Finite Group Databasebeta

← #11774699

Subgroups — #11774699

Order 1152.

756 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C4×Q8×C3⋊D3C211yesmaximal, characteristic
2#10592534C211yesmaximal, characteristic
3C4×#3743C211yesmaximal, characteristic
4C32×Q8⋊C8C211yesmaximal, characteristic, complemented
5#10589401C211yesmaximal, characteristic, complemented
6#10589350C211yesmaximal, characteristic, complemented
7#10589322C211yesmaximal, characteristic, complemented
8#68239—34nomaximal
9C2×Q8×C3⋊D3C411yescharacteristic
10C32×C4×Q8C2211yescharacteristic
11Q8×C3⋊Dic3C411yescharacteristic
12C2×#3743—21no—
13C4⋊C4×C3⋊D3—21no—
14C4⋊C4×C3⋊D3C411yescharacteristic
15#64096—21no—
16#64096C411yescharacteristic
17C42×C3⋊D3—21no—
18C42×C3⋊D3C2211yescharacteristic
19C4×#634—21no—
20C4×#634C2211yescharacteristic
21C32×C4⋊C8C2211yescharacteristic
22C32×C4×C8C2211yescharacteristic
23#63647—21no—
24#63642—21no—
25#63642—21no—
26C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
27C4×C3⋊(C3⋊C8)C2211yescharacteristic
28C4×D3×Q8—34no—
29#4835—34no—
30C4×ΓL(1,𝔽25)—34no—
31C3×Q8⋊C8D314yescomplemented
32#4535—34no—
33#4484—34no—
34#4456—34no—
35C2×C32×Q8C2×C411yescharacteristic
36Q8×C3⋊D3—22no—
37Q8×C3⋊D3—22no—
38C2×C4×C3⋊D3—21no—
39C2×C4×C3⋊D3—21no—
40C2×C4×C3⋊D3C2×C411yescharacteristic
41C2×C4×C3⋊D3D411yescharacteristic
42C2×C4×C3⋊D3D411yescharacteristic
43C2×#634—21no—
44C2×#634C2×C411yescharacteristic
45C2×C32×C8—21no—
46C2×C32×C8—21no—
47C32×C4⋊C4—21no—
48C32×C4⋊C4C2×C411yescharacteristic
49C32×C42—21no—
50C32×C42C2311yescharacteristic
51C4⋊(C3⋊Dic3)—21no—
52C4⋊(C3⋊Dic3)C2×C411yescharacteristic
53#3750—21no—
54#3750—21no—
55#3750—21no—
56C4×C3⋊Dic3—21no—
57C4×C3⋊Dic3—21no—
58C4×C3⋊Dic3—21no—
59C4×C3⋊Dic3C2×C411yescharacteristic
60C4×C3⋊Dic3D411yescharacteristic
61C4×C3⋊Dic3D411yescharacteristic
62C2×C3⋊(C3⋊C8)—21no—
63C2×C3⋊(C3⋊C8)—21no—
64#3743—24no—
65#1491—91noSylow
66C2×D3×Q8—34no—
67C3×C4×Q8C2×D314yes—
68Q8×Dic3—34no—
69C2×ΓL(1,𝔽25)—64no—
70D3×C4⋊C4—34no—
71D3×C4⋊C4—64no—
72#887—34no—
73#887—64no—
74C42×D3—34no—
75C42×D3—64no—
76C4×Dic6—34no—
77C4×Dic6—64no—
78C3×C4⋊C8C2×D314yes—
79C3×C4×C8C2×D314yes—
80(C2×D3)⋊C8—64no—
81#815—64no—
82#815—64no—
83C4⋊(C3⋊C8)—34no—
84C4×C3⋊C8—34no—
85C22×C3⋊D3C22⋊C411yescharacteristic
86C32×Q8—21no—
87C32×Q8M1612yescomplemented
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4C22×C411yescharacteristic
91C2×C32×C4C2×D411yescharacteristic
92C2×C32×C4C2×D411yescharacteristic
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—21no—
95C2×C3⋊Dic3C22⋊C411yescharacteristic
96C2×C3⋊Dic3C22⋊C411yescharacteristic
97C2×C3⋊Dic3C22⋊C411yescharacteristic
98C4×C3⋊D3—24no—
99C4×C3⋊D3—22no—
100C4×C3⋊D3—21no—
101C4×C3⋊D3—21no—
102C4×C3⋊D3—21no—
103C4×C3⋊D3—21no—
104C4×C3⋊D3—42no—
105#634—24no—
106#634—21no—
107#634M1612yescomplemented
108C32×C8—22no—
109C32×C8—41no—
110C3⋊(C3⋊C8)—22no—
111C3⋊(C3⋊C8)—41no—
112C2×C4×Q8—91no—
113#432—91no—
114C4×M16—91no—
115Q8⋊C8—91no—
116Q8⋊C8—91no—
117Q8⋊C8—91no—
118Q8⋊C8C3⋊D311yescharacteristic, complemented
119C2×C3×Q8C4×D314yes—
120D3×Q8—68no—
121D3×Q8—68no—
122C2×C4×D3—34no—
123C2×C4×D3—34no—
124C2×C4×D3—34no—
125C2×C4×D3—64no—
126C2×C4×D3—64no—
127C2×Dic6—34no—
128C2×Dic6—64no—
129C2×C3×C8—24no—
130C2×C3×C8—24no—
131C3×C4⋊C4—24no—
132C3×C4⋊C4C4×D314yes—
133C3×C42—24no—
134C3×C42C22×D314yes—
135C4⋊Dic3—34no—
136C4⋊Dic3—64no—
137Dic3⋊C4—64no—
138Dic3⋊C4—64no—
139Dic3⋊C4—64no—
140C4×Dic3—34no—
141C4×Dic3—34no—
142C4×Dic3—34no—
143C4×Dic3—64no—
144C4×Dic3—64no—
145C4×Dic3—64no—
146C2×C3⋊C8—64no—
147C2×C3⋊C8—64no—
148ΓL(1,𝔽25)—616no—
149C22×C32C2×C22⋊C411yescharacteristic
150C2×C3⋊D3—21no—
151C2×C3⋊D3—21no—
152C2×C3⋊D3#10412yes—
153C32×C4—22no—
154C32×C4—21no—
155C32×C4—41no—
156C32×C4C2×M1611yescharacteristic
157C32×C4C2×M1611yescharacteristic
158C32×C4#13711yescharacteristic
159C32×C4#13711yescharacteristic
160C3⋊Dic3—21no—
161C3⋊Dic3—22no—
162C3⋊Dic3—21no—
163C3⋊Dic3—21no—
164C3⋊Dic3—41no—
165C22×Q8—91no—
166C2×M16—181no—
167C4×Q8—91no—
168C4×Q8—91no—
169C4×Q8—91no—
170C4×Q8—181no—
171C4×Q8—181no—
172C4×Q8C2×C3⋊D311yescharacteristic
173C2×C4⋊C4—91no—
174C2×C4⋊C4—181no—
175C2×C42—91no—
176C2×C42—181no—
177C4⋊C8—91no—
178C4⋊C8C2×C3⋊D311yescharacteristic
179C22⋊C8—181no—
180#97—181no—
181#97—181no—
182C4×C8—91no—
183C4×C8C2×C3⋊D311yescharacteristic
184C22×D3—34no—
185C3×Q8—24no—
186C3×Q8ΓL(1,𝔽25)18yescomplemented
187C2×C3×C4—24no—
188C2×C3×C4—24no—
189C2×C3×C4C2×C4×D314yes—
190C2×C3×C4D3×D414yes—
191C2×C3×C4D3×D414yes—
192C2×Dic3—34no—
193C2×Dic3—34no—
194C2×Dic3—34no—
195C2×Dic3—64no—
196C2×Dic3—64no—
197C4×D3—68no—
198C4×D3—616no—
199C4×D3—64no—
200C4×D3—64no—
201C4×D3—64no—
202C4×D3—64no—
203C4×D3—128no—
204Dic6—38no—
205Dic6—616no—
206Dic6—64no—
207C3×C8—28no—
208C3×C8—44no—
209C3⋊C8—68no—
210C3⋊C8—124no—
211C2×C32#40711yescharacteristic
212C2×C32#41911yescharacteristic
213C2×C32C2×#10411yescharacteristic
214C3⋊D3—22no—
215C2×Q8—91no—
216C2×Q8—181no—
217C2×Q8—182no—
218C2×Q8—182no—
219C2×Q8C4×C3⋊D311yescharacteristic
220C22×C4—91no—
221C22×C4—91no—
222C22×C4—91no—
223C22×C4—181no—
224C22×C4—181no—
225M16—184no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—181no—
229C2×C8—181no—
230C4⋊C4—21no—
231C4⋊C4—91no—
232C4⋊C4—181no—
233C4⋊C4—181no—
234C4⋊C4—181no—
235C4⋊C4—181no—
236C4⋊C4C4×C3⋊D311yescharacteristic
237C42—21no—
238C42—91no—
239C42—91no—
240C42—91no—
241C42—181no—
242C42—181no—
243C42—181no—
244C42C22×C3⋊D311yescharacteristic
245C22×C3D3×C22⋊C414yes—
246C2×D3—38no—
247C2×D3—64no—
248C2×D3—64no—
249C3×C4—28no—
250C3×C4—24no—
251C3×C4—44no—
252C3×C4C2×ΓL(1,𝔽25)14yes—
253C3×C4D3×M1614yes—
254C3×C4#91514yes—
255C3×C4#91814yes—
256Dic3—68no—
257Dic3—64no—
258Dic3—64no—
259Dic3—64no—
260Dic3—124no—
261C32#149111yescharacteristic, complemented, Sylow
262C23—91no—
263Q8—21no—
264Q8—92no—
265Q8—184no—
266Q8—181no—
267Q8#374312yescomplemented
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—91no—
273C2×C4—182no—
274C2×C4—184no—
275C2×C4—181no—
276C2×C4—181no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4—362no—
282C2×C4C2×C4×C3⋊D311yescharacteristic
283C2×C4D4×C3⋊D311yescharacteristic
284C2×C4D4×C3⋊D311yescharacteristic
285C8—22no—
286C8—41no—
287C8—182no—
288C8—361no—
289C2×C3#472614yes—
290C2×C3#480214yes—
291C2×C3D3×#10414yes—
292D3—68no—
293C22—92no—
294C22—181no—
295C22—181no—
296C22C22⋊C4×C3⋊D311yescharacteristic
297C4—21no—
298C4—22no—
299C4—41no—
300C4—181no—
301C4—181no—
302C4—182no—
303C4—181no—
304C4—361no—
305C4C2×#374311yescharacteristic
306C4M16×C3⋊D311yescharacteristic
307C4#6412411yescharacteristic
308C4#6412711yescharacteristic
309C3#6823914yescomplemented
310C2—182no—
311C2#1059242511yescharacteristic
312C2#1059250111yescharacteristic
313C2C3⋊D3×#10411yescharacteristic
subgroupG/H
centerC2×C4D4×C3⋊D3
commutatorC2×C32×C4C22×C4
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32C2×C22⋊C4
prime radicalC22×C3⋊D3C22⋊C4
cosocleC32×C42C23
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1491
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. C32×Q8⋊C8C2semidirect
5. #10589401C2semidirect
6. #10589350C2semidirect
7. #10589322C2semidirect
31. C3×Q8⋊C8D3semidirect
87. C32×Q8M16semidirect
107. #634M16semidirect
118. Q8⋊C8C3⋊D3semidirect
186. C3×Q8ΓL(1,𝔽25)semidirect
261. C32#1491semidirect
267. Q8#3743semidirect
309. C3#68239semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×C3⋊D32
#105925342
C4×#37432
C32×Q8⋊C82
#105894012
#105893502
#105893222
#682393