
30M16×D151C2212yes—
31#18012739C2211yescharacteristic, complemented
32#18012738—22no—
33#18012737C2211yescharacteristic, complemented
34#18012737C2211yescharacteristic, complemented
35C4⋊C4×D151—22no—
36C42×D151C2211yescharacteristic
37#18012667C2212yescomplemented
38#18012666C2212yescomplemented
39#18012665C2212yescomplemented
40#18012658C2211yescharacteristic, complemented
41#18012657C2212yescomplemented
422−1+4×C151C2211yescharacteristic
43#137×C151C2212yescomplemented
44C4⋊Q8×C151C2211yescharacteristic, complemented
45#104×C151C2212yescomplemented
46#101×C151C2211yescharacteristic, complemented
47#14559005—22no—
48#14559005—22no—
49#14559005D412yes—
50#14559004—24no—
51#14559004—24no—
52#14559004—22no—
53#14559004—22no—
54#14559004—22no—
55#14559004C2312yes—
56#14559003—24no—
57#14559003—22no—
58#14559003—24no—
59#14559003—22no—
60#14559003—22no—
61#14559003D412yes—
62C2×Dic302—24no—
63C2×Dic302—22no—
64C2×Dic302—21no—
65C2×Dic302D412yes—
66C2×Dic302C2311yescharacteristic
67C2×Dic302C2311yescharacteristic
68Q8×D151—22no—
69Q8×D151—24no—
70Q8×D151—22no—
71Q8×D151—22no—
72Q8×D151—22no—
73Q8×D151—22no—
74Q8×D151—22no—
75Q8×D151—22no—
76Q8×D151D412yes—
77D4×D151—22no—
78D4×D151—22no—
79D4×D151D412yes—
80C2×C4×D151—24no—
81C2×C4×D151—21no—
82C2×C4×D151—22no—
83C2×C4×D151—21no—
84C2×C4×D151D412yes—
85C2×C4×D151C2311yescharacteristic
86Dic604—22no—
87#14558994—22no—
88#14558994—22no—
89#14558993—22no—
90#14558992—22no—
91#14558991—22no—
92#14558988C2312yes—
93#14558987—22no—
94C8×D151—22no—
95C4⋊Dic151—22no—
96Dic151⋊C4—22no—
97Dic151⋊C4—22no—
98C4×Dic151—21no—
99C4×Dic151C2311yescharacteristic
100#41×C151—22no—
101#41×C151—22no—
102#41×C151C2312yes—
103C2×Q8×C151—22no—
104C2×Q8×C151C2311yescharacteristic
105C2×Q8×C151C2311yescharacteristic
106Q16×C151—22no—
107SD16×C151—22no—
108M16×C151C2312yes—
109C4⋊C4×C151—22no—
110C42×C151C2311yescharacteristic
111C22×D151—22no—
112C22×D151C2×D411yescharacteristic
113Dic302—22no—
114Dic302—24no—
115Dic302—22no—
116Dic302—24no—
117Dic302—22no—
118Dic302—22no—
119Dic302—42no—
120Dic302C2×D412yes—
121D604—22no—
122D604—22no—
123D604C2×D412yes—
124C22⋊D151—24no—
125C22⋊D151—22no—
126C22⋊D151—22no—
127C22⋊D151—22no—
128C2×Dic151—24no—
129C2×Dic151—21no—
130C2×Dic151—22no—
131C2×Dic151—21no—
132C2×Dic151C2×D412yes—
133C2×Dic151C2×D411yescharacteristic
134C4×D151—24no—
135C4×D151—24no—
136C4×D151—24no—
137C4×D151—22no—
138C4×D151—22no—
139C4×D151—22no—
140C4×D151—22no—
141C4×D151—22no—
142C4×D151—41no—
143C4×D151—41no—
144C4×D151C2×D412yes—
145C4×D151C2×D412yes—
146C151⋊C8—22no—
147Q8×C151—22no—
148Q8×C151—22no—
149Q8×C151—22no—
150Q8×C151C2×D412yes—
151D4×C151—22no—
152D4×C151—22no—
153D4×C151C2×D412yes—
154C2×C4×C151—24no—
155C2×C4×C151—21no—
156C2×C4×C151—22no—
157C2×C4×C151—21no—
158C2×C4×C151C2×D412yes—
159C2×C4×C151C2411yescharacteristic
160C8×C151—22no—
161C22×C151—22no—
162C22×C151C22×D411yescharacteristic
163C2×D151—22no—
164C2×D151—22no—
165C2×D151—44no—
166C2×D151#12011yescharacteristic
167C2×D151#12011yescharacteristic
168C4×C151—22no—
169C4×C151—22no—
170C4×C151—22no—
171C4×C151—41no—
172C4×C151C22×D412yes—
173Dic151—22no—
174Dic151—22no—
175Dic151—22no—
176Dic151—41no—
177Dic151#12012yes—
178C2×C151—21no—
179C2×C151—42no—
180C2×C151C2×#12011yescharacteristic
181D151—21no—
182D151—42no—
183D151#45612yesdirect product factor
184C151C2×#45611yescharacteristic, complemented, Sylow
185C2×#456—1511nomaximal, Sylow
186C2×2−1+4—1511no—
187C2×#137—1512no—
188C2×C4⋊Q8—1511no—
189#456—1511no—
190#456—1512no—
191#456—1511no—
192#456—1512no—
193#456—1511no—
194#456D15111yescharacteristic, direct product factor
195C2×#104—1512no—
196C2×#101—1511no—
1972−1+4—3022no—
1982−1+4—3022no—
1992−1+4—3022no—
2002−1+4—1512no—
2012−1+4—1511no—
2022−1+4C2×D15111yescharacteristic
203C2×#41—3022no—
204C2×#41—3022no—
205C2×#41—1512no—
206C22×Q8—3022no—
207C22×Q8—1511no—
208C22×Q8—1511no—
209#137—3022no—
210#137—3022no—
211#137—1512no—
212#137—1512no—
213#137—1512no—
214#137C2×D15112yescomplemented
215C2×Q16—3022no—
216C2×SD16—3022no—
217C2×M16—1512no—
218C4⋊Q8—3022no—
219C4⋊Q8—1511no—
220C4⋊Q8—1511no—
221C4⋊Q8—1511no—
222C4⋊Q8C2×D15111yescharacteristic, complemented
223C2×C4⋊C4—3022no—
224C2×C42—1511no—
225#104—1512no—
226#104—1512no—
227#104—1512no—
228#104C2×D15112yescomplemented
229#101—1511no—
230#101—1512no—
231#101C2×D15111yescharacteristic, complemented
232#41—22no—
233#41—22no—
234#41—3024no—
235#41—3024no—
236#41—3022no—
237#41—3022no—
238#41—3024no—
239#41—3022no—
240#41—3022no—
241#41—3022no—
242#41—3022no—
243#41—3024no—
244#41—3022no—
245#41—3022no—
246#41—1512no—
247#41—1512no—
248#41—1512no—
249#41C22×D15112yes—
250C2×Q8—22no—
251C2×Q8—3024no—
252C2×Q8—3022no—
253C2×Q8—3021no—
254C2×Q8—3022no—
255C2×Q8—3024no—
256C2×Q8—3022no—
257C2×Q8—3022no—
258C2×Q8—3022no—
259C2×Q8—3022no—
260C2×Q8—3022no—
261C2×Q8—3022no—
262C2×Q8—1512no—
263C2×Q8—1511no—
264C2×Q8—1512no—
265C2×Q8—1511no—
266C2×Q8C22×D15111yescharacteristic
267C2×Q8C22×D15111yescharacteristic
268C2×D4—3022no—
269C2×D4—3022no—
270C2×D4—1512no—
271C22×C4—3024no—
272C22×C4—3021no—
273C22×C4—3022no—
274C22×C4—3021no—
275C22×C4—1512no—
276C22×C4—1511no—
277Q16—22no—
278Q16—3022no—
279Q16—3022no—
280Q16—3022no—
281SD16—22no—
282SD16—3022no—
283SD16—3022no—
284SD16—3022no—
285M16—3022no—
286M16—1512no—
287M16C22×D15112yes—
288C2×C8—3022no—
289C4⋊C4—22no—
290C4⋊C4—3022no—
291C4⋊C4—3022no—
292C4⋊C4—3022no—
293C42—3021no—
294C42—1511no—
295C42C22×D15111yescharacteristic
296C23—3022no—
297C23—1511no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—3024no—
302Q8—3022no—
303Q8—3022no—
304Q8—3024no—
305Q8—3022no—
306Q8—3022no—
307Q8—6042no—
308Q8—1512no—
309Q8D4×D15112yes—
310D4—22no—
311D4—22no—
312D4—3022no—
313D4—3022no—
314D4—3024no—
315D4—3022no—
316D4—3022no—
317D4—3022no—
318D4—1512no—
319D4D4×D15112yes—
320C2×C4—24no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—3024no—
325C2×C4—3024no—
326C2×C4—3024no—
327C2×C4—3022no—
328C2×C4—3022no—
329C2×C4—3022no—
330C2×C4—3022no—
331C2×C4—3024no—
332C2×C4—3021no—
333C2×C4—3022no—
334C2×C4—3022no—
335C2×C4—3021no—
336C2×C4—6041no—
337C2×C4—6041no—
338C2×C4—1512no—
339C2×C4—1511no—
340C2×C4—1512no—
341C2×C4—1512no—
342C2×C4D4×D15112yes—
343C2×C4C23×D15111yescharacteristic
344C8—22no—
345C8—3022no—
346C22—22no—
347C22—3022no—
348C22—3022no—
349C22—6044no—
350C22—1511no—
351C22—1511no—
352C22C2×D4×D15111yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—3022no—
358C4—3022no—
359C4—3022no—
360C4—6041no—
361C4—1512no—
362C4C2×D4×D15112yes—
363C2—21no—
364C2—42no—
365C2—3021no—
366C2—6042no—
367C2—1512no—
368C2#120×D15111yescharacteristic
subgroupG/H
centerC2#120×D151
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#456×C151C2
socleC2×C151C2×#120
prime radical2−1+4×D151C2
hypercenter#456D151
hypocenterC151C2×#456
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
2. #137×D151C2semidirect
3. C4⋊Q8×D151C2semidirect
4. #23752397C2semidirect
5. #23752396C2semidirect
6. #23752395C2semidirect
7. #23752394C2semidirect
8. #23752393C2semidirect
9. #104×D151C2semidirect
10. #101×D151C2semidirect
11. #456×C151C2semidirect
23. #18012772C22semidirect
25. #18012770C22semidirect
26. #18012769C22semidirect
31. #18012739C22semidirect
33. #18012737C22semidirect
34. #18012737C22semidirect
37. #18012667C22semidirect
38. #18012666C22semidirect
39. #18012665C22semidirect
40. #18012658C22semidirect
41. #18012657C22semidirect
43. #137×C151C22semidirect
44. C4⋊Q8×C151C22semidirect
45. #104×C151C22semidirect
46. #101×C151C22semidirect
183. D151#456direct (N × H)
184. C151C2×#456semidirect
194. #456D151direct (N × H)
214. #137C2×D151semidirect
222. C4⋊Q8C2×D151semidirect
228. #104C2×D151semidirect
231. #101C2×D151semidirect

Maximal subgroups

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

subgroupindex
2−1+4×D1512
#137×D1512
C4⋊Q8×D1512
#237523972
#237523962
#237523952
#237523942
#237523932
#104×D1512
#101×D1512
#456×C1512
C2×#456151