Finite Group Databasebeta

← #11808780

Subgroups — #11808780

Order 1152.

1704 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊Dic3C211yesmaximal, characteristic
2C2×#64096C211yesmaximal, characteristic, complemented
3C32×#394C211yesmaximal, characteristic
4#10592921C211yesmaximal, characteristic
5#10592685C211yesmaximal, characteristic, complemented
6#10592651C212yesmaximal, complemented
7#10592646C212yesmaximal, complemented
8C22⋊C4×C3⋊Dic3C212yesmaximal, complemented
9C22⋊C4×C3⋊Dic3C212yesmaximal, complemented
10#10592358C211yesmaximal, characteristic, complemented
11#10592347C211yesmaximal, characteristic, complemented
12#76893—34nomaximal
13C22×C32×D4C2211yescharacteristic
14C23×C3⋊Dic3C2212yes—
15C22×#634C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C22⋊C4C2212yes—
18C2×C32×C22⋊C4C2212yes—
19C2×#3757C2212yes—
20C2×#3757C2212yes—
21C2×#3757C2212yes—
22D4×C3⋊Dic3—24no—
23C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C2×#3750C2211yescharacteristic
25C2×#3750C2211yescharacteristic
26C2×C4×C3⋊Dic3C2212yes—
27C2×C4×C3⋊Dic3C2211yescharacteristic
28C2×C4×C3⋊Dic3C2211yescharacteristic
29C2×C4×C3⋊Dic3C2211yescharacteristic
30#64096—24no—
31C32×#95C2211yescharacteristic
32#63658C2212yes—
33#63658C2211yescharacteristic
34#63658C2212yes—
35#63658C2212yes—
36#63658C2212yes—
37#63658C2212yes—
38C2×D4×Dic3—34no—
39C2×#887—34no—
40C3×#394D314yes—
41#5222—34no—
42#4986—34no—
43#4952—38no—
44#4947—38no—
45Dic3×C22⋊C4—38no—
46Dic3×C22⋊C4—38no—
47#4659—34no—
48#4648—34no—
49C24×C32C2312yes—
50C2×C32×D4—24no—
51C2×C32×D4—22no—
52C22×C32×C4C2311yescharacteristic
53C22×C32×C4C2312yes—
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2311yescharacteristic
56C22×C3⋊Dic3—24no—
57C22×C3⋊Dic3—24no—
58C22×C3⋊Dic3—24no—
59C22×C3⋊Dic3C2312yes—
60C22×C3⋊Dic3C2311yescharacteristic
61C22×C3⋊Dic3C2312yes—
62C22×C3⋊Dic3C2311yescharacteristic
63C22×C3⋊Dic3C2311yescharacteristic
64C22×C3⋊Dic3C2311yescharacteristic
65C2×#634—22no—
66C2×#634—22no—
67C2×#634—22no—
68C32×C4⋊C4—22no—
69C32×C22⋊C4—24no—
70C32×C22⋊C4—24no—
71#3757—24no—
72#3757—24no—
73#3757—24no—
74C4⋊(C3⋊Dic3)—22no—
75#3750—22no—
76#3750—22no—
77C4×C3⋊Dic3—22no—
78C4×C3⋊Dic3—22no—
79C4×C3⋊Dic3—24no—
80C4×C3⋊Dic3D414yes—
81#2398—91noSylow
82C22×C3×D4C2×D314yes—
83C23×Dic3—38no—
84C22×Dic6—34no—
85C2×C3×C4⋊C4C2×D314yes—
86C2×C3×C22⋊C4C2×D318yes—
87C2×C3×C22⋊C4C2×D318yes—
88C2×#217—38no—
89C2×#217—38no—
90C2×#217—38no—
91D4×Dic3—616no—
92C2×C4⋊Dic3—34no—
93C2×Dic3⋊C4—34no—
94C2×Dic3⋊C4—34no—
95C2×C4×Dic3—34no—
96C2×C4×Dic3—38no—
97C2×C4×Dic3—34no—
98C2×C4×Dic3—34no—
99#887—616no—
100C3×#95C2×D314yes—
101C3⋊#95—38no—
102C3⋊#95—34no—
103C3⋊#95—38no—
104C3⋊#95—38no—
105C3⋊#95—38no—
106C3⋊#95—38no—
107C23×C32—22no—
108C23×C32—22no—
109C23×C32—22no—
110C23×C32—22no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32#4114yes—
114C23×C32C2411yescharacteristic
115C32×D4—28no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4C2×D412yes—
125C2×C32×C4#4112yes—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—22no—
134C2×C3⋊Dic3—22no—
135C2×C3⋊Dic3—22no—
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3—48no—
139C2×C3⋊Dic3C2×D412yes—
140C2×C3⋊Dic3C2×D412yes—
141C2×C3⋊Dic3#4112yes—
142C2×C3⋊Dic3#4114yes—
143#634—28no—
144C2×C4×Q8—91no—
145C2×C4×D4—91no—
146#400—91no—
147#400—91no—
148#397—92no—
149#394—91no—
150#394C3⋊D311yescharacteristic
151#393—92no—
152C4×C22⋊C4—92no—
153C4×C22⋊C4—92no—
154#376—91no—
155C24×C3C22×D318yes—
156C2×C3×D4—216no—
157C2×C3×D4—28no—
158C22×C3×C4C22×D318yes—
159C22×C3×C4C22×D314yes—
160C22×C3×C4C22×D314yes—
161C22×C3×C4C22×D314yes—
162C22×Dic3—34no—
163C22×Dic3—34no—
164C22×Dic3—34no—
165C22×Dic3—38no—
166C22×Dic3—38no—
167C22×Dic3—34no—
168C22×Dic3—616no—
169C22×Dic3—616no—
170C22×Dic3—616no—
171C2×Dic6—68no—
172C2×Dic6—68no—
173C2×Dic6—68no—
174C3×C4⋊C4—28no—
175C3×C22⋊C4—216no—
176C3×C22⋊C4—216no—
177#217—616no—
178#217—616no—
179#217—616no—
180C4⋊Dic3—68no—
181Dic3⋊C4—68no—
182Dic3⋊C4—68no—
183C4×Dic3—316no—
184C4×Dic3—68no—
185C4×Dic3—68no—
186C4×Dic3—616no—
187C22×C32—24no—
188C22×C32—24no—
189C22×C32—24no—
190C22×C32—42no—
191C22×C32—42no—
192C22×C32—42no—
193C22×C32—42no—
194C22×C32C22×D411yescharacteristic
195C22×C32C2×#4112yes—
196C22×C32C2×#4111yescharacteristic
197C22×C32C2×#4111yescharacteristic
198C22×C32C2×#4112yes—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—41no—
202C32×C4—42no—
203C3⋊Dic3—22no—
204C3⋊Dic3—22no—
205C3⋊Dic3—41no—
206C3⋊Dic3—42no—
207C3⋊Dic3—42no—
208C3⋊Dic3#12414yes—
209C22×Q8—91no—
210C22×D4C2×C3⋊D311yescharacteristic
211C23×C4—92no—
212C4×Q8—184no—
213C4×D4—184no—
214C2×C4⋊C4—91no—
215C2×C4⋊C4—91no—
216C2×C4⋊C4—91no—
217C2×C4⋊C4C2×C3⋊D311yescharacteristic
218C2×C22⋊C4—92no—
219C2×C22⋊C4—92no—
220C2×C22⋊C4—92no—
221C2×C22⋊C4C2×C3⋊D312yes—
222C2×C22⋊C4C2×C3⋊D312yes—
223C2×C42—91no—
224C2×C42—91no—
225C2×C42—91no—
226C2×C42—92no—
227#95—92no—
228#95—91no—
229#95—92no—
230#95—92no—
231#95—92no—
232#95—92no—
233#95C2×C3⋊D311yescharacteristic
234C23×C3—28no—
235C23×C3—28no—
236C23×C3—28no—
237C23×C3—28no—
238C23×C3—28no—
239C23×C3—28no—
240C23×C3#237116yes—
241C23×C3C23×D314yes—
242C3×D4—232no—
243C2×C3×C4—24no—
244C2×C3×C4—28no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—28no—
248C2×C3×C4—28no—
249C2×C3×C4—28no—
250C2×C3×C4—28no—
251C2×C3×C4D3×D418yes—
252C2×C3×C4#23718yes—
253C2×Dic3—38no—
254C2×Dic3—316no—
255C2×Dic3—38no—
256C2×Dic3—38no—
257C2×Dic3—64no—
258C2×Dic3—64no—
259C2×Dic3—64no—
260C2×Dic3—64no—
261C2×Dic3—64no—
262C2×Dic3—68no—
263C2×Dic3—68no—
264C2×Dic3—68no—
265C2×Dic3—68no—
266C2×Dic3—68no—
267C2×Dic3—68no—
268C2×Dic3—68no—
269C2×Dic3—1232no—
270Dic6—632no—
271C2×C32—44no—
272C2×C32#52511yescharacteristic
273C2×C32#52511yescharacteristic
274C2×C32C2×#12411yescharacteristic
275C2×C32#52912yes—
276C2×C32#52912yes—
277C24C22×C3⋊D312yes—
278C2×Q8—182no—
279C2×Q8—182no—
280C2×Q8—182no—
281C2×D4—24no—
282C2×D4—22no—
283C22×C4—91no—
284C22×C4—91no—
285C22×C4—91no—
286C22×C4—92no—
287C22×C4—92no—
288C22×C4—91no—
289C22×C4—184no—
290C22×C4—184no—
291C22×C4—184no—
292C22×C4C22×C3⋊D312yes—
293C22×C4C22×C3⋊D311yescharacteristic
294C22×C4C22×C3⋊D311yescharacteristic
295C22×C4C22×C3⋊D311yescharacteristic
296C4⋊C4—22no—
297C4⋊C4—182no—
298C4⋊C4—182no—
299C4⋊C4—182no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—184no—
303C22⋊C4—184no—
304C22⋊C4—184no—
305C42—94no—
306C42—182no—
307C42—182no—
308C42—184no—
309C22×C3—216no—
310C22×C3—216no—
311C22×C3—216no—
312C22×C3—48no—
313C22×C3—48no—
314C22×C3—48no—
315C22×C3—48no—
316C22×C3C2×D3×D414yes—
317C22×C3C2×#23718yes—
318C22×C3C2×#23714yes—
319C22×C3D3×#4114yes—
320C22×C3D3×#4118yes—
321C3×C4—28no—
322C3×C4—28no—
323C3×C4—44no—
324C3×C4—48no—
325Dic3—316no—
326Dic3—68no—
327Dic3—68no—
328Dic3—128no—
329Dic3—128no—
330Dic3—124no—
331C32#239811yescharacteristic, complemented, Sylow
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23#383014yes—
339C23C23×C3⋊D311yescharacteristic
340Q8—188no—
341D4—28no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—92no—
351C2×C4—94no—
352C2×C4—92no—
353C2×C4—92no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4—181no—
358C2×C4—181no—
359C2×C4—182no—
360C2×C4—182no—
361C2×C4—182no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4—182no—
365C2×C4—182no—
366C2×C4—368no—
367C2×C4D4×C3⋊D312yes—
368C2×C4#383012yes—
369C2×C3—416no—
370C2×C3#558614yes—
371C2×C3#560018yes—
372C2×C3#560614yes—
373C2×C3#564918yes—
374C2×C3D3×#12414yes—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22C2×D4×C3⋊D311yescharacteristic
383C22C2×#383012yes—
384C22C2×#383011yescharacteristic
385C22#41×C3⋊D311yescharacteristic
386C22#41×C3⋊D312yes—
387C4—22no—
388C4—22no—
389C4—41no—
390C4—42no—
391C4—94no—
392C4—182no—
393C4—182no—
394C4—362no—
395C4—362no—
396C4—361no—
397C3#7689314yescomplemented
398C2—44no—
399C2#1059605711yescharacteristic
400C2#1059607112yes—
401C2#1059607711yescharacteristic
402C2#1059612012yes—
403C2C3⋊D3×#12411yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutator, socle, cosocleC23×C32C24
FittingC32×#394C2
prime radicalC22×C32×D4C22
hypercenter#394C3⋊D3
hypocenterC32#2398
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#64096C2semidirect
5. #10592685C2semidirect
6. #10592651C2semidirect
7. #10592646C2semidirect
8. C22⋊C4×C3⋊Dic3C2semidirect
9. C22⋊C4×C3⋊Dic3C2semidirect
10. #10592358C2semidirect
11. #10592347C2semidirect
331. C32#2398semidirect
397. C3#76893semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊Dic32
C2×#640962
C32×#3942
#105929212
#105926852
#105926512
#105926462
C22⋊C4×C3⋊Dic32
#105923582
#105923472
#768933