Finite Group Databasebeta

← #13166086

Subgroups — #13166086

Order 1728.

312 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 44 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
1SL(2,𝔽3)×C3⋊Dic3C211yesmaximal, characteristic
2C3×C3⋊#859C211yesmaximal, characteristic
3C3×C3⋊#859C211yesmaximal, characteristic
4#10593199—31nomaximal
5#10593196—31nomaximal
6#10591615—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8#88815—41nomaximal
9Q8×C3⋊Dic3D311yescharacteristic
10#63872—31no—
11Dic3×SL(2,𝔽3)—31no—
12Dic3×SL(2,𝔽3)—31no—
13Dic3×SL(2,𝔽3)—61no—
14C3⋊#859D311yescharacteristic
15C3⋊#859D311yescharacteristic
16C3×#859—31no—
17C3×#859—31no—
18C3×#859—61no—
19C3×#859—61no—
20C3×#835—31no—
21C3×#835—31no—
22#63575—31no—
23#63573—31no—
24#63572—31no—
25C2×C3×C3⋊Dic3—41no—
26C2×C3×C3⋊Dic3—41no—
27C2×C3×C3⋊Dic3—41no—
28C32×SL(2,𝔽3)Q811yescharacteristic
29#5389—91no—
30#4799—91no—
31#4796—91no—
32C2×C32×Q8C2×D311yescharacteristic
33C2×C3×SL(2,𝔽3)—21no—
34C2×C3×SL(2,𝔽3)—21no—
35C2×C3×SL(2,𝔽3)—21no—
36C2×C3×SL(2,𝔽3)—41no—
37C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
38C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
39C4⋊(C3⋊Dic3)—31no—
40C4×C3⋊Dic3—31no—
41C3×C4⋊Dic3—31no—
42C3×C4⋊Dic3—31no—
43C2×C3×C3⋊C8—31no—
44C2×C3×C3⋊C8—31no—
45#3724—121no—
46Dic3⋊Dic3—61no—
47Dic3⋊Dic3—61no—
48Dic3⋊Dic3—121no—
49C22×C33—41no—
50C3×C3⋊Dic3—42no—
51C3×C3⋊Dic3—81no—
52C3×C3⋊Dic3—81no—
53Q8×Dic3—31no—
54Q8×Dic3—31no—
55Q8×Dic3—61no—
56#890—91no—
57#890—91no—
58C4×SL(2,𝔽3)—91no—
59#859—31no—
60#859—31no—
61#859—61no—
62#859—61no—
63C3×#103—91no—
64C3×#103—91no—
65#835—31no—
66#835—31no—
67#818—91no—
68#817—91no—
69Dic3⋊C8—91no—
70Dic3⋊C8—91no—
71#808—91no—
72#807—91no—
73C32×Q8—31no—
74C32×Q8Dic611yescharacteristic
75C2×C32×C4—31no—
76C2×C3⋊Dic3—31no—
77C2×C3⋊Dic3—41no—
78C2×C3⋊Dic3—41no—
79C2×C3⋊Dic3S411yescharacteristic
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—121no—
83C2×C3×Dic3—121no—
84C2×C3×Dic3—121no—
85C2×C3×Dic3—121no—
86C2×C3×Dic3—241no—
87C2×C3×Dic3—241no—
88C2×C3×Dic3—241no—
89C3×SL(2,𝔽3)—21no—
90C3×SL(2,𝔽3)—21no—
91C3×SL(2,𝔽3)—21no—
92C3×SL(2,𝔽3)—41no—
93C3×SL(2,𝔽3)Dic611yescharacteristic
94C3×SL(2,𝔽3)Dic611yescharacteristic
95C3×C3⋊C8—61no—
96C3×C3⋊C8—61no—
97#479—271noSylow
98C2×C33—41no—
99C2×C33—41no—
100C2×C33—41no—
101C2×C3×Q8—21no—
102C2×C3×Q8D3211yescharacteristic
103C2×C3×Q8D3211yescharacteristic
104C2×SL(2,𝔽3)—21no—
105C2×SL(2,𝔽3)—21no—
106C2×SL(2,𝔽3)—41no—
107C2×SL(2,𝔽3)D3211yescharacteristic
108C2×C3×C8—91no—
109C2×C3×C8—91no—
110C3×C4⋊C4—91no—
111C3×C4⋊C4—91no—
112C4⋊Dic3—31no—
113C4⋊Dic3—31no—
114C4⋊Dic3—91no—
115C4⋊Dic3—91no—
116C4⋊Dic3—181no—
117C4⋊Dic3—181no—
118C4⋊Dic3—181no—
119Dic3⋊C4—181no—
120Dic3⋊C4—181no—
121Dic3⋊C4—361no—
122C4×Dic3—91no—
123C4×Dic3—91no—
124C4×Dic3—181no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—31no—
127C22×C32—41no—
128C22×C32—41no—
129C22×C32—81no—
130C22×C32—81no—
131C22×C32—81no—
132C22×C32—161no—
133C22×C32C2×S411yescharacteristic
134C32×C4—31no—
135C32×C4—31no—
136C3⋊Dic3—21no—
137C3⋊Dic3—42no—
138C3⋊Dic3—61no—
139C3⋊Dic3—81no—
140C3×Dic3—122no—
141C3×Dic3—121no—
142C3×Dic3—121no—
143C3×Dic3—241no—
144C3×Dic3—241no—
145C3×Dic3—242no—
146C3×Dic3—241no—
147C3×Dic3—242no—
148C3×Dic3—242no—
149#125—271no—
150C4×Q8—91no—
151#107—271no—
152#106—271no—
153C4⋊C8—271no—
154#103—91no—
155#103—91no—
156C33—41noSylow
157C3×Q8—21no—
158C3×Q8—31no—
159C3×Q8—31no—
160C3×Q8—61no—
161C3×Q8#62711yescharacteristic
162C3×Q8#62711yescharacteristic
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—61no—
166C2×C3×C4—181no—
167C2×C3×C4—181no—
168C2×C3×C4—361no—
169C2×Dic3—31no—
170C2×Dic3—31no—
171C2×Dic3—61no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—91no—
175C2×Dic3—91no—
176C2×Dic3—121no—
177C2×Dic3—121no—
178C2×Dic3—181no—
179C2×Dic3—241no—
180C2×Dic3—241no—
181SL(2,𝔽3)—21no—
182SL(2,𝔽3)—21no—
183SL(2,𝔽3)—41no—
184SL(2,𝔽3)#62711yescharacteristic
185C3×C8—181no—
186C3×C8—181no—
187C3⋊C8—61no—
188C3⋊C8—61no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—41no—
194C2×C32—41no—
195C2×C32—81no—
196C2×C32—81no—
197C2×C32—81no—
198C2×C32—81no—
199C2×C32—81no—
200C2×C32—81no—
201C2×C32—81no—
202C2×C32—81no—
203C2×C32—81no—
204C2×C32—161no—
205C2×C32—161no—
206C2×C32—161no—
207C2×C32A4⋊Q811yescharacteristic
208C2×C32#98311yescharacteristic
209C2×C32#98511yescharacteristic
210C2×Q8#111311yescharacteristic
211C2×C8—91no—
212C2×C8—91no—
213C4⋊C4—91no—
214C4⋊C4—91no—
215C4⋊C4—271no—
216C4⋊C4—541no—
217C4⋊C4—541no—
218C42—271no—
219C22×C3—21no—
220C22×C3—41no—
221C22×C3—81no—
222C22×C3—81no—
223C22×C3—161no—
224C22×C3D3×S411yescharacteristic
225C22×C3D3×S411yescharacteristic
226C3×C4—31no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4—361no—
233C3×C4—361no—
234C3×C4—721no—
235Dic3—62no—
236Dic3—61no—
237Dic3—61no—
238Dic3—122no—
239Dic3—121no—
240Dic3—121no—
241Dic3—181no—
242Dic3—181no—
243Dic3—241no—
244Dic3—242no—
245Dic3—242no—
246Dic3—361no—
247C32—41no—
248C32—41no—
249C32—81no—
250C32—81no—
251C32—81no—
252C32—161no—
253C32#538911yescharacteristic, complemented
254Q8—31no—
255Q8#629611yescharacteristic
256C2×C4—31no—
257C2×C4—91no—
258C2×C4—181no—
259C2×C4—181no—
260C2×C4—271no—
261C8—181no—
262C8—181no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—41no—
267C2×C3—41no—
268C2×C3—41no—
269C2×C3—81no—
270C2×C3—81no—
271C2×C3—81no—
272C2×C3—81no—
273C2×C3—81no—
274C2×C3—81no—
275C2×C3—161no—
276C2×C3—161no—
277C2×C3—161no—
278C2×C3#6419711yescharacteristic
279C2×C3#6419811yescharacteristic
280C2×C3#6420111yescharacteristic
281C2×C3#6420211yescharacteristic
282C2×C3#6420411yescharacteristic
283C2×C3#6420411yescharacteristic
284C22#8910111yescharacteristic
285C4—31no—
286C4—31no—
287C4—181no—
288C4—361no—
289C4—361no—
290C4—541no—
291C3—21no—
292C3—41no—
293C3—81no—
294C3—81no—
295C3—161no—
296C3#1059319611yescharacteristic, complemented
297C3#1059319911yescharacteristic, complemented
298C2#1172420711yescharacteristic
299C2#1172421011yescharacteristic
300C2#1172421211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, prime radical, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)Q8
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
253. C32#5389semidirect
296. C3#10593196semidirect
297. C3#10593199semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×C3⋊Dic32
C3×C3⋊#8592
#105931993
#105931963
#105916153
#888154