Finite Group Databasebeta

← #10593659⋊C3 #13146923

Subgroups — #10593659⋊C3 #13146923

Order 1728.

324 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 39 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
1C2×#41⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×(C3⋊#226)⋊C3C211yesmaximal, characteristic, complemented
3(C3⋊#859)⋊C3C211yesmaximal, characteristic, complemented
4#10593659C311yesmaximal, characteristic, complemented
5C3×#5239—31nomaximal
6#41⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8(C3⋊#226)⋊C3—21no—
9#3750⋊C3—41nomaximal
10C2×C32×#41D311yescharacteristic
11C2×C3×#41⋊C3—21no—
12C2×C3×#41⋊C3C2×C311yescharacteristic
13C2×C3⋊#226C2×C311yescharacteristic, complemented
14C2×C3×#216—31no—
15C2×C3×#214—31no—
16C3×#924—31no—
17C3⋊#859C2×C311yescharacteristic, complemented
18C3×#835—31no—
19C3×#832—31no—
20C3×#830—31no—
21C2×C4×H(3,𝔽3)—41no—
22C2×(C3⋊Dic3)⋊C3—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24Q8⋊H(3,𝔽3)D411yescharacteristic
25#5425—91no—
26C3×#452—91no—
27#5239—31no—
28C32×#41—21no—
29C32×#41—61no—
30C2×C32×Q8C2×D311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×C3×Dic6—31no—
34C3×#41⋊C3—22no—
35C3×#41⋊C3—21no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
38C3⋊#226—21no—
39#3750—41no—
40C3×#217—61no—
41C3×#216—61no—
42C3×#216—61no—
43C3×#214—62no—
44C3×C4⋊Dic3—31no—
45C3×Dic3⋊C4—61no—
46C2×C3×C3⋊C8—31no—
47C2×C3×C3⋊C8—31no—
48C22×H(3,𝔽3)—41no—
49C4×H(3,𝔽3)—81no—
50(C3⋊Dic3)⋊C3—42no—
51(C3⋊Dic3)⋊C3—81no—
52C2×C3×#41—31no—
53C2×C3×#41C3×D311yescharacteristic
54C2×#41⋊C3—31no—
55C2×#41⋊C3—61no—
56C2×#226—91no—
57C2×C3×Q16—91no—
58C2×C3×SD16—91no—
59C3×C22⋊Q8—91no—
60C2×#216—31no—
61C2×#214—31no—
62#924—31no—
63#859—91no—
64C3×#103—91no—
65C3×#102—91no—
66C3×C22⋊C8—91no—
67#835—31no—
68#832—31no—
69#830—31no—
70C23×C32—31no—
71C32×Q8—31no—
72C32×Q8#6711yescharacteristic
73C32×D4—32no—
74C32×D4—61no—
75C2×C32×C4—31no—
76C2×C32×C4—41no—
77C2×C32×C4—61no—
78C2×C32×C4—61no—
79C2×C32×C4—81no—
80C2×C32×C4S411yescharacteristic
81C2×C3⋊Dic3—41no—
82C2×C3⋊Dic3—41no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C3×Dic6—61no—
86C3×Dic6—61no—
87C3×SL(2,𝔽3)—21no—
88C3×SL(2,𝔽3)C3×D411yescharacteristic
89C3×C3⋊C8—61no—
90C3×C3⋊C8—61no—
91#452—91noSylow
92C2×H(3,𝔽3)—41no—
93C2×H(3,𝔽3)—41no—
94C2×H(3,𝔽3)—41no—
95C3×#41—21no—
96C3×#41—61no—
97C3×#41—62no—
98C3×#41—61no—
99C3×#41—61no—
100C2×C3×Q8—31no—
101C2×C3×Q8—91no—
102C2×C3×Q8C2×C3×D311yescharacteristic
103C2×C3×D4—31no—
104C2×C3×D4—31no—
105C2×C3×D4—61no—
106C22×C3×C4—31no—
107C22×C3×C4—31no—
108C22×C3×C4—61no—
109C2×Dic6—31no—
110#41⋊C3—61no—
111#41⋊C3—62no—
112C2×SL(2,𝔽3)—31no—
113C2×SL(2,𝔽3)—61no—
114#226—181no—
115C3×Q16—181no—
116C3×Q16—181no—
117C3×SD16—182no—
118C2×C3×C8—91no—
119C2×C3×C8—91no—
120C3×C4⋊C4—91no—
121C3×C4⋊C4—181no—
122C3×C22⋊C4—181no—
123#217—61no—
124#216—61no—
125#216—61no—
126#214—62no—
127C4⋊Dic3—31no—
128Dic3⋊C4—61no—
129Dic3⋊C4—361no—
130C2×C3⋊C8—31no—
131C2×C3⋊C8—31no—
132C22×C32—41no—
133C22×C32—61no—
134C22×C32—61no—
135C22×C32—61no—
136C22×C32—81no—
137C22×C32C2×S411yescharacteristic
138C32×C4—21no—
139C32×C4—31no—
140C32×C4—31no—
141C32×C4—81no—
142C32×C4—82no—
143C3⋊Dic3—42no—
144C3⋊Dic3—81no—
145C3×Dic3—121no—
146C3×Dic3—121no—
147C2×#41(C3⋊D3)⋊C311yescharacteristic
148C2×Q16—91no—
149C2×SD16—91no—
150C22⋊Q8—91no—
151#103—91no—
152#102—91no—
153C22⋊C8—91no—
154H(3,𝔽3)—41noSylow
155C23×C3—31no—
156C23×C3—31no—
157C23×C3—61no—
158C3×Q8—31no—
159C3×Q8—31no—
160C3×Q8—31no—
161C3×Q8—61no—
162C3×Q8—181no—
163C3×Q8—181no—
164C3×Q8C3×#6711yescharacteristic
165C3×D4—32no—
166C3×D4—32no—
167C3×D4—62no—
168C3×D4—61no—
169C3×D4—61no—
170C3×D4—62no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—121no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184C2×C3×C4—241no—
185C2×C3×C4C3×S411yescharacteristic
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—361no—
189C2×Dic3—361no—
190Dic6—61no—
191Dic6—61no—
192SL(2,𝔽3)—31no—
193SL(2,𝔽3)—61no—
194C3×C8—181no—
195C3×C8—181no—
196C3⋊C8—61no—
197C3⋊C8—61no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—41no—
201C2×C32—81no—
202C2×C32—81no—
203C2×C32—81no—
204C2×C32—121no—
205C2×C32#98411yescharacteristic
206C2×C32#98511yescharacteristic
207C2×C32GL(2,ℤ/4)11yescharacteristic
208#41—21no—
209#41—61no—
210C2×Q8—91no—
211C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
212C2×D4—31no—
213C22×C4—31no—
214Q16—181no—
215Q16—181no—
216SD16—182no—
217C2×C8—91no—
218C2×C8—91no—
219C4⋊C4—91no—
220C4⋊C4—181no—
221C22⋊C4—181no—
222C22×C3—31no—
223C22×C3—62no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3—61no—
227C22×C3—61no—
228C22×C3—61no—
229C22×C3—61no—
230C22×C3—121no—
231C22×C3—121no—
232C22×C3—121no—
233C22×C3—241no—
234C22×C3C2×C3×S411yescharacteristic
235C3×C4—21no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—31no—
239C3×C4—31no—
240C3×C4—61no—
241C3×C4—61no—
242C3×C4—61no—
243C3×C4—241no—
244C3×C4—242no—
245C3×C4—361no—
246C3×C4—361no—
247Dic3—121no—
248Dic3—121no—
249Dic3—362no—
250Dic3—721no—
251C32—41no—
252C32—81no—
253C32#542511yescharacteristic, complemented
254C23—31no—
255Q8—31no—
256Q8—181no—
257Q8—181no—
258Q8#638⋊C311yescharacteristic
259D4—32no—
260D4—61no—
261C2×C4—31no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4(C3⋊S4)⋊C311yescharacteristic
267C8—181no—
268C8—181no—
269C2×C3—31no—
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—121no—
273C2×C3—122no—
274C2×C3—121no—
275C2×C3—121no—
276C2×C3—121no—
277C2×C3—121no—
278C2×C3—241no—
279C2×C3—241no—
280C2×C3—241no—
281C2×C3C3×#98411yescharacteristic
282C2×C3C3×#98511yescharacteristic
283C2×C3C3×GL(2,ℤ/4)11yescharacteristic
284C22—61no—
285C22—61no—
286C22—61no—
287C22C2×(C3⋊S4)⋊C311yescharacteristic
288C4—21no—
289C4—31no—
290C4—31no—
291C4—361no—
292C4—361no—
293C3—31no—
294C3—121no—
295C3—241no—
296C3C3×#542511yescharacteristic
297C2—121no—
298C2#64265⋊C311yescharacteristic
299C2#64266⋊C311yescharacteristic
300C2#64269⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
Frattini, socleC22×C3C2×C3×S4
FittingC2×C32×#41D3
prime radicalC2×#41⋊H(3,𝔽3)C2
hypercenterC2×C4(C3⋊S4)⋊C3
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊#226)⋊C3C2semidirect
3. (C3⋊#859)⋊C3C2semidirect
4. #10593659C3semidirect
13. C2×C3⋊#226C2×C3semidirect
17. C3⋊#859C2×C3semidirect
253. C32#5425semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
C2×(C3⋊#226)⋊C32
(C3⋊#859)⋊C32
#105936593
C3×#52393
#3750⋊C34