Finite Group Databasebeta

← #11783656

Subgroups — #11783656

Order 1152.

937 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C3×C4×#217C211yesmaximal, characteristic
3C2×C3×#830C211yesmaximal, characteristic
4C2×Dic3×C3⋊C8C212yesmaximal, complemented
5#10589155C212yesmaximal, complemented
6#69947—31nomaximal
7C22⋊C4×C3⋊C8—31nomaximal
8C23×C32×C4C2211yescharacteristic
9C22×C3⋊(C3⋊C8)—24no—
10C22×C3⋊(C3⋊C8)C414yescomplemented
11C22×C3⋊(C3⋊C8)C2211yescharacteristic
12C22×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C3×#217C412yes—
14C2×C3×C4×Dic3C2212yes—
15C22×C3×C3⋊C8C2212yes—
16C3×C3⋊#95C412yes—
17C3×#830C414yescomplemented
18Dic3×C3⋊C8—24no—
19C23×C3⋊C8—31no—
20C23×C3⋊C8—31no—
21C23×C3⋊C8—61no—
22C2×C3×C22⋊C8—31no—
23C3×C4×C22⋊C4—31no—
24C4×#217D311yescharacteristic
25C2×#830D311yescharacteristic
26C2×C8×Dic3—32no—
27C2×C4×C3⋊C8—32no—
28#4550—32no—
29#4524—32no—
30C24×C32C2×C411yescharacteristic
31C22×C32×C4—22no—
32C22×C32×C4—22no—
33C22×C32×C4C2×C411yescharacteristic
34C22×C32×C4C2×C412yes—
35C22×C32×C4C2×C412yes—
36C22×C32×C4C2311yescharacteristic
37C22×C3×Dic3C2×C414yes—
38C2×C3⋊(C3⋊C8)—22no—
39C2×C3⋊(C3⋊C8)—28no—
40C2×C3⋊(C3⋊C8)C2×C412yes—
41C2×C3⋊(C3⋊C8)C2×C412yes—
42C2×C3⋊(C3⋊C8)D414yes—
43C3×#217C818yescomplemented
44C3×C4×Dic3—22no—
45C3×C4×Dic3—22no—
46C2×C3×C3⋊C8—22no—
47C2×C3×C3⋊C8—22no—
48C2×C3×C3⋊C8C2×C414yes—
49C8×C22⋊C4—91noSylow
50C23×C3×C4—21no—
51C23×C3×C4C2×D311yescharacteristic
52C23×C3×C4C2×D311yescharacteristic
53C22×C3×C8—32no—
54C2×C3×C22⋊C4—32no—
55C2×C3×C42—32no—
56C2×#217Dic312yes—
57C2×C4×Dic3C2×D312yes—
58C22×C3⋊C8—31no—
59C22×C3⋊C8—31no—
60C22×C3⋊C8—34no—
61C22×C3⋊C8—34no—
62C22×C3⋊C8—31no—
63C22×C3⋊C8—31no—
64C22×C3⋊C8—64no—
65C22×C3⋊C8—61no—
66C22×C3⋊C8—61no—
67C22×C3⋊C8—64no—
68C22×C3⋊C8—64no—
69C22×C3⋊C8—68no—
70C22×C3⋊C8C2×D312yes—
71C3×C22⋊C8—34no—
72C3×#95—32no—
73C3⋊#95Dic312yes—
74#830Dic314yescomplemented
75C8×Dic3—64no—
76C4×C3⋊C8—64no—
77C23×C32—22no—
78C23×C32—22no—
79C23×C32C4212yes—
80C23×C32C2×C814yes—
81C23×C32C22×C411yescharacteristic
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—24no—
86C2×C32×C4C4212yes—
87C2×C32×C4C22×C411yescharacteristic
88C2×C32×C4C22×C411yescharacteristic
89C2×C32×C4C2×D412yes—
90C2×C32×C4#4112yes—
91C2×C3×Dic3—24no—
92C2×C3×Dic3—24no—
93C2×C3×Dic3C2×C818yescomplemented
94C3⋊(C3⋊C8)—22no—
95C3⋊(C3⋊C8)C22⋊C414yescomplemented
96C3×C3⋊C8—24no—
97C23×C8—91no—
98C2×C22⋊C8—31no—
99C2×C4×C8—92no—
100C4×C22⋊C4—31no—
101#336—92no—
102C24×C3—21no—
103C24×C3C4×D311yescharacteristic
104C24×C3C2×Dic311yescharacteristic
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—34no—
116C22×C3×C4C4×D311yescharacteristic
117C22×C3×C4C4×D312yes—
118C22×C3×C4C4×D312yes—
119C22×C3×C4C4×D312yes—
120C22×C3×C4C2×Dic311yescharacteristic
121C22×C3×C4C2×Dic312yes—
122C22×C3×C4C22×D311yescharacteristic
123C22×C3×C4C22×D311yescharacteristic
124C22×Dic3C2×Dic314yes—
125C2×C3×C8—34no—
126C2×C3×C8—62no—
127C2×C3×C8—62no—
128C3×C22⋊C4—38no—
129C3×C42—62no—
130C3×C42—62no—
131#217C3⋊C818yescomplemented
132C4×Dic3—22no—
133C4×Dic3—22no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—32no—
137C2×C3⋊C8—32no—
138C2×C3⋊C8—34no—
139C2×C3⋊C8—34no—
140C2×C3⋊C8—32no—
141C2×C3⋊C8—32no—
142C2×C3⋊C8—62no—
143C2×C3⋊C8—62no—
144C2×C3⋊C8—64no—
145C2×C3⋊C8—68no—
146C2×C3⋊C8—68no—
147C2×C3⋊C8—616no—
148C2×C3⋊C8—64no—
149C2×C3⋊C8—62no—
150C2×C3⋊C8—62no—
151C2×C3⋊C8C2×Dic314yes—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—28no—
155C22×C32C4×C814yescomplemented
156C22×C32C2×C4211yescharacteristic
157C22×C32C4×D412yes—
158C22×C32C22×C812yes—
159C22×C32#13112yes—
160C32×C4—22no—
161C32×C4C2×C22⋊C411yescharacteristic
162C32×C4#11711yescharacteristic
163C32×C4C4×D412yes—
164C3×Dic3—28no—
165C23×C4D3211yescharacteristic
166C22×C8—32no—
167C22×C8—91no—
168C22×C8—94no—
169C22×C8—91no—
170C22×C8—184no—
171C2×C22⋊C4—32no—
172C2×C42—32no—
173C22⋊C8—34no—
174C4×C8—184no—
175#95—32no—
176C23×C3—24no—
177C23×C3—24no—
178C23×C3—22no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—24no—
183C23×C3—22no—
184C23×C3—21no—
185C23×C3D3×C814yes—
186C23×C3C2×C3⋊C814yes—
187C23×C3C4×Dic312yes—
188C23×C3C4×Dic312yes—
189C23×C3C2×C4×D311yescharacteristic
190C23×C3C22×Dic311yescharacteristic
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—28no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—22no—
206C2×C3×C4—21no—
207C2×C3×C4—22no—
208C2×C3×C4—38no—
209C2×C3×C4—64no—
210C2×C3×C4—64no—
211C2×C3×C4C4×Dic312yes—
212C2×C3×C4C4×Dic312yes—
213C2×C3×C4C2×C4×D311yescharacteristic
214C2×C3×C4C2×C4×D311yescharacteristic
215C2×C3×C4C2×C4×D311yescharacteristic
216C2×C3×C4D3×D412yes—
217C2×C3×C4D3×D412yes—
218C2×C3×C4#23712yes—
219C2×C3×C4#23712yes—
220C2×C3×C4C22×Dic311yescharacteristic
221C2×Dic3—24no—
222C2×Dic3—24no—
223C2×Dic3C2×C3⋊C818yescomplemented
224C3×C8—64no—
225C3⋊C8—24no—
226C3⋊C8—34no—
227C3⋊C8—34no—
228C3⋊C8—64no—
229C3⋊C8—64no—
230C3⋊C8—62no—
231C3⋊C8—62no—
232C2×C32—24no—
233C2×C32C4×C22⋊C411yescharacteristic
234C2×C32C2×C4×C811yescharacteristic
235C2×C32#40511yescharacteristic
236C2×C32C8×D414yes—
237C24D3×Dic311yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—34no—
241C22×C4D3×Dic311yescharacteristic
242C22×C4D3×Dic312yes—
243C22×C4#62412yes—
244C22×C4C2×D3211yescharacteristic
245C2×C8—34no—
246C2×C8—62no—
247C2×C8—62no—
248C2×C8—92no—
249C2×C8—94no—
250C2×C8—92no—
251C2×C8—188no—
252C2×C8—182no—
253C22⋊C4—38no—
254C42—62no—
255C42—62no—
256C22×C3—24no—
257C22×C3—24no—
258C22×C3—216no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—28no—
263C22×C3—22no—
264C22×C3—28no—
265C22×C3—24no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3C4×C3⋊C814yescomplemented
271C22×C3C8×Dic314yescomplemented
272C22×C3#67⋊C412yes—
273C22×C3C2×D3×C812yes—
274C22×C3#90712yes—
275C22×C3C22×C3⋊C812yes—
276C22×C3C2×C4×Dic311yescharacteristic
277C22×C3C2×C4×Dic311yescharacteristic
278C22×C3D4×Dic312yes—
279C22×C3#94812yes—
280C3×C4—24no—
281C3×C4—22no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—22no—
285C3×C4—21no—
286C3×C4—68no—
287C3×C4#87711yescharacteristic
288C3×C4#87711yescharacteristic
289C3×C4D3×C22⋊C411yescharacteristic
290C3×C4D3×C22⋊C411yescharacteristic
291C3×C4#67⋊C412yes—
292C3×C4D4×Dic312yes—
293Dic3—28no—
294C32C8×C22⋊C411yescharacteristic, complemented, Sylow
295C23—22no—
296C23—22no—
297C23D3×C3⋊C814yes—
298C23Dic3212yes—
299C23C2×D3×Dic311yescharacteristic
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—38no—
305C2×C4—64no—
306C2×C4—64no—
307C2×C4Dic3212yes—
308C2×C4C2×D3×Dic311yescharacteristic
309C2×C4#380512yes—
310C2×C4C2×#62411yescharacteristic
311C2×C4#381112yes—
312C8—64no—
313C8—94no—
314C8—182no—
315C2×C3—28no—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—24no—
321C2×C3—21no—
322C2×C3#67⋊C814yes—
323C2×C3C2×C4×C3⋊C811yescharacteristic
324C2×C3Dic3×C22⋊C411yescharacteristic
325C2×C3Dic3×C22⋊C411yescharacteristic
326C2×C3#499711yescharacteristic
327C2×C3D4×C3⋊C814yes—
328C2×C3C2×C8×Dic311yescharacteristic
329C2×C3#512211yescharacteristic
330C22—22no—
331C22—28no—
332C22—22no—
333C22Dic3×C3⋊C814yescomplemented
334C22C2×D3×C3⋊C812yes—
335C22#6381512yes—
336C22C2×Dic3211yescharacteristic
337C22#67⋊Dic312yes—
338C4—22no—
339C4—68no—
340C4#6395711yescharacteristic
341C4#67⋊Dic312yes—
342C4#6397411yescharacteristic
343C3—21no—
344C3C22⋊C4×C3⋊C811yescharacteristic, complemented
345C3#6994711yescharacteristic, complemented
346C2—24no—
347C2C2×Dic3×C3⋊C811yescharacteristic
348C2#1059105711yescharacteristic
349C2#67⋊(C3⋊C8)14yes—
350C2#1059170511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C2×D32
commutatorC2×C32C2×C4×C8
FittingC23×C32×C4C22
socleC23×C32C22×C4
prime radicalC24×C32C2×C4
cosocleC22×C32×C4C23
hypercenterC23×C4D32
hypocenterC32C8×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
4. C2×Dic3×C3⋊C8C2semidirect
5. #10589155C2semidirect
10. C22×C3⋊(C3⋊C8)C4semidirect
17. C3×#830C4semidirect
43. C3×#217C8semidirect
74. #830Dic3semidirect
93. C2×C3×Dic3C2×C8semidirect
95. C3⋊(C3⋊C8)C22⋊C4semidirect
131. #217C3⋊C8semidirect
155. C22×C32C4×C8semidirect
223. C2×Dic3C2×C3⋊C8semidirect
270. C22×C3C4×C3⋊C8semidirect
271. C22×C3C8×Dic3semidirect
294. C32C8×C22⋊C4semidirect
333. C22Dic3×C3⋊C8semidirect
344. C3C22⋊C4×C3⋊C8semidirect
345. C3#69947semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊(C3⋊C8)2
C3×C4×#2172
C2×C3×#8302
C2×Dic3×C3⋊C82
#105891552
#699473
C22⋊C4×C3⋊C83