Finite Group Databasebeta

← #11896025

Subgroups — #11896025

Order 1152.

1014 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
1C32×Q82C211yesmaximal, characteristic, complemented
2#10596055C211yesmaximal, characteristic
3#10592793C212yesmaximal, complemented
4#10592792C211yesmaximal, characteristic, complemented
5#10592788C212yesmaximal
6#10592783C211yesmaximal, characteristic
7#10592730C211yesmaximal, characteristic, complemented
8#10592729C212yesmaximal, complemented
9#10592727C212yesmaximal
10C4×#3755C211yesmaximal, characteristic
11#10592723C211yesmaximal, characteristic, complemented
12#83704—34nomaximal
13C32×C4⋊Q8—21no—
14C32×C4⋊Q8—22no—
15C32×C4⋊Q8C2211yescharacteristic
16C32×C4⋊Q8C2212yes—
17C32×C4×Q8—21no—
18C32×C4×Q8C2211yescharacteristic
19C32×C4×Q8C2212yes—
20C32×C4×Q8C2211yescharacteristic
21#64155—21no—
22C2×#3756C2212yes—
23C2×#3755C2211yescharacteristic
24#64107—22no—
25#64101—21no—
26#64084C2212yes—
27C4×C4⋊(C3⋊D3)C2211yescharacteristic
28#64078C2211yescharacteristic
29C4×#634C2212yes—
30#63662C2211yescharacteristic
31#63637C2212yes—
32#63637C2212yes—
33#63637C2212yes—
34#63636C2212yes—
35#63636C2211yescharacteristic
36#63635C2212yes—
37#63635C2211yescharacteristic
38C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
39C4⋊(C3⋊(C3⋊C8))C2212yes—
40#63630C2212yes—
41C4×C3⋊(C3⋊C8)C2211yescharacteristic
42C3×Q82D314yescomplemented
43#5584—34no—
44#5094—38no—
45#5093—34no—
46#5089—38no—
47#5084—34no—
48#5031—34no—
49#5030—38no—
50#5028—38no—
51C4×#215—34no—
52#5024—34no—
53C2×C32×Q8—22no—
54C2×C32×Q8—21no—
55C2×C32×Q8D411yescharacteristic
56C2×C32×Q8C2311yescharacteristic
57C2×C4⋊(C3⋊D3)C2311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×#634C2312yes—
60C32×C4⋊C4—22no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4—22no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4D411yescharacteristic
65C32×C4⋊C4D412yes—
66C32×C4⋊C4C2312yes—
67C32×C4⋊C4C2311yescharacteristic
68C32×C42—22no—
69C32×C42—21no—
70C32×C42C2312yes—
71C32×C42C2311yescharacteristic
72#3756—24no—
73#3755—22no—
74#3752—21no—
75#3752—22no—
76#3752—22no—
77C4⋊(C3⋊Dic3)—21no—
78C4⋊(C3⋊Dic3)—21no—
79C4⋊(C3⋊Dic3)C2311yescharacteristic
80#3750—22no—
81C4×C3⋊Dic3—22no—
82C2×C3⋊(C3⋊C8)C2311yescharacteristic
83C2×C3⋊(C3⋊C8)C2311yescharacteristic
84C2×C3⋊(C3⋊C8)C2312yes—
85#3369—91noSylow
86C3×C4⋊Q8—24no—
87C3×C4⋊Q8—28no—
88C3×C4⋊Q8C2×D314yes—
89C3×C4⋊Q8C2×D318yes—
90C3×C4×Q8—24no—
91C3×C4×Q8C2×D314yes—
92C3×C4×Q8C2×D318yes—
93C3×C4×Q8C2×D314yes—
94#946—64no—
95C2×#216—38no—
96C2×#215—34no—
97#898—68no—
98#892—64no—
99#875—38no—
100C4×D12—34no—
101#869—34no—
102C4×Dic6—38no—
103#835—34no—
104#810—38no—
105#810—38no—
106#810—38no—
107#809—34no—
108#809—38no—
109#808—34no—
110#808—38no—
111C4⋊(C3⋊C8)—38no—
112C4⋊(C3⋊C8)—34no—
113#803—38no—
114C4×C3⋊C8—34no—
115C22×C3⋊D3—21no—
116C32×Q8—22no—
117C32×Q8—21no—
118C32×Q8—82no—
119C32×Q8—81no—
120C32×Q8#4112yes—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4C2×D412yes—
125C2×C32×C4C2×D412yes—
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2411yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—22no—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133C4×C3⋊D3—42no—
134#634—22no—
135#634—22no—
136C3⋊(C3⋊C8)—21no—
137C3⋊(C3⋊C8)—22no—
138C3⋊(C3⋊C8)—21no—
139Q82C3⋊D311yescharacteristic, complemented
140#556—91no—
141#489—92no—
142#487—91no—
143#476—91no—
144#475—92no—
145#464—92no—
146#462—91no—
147#446—91no—
148#441—92no—
149C4×SD16—91no—
150C2×C3×Q8—28no—
151C2×C3×Q8—24no—
152C2×C3×Q8#6714yes—
153C2×C3×Q8C22×D314yes—
154C2×D12—34no—
155C2×C4×D3—64no—
156C2×Dic6—38no—
157C3×C4⋊C4—28no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4—28no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4#6714yes—
162C3×C4⋊C4#6718yes—
163C3×C4⋊C4C22×D314yes—
164C3×C4⋊C4C22×D318yes—
165C3×C42—28no—
166C3×C42—24no—
167C3×C42C22×D314yes—
168C3×C42C22×D318yes—
169#216—616no—
170#215—68no—
171(C2×D3)⋊C4—64no—
172(C2×D3)⋊C4—68no—
173(C2×D3)⋊C4—68no—
174C4⋊Dic3—34no—
175C4⋊Dic3—64no—
176C4⋊Dic3—64no—
177Dic3⋊C4—68no—
178C4×Dic3—68no—
179C2×C3⋊C8—34no—
180C2×C3⋊C8—34no—
181C2×C3⋊C8—38no—
182C22×C32C22×D411yescharacteristic
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C32×C4—22no—
187C32×C4—22no—
188C32×C4—22no—
189C32×C4—21no—
190C32×C4—42no—
191C32×C4—41no—
192C32×C4#13712yes—
193C32×C4C2×#4111yescharacteristic
194C32×C42−1+411yescharacteristic
195C3⋊Dic3—42no—
196C3⋊Dic3—41no—
197C2×Q16—92no—
198C2×SD16—91no—
199C4⋊Q8—21no—
200C4⋊Q8—22no—
201C4⋊Q8—91no—
202C4⋊Q8C2×C3⋊D312yes—
203C4⋊Q8C2×C3⋊D311yescharacteristic
204#126—182no—
205#124—92no—
206C22⋊Q8—181no—
207C4×Q8—21no—
208C4×Q8—92no—
209C4×Q8C2×C3⋊D311yescharacteristic
210C4×Q8C2×C3⋊D312yes—
211C4×Q8C2×C3⋊D311yescharacteristic
212C4×D4—91no—
213#117—181no—
214#106—91no—
215#106—92no—
216C4⋊C8—91no—
217C4⋊C8—92no—
218#103—91no—
219#103—92no—
220#103—92no—
221#103—92no—
222#102—91no—
223#102—92no—
224#97—92no—
225C4×C8—91no—
226C22×D3—64no—
227C3×Q8—28no—
228C3×Q8—24no—
229C3×Q8—88no—
230C3×Q8—84no—
231C3×Q8#23918yes—
232C2×C3×C4—24no—
233C2×C3×C4—28no—
234C2×C3×C4—24no—
235C2×C3×C4C2×#6718yes—
236C2×C3×C4C2×#6718yes—
237C2×C3×C4C2×#6714yes—
238C2×C3×C4C2×#6714yes—
239C2×C3×C4C23×D314yes—
240C2×Dic3—64no—
241C2×Dic3—68no—
242D12—64no—
243D12—64no—
244C4×D3—128no—
245Dic6—68no—
246Dic6—68no—
247C3⋊C8—64no—
248C3⋊C8—68no—
249C3⋊C8—64no—
250C2×C32#54811yescharacteristic
251C2×C32C2×#13711yescharacteristic
252C2×C32#57711yescharacteristic
253C3⋊D3—81no—
254C2×Q8—22no—
255C2×Q8—21no—
256C2×Q8—92no—
257C2×Q8#63811yescharacteristic
258C2×Q8C22×C3⋊D311yescharacteristic
259C2×D4—91no—
260C22×C4—181no—
261Q16—184no—
262SD16—182no—
263C2×C8—91no—
264C2×C8—91no—
265C2×C8—92no—
266C4⋊C4—22no—
267C4⋊C4—21no—
268C4⋊C4—22no—
269C4⋊C4—21no—
270C4⋊C4—91no—
271C4⋊C4—181no—
272C4⋊C4—182no—
273C4⋊C4—181no—
274C4⋊C4#63811yescharacteristic
275C4⋊C4#63812yes—
276C4⋊C4C22×C3⋊D311yescharacteristic
277C4⋊C4C22×C3⋊D312yes—
278C22⋊C4—182no—
279C22⋊C4—181no—
280C22⋊C4—182no—
281C42—22no—
282C42—21no—
283C42—182no—
284C42C22×C3⋊D311yescharacteristic
285C42C22×C3⋊D312yes—
286C22×C3C22×#6714yes—
287C2×D3—124no—
288C2×D3—124no—
289C2×D3—124no—
290C3×C4—28no—
291C3×C4—28no—
292C3×C4—28no—
293C3×C4—24no—
294C3×C4—48no—
295C3×C4—44no—
296C3×C4#94218yes—
297C3×C4C2×#23914yes—
298C3×C4#100714yes—
299Dic3—124no—
300Dic3—128no—
301C32#336911yescharacteristic, complemented, Sylow
302C23—181no—
303Q8—21no—
304Q8—22no—
305Q8—82no—
306Q8—81no—
307Q8—182no—
308Q8—182no—
309Q8#383212yes—
310D4—181no—
311D4—181no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—181no—
316C2×C4—182no—
317C2×C4—362no—
318C2×C4C2×#63812yes—
319C2×C4C2×#63812yes—
320C2×C4C2×#63811yescharacteristic
321C2×C4C2×#63811yescharacteristic
322C2×C4C23×C3⋊D311yescharacteristic
323C8—181no—
324C8—182no—
325C8—181no—
326C2×C3C2×#94214yes—
327C2×C3#581714yes—
328C2×C3#583614yes—
329D3—244no—
330C22—361no—
331C22—361no—
332C22—361no—
333C22C22×#63811yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—21no—
338C4—42no—
339C4—41no—
340C4—361no—
341C4—362no—
342C4#6415112yes—
343C4C2×#383211yescharacteristic
344C4#6436411yescharacteristic
345C3#8370414yescomplemented
346C2—721no—
347C2C2×#6415111yescharacteristic
348C2#1059628811yescharacteristic
349C2#1059630711yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×Q82C2
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenterQ82C3⋊D3
hypocenterC32#3369
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q82

Complements

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

NH ≅ G/Ntype
1. C32×Q82C2semidirect
3. #10592793C2semidirect
4. #10592792C2semidirect
7. #10592730C2semidirect
8. #10592729C2semidirect
11. #10592723C2semidirect
42. C3×Q82D3semidirect
139. Q82C3⋊D3semidirect
301. C32#3369semidirect
345. C3#83704semidirect

Maximal subgroups

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

subgroupindex
C32×Q822
#105960552
#105927932
#105927922
#105927882
#105927832
#105927302
#105927292
#105927272
C4×#37552
#105927232
#837043