Finite Group Databasebeta

← #11783822

Subgroups — #11783822

Order 1152.

852 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64116C211yesmaximal, characteristic
2#10592831C211yesmaximal, characteristic
3#10592821C211yesmaximal, characteristic, complemented
4#10592705C211yesmaximal, characteristic
5#10592696C211yesmaximal, characteristic, complemented
6C32×#344C211yesmaximal, characteristic, complemented
7#10589422C211yesmaximal, characteristic, complemented
8#69014—34nomaximal
9C2×#3828C2211yescharacteristic
10C2×C32×M16C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C32×#117C2211yescharacteristic
13#64135C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C22×C3⋊(C3⋊C8)C2211yescharacteristic
16#64116—21no—
17#64116—21no—
18#64116—21no—
19#64116—21no—
20C2×#3743C412yescomplemented
21C2×C8×C3⋊D3C412yescomplemented
22#63651C412yescomplemented
23#63650C412yescomplemented
24#63642—21no—
25C8×C3⋊Dic3—21no—
26#63633C414yes—
27#63630—21no—
28C4×C3⋊(C3⋊C8)—21no—
29C2×#907—34no—
30#5132—34no—
31#5122—34no—
32#5006—34no—
33#4997—34no—
34C3×#344D314yescomplemented
35#4556—34no—
36C22×C32×C4C2311yescharacteristic
37C2×#638C2×C412yes—
38#3828—21no—
39#3828—21no—
40#3828—21no—
41#3828—21no—
42C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
43C2×C4×C3⋊D3C2×C412yes—
44C2×#634C2×C411yescharacteristic
45C32×M16—21no—
46C32×M16—21no—
47C32×M16—21no—
48C32×M16C2×C412yes—
49C2×C32×C8—21no—
50C2×C32×C8C2×C412yes—
51C32×C4⋊C4—21no—
52C32×C22⋊C4C2×C412yes—
53C32×C42—21no—
54#3757C2×C412yes—
55C4⋊(C3⋊Dic3)—21no—
56C4×C3⋊Dic3—21no—
57#3748—21no—
58#3748C2×C412yes—
59C2×C3⋊(C3⋊C8)—21no—
60C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
61C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
62C2×C3⋊(C3⋊C8)D411yescharacteristic
63C2×C3⋊(C3⋊C8)D411yescharacteristic
64C2×C3⋊(C3⋊C8)D411yescharacteristic
65C2×C3⋊(C3⋊C8)D411yescharacteristic
66#3743—22no—
67#3743—22no—
68C8×C3⋊D3—22no—
69C8×C3⋊D3—22no—
70#1757—91noSylow
71C2×#235—34no—
72C2×C3×M16C2×D314yes—
73C2×C3×M16C2×D314yes—
74C3×#117C2×D314yes—
75#926—34no—
76C2×#208—34no—
77C22×C3⋊C8—34no—
78#907—64no—
79#907—64no—
80#907—64no—
81#907—64no—
82C2×ΓL(1,𝔽25)—38no—
83C2×D3×C8—38no—
84#824—38no—
85#823—38no—
86#815—64no—
87C8×Dic3—64no—
88#806—316no—
89#803—64no—
90C4×C3⋊C8—64no—
91C23×C32C22×C411yescharacteristic
92C22×C3⋊D3C4212yes—
93C2×C32×C4—22no—
94C2×C32×C4C22×C411yescharacteristic
95C2×C32×C4C22×C411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4#4111yescharacteristic
99C2×C32×C4#4111yescharacteristic
100#638—22no—
101#638—22no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3C4212yes—
104C4⋊(C3⋊D3)—21no—
105C4⋊(C3⋊D3)—21no—
106C4×C3⋊D3—22no—
107C4×C3⋊D3—22no—
108#634—21no—
109#634—21no—
110C32×C8—22no—
111C32×C8—22no—
112C3⋊(C3⋊C8)—22no—
113C3⋊(C3⋊C8)C22⋊C411yescharacteristic
114C3⋊(C3⋊C8)C22⋊C411yescharacteristic
115C3⋊(C3⋊C8)C22⋊C411yescharacteristic
116C3⋊(C3⋊C8)C22⋊C411yescharacteristic
117C2×#131—91no—
118#413—91no—
119#410—91no—
120#405—91no—
121#405—91no—
122#344—91no—
123#344C3⋊D311yescharacteristic, complemented
124C22×C3×C4C22×D314yes—
125C2×#67—38no—
126#235—64no—
127#235—64no—
128#235—64no—
129#235—64no—
130C2×D12—34no—
131C2×C4×D3—38no—
132C2×Dic6—34no—
133C3×M16—24no—
134C3×M16—24no—
135C3×M16—24no—
136C3×M16C4×D318yes—
137C2×C3×C8—24no—
138C2×C3×C8C4×D318yes—
139C3×C4⋊C4—24no—
140C3×C22⋊C4C4×D318yes—
141C3×C42—24no—
142#217—38no—
143C4⋊Dic3—64no—
144C4×Dic3—64no—
145#208—38no—
146#208—64no—
147C2×C3⋊C8—34no—
148C2×C3⋊C8—34no—
149C2×C3⋊C8—34no—
150C2×C3⋊C8—34no—
151C2×C3⋊C8—34no—
152C2×C3⋊C8—34no—
153C2×C3⋊C8—64no—
154ΓL(1,𝔽25)—68no—
155ΓL(1,𝔽25)—68no—
156D3×C8—68no—
157D3×C8—68no—
158C22×C32—41no—
159C22×C32C2×C4211yescharacteristic
160C22×C32C4×D411yescharacteristic
161C22×C32C4×D411yescharacteristic
162C2×C3⋊D3—22no—
163C2×C3⋊D3—42no—
164C32×C4—42no—
165C32×C4C2×C22⋊C411yescharacteristic
166C32×C4#11711yescharacteristic
167C32×C4C4×D411yescharacteristic
168C32×C4C4×D411yescharacteristic
169C3⋊Dic3—22no—
170C3⋊Dic3—42no—
171C2×#41—91no—
172#131—181no—
173#131—181no—
174#131—181no—
175#131—181no—
176C2×M16—92no—
177C2×M16—91no—
178C2×M16C2×C3⋊D311yescharacteristic
179C2×M16C2×C3⋊D311yescharacteristic
180C22×C8—91no—
181C22×C8—92no—
182#117—91no—
183#117C2×C3⋊D311yescharacteristic
184#101—92no—
185#100—92no—
186#99—94no—
187#97—181no—
188#97—181no—
189C4×C8—181no—
190C4×C8—181no—
191C23×C3C2×C4×D314yes—
192C22×D3—38no—
193C2×C3×C4—28no—
194C2×C3×C4C2×C4×D314yes—
195C2×C3×C4C2×C4×D314yes—
196C2×C3×C4D3×D414yes—
197C2×C3×C4D3×D414yes—
198C2×C3×C4#23714yes—
199C2×C3×C4#23914yes—
200#67—68no—
201#67—68no—
202C2×Dic3—38no—
203C2×Dic3—68no—
204D12—64no—
205D12—64no—
206C4×D3—68no—
207C4×D3—68no—
208Dic6—64no—
209Dic6—64no—
210C3×C8—28no—
211C3×C8—28no—
212C3⋊C8—34no—
213C3⋊C8—34no—
214C3⋊C8—34no—
215C3⋊C8—34no—
216C3⋊C8—68no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32C4×C22⋊C411yescharacteristic
221C3⋊D3—42no—
222#41—181no—
223#41—181no—
224#41—181no—
225#41—181no—
226C2×Q8—91no—
227C2×D4—92no—
228C2×D4—91no—
229C22×C4—92no—
230C22×C4C22×C3⋊D311yescharacteristic
231M16—21no—
232M16—21no—
233M16—21no—
234M16—92no—
235M16—182no—
236M16—182no—
237M16—181no—
238M16C4×C3⋊D312yes—
239C2×C8—21no—
240C2×C8—91no—
241C2×C8—91no—
242C2×C8—91no—
243C2×C8—91no—
244C2×C8—91no—
245C2×C8—91no—
246C2×C8—182no—
247C2×C8—182no—
248C2×C8—181no—
249C2×C8C4×C3⋊D312yes—
250C4⋊C4—21no—
251C4⋊C4—181no—
252C22⋊C4—92no—
253C22⋊C4C4×C3⋊D312yes—
254C42—21no—
255C42—181no—
256C22×C3—44no—
257C22×C3C42×D314yes—
258C22×C3#67⋊C414yes—
259C22×C3#89314yes—
260C2×D3—68no—
261C2×D3—128no—
262C3×C4—48no—
263C3×C4D3×C22⋊C414yes—
264C3×C4#67⋊C414yes—
265C3×C4#89214yes—
266C3×C4#89314yes—
267Dic3—68no—
268Dic3—128no—
269C32#175711yescharacteristic, complemented, Sylow
270C23—92no—
271C23C2×C4×C3⋊D311yescharacteristic
272Q8—181no—
273Q8—181no—
274D4—182no—
275D4—182no—
276D4—181no—
277D4—181no—
278C2×C4—22no—
279C2×C4—92no—
280C2×C4—182no—
281C2×C4—182no—
282C2×C4—182no—
283C2×C4C2×C4×C3⋊D311yescharacteristic
284C2×C4C2×C4×C3⋊D311yescharacteristic
285C2×C4D4×C3⋊D311yescharacteristic
286C2×C4D4×C3⋊D311yescharacteristic
287C2×C4#383011yescharacteristic
288C2×C4#383211yescharacteristic
289C8—22no—
290C8—22no—
291C8—91no—
292C8—91no—
293C8—91no—
294C8—91no—
295C8—182no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3#466614yes—
300D3—128no—
301C22—41no—
302C22—182no—
303C22—362no—
304C22C42×C3⋊D311yescharacteristic
305C22#6409011yescharacteristic
306C22#6410211yescharacteristic
307C4—42no—
308C4—182no—
309C4—362no—
310C4C22⋊C4×C3⋊D311yescharacteristic
311C4#6409011yescharacteristic
312C4#6410111yescharacteristic
313C4#6410211yescharacteristic
314C3#6901414yescomplemented
315C2—21no—
316C2—21no—
317C2—21no—
318C2—362no—
319C2#1059236511yescharacteristic
subgroupG/H
centerC4C22⋊C4×C3⋊D3
commutatorC22×C32C2×C42
FrattiniC22×C4C22×C3⋊D3
FittingC32×#344C2
socleC2×C32C4×C22⋊C4
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenter#344C3⋊D3
hypocenterC32#1757
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
3. #10592821C2semidirect
5. #10592696C2semidirect
6. C32×#344C2semidirect
7. #10589422C2semidirect
20. C2×#3743C4semidirect
21. C2×C8×C3⋊D3C4semidirect
22. #63651C4semidirect
23. #63650C4semidirect
34. C3×#344D3semidirect
123. #344C3⋊D3semidirect
269. C32#1757semidirect
314. C3#69014semidirect

Maximal subgroups

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

subgroupindex
C2×#641162
#105928312
#105928212
#105927052
#105926962
C32×#3442
#105894222
#690143