Finite Group Databasebeta

← #11808840

Subgroups — #11808840

Order 1152.

1512 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#634C212yesmaximal
2C32×C4×C4⋊C4C211yesmaximal, characteristic
3#10592681C212yesmaximal
4#10592678C212yesmaximal
5C4×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
6#10592631C211yesmaximal, characteristic
7C4×#3750C212yesmaximal
8#10592356C212yesmaximal
9#10592355C212yesmaximal
10#73669—34nomaximal
11C22×#634C2212yes—
12C2×C32×C4⋊C4C2212yes—
13C2×C32×C42C2212yes—
14C2×C32×C42C2211yescharacteristic
15C2×C4⋊(C3⋊Dic3)C2212yes—
16C2×C4⋊(C3⋊Dic3)C2212yes—
17C2×#3750C2212yes—
18C2×#3750C2212yes—
19C2×#3750C2212yes—
20C2×#3750C2212yes—
21C2×C4×C3⋊Dic3C2212yes—
22C2×C4×C3⋊Dic3C2211yescharacteristic
23C2×C4×C3⋊Dic3C2211yescharacteristic
24C4×#634—28no—
25C32×#95C2211yescharacteristic
26C32×#95C2211yescharacteristic
27#63658C2212yes—
28#63658C2212yes—
29#63658C2212yes—
30#63658C2212yes—
31#63658C2211yescharacteristic
32#63658C2211yescharacteristic
33C2×C4×Dic6—38no—
34C3×C4×C4⋊C4D314yes—
35#4982—38no—
36#4979—38no—
37C4×C4⋊Dic3—34no—
38#4932—34no—
39C4×Dic3⋊C4—38no—
40#4657—38no—
41#4656—38no—
42C22×C32×C4C2312yes—
43C22×C32×C4C2311yescharacteristic
44C22×C32×C4C2311yescharacteristic
45C22×C32×C4C2312yes—
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3C2312yes—
48C22×C3⋊Dic3C2312yes—
49C22×C3⋊Dic3C2312yes—
50C22×C3⋊Dic3C2312yes—
51C2×#634—24no—
52C2×#634—24no—
53C2×#634—24no—
54C32×C4⋊C4—24no—
55C32×C42—24no—
56C32×C42Q814yescomplemented
57C4⋊(C3⋊Dic3)—24no—
58C4⋊(C3⋊Dic3)—24no—
59#3750—24no—
60#3750—24no—
61#3750—24no—
62#3750—24no—
63C4×C3⋊Dic3—24no—
64C4×C3⋊Dic3D414yes—
65C4×C3⋊Dic3Q814yescomplemented
66#2399—91noSylow
67C22×Dic6—38no—
68C2×C3×C4⋊C4C2×D318yes—
69C2×C3×C42C2×D318yes—
70C2×C3×C42C2×D314yes—
71C2×C4⋊Dic3—38no—
72C2×C4⋊Dic3—38no—
73C2×Dic3⋊C4—38no—
74C2×Dic3⋊C4—38no—
75C2×Dic3⋊C4—38no—
76C2×Dic3⋊C4—38no—
77C2×C4×Dic3—34no—
78C2×C4×Dic3—34no—
79C2×C4×Dic3—38no—
80C4×Dic6—632no—
81C3×#95C2×D314yes—
82C3×#95C2×D314yes—
83C3⋊#95—38no—
84C3⋊#95—38no—
85C3⋊#95—38no—
86C3⋊#95—38no—
87C3⋊#95—34no—
88C3⋊#95—34no—
89C23×C32C2411yescharacteristic
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4C2×D412yes—
98C2×C32×C4C2×Q812yes—
99C2×C32×C4C2×Q812yes—
100C2×C32×C4C2×Q812yes—
101C2×C32×C4C2×Q812yes—
102C2×C32×C4#4114yes—
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3C2×D414yes—
113C2×C3⋊Dic3C2×Q814yes—
114C2×C3⋊Dic3#4114yes—
115#634—216no—
116C2×C4×Q8—92no—
117#398—92no—
118#398—92no—
119#395—92no—
120#395—92no—
121C4×C4⋊C4—92no—
122C4×C4⋊C4—91no—
123C4×C4⋊C4C3⋊D311yescharacteristic
124#376—91no—
125C22×C3×C4C22×D318yes—
126C22×C3×C4C22×D314yes—
127C22×C3×C4C22×D314yes—
128C22×C3×C4C22×D318yes—
129C22×C3×C4C22×D314yes—
130C22×Dic3—38no—
131C22×Dic3—38no—
132C22×Dic3—38no—
133C22×Dic3—38no—
134C2×Dic6—616no—
135C2×Dic6—616no—
136C2×Dic6—616no—
137C3×C4⋊C4—216no—
138C3×C42—216no—
139C3×C42Dic6116yescomplemented
140C4⋊Dic3—616no—
141C4⋊Dic3—616no—
142Dic3⋊C4—616no—
143Dic3⋊C4—616no—
144Dic3⋊C4—616no—
145Dic3⋊C4—616no—
146C4×Dic3—316no—
147C4×Dic3—316no—
148C4×Dic3—616no—
149C22×C32C22×D411yescharacteristic
150C22×C32C22×Q811yescharacteristic
151C22×C32C22×Q811yescharacteristic
152C22×C32C2×#4112yes—
153C22×C32C2×#4112yes—
154C32×C4—24no—
155C32×C4—22no—
156C32×C4—22no—
157C32×C4—42no—
158C32×C4C4⋊Q814yes—
159C3⋊Dic3—24no—
160C3⋊Dic3—24no—
161C3⋊Dic3—24no—
162C3⋊Dic3—42no—
163C22×Q8—92no—
164C4×Q8—188no—
165C2×C4⋊C4—92no—
166C2×C4⋊C4—92no—
167C2×C4⋊C4—92no—
168C2×C4⋊C4—92no—
169C2×C4⋊C4—92no—
170C2×C4⋊C4—92no—
171C2×C4⋊C4C2×C3⋊D312yes—
172C2×C42—91no—
173C2×C42—91no—
174C2×C42—92no—
175C2×C42C2×C3⋊D312yes—
176C2×C42C2×C3⋊D311yescharacteristic
177#95—92no—
178#95—92no—
179#95—92no—
180#95—91no—
181#95—92no—
182#95—91no—
183#95C2×C3⋊D311yescharacteristic
184#95C2×C3⋊D311yescharacteristic
185C23×C3C23×D314yes—
186C2×C3×C4—28no—
187C2×C3×C4—28no—
188C2×C3×C4—28no—
189C2×C3×C4—28no—
190C2×C3×C4—28no—
191C2×C3×C4—28no—
192C2×C3×C4—28no—
193C2×C3×C4C2×Dic618yes—
194C2×C3×C4C2×Dic618yes—
195C2×C3×C4C2×Dic618yes—
196C2×C3×C4#235116yes—
197C2×C3×C4D3×D418yes—
198C2×C3×C4D3×Q818yes—
199C2×Dic3—316no—
200C2×Dic3—316no—
201C2×Dic3—316no—
202C2×Dic3—68no—
203C2×Dic3—68no—
204C2×Dic3—68no—
205C2×Dic3—68no—
206C2×Dic3—68no—
207C2×Dic3—68no—
208C2×Dic3—68no—
209C2×Dic3—68no—
210C2×Dic3—68no—
211Dic6—664no—
212C2×C32#52512yes—
213C2×C32C2×C4⋊Q811yescharacteristic
214C2×C32#53311yescharacteristic
215C2×C32#53311yescharacteristic
216C2×C32#53312yes—
217C2×Q8—184no—
218C2×Q8—184no—
219C2×Q8—184no—
220C22×C4—92no—
221C22×C4—92no—
222C22×C4—92no—
223C22×C4—92no—
224C22×C4C22×C3⋊D312yes—
225C22×C4C22×C3⋊D311yescharacteristic
226C22×C4C22×C3⋊D311yescharacteristic
227C22×C4C22×C3⋊D312yes—
228C22×C4C22×C3⋊D311yescharacteristic
229C4⋊C4—24no—
230C4⋊C4—184no—
231C4⋊C4—184no—
232C4⋊C4—184no—
233C4⋊C4—184no—
234C4⋊C4—184no—
235C4⋊C4—184no—
236C42—24no—
237C42—94no—
238C42—94no—
239C42—184no—
240C42#63414yescomplemented
241C22×C3C22×Dic614yes—
242C22×C3C2×#23518yes—
243C22×C3C2×D3×D414yes—
244C22×C3C2×D3×Q814yes—
245C22×C3D3×#4118yes—
246C3×C4—28no—
247C3×C4—28no—
248C3×C4—216no—
249C3×C4—48no—
250C3×C4#888116yes—
251Dic3—616no—
252Dic3—616no—
253Dic3—616no—
254Dic3—128no—
255C32#239911yescharacteristic, complemented, Sylow
256C23C23×C3⋊D311yescharacteristic
257Q8—1816no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—94no—
266C2×C4—94no—
267C2×C4—94no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4—182no—
271C2×C4—182no—
272C2×C4—182no—
273C2×C4—182no—
274C2×C4—182no—
275C2×C4—182no—
276C2×C4—182no—
277C2×C4C2×#63412yes—
278C2×C4C2×#63412yes—
279C2×C4C2×#63412yes—
280C2×C4#382814yes—
281C2×C4D4×C3⋊D312yes—
282C2×C4Q8×C3⋊D312yes—
283C2×C3#547014yes—
284C2×C3C2×#88814yes—
285C2×C3#551714yes—
286C2×C3#554718yes—
287C2×C3#557818yes—
288C22C22×#63411yescharacteristic
289C22C2×#382812yes—
290C22C2×D4×C3⋊D311yescharacteristic
291C22C2×Q8×C3⋊D311yescharacteristic
292C22#41×C3⋊D312yes—
293C4—22no—
294C4—22no—
295C4—24no—
296C4—42no—
297C4—184no—
298C4—184no—
299C4—184no—
300C4—362no—
301C4#6409714yes—
302C3#7366914yescomplemented
303C2#1059594111yescharacteristic
304C2C2×#6409711yescharacteristic
305C2#1059598811yescharacteristic
306C2#1059601812yes—
307C2#1059604912yes—
subgroupG/H
centerC22×C4C22×C3⋊D3
commutator, socle, prime radical, cosocleC23×C32C24
FrattiniC23C23×C3⋊D3
FittingC32×C4×C4⋊C4C2
hypercenterC4×C4⋊C4C3⋊D3
hypocenterC32#2399
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
56. C32×C42Q8semidirect
65. C4×C3⋊Dic3Q8semidirect
139. C3×C42Dic6semidirect
240. C42#634semidirect
255. C32#2399semidirect
302. C3#73669semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#6342
C32×C4×C4⋊C42
#105926812
#105926782
C4×C4⋊(C3⋊Dic3)2
#105926312
C4×#37502
#105923562
#105923552
#736693