Finite Group Databasebeta

← #11778503

Subgroups — #11778503

Order 1152.

648 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C32×#539C211yesmaximal, characteristic
2#10592742C211yesmaximal, characteristic, complemented
3#10592732C211yesmaximal, characteristic
4#10592680C211yesmaximal, characteristic, complemented
5#10589404C211yesmaximal, characteristic, complemented
6#10589403C211yesmaximal, characteristic
7#10589394C211yesmaximal, characteristic, complemented
8#71886—34nomaximal
9C32×#126—21no—
10C32×#125—21no—
11C32×#123—21no—
12C32×#123—21no—
13C32×C22⋊Q8—21no—
14C32×C22⋊Q8C2211yescharacteristic
15C32×#121C2211yescharacteristic
16C32×C4×D4—21no—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×#3750—21no—
20#63662—21no—
21#63659—21no—
22#63658—21no—
23#63658C2211yescharacteristic
24#63657C2211yescharacteristic
25#63657C2211yescharacteristic
26#63635—21no—
27#63634—21no—
28C3×#539D314yes—
29#5043—34no—
30#5033—34no—
31#4981—34no—
32#4538—34no—
33#4537—34no—
34#4528—34no—
35C2×C32×Q8D411yescharacteristic
36C2×C32×D4—21no—
37C2×C32×D4D411yescharacteristic
38C22×C32×C4—21no—
39C22×C32×C4D411yescharacteristic
40C22×C32×C4D411yescharacteristic
41C22×C32×C4C2311yescharacteristic
42C22×C3⋊Dic3—21no—
43C22×C3⋊Dic3—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4—21no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4D411yescharacteristic
51C32×C4⋊C4D411yescharacteristic
52C32×C22⋊C4—21no—
53C32×C22⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C22⋊C4—21no—
56C32×C22⋊C4—21no—
57C32×C42—21no—
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)—21no—
60#3750—42no—
61C2×C3⋊(C3⋊C8)—21no—
62C2×C3⋊(C3⋊C8)—21no—
63#1609—91noSylow
64C3×#126—24no—
65C3×#125—24no—
66C3×#123—24no—
67C3×#123—24no—
68C3×C22⋊Q8—24no—
69C3×C22⋊Q8C2×D314yes—
70C3×#121C2×D314yes—
71C3×C4×D4—24no—
72C2×C3×C4⋊C4C2×D314yes—
73C2×C4⋊Dic3—34no—
74C2×Dic3⋊C4—64no—
75#835—64no—
76#832—64no—
77C3⋊#95—34no—
78C3⋊#95—64no—
79#830—34no—
80#830—34no—
81#808—64no—
82#807—64no—
83C23×C32—21no—
84C23×C32C2×D411yescharacteristic
85C32×Q8—41no—
86C32×D4—41no—
87C32×D4—41no—
88C32×D4—41no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—81no—
99C2×C32×C4C2×D411yescharacteristic
100C2×C32×C4C2×D411yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3—41no—
104C2×C3⋊Dic3—41no—
105C2×C3⋊Dic3—41no—
106C2×C3⋊Dic3—41no—
107C3⋊(C3⋊C8)—41no—
108C3⋊(C3⋊C8)—41no—
109#539C3⋊D311yescharacteristic
110#483—91no—
111#480—91no—
112#398—91no—
113#328—91no—
114#327—91no—
115#324—91no—
116C2×C3×Q8#6714yes—
117C2×C3×D4—24no—
118C2×C3×D4#6714yes—
119C22×C3×C4—24no—
120C22×C3×C4#6714yes—
121C22×C3×C4#6714yes—
122C22×C3×C4C22×D314yes—
123C22×Dic3—64no—
124C22×Dic3—64no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4—24no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4#6714yes—
132C3×C4⋊C4#6714yes—
133C3×C22⋊C4—24no—
134C3×C22⋊C4—24no—
135C3×C22⋊C4—24no—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—24no—
138C3×C42—24no—
139C4⋊Dic3—64no—
140C4⋊Dic3—64no—
141Dic3⋊C4—128no—
142C2×C3⋊C8—64no—
143C2×C3⋊C8—64no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32—41no—
147C22×C32—41no—
148C22×C32—41no—
149C22×C32#12011yescharacteristic
150C22×C32#13511yescharacteristic
151C22×C32AGL(1,ℤ/8)11yescharacteristic
152C32×C4—21no—
153C32×C4—21no—
154C32×C4—41no—
155C32×C4—41no—
156C32×C4—41no—
157C32×C4—41no—
158C32×C4—41no—
159C3⋊Dic3—42no—
160C3⋊Dic3—81no—
161#126—21no—
162#125—21no—
163#123—21no—
164#123—21no—
165C22⋊Q8—21no—
166C22⋊Q8C2×C3⋊D311yescharacteristic
167#121C2×C3⋊D311yescharacteristic
168C4×D4—21no—
169C2×C4⋊C4—91no—
170C2×C4⋊C4—181no—
171C2×C4⋊C4C2×C3⋊D311yescharacteristic
172#107—181no—
173#106—181no—
174#103—181no—
175#102—181no—
176C22⋊C8—91no—
177C22⋊C8—91no—
178#95—91no—
179#95—181no—
180C23×C3—24no—
181C23×C3C2×#6714yes—
182C3×Q8—44no—
183C3×D4—44no—
184C3×D4—44no—
185C3×D4—44no—
186C2×C3×C4—24no—
187C2×C3×C4—24no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—24no—
191C2×C3×C4—24no—
192C2×C3×C4—24no—
193C2×C3×C4—44no—
194C2×C3×C4—44no—
195C2×C3×C4—84no—
196C2×C3×C4C2×#6714yes—
197C2×C3×C4C2×#6714yes—
198C2×Dic3—68no—
199C2×Dic3—124no—
200C2×Dic3—124no—
201C2×Dic3—124no—
202C2×Dic3—124no—
203C2×Dic3—124no—
204C3⋊C8—124no—
205C3⋊C8—124no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—81no—
209C2×C32#44911yescharacteristic
210C2×C32#45611yescharacteristic
211C2×C32#45811yescharacteristic
212C2×Q8#63811yescharacteristic
213C2×D4—21no—
214C2×D4#63811yescharacteristic
215C22×C4—21no—
216C22×C4—181no—
217C22×C4—181no—
218C22×C4#63811yescharacteristic
219C22×C4#63811yescharacteristic
220C22×C4C22×C3⋊D311yescharacteristic
221C2×C8—181no—
222C2×C8—181no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—181no—
230C4⋊C4—181no—
231C4⋊C4—362no—
232C4⋊C4#63811yescharacteristic
233C4⋊C4#63811yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—21no—
239C42—21no—
240C22×C3—24no—
241C22×C3—24no—
242C22×C3—44no—
243C22×C3—44no—
244C22×C3—44no—
245C22×C3#94914yes—
246C22×C3#95014yes—
247C22×C3#95314yes—
248C3×C4—24no—
249C3×C4—24no—
250C3×C4—44no—
251C3×C4—44no—
252C3×C4—44no—
253C3×C4—44no—
254C3×C4—44no—
255Dic3—128no—
256Dic3—244no—
257C32#160911yescharacteristic, complemented, Sylow
258C23—21no—
259C23C2×#63811yescharacteristic
260Q8—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—182no—
275C2×C4—361no—
276C2×C4—361no—
277C2×C4—361no—
278C2×C4—361no—
279C2×C4—361no—
280C2×C4C2×#63811yescharacteristic
281C2×C4C2×#63811yescharacteristic
282C8—361no—
283C8—361no—
284C2×C3—24no—
285C2×C3—24no—
286C2×C3—84no—
287C2×C3#523814yes—
288C2×C3#524314yes—
289C2×C3#524614yes—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22#6415811yescharacteristic
296C22#6415911yescharacteristic
297C22#6416211yescharacteristic
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—362no—
306C4—721no—
307C3#7188614yescomplemented
308C2—21no—
309C2—21no—
310C2—81no—
311C2#1059293711yescharacteristic
312C2#1059294211yescharacteristic
313C2#1059294511yescharacteristic
subgroupG/H
centerC22#64162
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#539C2
socleC22×C32#120
prime radicalC32×#121C22
hypercenter#539C3⋊D3
hypocenterC32#1609
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
2. #10592742C2semidirect
4. #10592680C2semidirect
5. #10589404C2semidirect
7. #10589394C2semidirect
257. C32#1609semidirect
307. C3#71886semidirect

Maximal subgroups

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

subgroupindex
C32×#5392
#105927422
#105927322
#105926802
#105894042
#105894032
#105893942
#718863