Finite Group Databasebeta

← #11785539

Subgroups — #11785539

Order 1152.

768 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 173 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64116C211yesmaximal, characteristic
2#10592838C211yesmaximal, characteristic
3C2×#63649C211yesmaximal, characteristic, complemented
4#10592704C211yesmaximal, characteristic
5#10592697C211yesmaximal, characteristic, complemented
6C32×#344C211yesmaximal, characteristic, complemented
7#10589425C211yesmaximal, characteristic, complemented
8#69032—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#64116C412yescomplemented
19#64116C412yescomplemented
20C2×#3743—21no—
21C2×C8×C3⋊D3—21no—
22#63652—22no—
23#63649—21no—
24#63649—21no—
25#63635—21no—
26#63634—21no—
27#63632—22no—
28C2×#907—34no—
29#5139—34no—
30C2×#822—34no—
31#5005—34no—
32#4998—34no—
33C3×#344D314yescomplemented
34#4559—34no—
35C22×C32×C4C2311yescharacteristic
36C2×#638—21no—
37#3828—21no—
38#3828—21no—
39#3828C2×C412yes—
40#3828D412yes—
41C2×C4⋊(C3⋊D3)Q811yescharacteristic
42C2×C4×C3⋊D3—21no—
43C2×#634Q811yescharacteristic
44C32×M16—21no—
45C32×M16—21no—
46C32×M16—21no—
47C32×M16C2×C412yes—
48C2×C32×C8—21no—
49C2×C32×C8—21no—
50C32×C4⋊C4—21no—
51C32×C22⋊C4—21no—
52C32×C42—21no—
53#3757—21no—
54C4⋊(C3⋊Dic3)—21no—
55C4×C3⋊Dic3—21no—
56#3748—21no—
57#3748—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
60C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
61C2×C3⋊(C3⋊C8)D411yescharacteristic
62C2×C3⋊(C3⋊C8)D411yescharacteristic
63C2×C3⋊(C3⋊C8)D411yescharacteristic
64C2×C3⋊(C3⋊C8)D411yescharacteristic
65#3743—22no—
66#3743—22no—
67C8×C3⋊D3—22no—
68C8×C3⋊D3—22no—
69#1814—91noSylow
70C2×#235—34no—
71C2×C3×M16C2×D314yes—
72C2×C3×M16C2×D314yes—
73C3×#117C2×D314yes—
74#926—34no—
75C2×#208—34no—
76C22×C3⋊C8—34no—
77#907—38no—
78#907—38no—
79#907—64no—
80#907—64no—
81C2×ΓL(1,𝔽25)—64no—
82C2×D3×C8—64no—
83#825—68no—
84#822—64no—
85#822—64no—
86#808—64no—
87#807—64no—
88#805—68no—
89C23×C32#4111yescharacteristic
90C22×C3⋊D3—21no—
91C2×C32×C4—41no—
92C2×C32×C4C22×C411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×Q811yescharacteristic
97C2×C32×C4#4111yescharacteristic
98#638—22no—
99#638—22no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—41no—
102C4⋊(C3⋊D3)—21no—
103C4⋊(C3⋊D3)C4⋊C412yes—
104C4×C3⋊D3—22no—
105C4×C3⋊D3—22no—
106#634—21no—
107#634C4⋊C412yes—
108C32×C8—22no—
109C32×C8—41no—
110C3⋊(C3⋊C8)—41no—
111C3⋊(C3⋊C8)C22⋊C411yescharacteristic
112C3⋊(C3⋊C8)C22⋊C411yescharacteristic
113C3⋊(C3⋊C8)C22⋊C411yescharacteristic
114C3⋊(C3⋊C8)C22⋊C411yescharacteristic
115C2×#131—91no—
116C2×#108—91no—
117#427—91no—
118#421—91no—
119#421—91no—
120#344—91no—
121#344C3⋊D311yescharacteristic, complemented
122C22×C3×C4C22×D314yes—
123C2×#67—64no—
124#235—38no—
125#235—38no—
126#235—64no—
127#235—64no—
128C2×D12—34no—
129C2×C4×D3—64no—
130C2×Dic6—34no—
131C3×M16—24no—
132C3×M16—24no—
133C3×M16—24no—
134C3×M16C4×D318yes—
135C2×C3×C8—24no—
136C2×C3×C8—24no—
137C3×C4⋊C4—24no—
138C3×C22⋊C4—24no—
139C3×C42—24no—
140#217—64no—
141C4⋊Dic3—64no—
142C4×Dic3—64no—
143#208—64no—
144#208—64no—
145C2×C3⋊C8—34no—
146C2×C3⋊C8—34no—
147C2×C3⋊C8—34no—
148C2×C3⋊C8—34no—
149C2×C3⋊C8—34no—
150C2×C3⋊C8—34no—
151C2×C3⋊C8—64no—
152ΓL(1,𝔽25)—68no—
153ΓL(1,𝔽25)—68no—
154D3×C8—68no—
155D3×C8—68no—
156C22×C32—41no—
157C22×C32#11711yescharacteristic
158C22×C32#12111yescharacteristic
159C22×C32C22⋊Q811yescharacteristic
160C2×C3⋊D3—22no—
161C2×C3⋊D3—42no—
162C32×C4—81no—
163C32×C4C2×C22⋊C411yescharacteristic
164C32×C4C2×C4⋊C411yescharacteristic
165C32×C4#12111yescharacteristic
166C32×C4C22⋊Q811yescharacteristic
167C3⋊Dic3—22no—
168C3⋊Dic3—81no—
169C2×#41—91no—
170#131—92no—
171#131—92no—
172#131—181no—
173#131—181no—
174C2×M16—91no—
175C2×M16—181no—
176C2×M16C2×C3⋊D311yescharacteristic
177C2×M16C2×C3⋊D311yescharacteristic
178C22×C8—91no—
179C22×C8—181no—
180#117—91no—
181#117C2×C3⋊D311yescharacteristic
182#108—181no—
183#108—181no—
184#107—181no—
185#106—181no—
186#104—182no—
187#104—182no—
188C23×C3#23514yes—
189C22×D3—64no—
190C2×C3×C4—44no—
191C2×C3×C4C2×C4×D314yes—
192C2×C3×C4#23514yes—
193C2×C3×C4D3×D414yes—
194C2×C3×C4D3×D414yes—
195C2×C3×C4D3×D414yes—
196C2×C3×C4D3×Q814yes—
197#67—68no—
198#67—68no—
199C2×Dic3—64no—
200C2×Dic3—124no—
201D12—38no—
202D12—64no—
203C4×D3—68no—
204C4×D3—68no—
205Dic6—38no—
206Dic6—64no—
207C3×C8—28no—
208C3×C8—44no—
209C3⋊C8—34no—
210C3⋊C8—34no—
211C3⋊C8—34no—
212C3⋊C8—34no—
213C3⋊C8—124no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32#38011yescharacteristic
218C3⋊D3—42no—
219#41—92no—
220#41—92no—
221#41—181no—
222#41—181no—
223C2×Q8—91no—
224C2×D4—91no—
225C2×D4—181no—
226C22×C4—181no—
227C22×C4C22×C3⋊D311yescharacteristic
228M16—21no—
229M16—21no—
230M16—21no—
231M16—182no—
232M16—182no—
233M16—181no—
234M16—181no—
235M16C4×C3⋊D312yes—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—91no—
239C2×C8—91no—
240C2×C8—91no—
241C2×C8—91no—
242C2×C8—91no—
243C2×C8—91no—
244C2×C8—182no—
245C2×C8—182no—
246C2×C8—181no—
247C4⋊C4—21no—
248C4⋊C4—181no—
249C22⋊C4—21no—
250C22⋊C4—181no—
251C42—21no—
252C42—181no—
253C22×C3—44no—
254C22×C3#87214yes—
255C22×C3#88414yes—
256C22×C3#89614yes—
257C2×D3—68no—
258C2×D3—128no—
259C3×C4—84no—
260C3×C4D3×C22⋊C414yes—
261C3×C4#88414yes—
262C3×C4D3×C4⋊C414yes—
263C3×C4#89614yes—
264Dic3—68no—
265Dic3—244no—
266C32#181411yescharacteristic, complemented, Sylow
267C23—181no—
268C23#382811yescharacteristic
269Q8—92no—
270Q8—181no—
271D4—92no—
272D4—182no—
273D4—182no—
274D4—181no—
275C2×C4—41no—
276C2×C4—181no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4—361no—
280C2×C4C2×C4×C3⋊D311yescharacteristic
281C2×C4#382811yescharacteristic
282C2×C4D4×C3⋊D311yescharacteristic
283C2×C4D4×C3⋊D311yescharacteristic
284C2×C4D4×C3⋊D311yescharacteristic
285C2×C4Q8×C3⋊D311yescharacteristic
286C8—22no—
287C8—41no—
288C8—91no—
289C8—91no—
290C8—91no—
291C8—91no—
292C8—361no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3#466714yes—
297D3—128no—
298C22—41no—
299C22—182no—
300C22—362no—
301C22#6408111yescharacteristic
302C22#6409311yescharacteristic
303C22#6410511yescharacteristic
304C4—81no—
305C4—182no—
306C4—721no—
307C4C22⋊C4×C3⋊D311yescharacteristic
308C4#6409311yescharacteristic
309C4C4⋊C4×C3⋊D311yescharacteristic
310C4#6410511yescharacteristic
311C3#6903214yescomplemented
312C2—21no—
313C2—21no—
314C2—21no—
315C2—362no—
316C2#1059236611yescharacteristic
subgroupG/H
centerC4#64093
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×#344C2
socleC2×C32#380
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenter#344C3⋊D3
hypocenterC32#1814
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
3. C2×#63649C2semidirect
5. #10592697C2semidirect
6. C32×#344C2semidirect
7. #10589425C2semidirect
18. #64116C4semidirect
19. #64116C4semidirect
33. C3×#344D3semidirect
121. #344C3⋊D3semidirect
266. C32#1814semidirect
311. C3#69032semidirect

Maximal subgroups

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

subgroupindex
C2×#641162
#105928382
C2×#636492
#105927042
#105926972
C32×#3442
#105894252
#690323