Finite Group Databasebeta

← #11790746

Subgroups — #11790746

Order 1152.

1236 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64149C211yesmaximal, characteristic
2C2×C32×#102C211yesmaximal, characteristic, complemented
3C2×#63659C211yesmaximal, characteristic, complemented
4C2×#63648C211yesmaximal, characteristic, complemented
5C2×#63642C211yesmaximal, characteristic, complemented
6#10592676C211yesmaximal, characteristic, complemented
7C2×#63636C211yesmaximal, characteristic, complemented
8#70160—34nomaximal
9C22×C32×D4C2211yescharacteristic
10C22×#638—22no—
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14#64149—22no—
15#64149C414yescomplemented
16C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C4×C3⋊Dic3C2211yescharacteristic
18C22×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#102—22no—
20#63659—22no—
21#63658—22no—
22#63648—22no—
23#63642—22no—
24#63636—22no—
25C2×#940—34no—
26C2×C3×#102D314yescomplemented
27C2×#832—34no—
28C2×#821—34no—
29C2×#815—34no—
30#4977—34no—
31C2×#809—34no—
32C24×C32—21no—
33C23×C3⋊D3—21no—
34C2×C32×D4—21no—
35C2×C32×D4—21no—
36C2×C32×D4—21no—
37C2×C32×D4—21no—
38C2×C32×D4—21no—
39C2×C32×D4C2×C412yes—
40C22×C32×C4—21no—
41C22×C32×C4C2311yescharacteristic
42C2×#638—28no—
43C2×#638—44no—
44C2×#638—44no—
45C22×C3⋊Dic3—21no—
46C22×C3⋊Dic3D412yes—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)—21no—
51C2×C4⋊(C3⋊D3)—21no—
52C2×C4⋊(C3⋊D3)C2×C412yes—
53C2×C32×C8—21no—
54C2×C32×C8—21no—
55C2×C32×C8D412yescomplemented
56C32×C4⋊C4—21no—
57C32×C4⋊C4—21no—
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)—21no—
60C4×C3⋊Dic3C2×C412yes—
61C4×C3⋊Dic3D412yes—
62C2×C3⋊(C3⋊C8)—21no—
63C2×C3⋊(C3⋊C8)—21no—
64C2×C3⋊(C3⋊C8)D412yescomplemented
65#1961—91noSylow
66C22×C3×D4C2×D314yes—
67C22×#67—68no—
68C22×D12—34no—
69C22×C3×C8C2×D314yes—
70C2×C3×C4⋊C4C2×D314yes—
71#940—316no—
72#940—68no—
73C2×C4⋊Dic3—34no—
74C2×C4×Dic3—34no—
75C22×C3⋊C8—34no—
76C3×#102—28no—
77#832—68no—
78C3⋊#95—68no—
79#821—68no—
80#815—68no—
81#809—68no—
82C23×C32—22no—
83C23×C32—41no—
84C23×C32—41no—
85C23×C32—42no—
86C23×C32—42no—
87C23×C32C2×D411yescharacteristic
88C22×C3⋊D3—22no—
89C22×C3⋊D3—41no—
90C22×C3⋊D3—41no—
91C22×C3⋊D3—42no—
92C22×C3⋊D3—42no—
93C32×D4—22no—
94C32×D4—22no—
95C32×D4—41no—
96C32×D4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—42no—
99C2×C32×C4C22×C411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C32×C4C2×D411yescharacteristic
102C2×C32×C4C2×D411yescharacteristic
103C2×C32×C4#4111yescharacteristic
104C2×C32×C4#4111yescharacteristic
105#638—416no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—41no—
109C2×C3⋊Dic3—42no—
110C2×C3⋊Dic3C22⋊C414yes—
111C4⋊(C3⋊D3)—22no—
112C4⋊(C3⋊D3)—22no—
113C4⋊(C3⋊D3)—41no—
114C4⋊(C3⋊D3)—41no—
115C32×C8—22no—
116C3⋊(C3⋊C8)—22no—
117C2×#127—91no—
118C2×#102—91no—
119C2×#102—91no—
120C2×#102—91no—
121C2×#102C3⋊D311yescharacteristic, complemented
122C2×#97—91no—
123#382—91no—
124C24×C3—24no—
125C23×D3—64no—
126C2×C3×D4—24no—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—24no—
130C2×C3×D4—24no—
131C2×C3×D4C4×D318yes—
132C22×C3×C4—24no—
133C22×C3×C4C22×D314yes—
134C2×#67—632no—
135C2×#67—1216no—
136C2×#67—1216no—
137C22×Dic3—38no—
138C22×Dic3—64no—
139C2×D12—38no—
140C2×D12—64no—
141C2×D12—64no—
142C2×D12—64no—
143C2×D12—64no—
144C2×D12—64no—
145C2×C3×C8—24no—
146C2×C3×C8—24no—
147C2×C3×C8#6718yescomplemented
148C3×C4⋊C4—24no—
149C3×C4⋊C4—24no—
150C4⋊Dic3—64no—
151C4⋊Dic3—64no—
152C4×Dic3—38no—
153C4×Dic3—38no—
154C2×C3⋊C8—38no—
155C2×C3⋊C8—64no—
156C2×C3⋊C8—64no—
157C22×C32—42no—
158C22×C32—44no—
159C22×C32—82no—
160C22×C32—81no—
161C22×C32—81no—
162C22×C32C2×C22⋊C411yescharacteristic
163C22×C32#12411yescharacteristic
164C22×C32#12711yescharacteristic
165C22×C32AGL(1,ℤ/8)12yes—
166C22×C32AGL(1,ℤ/8)12yes—
167C2×C3⋊D3—42no—
168C2×C3⋊D3—44no—
169C2×C3⋊D3—82no—
170C2×C3⋊D3—81no—
171C2×C3⋊D3—81no—
172C32×C4—81no—
173C32×C4C4×D411yescharacteristic
174C32×C4C4×D411yescharacteristic
175C32×C4#12111yescharacteristic
176C32×C4#12111yescharacteristic
177C3⋊Dic3—24no—
178C3⋊Dic3—81no—
179C22×D4—91no—
180C22×D4—182no—
181C22×D4C2×C3⋊D311yescharacteristic
182C22×C8—91no—
183C22×C8C2×C3⋊D311yescharacteristic
184#127—94no—
185#127—182no—
186C2×C4⋊C4—91no—
187C2×C4⋊C4C2×C3⋊D311yescharacteristic
188C2×C42—91no—
189#102—22no—
190#102—182no—
191#102—182no—
192#102—182no—
193#97—182no—
194#95—182no—
195C23×C3—28no—
196C23×C3—44no—
197C23×C3—48no—
198C23×C3—48no—
199C23×C3—44no—
200C23×C3D3×D414yes—
201C22×D3—68no—
202C22×D3—124no—
203C22×D3—128no—
204C22×D3—128no—
205C22×D3—124no—
206C3×D4—28no—
207C3×D4—28no—
208C3×D4—44no—
209C3×D4—44no—
210C2×C3×C4—48no—
211C2×C3×C4—44no—
212C2×C3×C4C2×C4×D314yes—
213C2×C3×C4#23514yes—
214C2×C3×C4D3×D414yes—
215C2×C3×C4D3×D414yes—
216C2×C3×C4#23714yes—
217C2×C3×C4C2×#6714yes—
218#67—1264no—
219C2×Dic3—316no—
220C2×Dic3—68no—
221C2×Dic3—68no—
222C2×Dic3—128no—
223C2×Dic3—124no—
224D12—68no—
225D12—68no—
226D12—124no—
227D12—124no—
228C3×C8—28no—
229C3⋊C8—68no—
230C2×C32—82no—
231C2×C32#39011yescharacteristic
232C2×C32#41812yes—
233C2×C32#49012yes—
234C2×C32#49611yescharacteristic
235C2×C32#49611yescharacteristic
236C3⋊D3—82no—
237C24—21no—
238C24—181no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—92no—
245C2×D4—188no—
246C2×D4—181no—
247C2×D4—181no—
248C2×D4—181no—
249C2×D4—181no—
250C2×D4—181no—
251C2×D4—364no—
252C2×D4—364no—
253C2×D4C4×C3⋊D312yes—
254C22×C4—21no—
255C22×C4—92no—
256C22×C4—181no—
257C22×C4C22×C3⋊D311yescharacteristic
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—92no—
261C2×C8—181no—
262C2×C8—181no—
263C2×C8#63812yescomplemented
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—181no—
267C4⋊C4—181no—
268C42—92no—
269C42—92no—
270C22×C3—48no—
271C22×C3—416no—
272C22×C3—84no—
273C22×C3—84no—
274C22×C3—88no—
275C22×C3D3×C22⋊C414yes—
276C22×C3#88514yes—
277C22×C3#91118yes—
278C22×C3#91418yes—
279C22×C3#94014yes—
280C2×D3—128no—
281C2×D3—1216no—
282C2×D3—244no—
283C2×D3—244no—
284C2×D3—248no—
285C3×C4—84no—
286C3×C4#67⋊C414yes—
287C3×C4#88414yes—
288C3×C4C4×#6714yes—
289C3×C4#93914yes—
290Dic3—616no—
291Dic3—244no—
292C32#196111yescharacteristic, complemented, Sylow
293C23—22no—
294C23—41no—
295C23—42no—
296C23—42no—
297C23—41no—
298C23—182no—
299C23—361no—
300C23—362no—
301C23—362no—
302C23—361no—
303C23D4×C3⋊D311yescharacteristic
304D4—22no—
305D4—22no—
306D4—41no—
307D4—41no—
308D4—182no—
309D4—182no—
310D4—3616no—
311D4—361no—
312D4—361no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—94no—
316C2×C4—182no—
317C2×C4—182no—
318C2×C4—362no—
319C2×C4—361no—
320C2×C4C2×C4×C3⋊D311yescharacteristic
321C2×C4#382811yescharacteristic
322C2×C4D4×C3⋊D311yescharacteristic
323C2×C4D4×C3⋊D311yescharacteristic
324C2×C4#383011yescharacteristic
325C2×C4C2×#63811yescharacteristic
326C8—22no—
327C8—182no—
328C2×C3—88no—
329C2×C3#476418yes—
330C2×C3#477018yes—
331C2×C3#495714yes—
332C2×C3#515414yes—
333C2×C3#517614yes—
334D3—248no—
335C22—42no—
336C22—44no—
337C22—81no—
338C22—81no—
339C22—82no—
340C22—362no—
341C22—364no—
342C22—721no—
343C22—721no—
344C22—722no—
345C22C22⋊C4×C3⋊D311yescharacteristic
346C22#6409411yescharacteristic
347C22#6412012yes—
348C22#6412312yes—
349C22#6414911yescharacteristic
350C4—81no—
351C4—184no—
352C4—721no—
353C4#6409011yescharacteristic
354C4#6409311yescharacteristic
355C4C4×#63811yescharacteristic
356C4#6414811yescharacteristic
357C3#7016014yescomplemented
358C2—82no—
359C2—722no—
360C2#1059246312yes—
361C2#1059246912yes—
362C2#1059265611yescharacteristic
363C2#1059285311yescharacteristic
364C2#1059287511yescharacteristic
subgroupG/H
centerC23D4×C3⋊D3
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC2×C32×#102C2
socleC23×C32C2×D4
prime radicalC2×#64149C2
cosocleC22×C32×C4C23
hypercenterC2×#102C3⋊D3
hypocenterC32#1961
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. C2×C32×#102C2semidirect
3. C2×#63659C2semidirect
4. C2×#63648C2semidirect
5. C2×#63642C2semidirect
6. #10592676C2semidirect
7. C2×#63636C2semidirect
15. #64149C4semidirect
26. C2×C3×#102D3semidirect
55. C2×C32×C8D4semidirect
64. C2×C3⋊(C3⋊C8)D4semidirect
121. C2×#102C3⋊D3semidirect
147. C2×C3×C8#67semidirect
263. C2×C8#638semidirect
292. C32#1961semidirect
357. C3#70160semidirect

Maximal subgroups

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

subgroupindex
C2×#641492
C2×C32×#1022
C2×#636592
C2×#636482
C2×#636422
#105926762
C2×#636362
#701603