Finite Group Databasebeta

← #11874467

Subgroups — #11874467

Order 1152.

1968 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 1968 conjugacy classes of subgroups — too many to draw. Only its 277 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
1C2×#64120C214yesmaximal, complemented
2#10596187C211yesmaximal, characteristic, complemented
3C3⋊D3×#127C211yesmaximal, characteristic, complemented
4C32×#493C211yesmaximal, characteristic, complemented
5#10592853C214yesmaximal, complemented
6#10592775C211yesmaximal, characteristic, complemented
7#10592772C211yesmaximal, characteristic, complemented
8#10592392C211yesmaximal, characteristic, complemented
9C4×#3743C211yesmaximal, characteristic, complemented
10#81783—34nomaximal
11C2×#3830C2212yes—
12C2×D4×C3⋊D3—22no—
13C2×D4×C3⋊D3C2212yes—
14C2×C32×D8C2214yescomplemented
15C32×#127C2211yescharacteristic
16C32×#127C2211yescharacteristic, complemented
17#64149—22no—
18#64149C2212yes—
19#64147—22no—
20#64145C2212yescomplemented
21D4×C3⋊Dic3—22no—
22C2×#3754C2214yescomplemented
23C2×#3753C2214yescomplemented
24#64120—216no—
25C2×#3744C2214yescomplemented
26C2×#3743C2212yes—
27#64083C2211yescharacteristic
28C42×C3⋊D3C2211yescharacteristic
29#64078C2211yescharacteristic, complemented
30C32×C4×C8C2211yescharacteristic, complemented
31#63642C2212yescomplemented
32C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
33C2×#911—316no—
34#5716—34no—
35D3×#127—34no—
36C3×#493D314yescomplemented
37#5154—316no—
38#5076—34no—
39#5073—34no—
40#4693—34no—
41C4×ΓL(1,𝔽25)—34no—
42C23×C3⋊D3—22no—
43C2×C32×D4—22no—
44C2×C32×D4—22no—
45C2×C32×D4C2312yes—
46C2×C32×D4C2312yes—
47C2×#638—22no—
48C2×#638—24no—
49C2×#638—22no—
50C22×C3⋊Dic3—22no—
51#3830—28no—
52D4×C3⋊D3—28no—
53D4×C3⋊D3—48no—
54C2×C4⋊(C3⋊D3)—22no—
55C2×C4⋊(C3⋊D3)C2312yes—
56C2×C4×C3⋊D3D412yes—
57C2×C4×C3⋊D3C2311yescharacteristic
58C2×#634C2312yes—
59C32×D8—28no—
60C2×C32×C8C2312yes—
61C32×C42C2311yescharacteristic
62#3757—24no—
63#3754—28no—
64#3753—28no—
65C4⋊(C3⋊Dic3)—22no—
66C4×C3⋊Dic3D412yes—
67C4×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊(C3⋊C8)C2312yes—
69#3744—28no—
70#3743D418yescomplemented
71#3151—91noSylow
72C2×#237—38no—
73C2×D3×D4—38no—
74C2×D3×D4—68no—
75C2×C3×D8C2×D3116yescomplemented
76C3×#127C2×D314yes—
77C3×#127C2×D314yescomplemented
78#940—38no—
79#940—68no—
80#938—68no—
81#936—38no—
82D4×Dic3—68no—
83C2×#214—316no—
84C2×#213—316no—
85#911—664no—
86C2×#204—316no—
87C2×ΓL(1,𝔽25)—38no—
88#874—34no—
89C42×D3—34no—
90#869—34no—
91C3×C4×C8C2×D314yescomplemented
92#815—38no—
93C4×C3⋊C8—34no—
94C23×C32—22no—
95C23×C32—22no—
96C22×C3⋊D3—22no—
97C22×C3⋊D3—48no—
98C22×C3⋊D3—44no—
99C22×C3⋊D3C2×D411yescharacteristic
100C32×D4—24no—
101C32×D4—24no—
102C32×D4—44no—
103C32×D4—44no—
104C2×C32×C4C2×D412yes—
105C2×C32×C4C2411yescharacteristic
106#638—48no—
107#638—44no—
108#638—44no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—44no—
111C2×C3⋊Dic3C2×D412yes—
112C2×C3⋊Dic3C2×D411yescharacteristic
113C4⋊(C3⋊D3)—24no—
114C4⋊(C3⋊D3)—44no—
115C4×C3⋊D3—24no—
116C4×C3⋊D3C2×D414yes—
117#634—24no—
118C32×C8C2×D414yescomplemented
119C3⋊(C3⋊C8)C2×D414yescomplemented
120C2×AGL(1,ℤ/8)—94no—
121#532—91no—
122C2×#127—91no—
123#496—94no—
124#493—91no—
125#493C3⋊D311yescharacteristic, complemented
126#492—91no—
127#492—91no—
128C4×M16—91no—
129C23×D3—68no—
130C2×C3×D4—28no—
131C2×C3×D4—28no—
132C2×C3×D4C22×D318yes—
133C2×C3×D4C22×D318yes—
134C2×#67—616no—
135C2×#67—68no—
136C2×#67—68no—
137C22×Dic3—68no—
138#237—632no—
139D3×D4—632no—
140D3×D4—1232no—
141C2×D12—38no—
142C2×D12—68no—
143C2×C4×D3—34no—
144C2×C4×D3—38no—
145C2×Dic6—38no—
146C3×D8—232no—
147C2×C3×C8C22×D318yes—
148C3×C42C22×D314yes—
149#217—616no—
150#214—632no—
151#213—632no—
152C4⋊Dic3—68no—
153C4×Dic3—38no—
154C4×Dic3—34no—
155C2×C3⋊C8—38no—
156#204—632no—
157ΓL(1,𝔽25)—332no—
158C22×C32—42no—
159C22×C32—42no—
160C22×C32—44no—
161C22×C32—44no—
162C22×C32C22×D411yescharacteristic
163C2×C3⋊D3—22no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3—44no—
166C2×C3⋊D3—88no—
167C2×C3⋊D3#12712yes—
168C32×C4AGL(1,ℤ/8)14yescomplemented
169C32×C4C22×D412yes—
170C3⋊Dic3—22no—
171C3⋊Dic3—42no—
172C3⋊Dic3#12712yes—
173C2×#41—92no—
174C22×D4—92no—
175C22×D4—182no—
176AGL(1,ℤ/8)—1816no—
177C2×SD16—94no—
178C2×SD16—94no—
179C2×D8—94no—
180C2×D8C2×C3⋊D314yescomplemented
181C2×M16—92no—
182C4⋊Q8—91no—
183#127—92no—
184#127—91no—
185#127—182no—
186#127C2×C3⋊D311yescharacteristic
187#127C2×C3⋊D311yescharacteristic, complemented
188#124—92no—
189#121—182no—
190C4×D4—182no—
191C2×C42—91no—
192#97—92no—
193C4×C8—91no—
194C4×C8C2×C3⋊D311yescharacteristic, complemented
195C23×C3—28no—
196C23×C3—28no—
197C22×D3—34no—
198C22×D3—68no—
199C22×D3—1216no—
200C22×D3—1232no—
201C3×D4—216no—
202C3×D4—216no—
203C3×D4—416no—
204C3×D4—416no—
205C2×C3×C4D3×D418yes—
206C2×C3×C4C23×D314yes—
207#67—1216no—
208#67—1232no—
209#67—1216no—
210C2×Dic3—34no—
211C2×Dic3—38no—
212C2×Dic3—68no—
213C2×Dic3—1216no—
214D12—616no—
215D12—1216no—
216C4×D3—316no—
217C4×D3—616no—
218Dic6—616no—
219C3×C8D3×D4116yescomplemented
220C3⋊C8—316no—
221C2×C32—82no—
222C2×C32—82no—
223C2×C32C2×#12711yescharacteristic
224C2×C32C2×AGL(1,ℤ/8)12yes—
225C3⋊D3—22no—
226C3⋊D3—82no—
227C24—182no—
228#41—188no—
229C2×Q8—92no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—92no—
233C2×D4—188no—
234C2×D4—184no—
235C2×D4—182no—
236C2×D4—182no—
237C2×D4—182no—
238C2×D4—368no—
239C2×D4C22×C3⋊D312yes—
240C2×D4C22×C3⋊D312yes—
241C22×C4—91no—
242C22×C4—92no—
243C22×C4—182no—
244SD16—188no—
245SD16—188no—
246D8—28no—
247D8—188no—
248M16—98no—
249C2×C8—92no—
250C2×C8C22×C3⋊D312yes—
251C4⋊C4—182no—
252C22⋊C4—184no—
253C42—91no—
254C42—92no—
255C42C22×C3⋊D311yescharacteristic
256C22×C3—48no—
257C22×C3—48no—
258C22×C3—416no—
259C22×C3—416no—
260C22×C3C2×D3×D414yes—
261C2×D3—38no—
262C2×D3—68no—
263C2×D3—128no—
264C2×D3—1216no—
265C2×D3—2432no—
266C3×C4#911116yescomplemented
267C3×C4C2×D3×D418yes—
268Dic3—38no—
269Dic3—68no—
270Dic3—128no—
271C32#315111yescharacteristic, complemented, Sylow
272C23—22no—
273C23—22no—
274C23—91no—
275C23—182no—
276C23—364no—
277C23—368no—
278Q8—184no—
279D4—24no—
280D4—24no—
281D4—44no—
282D4—44no—
283D4—184no—
284D4—364no—
285D4—368no—
286D4—364no—
287D4—364no—
288C2×C4—91no—
289C2×C4—94no—
290C2×C4—92no—
291C2×C4—184no—
292C2×C4—182no—
293C2×C4—364no—
294C2×C4D4×C3⋊D312yes—
295C2×C4C23×C3⋊D311yescharacteristic
296C8—94no—
297C8D4×C3⋊D314yescomplemented
298C2×C3—88no—
299C2×C3—88no—
300C2×C3D3×#12714yes—
301C2×C3C2×#91118yes—
302D3—68no—
303D3—248no—
304C22—42no—
305C22—42no—
306C22—44no—
307C22—44no—
308C22—92no—
309C22—182no—
310C22—362no—
311C22—364no—
312C22—728no—
313C22C2×D4×C3⋊D311yescharacteristic
314C4—92no—
315C4—182no—
316C4—362no—
317C4#6412014yescomplemented
318C4C2×D4×C3⋊D312yes—
319C3#8178314yescomplemented
320C2—82no—
321C2—82no—
322C2—182no—
323C2—722no—
324C2C3⋊D3×#12711yescharacteristic
325C2C2×#6412012yes—
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#493C2
socleC22×C32C22×D4
hypercenter#493C3⋊D3
hypocenterC32#3151
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#64120C2semidirect
2. #10596187C2semidirect
3. C3⋊D3×#127C2semidirect
4. C32×#493C2semidirect
5. #10592853C2semidirect
6. #10592775C2semidirect
7. #10592772C2semidirect
8. #10592392C2semidirect
9. C4×#3743C2semidirect
14. C2×C32×D8C22semidirect
16. C32×#127C22semidirect
20. #64145C22semidirect
22. C2×#3754C22semidirect
23. C2×#3753C22semidirect
25. C2×#3744C22semidirect
29. #64078C22semidirect
30. C32×C4×C8C22semidirect
31. #63642C22semidirect
32. C4×C3⋊(C3⋊C8)C22semidirect
36. C3×#493D3semidirect
70. #3743D4semidirect
75. C2×C3×D8C2×D3semidirect
77. C3×#127C2×D3semidirect
91. C3×C4×C8C2×D3semidirect
118. C32×C8C2×D4semidirect
119. C3⋊(C3⋊C8)C2×D4semidirect
125. #493C3⋊D3semidirect
168. C32×C4AGL(1,ℤ/8)semidirect
180. C2×D8C2×C3⋊D3semidirect
187. #127C2×C3⋊D3semidirect
194. C4×C8C2×C3⋊D3semidirect
219. C3×C8D3×D4semidirect
266. C3×C4#911semidirect
271. C32#3151semidirect
297. C8D4×C3⋊D3semidirect
317. C4#64120semidirect
319. C3#81783semidirect

Maximal subgroups

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

subgroupindex
C2×#641202
#105961872
C3⋊D3×#1272
C32×#4932
#105928532
#105927752
#105927722
#105923922
C4×#37432
#817833