Finite Group Databasebeta

← #10588469⋊C3 #13141944

Subgroups — #10588469⋊C3 #13141944

Order 1728.

345 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 112 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
2C2×#3674⋊C3C211yesmaximal, characteristic
3#63449⋊C3C211yesmaximal, characteristic
43−1+2×#103C211yesmaximal, characteristic
5#63376⋊C3C211yesmaximal, characteristic
6#63375⋊C3C211yesmaximal, characteristic
7#63367⋊C3C211yesmaximal, characteristic
8C3×#4789—31nomaximal
9#10588469C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×3−1+2⋊Q8C2211yescharacteristic
12C2×C8×3−1+2C2211yescharacteristic
13C4⋊C4×3−1+2C2211yescharacteristic
14#3674⋊C3C414yescomplemented
15C4⋊(3−1+2⋊C4)—21no—
16C4⋊(3−1+2⋊C4)C2211yescharacteristic
17(3−1+2⋊C4)⋊C4—21no—
18C4×3−1+2⋊C4—21no—
19C4×3−1+2⋊C4—21no—
20C4×3−1+2⋊C4C2211yescharacteristic
21C2×3−1+2⋊C8C2211yescharacteristic
22C3×Q8×Dic3—31no—
23C2×C3×#216—31no—
24C3×#887—31no—
25C32×#103D311yescharacteristic
26C3×#816—31no—
27C3×#815—31no—
28C3×#808—31no—
29Q8×Dic9C2×C311yescharacteristic
30C2×#3674C2×C311yescharacteristic
31#63449C2×C311yescharacteristic
32C9×#103—21no—
33C9×#103C2×C311yescharacteristic
34#63376C2×C311yescharacteristic
35#63375C2×C311yescharacteristic
36#63367C2×C311yescharacteristic
37Q8×3−1+2—21no—
38Q8×3−1+2C2×C412yes—
39C2×C4×3−1+2—21no—
40C2×C4×3−1+2—21no—
41C2×C4×3−1+2C2311yescharacteristic
42C2×3−1+2⋊C4—21no—
43C2×3−1+2⋊C4—21no—
44C2×3−1+2⋊C4D411yescharacteristic
45C2×3−1+2⋊C4D411yescharacteristic
463−1+2⋊Q8—21no—
473−1+2⋊Q8C2×C412yes—
48C8×3−1+2—21no—
493−1+2⋊C8C2×C412yes—
50C3×#441—91no—
51#4789—31no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C3×Dic6—31no—
54C2×C32×C8C2×D311yescharacteristic
55C32×C4⋊C4C2×D311yescharacteristic
56C3×#216—34no—
57C3×C4⋊Dic3—31no—
58C3×C4⋊Dic3—61no—
59C3×Dic3⋊C4—61no—
60C3×C4×Dic3—31no—
61C3×C4×Dic3—61no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—31no—
64C2×Q8×C9—21no—
65C2×Q8×C9C22×C311yescharacteristic
66C2×Dic18C22×C311yescharacteristic
67C2×C8×C9—21no—
68C2×C8×C9C22×C311yescharacteristic
69C9×C4⋊C4—21no—
70C9×C4⋊C4C22×C311yescharacteristic
71#3674C3×C414yescomplemented
72C4⋊Dic9—21no—
73C4⋊Dic9C22×C311yescharacteristic
74Dic9⋊C4—21no—
75C4×Dic9—21no—
76C4×Dic9—21no—
77C4×Dic9C22×C311yescharacteristic
78C2×C9⋊C8C22×C311yescharacteristic
79C22×3−1+2C2×D411yescharacteristic
80C4×3−1+2—22no—
81C4×3−1+2—41no—
82C4×3−1+2C22×C411yescharacteristic
83C4×3−1+2#4111yescharacteristic
843−1+2⋊C4—22no—
853−1+2⋊C4—21no—
863−1+2⋊C4—21no—
873−1+2⋊C4—41no—
88C2×C3×Q16—91no—
89C3×C4×Q8—91no—
90C3×C4×Q8—91no—
91Q8×Dic3—31no—
92C2×#216—31no—
93#887—31no—
94C3×#106—91no—
95C3×#103—31no—
96C3×#103—91no—
97C3×#103C3×D311yescharacteristic
98C3×#97—91no—
99#816—31no—
100#815—31no—
101#808—31no—
102C32×Q8—21no—
103C32×Q8C4×D312yes—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3×Dic3—31no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—61no—
110C2×C3×Dic3—61no—
111C3×Dic6—32no—
112C3×Dic6—61no—
113C32×C8—21no—
114C3×C3⋊C8—32no—
115Q8×C9—21no—
116Q8×C9—22no—
117Q8×C9—41no—
118Q8×C9C2×C3×C412yes—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—21no—
122C2×C4×C9—41no—
123C2×C4×C9—41no—
124C2×C4×C9C23×C311yescharacteristic
125C2×Dic9—21no—
126C2×Dic9—21no—
127C2×Dic9C3×D411yescharacteristic
128C2×Dic9C3×D411yescharacteristic
129Dic18—21no—
130Dic18C2×C3×C412yes—
131C8×C9—21no—
132C8×C9—41no—
133C9⋊C8C2×C3×C412yes—
134#441—91noSylow
135C2×3−1+2C4×D411yescharacteristic
136C2×3−1+2#13711yescharacteristic
137C2×3−1+2#13711yescharacteristic
138C2×C3×Q8—31no—
139C2×C3×Q8—91no—
140C2×C3×Q8C2×C3×D311yescharacteristic
141C2×Dic6—31no—
142C3×Q16—94no—
143C2×C3×C8—31no—
144C2×C3×C8—91no—
145C2×C3×C8C2×C3×D311yescharacteristic
146C3×C4⋊C4—31no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—181no—
149C3×C4⋊C4—181no—
150C3×C4⋊C4C2×C3×D311yescharacteristic
151C3×C42—91no—
152C3×C42—181no—
153C3×C42—181no—
154#216—34no—
155C4⋊Dic3—31no—
156C4⋊Dic3—61no—
157Dic3⋊C4—61no—
158C4×Dic3—31no—
159C4×Dic3—61no—
160C4×Dic3—61no—
161C2×C3⋊C8—31no—
162C22×C32D3×D411yescharacteristic
163C32×C4—22no—
164C32×C4—41no—
165C32×C4C2×C4×D311yescharacteristic
166C32×C4#23711yescharacteristic
167C3×Dic3—62no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C3×Dic3—121no—
171C22×C9—21no—
172C22×C9C2×C3×D411yescharacteristic
173C4×C9—22no—
174C4×C9—21no—
175C4×C9—21no—
176C4×C9—42no—
177C4×C9—41no—
178C4×C9—81no—
179C4×C9C22×C3×C411yescharacteristic
180C4×C9C3×#4111yescharacteristic
181Dic9—22no—
182Dic9—21no—
183Dic9—21no—
184Dic9—41no—
185C2×Q16—91no—
186C4×Q8—91no—
187C4×Q8—91no—
188#106—91no—
189#103—91no—
190#103AGL(1,ℤ/9)11yescharacteristic
191#97—91no—
1923−1+2#44111yescharacteristic, complemented, Sylow
193C3×Q8—21no—
194C3×Q8—32no—
195C3×Q8—61no—
196C3×Q8—92no—
197C3×Q8—181no—
198C3×Q8C3×C4×D312yes—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—31no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4C22×C3×D311yescharacteristic
209C2×Dic3—31no—
210C2×Dic3—31no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213Dic6—32no—
214Dic6—61no—
215C3×C8—21no—
216C3×C8—61no—
217C3×C8—92no—
218C3⋊C8—32no—
219C2×C32#67⋊C411yescharacteristic
220C2×C32#91511yescharacteristic
221C2×C32#91811yescharacteristic
222C2×C9—21no—
223C2×C9—21no—
224C2×C9—21no—
225C2×C9C3×C4×D411yescharacteristic
226C2×C9C3×#13711yescharacteristic
227C2×C9C3×#13711yescharacteristic
228C2×Q8—91no—
229C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
230Q16—94no—
231C2×C8—91no—
232C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
233C4⋊C4—91no—
234C4⋊C4—181no—
235C4⋊C4—181no—
236C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
237C42—91no—
238C42—181no—
239C42—181no—
240C22×C3—31no—
241C22×C3C3×D3×D411yescharacteristic
242C3×C4—22no—
243C3×C4—31no—
244C3×C4—31no—
245C3×C4—41no—
246C3×C4—62no—
247C3×C4—121no—
248C3×C4—181no—
249C3×C4—181no—
250C3×C4—182no—
251C3×C4—361no—
252C3×C4C2×C3×C4×D311yescharacteristic
253C3×C4C3×#23711yescharacteristic
254Dic3—62no—
255Dic3—61no—
256Dic3—61no—
257Dic3—121no—
258C32#478911yescharacteristic
259C9—21no—
260C9C3×#44111yescharacteristic, complemented
261Q8—21no—
262Q8—92no—
263Q8—181no—
264Q8C4×AGL(1,ℤ/9)12yes—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—91no—
268C2×C4—91no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
272C8—21no—
273C8—92no—
274C2×C3—31no—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3C3×#67⋊C411yescharacteristic
278C2×C3C3×#91511yescharacteristic
279C2×C3C3×#91811yescharacteristic
280C22D4×AGL(1,ℤ/9)11yescharacteristic
281C4—22no—
282C4—41no—
283C4—181no—
284C4—181no—
285C4—182no—
286C4—361no—
287C4C2×C4×AGL(1,ℤ/9)11yescharacteristic
288C4#3699⋊C311yescharacteristic
289C3—31no—
290C3C3×#478911yescharacteristic
291C2(#611⋊C3)⋊C411yescharacteristic
292C2#63477⋊C311yescharacteristic
293C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutatorC4×C9C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radicalC22×C32D3×D4
cosocleC2×C4×C9C23×C3
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#441
LCSG ▷ 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. #10588469C3semidirect
14. #3674⋊C3C4semidirect
71. #3674C3×C4semidirect
192. 3−1+2#441semidirect
260. C9C3×#441semidirect

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
C2×#3674⋊C32
#63449⋊C32
3−1+2×#1032
#63376⋊C32
#63375⋊C32
#63367⋊C32
C3×#47893
#105884693