Finite Group Databasebeta

← #11775110

Subgroups — #11775110

Order 1152.

1002 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#525C211yesmaximal, characteristic
2C2×#63658C211yesmaximal, characteristic, complemented
3#10592906C211yesmaximal, characteristic, complemented
4#10589404C212yesmaximal, complemented
5#10589403C212yesmaximal, complemented
6#71708—34nomaximal
7C2×C32×#41C411yescharacteristic, complemented
8C23×C32×C4C2211yescharacteristic
9C23×C3⋊Dic3—21no—
10C32×#123—21no—
11C32×C22⋊Q8C411yescharacteristic, complemented
12C32×C22⋊Q8C2211yescharacteristic
13C32×#121C411yescharacteristic, complemented
14C32×#121C2211yescharacteristic
15C32×#120—21no—
16C32×C4×D4—21no—
17C32×C4×D4—21no—
18C32×#117C411yescharacteristic, complemented
19C2×#3748—21no—
20#63658—21no—
21#63658C2212yes—
22#63657—21no—
23#63657C2212yes—
24C3×#525D314yes—
25C2×C3⋊#95—34no—
26#5207—34no—
27#4538—38no—
28#4537—38no—
29C24×C32D411yescharacteristic
30C32×#41—42no—
31C2×C32×Q8C2×C411yescharacteristic, complemented
32C2×C32×D4—21no—
33C2×C32×D4—21no—
34C2×C32×D4—21no—
35C2×C32×D4C2×C411yescharacteristic, complemented
36C22×C32×C4—21no—
37C22×C32×C4—42no—
38C22×C32×C4C2×C411yescharacteristic
39C22×C32×C4C2×C411yescharacteristic
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4D411yescharacteristic
43C22×C32×C4C2311yescharacteristic
44C22×C3⋊Dic3—24no—
45C22×C3⋊Dic3—22no—
46C22×C3⋊Dic3—42no—
47C22×C3⋊Dic3—42no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4C2×C411yescharacteristic, complemented
51C32×C4⋊C4C2×C411yescharacteristic, complemented
52C32×C22⋊C4—21no—
53C32×C22⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C22⋊C4—21no—
56C32×C22⋊C4—21no—
57C32×C42—21no—
58#3748—42no—
59C2×C3⋊(C3⋊C8)—22no—
60#1504—91noSylow
61C2×C3×#41Dic314yescomplemented
62C23×C3×C4C2×D314yes—
63C23×Dic3—64no—
64C3×#123—24no—
65C3×C22⋊Q8Dic314yescomplemented
66C3×C22⋊Q8C2×D314yes—
67C3×#121Dic314yescomplemented
68C3×#121C2×D314yes—
69C3×#120—24no—
70C3×C4×D4—24no—
71C3×C4×D4—24no—
72C3×#117Dic314yescomplemented
73C2×#208—64no—
74C3⋊#95—38no—
75C3⋊#95—64no—
76#830—38no—
77#830—64no—
78C23×C32—22no—
79C23×C32—22no—
80C23×C32—21no—
81C23×C32—21no—
82C23×C32—21no—
83C23×C32C22⋊C411yescharacteristic
84C23×C32C22⋊C411yescharacteristic
85C23×C32C2×D411yescharacteristic
86C32×Q8—41no—
87C32×D4—41no—
88C32×D4—42no—
89C32×D4—42no—
90C32×D4—41no—
91C32×D4—41no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—42no—
102C2×C32×C4C22⋊C411yescharacteristic
103C2×C32×C4C22⋊C411yescharacteristic
104C2×C32×C4C22×C411yescharacteristic
105C2×C32×C4C2×D411yescharacteristic
106C2×C3⋊Dic3—24no—
107C2×C3⋊Dic3—48no—
108C2×C3⋊Dic3—42no—
109C2×C3⋊Dic3—42no—
110C3⋊(C3⋊C8)—42no—
111#525C3⋊D311yescharacteristic
112#407—91no—
113C2×#95—91no—
114#328—92no—
115#327—92no—
116C24×C3#6714yes—
117C3×#41—48no—
118C2×C3×Q8C2×Dic314yescomplemented
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—24no—
122C2×C3×D4C2×Dic314yescomplemented
123C22×C3×C4—24no—
124C22×C3×C4—48no—
125C22×C3×C4C2×Dic314yes—
126C22×C3×C4C2×Dic314yes—
127C22×C3×C4#6714yes—
128C22×C3×C4#6714yes—
129C22×C3×C4#6714yes—
130C22×C3×C4C22×D314yes—
131C22×Dic3—616no—
132C22×Dic3—68no—
133C22×Dic3—128no—
134C22×Dic3—128no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—24no—
137C3×C4⋊C4C2×Dic314yescomplemented
138C3×C4⋊C4C2×Dic314yescomplemented
139C3×C22⋊C4—24no—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—24no—
144C3×C42—24no—
145#208—128no—
146C2×C3⋊C8—68no—
147C22×C32—24no—
148C22×C32—24no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—41no—
156C22×C32—41no—
157C22×C32—41no—
158C22×C32#9912yes—
159C22×C32#10412yes—
160C22×C32C2×C22⋊C411yescharacteristic
161C22×C32AGL(1,ℤ/8)11yescharacteristic
162C22×C32#13711yescharacteristic
163C32×C4—21no—
164C32×C4—21no—
165C32×C4—21no—
166C32×C4—21no—
167C32×C4—41no—
168C32×C4—41no—
169C32×C4—41no—
170C3⋊Dic3—44no—
171C2×#41C3⋊Dic311yescharacteristic, complemented
172C23×C4—181no—
173C23×C4C2×C3⋊D311yescharacteristic
174C2×M16—181no—
175#123—21no—
176C22⋊Q8C3⋊Dic311yescharacteristic, complemented
177C22⋊Q8C2×C3⋊D311yescharacteristic
178#121C3⋊Dic311yescharacteristic, complemented
179#121C2×C3⋊D311yescharacteristic
180#120—21no—
181C4×D4—21no—
182C4×D4—21no—
183#117C3⋊Dic311yescharacteristic, complemented
184C22⋊C8—92no—
185C22⋊C8—181no—
186#95—92no—
187#95—181no—
188C23×C3—28no—
189C23×C3—28no—
190C23×C3—24no—
191C23×C3—24no—
192C23×C3—24no—
193C23×C3#21714yes—
194C23×C3#21714yes—
195C23×C3C2×#6714yes—
196C3×Q8—44no—
197C3×D4—44no—
198C3×D4—48no—
199C3×D4—48no—
200C3×D4—44no—
201C3×D4—44no—
202C2×C3×C4—28no—
203C2×C3×C4—28no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—44no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—48no—
212C2×C3×C4#21714yes—
213C2×C3×C4#21714yes—
214C2×C3×C4C22×Dic314yes—
215C2×C3×C4C2×#6714yes—
216C2×Dic3—616no—
217C2×Dic3—1232no—
218C2×Dic3—128no—
219C2×Dic3—128no—
220C3⋊C8—128no—
221C2×C32—21no—
222C2×C32—22no—
223C2×C32—22no—
224C2×C32—21no—
225C2×C32—81no—
226C2×C32C2×#9911yescharacteristic
227C2×C32#41711yescharacteristic
228C2×C32C2×#10411yescharacteristic
229C24#63811yescharacteristic
230#41—42no—
231C2×Q8C2×C3⋊Dic311yescharacteristic, complemented
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4C2×C3⋊Dic311yescharacteristic, complemented
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—184no—
239C22×C4—182no—
240C22×C4—362no—
241C22×C4—362no—
242C22×C4C2×C3⋊Dic311yescharacteristic
243C22×C4C2×C3⋊Dic311yescharacteristic
244C22×C4#63811yescharacteristic
245C22×C4#63811yescharacteristic
246C22×C4#63811yescharacteristic
247C22×C4C22×C3⋊D311yescharacteristic
248M16—362no—
249C2×C8—182no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4C2×C3⋊Dic311yescharacteristic, complemented
253C4⋊C4C2×C3⋊Dic311yescharacteristic, complemented
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C42—21no—
260C22×C3—216no—
261C22×C3—216no—
262C22×C3—24no—
263C22×C3—24no—
264C22×C3—24no—
265C22×C3—24no—
266C22×C3—24no—
267C22×C3—24no—
268C22×C3—44no—
269C22×C3—44no—
270C22×C3—44no—
271C22×C3#83418yes—
272C22×C3#83718yes—
273C22×C3#94914yes—
274C22×C3#95114yes—
275C22×C3C2×#21714yes—
276C3×C4—24no—
277C3×C4—24no—
278C3×C4—24no—
279C3×C4—24no—
280C3×C4—44no—
281C3×C4—44no—
282C3×C4—44no—
283Dic3—1216no—
284C32#150411yescharacteristic, complemented, Sylow
285C23—21no—
286C23—21no—
287C23—22no—
288C23—22no—
289C23—21no—
290C23#375711yescharacteristic
291C23#375711yescharacteristic
292C23C2×#63811yescharacteristic
293Q8—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—42no—
298D4—42no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—184no—
310C2×C4—368no—
311C2×C4—362no—
312C2×C4—362no—
313C2×C4#375711yescharacteristic
314C2×C4#375711yescharacteristic
315C2×C4C22×C3⋊Dic311yescharacteristic
316C2×C4C2×#63811yescharacteristic
317C8—362no—
318C2×C3—24no—
319C2×C3—28no—
320C2×C3—28no—
321C2×C3—24no—
322C2×C3—84no—
323C2×C3C2×#83414yes—
324C2×C3#523214yes—
325C2×C3C2×#83714yes—
326C22—24no—
327C22—24no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22#6366112yes—
338C22#6366412yes—
339C22#6415811yescharacteristic
340C22#6416011yescharacteristic
341C22C2×#375711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—364no—
350C3#7170814yescomplemented
351C2—21no—
352C2—21no—
353C2—22no—
354C2—22no—
355C2—81no—
356C2C2×#6366111yescharacteristic
357C2#1059293111yescharacteristic
358C2C2×#6366411yescharacteristic
subgroupG/H
centerC22C2×#3757
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
Fitting, prime radicalC32×#525C2
socleC22×C32C2×C22⋊C4
cosocleC22×C32×C4C23
hypercenter#525C3⋊D3
hypocenterC32#1504
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
2. C2×#63658C2semidirect
3. #10592906C2semidirect
4. #10589404C2semidirect
5. #10589403C2semidirect
7. C2×C32×#41C4semidirect
11. C32×C22⋊Q8C4semidirect
13. C32×#121C4semidirect
18. C32×#117C4semidirect
31. C2×C32×Q8C2×C4semidirect
35. C2×C32×D4C2×C4semidirect
50. C32×C4⋊C4C2×C4semidirect
51. C32×C4⋊C4C2×C4semidirect
61. C2×C3×#41Dic3semidirect
65. C3×C22⋊Q8Dic3semidirect
67. C3×#121Dic3semidirect
72. C3×#117Dic3semidirect
118. C2×C3×Q8C2×Dic3semidirect
122. C2×C3×D4C2×Dic3semidirect
137. C3×C4⋊C4C2×Dic3semidirect
138. C3×C4⋊C4C2×Dic3semidirect
171. C2×#41C3⋊Dic3semidirect
176. C22⋊Q8C3⋊Dic3semidirect
178. #121C3⋊Dic3semidirect
183. #117C3⋊Dic3semidirect
231. C2×Q8C2×C3⋊Dic3semidirect
235. C2×D4C2×C3⋊Dic3semidirect
252. C4⋊C4C2×C3⋊Dic3semidirect
253. C4⋊C4C2×C3⋊Dic3semidirect
284. C32#1504semidirect
350. C3#71708semidirect

Maximal subgroups

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

subgroupindex
C32×#5252
C2×#636582
#105929062
#105894042
#105894032
#717083