Finite Group Databasebeta

← #10588473⋊C3 #13141936

Subgroups — #10588473⋊C3 #13141936

Order 1728.

309 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 100 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
1Q8×3−1+2⋊C4C211yesmaximal, characteristic
2#63450⋊C3C211yesmaximal, characteristic
33−1+2×#103C211yesmaximal, characteristic
4#63395⋊C3C211yesmaximal, characteristic
5#63378⋊C3C211yesmaximal, characteristic
6(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic
7#63366⋊C3C211yesmaximal, characteristic
8C3×#4793—31nomaximal
9#10588473C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×3−1+2⋊Q8—21no—
12C2×C8×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14C4⋊(3−1+2⋊C4)—21no—
15C4⋊(3−1+2⋊C4)C2211yescharacteristic
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18C4×3−1+2⋊C4—21no—
19C4×3−1+2⋊C4C2211yescharacteristic
20C2×3−1+2⋊C8C2211yescharacteristic
21C3×Q8×Dic3—31no—
22C3×#888—31no—
23C32×#103D311yescharacteristic
24C3×#835—31no—
25C3×#818—31no—
26C3×Dic3⋊C8—31no—
27C3×#807—31no—
28Q8×Dic9C2×C311yescharacteristic
29#63450C2×C311yescharacteristic
30C9×#103—21no—
31C9×#103C2×C311yescharacteristic
32#63395C2×C311yescharacteristic
33#63378C2×C311yescharacteristic
34Dic9⋊C8C2×C311yescharacteristic
35#63366C2×C311yescharacteristic
36Q8×3−1+2—21no—
37Q8×3−1+2Q812yescomplemented
38C2×C4×3−1+2—21no—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2C2311yescharacteristic
41C2×3−1+2⋊C4—21no—
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4D411yescharacteristic
44C2×3−1+2⋊C4D411yescharacteristic
453−1+2⋊Q8—42no—
46C8×3−1+2—21no—
473−1+2⋊C8—21no—
48C3×#477—91no—
49#4793—31no—
50C2×C32×Q8C2×D311yescharacteristic
51C2×C3×Dic6—61no—
52C2×C32×C8C2×D311yescharacteristic
53C32×C4⋊C4C2×D311yescharacteristic
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—61no—
57C3×Dic3⋊C4—61no—
58C3×C4×Dic3—31no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—31no—
61C2×Q8×C9—21no—
62C2×Q8×C9C22×C311yescharacteristic
63C2×Dic18—21no—
64C2×C8×C9—21no—
65C2×C8×C9C22×C311yescharacteristic
66C9×C4⋊C4—21no—
67C9×C4⋊C4C22×C311yescharacteristic
68C4⋊Dic9—21no—
69C4⋊Dic9C22×C311yescharacteristic
70C4⋊Dic9C22×C311yescharacteristic
71Dic9⋊C4—21no—
72C4×Dic9—21no—
73C4×Dic9C22×C311yescharacteristic
74C2×C9⋊C8C22×C311yescharacteristic
75C22×3−1+2C2×D411yescharacteristic
76C4×3−1+2—22no—
77C4×3−1+2—41no—
78C4×3−1+2C2×Q811yescharacteristic
79C4×3−1+2#4111yescharacteristic
803−1+2⋊C4—21no—
813−1+2⋊C4—41no—
823−1+2⋊C4—41no—
833−1+2⋊C4Q1612yes—
84C3×C4⋊Q8—91no—
85C3×C4×Q8—91no—
86Q8×Dic3—31no—
87#888—31no—
88C3×#107—91no—
89C3×#107—91no—
90C3×C4⋊C8—91no—
91C3×#103—31no—
92C3×#103—91no—
93C3×#103C3×D311yescharacteristic
94#835—31no—
95#818—31no—
96Dic3⋊C8—31no—
97#807—31no—
98C32×Q8—21no—
99C32×Q8Dic612yes—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3×Dic3—31no—
104C2×C3×Dic3—31no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C3×Dic6—122no—
108C32×C8—21no—
109C3×C3⋊C8—61no—
110Q8×C9—21no—
111Q8×C9—22no—
112Q8×C9—41no—
113Q8×C9C3×Q812yescomplemented
114C2×C4×C9—21no—
115C2×C4×C9—21no—
116C2×C4×C9—21no—
117C2×C4×C9—41no—
118C2×C4×C9—41no—
119C2×C4×C9C23×C311yescharacteristic
120C2×Dic9—21no—
121C2×Dic9—21no—
122C2×Dic9C3×D411yescharacteristic
123C2×Dic9C3×D411yescharacteristic
124Dic18—42no—
125C8×C9—21no—
126C8×C9—41no—
127C9⋊C8—21no—
128#477—91noSylow
129C2×3−1+2C22⋊Q811yescharacteristic
130C2×3−1+2C2×Q1611yescharacteristic
131C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
132C2×C3×Q8—31no—
133C2×C3×Q8—181no—
134C2×C3×Q8C2×C3×D311yescharacteristic
135C2×Dic6—61no—
136C2×C3×C8—31no—
137C2×C3×C8—91no—
138C2×C3×C8C2×C3×D311yescharacteristic
139C3×C4⋊C4—31no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—91no—
142C3×C4⋊C4—181no—
143C3×C4⋊C4—181no—
144C3×C4⋊C4C2×C3×D311yescharacteristic
145C3×C42—91no—
146C3×C42—181no—
147C4⋊Dic3—31no—
148C4⋊Dic3—31no—
149C4⋊Dic3—61no—
150Dic3⋊C4—61no—
151C4×Dic3—31no—
152C4×Dic3—61no—
153C2×C3⋊C8—31no—
154C22×C32D3×D411yescharacteristic
155C32×C4—22no—
156C32×C4—41no—
157C32×C4C2×Dic611yescharacteristic
158C32×C4#23711yescharacteristic
159C3×Dic3—32no—
160C3×Dic3—61no—
161C3×Dic3—121no—
162C3×Dic3—121no—
163C22×C9—21no—
164C22×C9C2×C3×D411yescharacteristic
165C4×C9—22no—
166C4×C9—21no—
167C4×C9—21no—
168C4×C9—42no—
169C4×C9—41no—
170C4×C9—81no—
171C4×C9C2×C3×Q811yescharacteristic
172C4×C9C3×#4111yescharacteristic
173Dic9—21no—
174Dic9—41no—
175Dic9—41no—
176Dic9C3×Q1612yes—
177C4⋊Q8—91no—
178C4×Q8—91no—
179#107—91no—
180#107—91no—
181C4⋊C8—91no—
182#103—91no—
183#103AGL(1,ℤ/9)11yescharacteristic
1843−1+2#47711yescharacteristic, complemented, Sylow
185C3×Q8—21no—
186C3×Q8—32no—
187C3×Q8—61no—
188C3×Q8—362no—
189C3×Q8C3×Dic612yes—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—31no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—91no—
196C2×C3×C4—91no—
197C2×C3×C4—181no—
198C2×C3×C4—181no—
199C2×C3×C4C22×C3×D311yescharacteristic
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204Dic6—122no—
205C3×C8—21no—
206C3×C8—61no—
207C3×C8—181no—
208C3⋊C8—61no—
209C2×C32#87811yescharacteristic
210C2×C32#91411yescharacteristic
211C2×C32D3×Q1611yescharacteristic
212C2×C9—21no—
213C2×C9—21no—
214C2×C9—21no—
215C2×C9C3×C22⋊Q811yescharacteristic
216C2×C9C2×C3×Q1611yescharacteristic
217C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
218C2×Q8—181no—
219C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
220C2×C8—91no—
221C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
222C4⋊C4—91no—
223C4⋊C4—91no—
224C4⋊C4—181no—
225C4⋊C4—181no—
226C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
227C42—91no—
228C42—181no—
229C22×C3—31no—
230C22×C3C3×D3×D411yescharacteristic
231C3×C4—22no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—41no—
235C3×C4—62no—
236C3×C4—92no—
237C3×C4—121no—
238C3×C4—181no—
239C3×C4—361no—
240C3×C4—361no—
241C3×C4C2×C3×Dic611yescharacteristic
242C3×C4C3×#23711yescharacteristic
243Dic3—32no—
244Dic3—61no—
245Dic3—121no—
246Dic3—121no—
247C32#479311yescharacteristic
248C9—21no—
249C9C3×#47711yescharacteristic, complemented
250Q8—21no—
251Q8—362no—
252Q83−1+2⋊Q812yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—91no—
256C2×C4—91no—
257C2×C4—181no—
258C2×C4—181no—
259C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
260C8—21no—
261C8—181no—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3C3×#87811yescharacteristic
266C2×C3C3×#91411yescharacteristic
267C2×C3C3×D3×Q1611yescharacteristic
268C22D4×AGL(1,ℤ/9)11yescharacteristic
269C4—22no—
270C4—41no—
271C4—92no—
272C4—181no—
273C4—361no—
274C4—361no—
275C4C2×3−1+2⋊Q811yescharacteristic
276C4#3699⋊C311yescharacteristic
277C3—31no—
278C3C3×#479311yescharacteristic
279C2#63440⋊C311yescharacteristic
280C2#63476⋊C311yescharacteristic
281C2Q16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#477
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
9. #10588473C3semidirect
37. Q8×3−1+2Q8semidirect
113. Q8×C9C3×Q8semidirect
184. 3−1+2#477semidirect
249. C9C3×#477semidirect
252. Q83−1+2⋊Q8semidirect

Maximal subgroups

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

subgroupindex
Q8×3−1+2⋊C42
#63450⋊C32
3−1+2×#1032
#63395⋊C32
#63378⋊C32
(3−1+2⋊C4)⋊C82
#63366⋊C32
C3×#47933
#105884733