Finite Group Databasebeta

← #11866251

Subgroups — #11866251

Order 1152.

1482 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#10596296C211yesmaximal, characteristic, complemented
3#10596269C211yesmaximal, characteristic, complemented
4C22×#3754C211yesmaximal, characteristic, complemented
5#10592932C211yesmaximal, characteristic, complemented
6#10592861C214yesmaximal, complemented
7#10592852C214yesmaximal, complemented
8#10592819C211yesmaximal, characteristic, complemented
9#10592818C211yesmaximal, characteristic, complemented
10#83935—34nomaximal
11C2×C32×#41C2211yescharacteristic
12C22×C32×D4C2211yescharacteristic
13C22×#634C2211yescharacteristic
14C32×AGL(1,ℤ/8)—24no—
15C2×C32×SD16C2212yes—
16C2×C32×D8C2212yes—
17C2×C32×M16C2211yescharacteristic
18C2×#3757—22no—
19#64153C2212yescomplemented
20#64145C2212yescomplemented
21#64144—22no—
22D4×C3⋊Dic3—22no—
23D4×C3⋊Dic3C2214yes—
24C2×#3754—24no—
25C2×#3754C2214yes—
26#64135C2211yescharacteristic
27#64135C2211yescharacteristic
28#64133—22no—
29C22×C3⋊(C3⋊C8)C2211yescharacteristic
30#63662C2212yescomplemented
31#63659C2212yescomplemented
32#63643C2214yescomplemented
33#63641C2214yescomplemented
34C2×C3×AGL(1,ℤ/8)D314yes—
35#5825—34no—
36#5798—34no—
37C22×#214—34no—
38#5233—34no—
39#5162—316no—
40#5153—316no—
41#5120—34no—
42#5119—34no—
43C24×C32—21no—
44C32×#41—22no—
45C32×#41—22no—
46C2×C32×Q8C2311yescharacteristic
47C2×C32×D4—21no—
48C2×C32×D4—21no—
49C2×C32×D4—22no—
50C2×C32×D4D414yes—
51C2×C32×D4C2311yescharacteristic
52C2×C32×D4C2312yes—
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C22×C3⋊Dic3—21no—
56C22×C3⋊Dic3—24no—
57C2×#634—22no—
58C2×#634—22no—
59C2×#634—21no—
60C2×#634C2312yes—
61C32×SD16—24no—
62C32×D8—24no—
63C32×M16—22no—
64C2×C32×C8C2312yes—
65#3757—21no—
66#3757—24no—
67#3757—24no—
68#3757D412yes—
69#3754—28no—
70C4⋊(C3⋊Dic3)D412yes—
71C4⋊(C3⋊Dic3)C2312yes—
72#3750—24no—
73C4×C3⋊Dic3—21no—
74C4×C3⋊Dic3C2312yes—
75C2×C3⋊(C3⋊C8)—22no—
76C2×C3⋊(C3⋊C8)C2312yes—
77#3042—91noSylow
78C2×C3×#41C2×D314yes—
79C22×C3×D4C2×D314yes—
80C22×Dic6—34no—
81C3×AGL(1,ℤ/8)—216no—
82C2×C3×SD16C2×D318yes—
83C2×C3×D8C2×D318yes—
84C2×C3×M16C2×D314yes—
85C2×#217—68no—
86#944—38no—
87#936—38no—
88#935—68no—
89D4×Dic3—316no—
90D4×Dic3—68no—
91C2×#214—316no—
92C2×#214—616no—
93#926—34no—
94#926—34no—
95#924—68no—
96C22×C3⋊C8—34no—
97#835—38no—
98#832—38no—
99#816—316no—
100Dic3⋊C8—316no—
101C23×C32—22no—
102C23×C32—24no—
103C23×C32—21no—
104C23×C32—82no—
105C23×C32C2×D411yescharacteristic
106C32×Q8—21no—
107C32×Q8—21no—
108C32×D4—24no—
109C32×D4—21no—
110C32×D4—22no—
111C32×D4—21no—
112C32×D4—42no—
113C32×D4C2×D414yes—
114C2×C32×C4—21no—
115C2×C32×C4—42no—
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D412yes—
118C2×C32×C4#4112yes—
119C2×C32×C4C2411yescharacteristic
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—42no—
123C2×C3⋊Dic3—84no—
124C2×C3⋊Dic3C2×D414yes—
125#634—22no—
126#634—22no—
127#634—42no—
128C32×C8—22no—
129C3⋊(C3⋊C8)—22no—
130C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic
131C22×SD16—91no—
132#536—91no—
133#518—91no—
134#463—94no—
135#461—94no—
136#424—91no—
137#419—91no—
138#416—91no—
139C24×C3—24no—
140C3×#41—28no—
141C3×#41—28no—
142C2×C3×Q8C22×D314yes—
143C2×C3×D4—24no—
144C2×C3×D4—24no—
145C2×C3×D4—28no—
146C2×C3×D4#67116yes—
147C2×C3×D4C22×D314yes—
148C2×C3×D4C22×D318yes—
149C22×C3×C4—24no—
150C22×C3×C4C22×D314yes—
151C22×Dic3—616no—
152C22×Dic3—64no—
153C2×Dic6—38no—
154C2×Dic6—68no—
155C2×Dic6—68no—
156C2×Dic6—64no—
157C3×SD16—216no—
158C3×D8—216no—
159C3×M16—28no—
160C2×C3×C8C22×D318yes—
161#217—38no—
162#217—616no—
163#217—616no—
164#217—64no—
165#214—632no—
166C4⋊Dic3—38no—
167C4⋊Dic3—38no—
168Dic3⋊C4—616no—
169C4×Dic3—38no—
170C4×Dic3—64no—
171C2×C3⋊C8—38no—
172C2×C3⋊C8—68no—
173C22×C32—22no—
174C22×C32—24no—
175C22×C32—44no—
176C22×C32—41no—
177C22×C32—42no—
178C22×C32—84no—
179C22×C32#12112yes—
180C22×C32C22×D411yescharacteristic
181C32×C4—41no—
182C32×C4#12112yes—
183C32×C4C22×D411yescharacteristic
184C32×C4C2×#4111yescharacteristic
185C3⋊Dic3—24no—
186C3⋊Dic3—42no—
187C3⋊Dic3—42no—
188C2×#41C2×C3⋊D311yescharacteristic
189C22×Q8—91no—
190C22×D4C2×C3⋊D311yescharacteristic
191AGL(1,ℤ/8)—24no—
192C2×SD16—94no—
193C2×SD16—184no—
194C2×SD16C2×C3⋊D312yes—
195C2×D8C2×C3⋊D312yes—
196C2×M16C2×C3⋊D311yescharacteristic
197C22×C8—91no—
198C4⋊Q8—92no—
199#124—92no—
200#123—182no—
201C22⋊Q8—182no—
202C4×D4—94no—
203C4×D4—182no—
204#117—91no—
205#117—91no—
206C2×C22⋊C4—182no—
207C4⋊C8—94no—
208#103—92no—
209#103—94no—
210#102—92no—
211C23×C3—28no—
212C23×C3—216no—
213C23×C3—24no—
214C23×C3—88no—
215C23×C3D3×D414yes—
216C3×Q8—24no—
217C3×Q8—24no—
218C3×D4—216no—
219C3×D4—28no—
220C3×D4—24no—
221C3×D4—24no—
222C3×D4—48no—
223C3×D4C2×#67116yes—
224C2×C3×C4—24no—
225C2×C3×C4—48no—
226C2×C3×C4D3×D414yes—
227C2×C3×C4#23718yes—
228C2×C3×C4C2×#6718yes—
229C2×C3×C4C23×D314yes—
230C2×Dic3—316no—
231C2×Dic3—68no—
232C2×Dic3—68no—
233C2×Dic3—128no—
234C2×Dic3—2416no—
235Dic6—68no—
236Dic6—68no—
237Dic6—128no—
238C3×C8—28no—
239C3⋊C8—68no—
240C2×C32—22no—
241C2×C32—44no—
242C2×C32—81no—
243C2×C32C2×#12111yescharacteristic
244C2×C32#57712yes—
245C24—21no—
246#41—22no—
247#41—22no—
248C2×Q8—92no—
249C2×Q8—181no—
250C2×Q8—182no—
251C2×Q8—182no—
252C2×Q8C22×C3⋊D311yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—22no—
256C2×D4#63814yes—
257C2×D4C22×C3⋊D311yescharacteristic
258C2×D4C22×C3⋊D312yes—
259C22×C4—21no—
260C22×C4—184no—
261C22×C4—181no—
262C22×C4C22×C3⋊D311yescharacteristic
263SD16—24no—
264SD16—188no—
265D8—24no—
266M16—22no—
267C2×C8—92no—
268C2×C8—182no—
269C2×C8C22×C3⋊D312yes—
270C4⋊C4—92no—
271C4⋊C4—92no—
272C4⋊C4—184no—
273C22⋊C4—92no—
274C22⋊C4—184no—
275C22⋊C4—184no—
276C22⋊C4—181no—
277C42—92no—
278C42—181no—
279C22×C3—28no—
280C22×C3—216no—
281C22×C3—416no—
282C22×C3—48no—
283C22×C3—44no—
284C22×C3—816no—
285C22×C3#93918yes—
286C22×C3C2×D3×D414yes—
287C3×C4—44no—
288C3×C4#93918yes—
289C3×C4C2×#23714yes—
290C3×C4C22×#6714yes—
291Dic3—616no—
292Dic3—128no—
293Dic3—128no—
294C32#304211yescharacteristic, complemented, Sylow
295C23—22no—
296C23—24no—
297C23—21no—
298C23—82no—
299C23D4×C3⋊D311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—182no—
303Q8—182no—
304Q8—362no—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—24no—
309D4—42no—
310D4C2×#63814yes—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—94no—
314C2×C4—182no—
315C2×C4—182no—
316C2×C4—362no—
317C2×C4—724no—
318C2×C4D4×C3⋊D311yescharacteristic
319C2×C4#383012yes—
320C2×C4C2×#63812yes—
321C2×C4C23×C3⋊D311yescharacteristic
322C8—22no—
323C8—182no—
324C2×C3—28no—
325C2×C3—416no—
326C2×C3—84no—
327C2×C3#577518yes—
328C2×C3C2×#93914yes—
329C22—22no—
330C22—24no—
331C22—44no—
332C22—42no—
333C22—41no—
334C22—84no—
335C22#6414812yes—
336C22C2×D4×C3⋊D311yescharacteristic
337C4—41no—
338C4—184no—
339C4—362no—
340C4—362no—
341C4#6414812yes—
342C4C2×#383011yescharacteristic
343C4C22×#63811yescharacteristic
344C3#8393514yescomplemented
345C2—22no—
346C2—44no—
347C2—81no—
348C2#1059624612yes—
349C2C2×#6414811yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
Fitting, prime radicalC2×C32×AGL(1,ℤ/8)C2
socleC22×C32C22×D4
hypercenterC2×AGL(1,ℤ/8)C3⋊D3
hypocenterC32#3042
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #10596296C2semidirect
3. #10596269C2semidirect
4. C22×#3754C2semidirect
5. #10592932C2semidirect
6. #10592861C2semidirect
7. #10592852C2semidirect
8. #10592819C2semidirect
9. #10592818C2semidirect
19. #64153C22semidirect
20. #64145C22semidirect
30. #63662C22semidirect
31. #63659C22semidirect
32. #63643C22semidirect
33. #63641C22semidirect
294. C32#3042semidirect
344. C3#83935semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
#105962962
#105962692
C22×#37542
#105929322
#105928612
#105928522
#105928192
#105928182
#839353