Finite Group Databasebeta

← #11808512

Subgroups — #11808512

Order 1152.

1417 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
2C2×#63883C211yesmaximal, characteristic
3C3×#4985C212yesmaximal, complemented
4#10591370C212yesmaximal
5#10591370C211yesmaximal, characteristic
6#10591370C212yesmaximal
7#10591226C212yesmaximal, complemented
8#10591210C214yesmaximal, complemented
9D3×#400—32nomaximal
10C22×C4×C3⋊D3C2212yes—
11C22×C4×C3⋊D3C2211yescharacteristic
12C22×#624C2211yescharacteristic
13C22×#624C2212yes—
14C22×#624C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×#3750C2212yes—
18C4⋊C4×C3⋊D3—24no—
19C2×C3×Dic3⋊C4C2212yes—
20C2×Dic3⋊Dic3C2212yes—
21#63883—24no—
22C3×C3⋊#95C2214yes—
23C3×C3⋊#95C2214yes—
24C3×C3⋊#95C2212yes—
25#63579C2212yes—
26#63579C2214yes—
27#63579C2214yes—
28C2×D3×C4⋊C4—32no—
29C2×D3×C4⋊C4—32no—
30C2×D3×C4⋊C4—61no—
31C3×#400—32no—
32#4986—32no—
33#4985D312yescomplemented
34D3×#95—34no—
35D3×#95—32no—
36D3×#95—34no—
37#4660—34no—
38#4659—32no—
39#4658—34no—
40C23×C3⋊D3C2311yescharacteristic
41C22×C32×C4C2312yes—
42C22×C32×C4C2311yescharacteristic
43C22×C3⋊Dic3C2312yes—
44C22×C3⋊Dic3C2311yescharacteristic
45C2×C4×C3⋊D3—24no—
46C2×C4×C3⋊D3—24no—
47C2×C4×C3⋊D3—28no—
48C2×C4×C3⋊D3D414yes—
49C22×C3×Dic3C2312yes—
50C22×C3×Dic3C2314yes—
51C22×C3×Dic3C2312yes—
52C2×#624—22no—
53C2×#624—22no—
54C2×#624—22no—
55C2×#624—24no—
56C2×#624—24no—
57C2×#624—24no—
58C2×#624—24no—
59C2×#624Q814yes—
60C32×C4⋊C4—22no—
61C4⋊(C3⋊Dic3)—22no—
62#3750—24no—
63C3×Dic3⋊C4—24no—
64Dic3⋊Dic3—24no—
65C2×#400—91noSylow
66C22×C4×D3—32no—
67C22×C4×D3—32no—
68C22×C4×D3—34no—
69C22×C4×D3—34no—
70C22×C4×D3—32no—
71C22×C4×D3—61no—
72C22×C4×D3—62no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—32no—
75C2×C3×C4⋊C4C2×D312yes—
76C2×C4⋊Dic3—32no—
77C2×C4⋊Dic3—32no—
78C2×C4⋊Dic3—61no—
79C2×Dic3⋊C4—34no—
80C2×Dic3⋊C4—32no—
81C2×Dic3⋊C4—62no—
82C2×Dic3⋊C4C2×D312yes—
83D3×C4⋊C4—68no—
84D3×C4⋊C4—68no—
85D3×C4⋊C4—68no—
86C3×#95—34no—
87C3×#95—32no—
88C3×#95—34no—
89C3⋊#95—34no—
90C3⋊#95—34no—
91C3⋊#95—34no—
92C3⋊#95—34no—
93C3⋊#95—34no—
94C3⋊#95C2×D314yes—
95C3⋊#95C2×D314yes—
96C3⋊#95C2×D312yes—
97C23×C32C2411yescharacteristic
98C22×C3⋊D3C2×D412yes—
99C22×C3⋊D3C2×Q812yes—
100C22×C3⋊D3#4112yes—
101C22×C3⋊D3#4114yes—
102C22×C3⋊D3#4114yes—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—24no—
106C2×C32×C4C2×D412yes—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—24no—
110C2×C3⋊Dic3C2×D412yes—
111C4×C3⋊D3—28no—
112C4×C3⋊D3—48no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—24no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—24no—
119C2×C3×Dic3—24no—
120C2×C3×Dic3C2×Q814yes—
121#624—28no—
122#624—44no—
123#624—48no—
124C22×C4⋊C4—91no—
125C22×C4⋊C4—91no—
126#400—32no—
127#400—94no—
128#400—92no—
129C2×#95—92no—
130C2×#95—92no—
131C2×#95—91no—
132C23×D3—32no—
133C23×D3—61no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—32no—
137C22×C3×C4—34no—
138C22×C3×C4—32no—
139C22×C3×C4C22×D314yes—
140C22×C3×C4C22×D312yes—
141C22×Dic3—32no—
142C22×Dic3—34no—
143C22×Dic3—61no—
144C22×Dic3—62no—
145C22×Dic3C22×D314yes—
146C22×Dic3C22×D312yes—
147C22×Dic3C22×D312yes—
148C2×C4×D3—38no—
149C2×C4×D3—38no—
150C2×C4×D3—64no—
151C2×C4×D3—64no—
152C2×C4×D3—64no—
153C2×C4×D3—68no—
154C2×C4×D3—68no—
155C2×C4×D3—64no—
156C2×C4×D3—68no—
157C2×C4×D3—68no—
158C2×C4×D3—68no—
159C2×C4×D3—68no—
160C2×C4×D3—68no—
161C2×C4×D3—616no—
162C2×C4×D3—68no—
163C2×C4×D3—128no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—64no—
167C4⋊Dic3—64no—
168C4⋊Dic3—64no—
169C4⋊Dic3—64no—
170Dic3⋊C4—24no—
171Dic3⋊C4—64no—
172Dic3⋊C4—68no—
173Dic3⋊C4—68no—
174C22×C32C22×D411yescharacteristic
175C22×C32C22×Q811yescharacteristic
176C22×C32C2×#4111yescharacteristic
177C22×C32C2×#4112yes—
178C22×C32C2×#4112yes—
179C2×C3⋊D3C22⋊Q814yes—
180C2×C3⋊D3#12314yes—
181C2×C3⋊D3#12414yes—
182C2×C3⋊D3#12518yes—
183C2×C3⋊D3#12618yes—
184C32×C4—22no—
185C32×C4—42no—
186C3⋊Dic3—22no—
187C3⋊Dic3—42no—
188C3×Dic3—24no—
189C3×Dic3—42no—
190C3×Dic3—44no—
191C23×C4—91no—
192C23×C4—91no—
193C23×C4—92no—
194C23×C4—91no—
195C23×C4—92no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4—92no—
198C2×C4⋊C4—92no—
199C2×C4⋊C4—91no—
200C2×C4⋊C4—184no—
201C2×C4⋊C4—184no—
202C2×C4⋊C4D3211yescharacteristic
203#95—34no—
204#95—32no—
205#95—34no—
206#95—94no—
207#95—92no—
208#95—94no—
209C23×C3—21no—
210C23×C3C23×D312yes—
211C22×D3—34no—
212C22×D3—34no—
213C22×D3—38no—
214C22×D3—38no—
215C22×D3—34no—
216C22×D3—62no—
217C22×D3—62no—
218C22×D3—64no—
219C22×D3—64no—
220C22×D3—62no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—28no—
226C2×C3×C4—22no—
227C2×C3×C4—34no—
228C2×C3×C4—44no—
229C2×C3×C4—62no—
230C2×C3×C4—62no—
231C2×C3×C4—62no—
232C2×C3×C4—64no—
233C2×C3×C4—64no—
234C2×C3×C4—64no—
235C2×C3×C4—64no—
236C2×C3×C4D3×D414yes—
237C2×Dic3—22no—
238C2×Dic3—22no—
239C2×Dic3—22no—
240C2×Dic3—24no—
241C2×Dic3—24no—
242C2×Dic3—24no—
243C2×Dic3—24no—
244C2×Dic3—34no—
245C2×Dic3—62no—
246C2×Dic3—64no—
247C2×Dic3—64no—
248C2×Dic3—64no—
249C2×Dic3—64no—
250C2×Dic3—68no—
251C2×Dic3—124no—
252C2×Dic3D3×Q814yes—
253C4×D3—616no—
254C4×D3—616no—
255C4×D3—616no—
256C4×D3—128no—
257C4×D3—1216no—
258C4×D3—1216no—
259C4×D3—1216no—
260C2×C32C2×C22⋊Q811yescharacteristic
261C2×C32C2×#12311yescharacteristic
262C2×C32C2×#12411yescharacteristic
263C2×C32C2×#12512yes—
264C2×C32C2×#12612yes—
265C3⋊D3#40018yescomplemented
266C24—91no—
267C22×C4—32no—
268C22×C4—34no—
269C22×C4—32no—
270C22×C4—94no—
271C22×C4—91no—
272C22×C4—94no—
273C22×C4—92no—
274C22×C4—184no—
275C22×C4—184no—
276C22×C4—188no—
277C22×C4—182no—
278C22×C4—182no—
279C22×C4—182no—
280C22×C4—184no—
281C22×C4—184no—
282C22×C4—184no—
283C22×C4—184no—
284C22×C4C2×D3212yes—
285C22×C4C2×D3211yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—64no—
288C4⋊C4—184no—
289C4⋊C4—184no—
290C4⋊C4—182no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3C2×D3×D412yes—
297C22×C3C2×D3×Q812yes—
298C22×C3D3×#4114yes—
299C22×C3D3×#4114yes—
300C22×C3D3×#4112yes—
301C2×D3—38no—
302C2×D3—38no—
303C2×D3—316no—
304C2×D3—316no—
305C2×D3—38no—
306C2×D3—64no—
307C2×D3—64no—
308C2×D3—64no—
309C2×D3—68no—
310C2×D3—68no—
311C3×C4—24no—
312C3×C4—24no—
313C3×C4—44no—
314C3×C4—44no—
315C3×C4—64no—
316C3×C4—122no—
317C3×C4—124no—
318Dic3—24no—
319Dic3—42no—
320Dic3—44no—
321Dic3—64no—
322Dic3—64no—
323Dic3—124no—
324Dic3—124no—
325C32C2×#40011yescharacteristic, complemented, Sylow
326C23—94no—
327C23—94no—
328C23—92no—
329C23—92no—
330C23—92no—
331C23C22×D3211yescharacteristic
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—34no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—64no—
339C2×C4—64no—
340C2×C4—64no—
341C2×C4—64no—
342C2×C4—62no—
343C2×C4—92no—
344C2×C4—188no—
345C2×C4—188no—
346C2×C4—182no—
347C2×C4—184no—
348C2×C4—182no—
349C2×C4—368no—
350C2×C4—364no—
351C2×C4—368no—
352C2×C4#381112yes—
353C2×C3—21no—
354C2×C3—22no—
355C2×C3—22no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3D3×C22⋊Q812yes—
359C2×C3D3×#12312yes—
360C2×C3D3×#12412yes—
361C2×C3D3×#12514yes—
362C2×C3D3×#12614yes—
363D3—316no—
364D3—68no—
365C22—94no—
366C22—98no—
367C22—98no—
368C22—94no—
369C22—94no—
370C22#6430612yes—
371C22#6431212yes—
372C22D32×Q811yescharacteristic
373C22#6431911yescharacteristic
374C22C2×#381111yescharacteristic
375C4—22no—
376C4—42no—
377C4—64no—
378C4—122no—
379C4—124no—
380C4—182no—
381C4—362no—
382C3—21no—
383C3D3×#40012yescomplemented
384C2—98no—
385C2#1059439912yes—
386C2#1059465612yes—
387C2#1059507911yescharacteristic
388C2#1059517011yescharacteristic
389C2#1059542911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, cosocleC23×C32C24
FittingC2×C32×C4⋊C4C22
prime radicalC23×C3⋊D3C23
hypercenterC2×C4⋊C4D32
hypocenterC32C2×#400
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#4985C2semidirect
7. #10591226C2semidirect
8. #10591210C2semidirect
33. #4985D3semidirect
265. C3⋊D3#400semidirect
325. C32C2×#400semidirect
383. C3D3×#400semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×C3⋊D32
C2×#638832
C3×#49852
#105913702
#105912262
#105912102
D3×#4003