Finite Group Databasebeta

← #11775347

Subgroups — #11775347

Order 1152.

666 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#10596095C211yesmaximal, characteristic
2#10592418C211yesmaximal, characteristic, complemented
3#10592349C211yesmaximal, characteristic, complemented
4C32×#328C211yesmaximal, characteristic, complemented
5#10589404C211yesmaximal, characteristic, complemented
6#10589336C211yesmaximal, characteristic
7#10589317C211yesmaximal, characteristic
8#68312—34nomaximal
9C32×C22⋊Q8C2211yescharacteristic
10#64156C411yescharacteristic, complemented
11Q8×C3⋊Dic3C411yescharacteristic, complemented
12#64139C2211yescharacteristic
13C2×#3750—21no—
14C2×C4×C3⋊Dic3C2211yescharacteristic
15#64107—21no—
16#64102C411yescharacteristic, complemented
17#64101—21no—
18#64099C411yescharacteristic, complemented
19#64095—21no—
20#64090—21no—
21C32×C22⋊C8C2211yescharacteristic
22C32×#95C2211yescharacteristic
23#63658—21no—
24#63658C2211yescharacteristic
25#63657C2211yescharacteristic
26#63642—21no—
27#63641—21no—
28#5624—34no—
29#4719—34no—
30#4650—34no—
31C3×#328D314yescomplemented
32#4538—34no—
33#4470—34no—
34#4451—34no—
35C2×C32×Q8C2×C411yescharacteristic
36C22×C32×C4—21no—
37C22×C32×C4C2311yescharacteristic
38C2×#638—21no—
39C22×C3⋊Dic3—21no—
40C22×C3⋊Dic3D411yescharacteristic
41C22×C3⋊Dic3D411yescharacteristic
42C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
43C2×C4×C3⋊D3—21no—
44C2×C32×C8—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4C2×C411yescharacteristic
47C32×C22⋊C4—21no—
48#3752—21no—
49#3752—21no—
50#3752—21no—
51#3752—21no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)C2×C411yescharacteristic
55#3750—21no—
56#3750—42no—
57C4×C3⋊Dic3—21no—
58C4×C3⋊Dic3C2×C411yescharacteristic
59C4×C3⋊Dic3C2×C411yescharacteristic
60C4×C3⋊Dic3D411yescharacteristic
61C4×C3⋊Dic3D411yescharacteristic
62C2×C3⋊(C3⋊C8)—21no—
63#1512—91noSylow
64C3×C22⋊Q8C2×D314yes—
65#947—34no—
66Q8×Dic3—34no—
67#930—34no—
68C2×Dic3⋊C4—64no—
69C2×C4×Dic3—34no—
70#898—64no—
71#893—34no—
72#892—64no—
73#890—34no—
74#886—64no—
75#67⋊C4—64no—
76C3×C22⋊C8C2×D314yes—
77C3×#95C2×D314yes—
78C3⋊#95—34no—
79C3⋊#95—64no—
80#830—34no—
81#815—64no—
82Dic3⋊C8—64no—
83C23×C32C2×D411yescharacteristic
84C22×C3⋊D3—21no—
85C32×Q8—41no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—41no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4C22×C411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94#638—41no—
95#638—41no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—21no—
99C2×C3⋊Dic3—41no—
100C2×C3⋊Dic3—41no—
101C2×C3⋊Dic3—41no—
102C2×C3⋊Dic3—41no—
103C2×C3⋊Dic3C22⋊C411yescharacteristic
104C2×C3⋊Dic3C22⋊C411yescharacteristic
105C2×C3⋊Dic3C22⋊C411yescharacteristic
106C2×C3⋊Dic3C22⋊C411yescharacteristic
107C4⋊(C3⋊D3)—41no—
108C4×C3⋊D3—42no—
109C32×C8—41no—
110C3⋊(C3⋊C8)—41no—
111#529—91no—
112#432—91no—
113#382—91no—
114#328—91no—
115#328C3⋊D311yescharacteristic, complemented
116#327—91no—
117#327—91no—
118C2×C3×Q8C4×D314yes—
119C22×C3×C4—24no—
120C22×C3×C4C22×D314yes—
121C2×#67—64no—
122C22×Dic3—34no—
123C22×Dic3—34no—
124C22×Dic3—64no—
125C2×D12—34no—
126C2×C4×D3—64no—
127C2×C3×C8—24no—
128C3×C4⋊C4—24no—
129C3×C4⋊C4C4×D314yes—
130C3×C22⋊C4—24no—
131(C2×D3)⋊C4—64no—
132(C2×D3)⋊C4—64no—
133(C2×D3)⋊C4—64no—
134(C2×D3)⋊C4—64no—
135C4⋊Dic3—34no—
136C4⋊Dic3—64no—
137C4⋊Dic3—64no—
138Dic3⋊C4—64no—
139Dic3⋊C4—128no—
140C4×Dic3—34no—
141C4×Dic3—34no—
142C4×Dic3—34no—
143C4×Dic3—34no—
144C4×Dic3—64no—
145C2×C3⋊C8—64no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32C2×C22⋊C411yescharacteristic
149C22×C32AGL(1,ℤ/8)11yescharacteristic
150C22×C32#13711yescharacteristic
151C2×C3⋊D3—41no—
152C2×C3⋊D3—41no—
153C2×C3⋊D3—41no—
154C32×C4—21no—
155C32×C4—21no—
156C32×C4—41no—
157C32×C4—41no—
158C32×C4—81no—
159C3⋊Dic3—21no—
160C3⋊Dic3—21no—
161C3⋊Dic3—21no—
162C3⋊Dic3—21no—
163C3⋊Dic3—41no—
164C3⋊Dic3—81no—
165#126—181no—
166#125—91no—
167#124—91no—
168#123—181no—
169C22⋊Q8C2×C3⋊D311yescharacteristic
170#121—91no—
171C4×Q8—91no—
172C4×D4—91no—
173C4×D4—181no—
174#117—181no—
175C2×C4⋊C4—181no—
176C2×C42—91no—
177C4⋊C8—181no—
178C22⋊C8—91no—
179C22⋊C8C2×C3⋊D311yescharacteristic
180#97—181no—
181#95—91no—
182#95—181no—
183#95C2×C3⋊D311yescharacteristic
184C23×C3D3×D414yes—
185C22×D3—64no—
186C3×Q8—44no—
187C2×C3×C4—24no—
188C2×C3×C4—24no—
189C2×C3×C4—44no—
190C2×C3×C4—44no—
191C2×C3×C4—44no—
192C2×C3×C4—44no—
193C2×C3×C4C2×C4×D314yes—
194C2×C3×C4D3×D414yes—
195#67—124no—
196#67—124no—
197C2×Dic3—34no—
198C2×Dic3—34no—
199C2×Dic3—34no—
200C2×Dic3—34no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204C2×Dic3—124no—
205C2×Dic3—124no—
206C2×Dic3—124no—
207C2×Dic3—124no—
208D12—124no—
209C4×D3—128no—
210C3×C8—44no—
211C3⋊C8—124no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32#41011yescharacteristic
215C2×C32#41711yescharacteristic
216C2×C32#42111yescharacteristic
217C3⋊D3—81no—
218C2×Q8C4×C3⋊D311yescharacteristic
219C2×D4—91no—
220C2×D4—181no—
221C22×C4—21no—
222C22×C4—91no—
223C22×C4—91no—
224C22×C4—181no—
225C22×C4—181no—
226C22×C4C22×C3⋊D311yescharacteristic
227C2×C8—21no—
228C2×C8—181no—
229C4⋊C4—21no—
230C4⋊C4—91no—
231C4⋊C4—181no—
232C4⋊C4—181no—
233C4⋊C4—181no—
234C4⋊C4—362no—
235C4⋊C4C4×C3⋊D311yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—181no—
238C22⋊C4—181no—
239C22⋊C4—181no—
240C22⋊C4—181no—
241C42—91no—
242C42—91no—
243C42—91no—
244C42—91no—
245C42—181no—
246C22×C3—24no—
247C22×C3—24no—
248C22×C3D3×C22⋊C414yes—
249C22×C3#91414yes—
250C22×C3#91814yes—
251C2×D3—124no—
252C2×D3—124no—
253C2×D3—124no—
254C3×C4—24no—
255C3×C4—24no—
256C3×C4—44no—
257C3×C4—44no—
258C3×C4—84no—
259Dic3—64no—
260Dic3—64no—
261Dic3—64no—
262Dic3—64no—
263Dic3—124no—
264Dic3—244no—
265C32#151211yescharacteristic, complemented, Sylow
266C23—181no—
267C23D4×C3⋊D311yescharacteristic
268Q8—41no—
269D4—361no—
270D4—361no—
271D4—361no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4—181no—
285C2×C4—362no—
286C2×C4—361no—
287C2×C4—361no—
288C2×C4—361no—
289C2×C4—361no—
290C2×C4C2×C4×C3⋊D311yescharacteristic
291C2×C4D4×C3⋊D311yescharacteristic
292C8—41no—
293C8—361no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3#474014yes—
297C2×C3#480314yes—
298C2×C3#482114yes—
299D3—244no—
300C22—21no—
301C22—21no—
302C22—361no—
303C22—361no—
304C22—361no—
305C22C22⋊C4×C3⋊D311yescharacteristic
306C22#6412311yescharacteristic
307C22#6412711yescharacteristic
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—181no—
314C4—181no—
315C4—181no—
316C4—181no—
317C4—361no—
318C4—721no—
319C3#6831214yescomplemented
320C2—21no—
321C2—21no—
322C2—721no—
323C2#1059243911yescharacteristic
324C2#1059250211yescharacteristic
325C2#1059252011yescharacteristic
subgroupG/H
centerC22C22⋊C4×C3⋊D3
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×#328C2
socleC22×C32C2×C22⋊C4
prime radical#64139C22
cosocleC22×C32×C4C23
hypercenter#328C3⋊D3
hypocenterC32#1512
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #10592418C2semidirect
3. #10592349C2semidirect
4. C32×#328C2semidirect
5. #10589404C2semidirect
10. #64156C4semidirect
11. Q8×C3⋊Dic3C4semidirect
16. #64102C4semidirect
18. #64099C4semidirect
31. C3×#328D3semidirect
115. #328C3⋊D3semidirect
265. C32#1512semidirect
319. C3#68312semidirect

Maximal subgroups

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

subgroupindex
#105960952
#105924182
#105923492
C32×#3282
#105894042
#105893362
#105893172
#683123