Finite Group Databasebeta

← #13166021

Subgroups — #13166021

Order 1728.

474 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 56 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×C3×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593659—31nomaximal
5#10593194—33nomaximal
6#10591597—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊(#41⋊C3)—21no—
9C3×C3⋊#226—21no—
10#88806—41nomaximal
11C2×C3×#239D311yescharacteristic
12C2×C3×#41⋊C3—31no—
13C2×C3⋊(#41⋊C3)D313yes—
14C2×C3⋊#226D311yescharacteristic, complemented
15C2×C3×#226—33no—
16C2×C3×#216—31no—
17#63856—31no—
18C2×#3715—31no—
19C3⋊#859—31no—
20C3⋊#859—33no—
21C3⋊#859—61no—
22C3⋊#859—63no—
23#63662—31no—
24#63563—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26C2×C3⋊(C3⋊Dic3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30#5425—93no—
31#5239—91no—
32#5187—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C3×#239—21no—
35C3×#239—61no—
36C2×C3×D12—31no—
37C2×C3×C4×D3—31no—
38C2×C3×Dic6—31no—
39C3×#41⋊C3—61no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—23no—
42C2×C3×SL(2,𝔽3)C2×D313yes—
43C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
44C3⋊(#41⋊C3)—23no—
45C3⋊#226—21no—
46C3×#226—63no—
47C4⋊(C3⋊Dic3)—31no—
48#3750—121no—
49C2×C3⋊(C3⋊C8)—31no—
50C3×#216—61no—
51C3×#216—61no—
52C2×C3×C3⋊C8—31no—
53#3724—61no—
54#3724—123no—
55(C2×D3)⋊Dic3—61no—
56#3715—62no—
57C22×C33—41no—
58C3⋊(C3⋊Dic3)—42no—
59C3×C3⋊Dic3—81no—
60C32×Dic3—81no—
61C2×C3×#41—31no—
62C2×#239C3⋊D311yescharacteristic
63C2×#41⋊C3—33no—
64C2×#226—33no—
65C2×C3×Q16—91no—
66#946—91no—
67C2×#216—31no—
68C2×#215—91no—
69C2×#214—91no—
70#924—91no—
71#859—93no—
72#859—183no—
73#835—91no—
74#835—91no—
75#835—181no—
76#832—91no—
77#830—91no—
78#821—91no—
79(C2×D3)⋊C8—91no—
80C22×C3×D3—31no—
81C32×Q8—31no—
82C32×Q8#6711yescharacteristic
83C2×C32×C4—31no—
84C2×C32×C4—121no—
85C2×C3⋊Dic3—41no—
86C2×C3⋊Dic3—61no—
87C2×C3⋊Dic3—121no—
88C2×C3⋊Dic3—123no—
89C2×C3⋊Dic3—241no—
90C2×C3⋊Dic3—243no—
91C2×C3×Dic3—43no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—123no—
94C2×C3×Dic3S411yescharacteristic
95C3×D12—32no—
96C3×D12—61no—
97C3×C4×D3—61no—
98C3×C4×D3—61no—
99C3×Dic6—61no—
100C3×Dic6—61no—
101C3×SL(2,𝔽3)—21no—
102C3×SL(2,𝔽3)—23no—
103C3×SL(2,𝔽3)#6713yes—
104C3×SL(2,𝔽3)#6711yescharacteristic
105C3⋊(C3⋊C8)—61no—
106C3×C3⋊C8—61no—
107#452—271noSylow
108C2×C33—41no—
109C2×C33—41no—
110C2×C33—41no—
111C3×#41—61no—
112C3×#41—181no—
113C2×C3×Q8—21no—
114C2×C3×Q8—91no—
115C2×C3×Q8D3211yescharacteristic
116C2×C3×Q8C2×C3⋊D311yescharacteristic
117C2×C3×D4—91no—
118C22×C3×C4—91no—
119#239—21no—
120#239—61no—
121C2×D12—31no—
122C2×C4×D3—31no—
123C2×Dic6—31no—
124#41⋊C3—63no—
125C2×SL(2,𝔽3)—23no—
126C2×SL(2,𝔽3)D3213yes—
127#226—63no—
128C3×Q16—181no—
129C3×Q16—181no—
130C2×C3×C8—91no—
131#217—181no—
132#216—61no—
133#216—61no—
134#215—182no—
135#214—182no—
136(C2×D3)⋊C4—181no—
137C4⋊Dic3—91no—
138C4⋊Dic3—91no—
139C4⋊Dic3—181no—
140Dic3⋊C4—181no—
141Dic3⋊C4—181no—
142Dic3⋊C4—363no—
143C2×C3⋊C8—31no—
144C2×C3⋊C8—91no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—181no—
147C22×C32—41no—
148C22×C32—43no—
149C22×C32—83no—
150C22×C32—81no—
151C22×C32C2×S411yescharacteristic
152C2×C3×D3—61no—
153C2×C3×D3—61no—
154C2×C3×D3—61no—
155C32×C4—31no—
156C32×C4—31no—
157C32×C4—241no—
158C3⋊Dic3—81no—
159C3⋊Dic3—122no—
160C3⋊Dic3—126no—
161C3⋊Dic3—121no—
162C3⋊Dic3—246no—
163C3⋊Dic3—242no—
164C3×Dic3—21no—
165C3×Dic3—83no—
166C3×Dic3—121no—
167C3×Dic3—243no—
168C2×#41—31no—
169C2×Q16—91no—
170C2×SD16—271no—
171C22⋊Q8—271no—
172#103—271no—
173#102—271no—
174C22⋊C8—271no—
175C33—41noSylow
176C23×C3—91no—
177C22×D3—31no—
178C3×Q8—21no—
179C3×Q8—31no—
180C3×Q8—31no—
181C3×Q8—61no—
182C3×Q8—181no—
183C3×Q8—181no—
184C3×Q8#62511yescharacteristic
185C3×Q8#63811yescharacteristic
186C3×D4—92no—
187C3×D4—181no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—61no—
192C2×C3×C4—123no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4—181no—
196C2×Dic3—61no—
197C2×Dic3—123no—
198C2×Dic3—181no—
199C2×Dic3—181no—
200C2×Dic3—363no—
201C2×Dic3—361no—
202C2×Dic3—723no—
203C2×Dic3C3⋊S411yescharacteristic
204D12—32no—
205D12—61no—
206C4×D3—61no—
207C4×D3—61no—
208Dic6—61no—
209Dic6—61no—
210SL(2,𝔽3)—23no—
211SL(2,𝔽3)#62513yes—
212C3×C8—181no—
213C3⋊C8—61no—
214C3⋊C8—181no—
215C3⋊C8—181no—
216C3⋊C8—361no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—43no—
221C2×C32—43no—
222C2×C32—43no—
223C2×C32—83no—
224C2×C32—83no—
225C2×C32—83no—
226C2×C32—81no—
227C2×C32—81no—
228C2×C32—81no—
229C2×C32#98411yescharacteristic
230C2×C32#98511yescharacteristic
231C2×C32GL(2,ℤ/4)11yescharacteristic
232C3×D3—121no—
233#41—61no—
234#41—181no—
235C2×Q8—91no—
236C2×Q8D3×C3⋊D311yescharacteristic
237C2×D4—91no—
238C22×C4—91no—
239Q16—181no—
240Q16—181no—
241SD16—542no—
242C2×C8—91no—
243C2×C8—271no—
244C4⋊C4—271no—
245C4⋊C4—541no—
246C22⋊C4—541no—
247C22×C3—21no—
248C22×C3—43no—
249C22×C3—83no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3D3×S411yescharacteristic
254C22×C3C2×C3⋊S411yescharacteristic
255C2×D3—61no—
256C2×D3—61no—
257C2×D3—61no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—243no—
266C3×C4—361no—
267Dic3—21no—
268Dic3—121no—
269Dic3—243no—
270Dic3—366no—
271Dic3—361no—
272Dic3—361no—
273Dic3—721no—
274Dic3—726no—
275C32—41no—
276C32—43no—
277C32—83no—
278C32—81no—
279C32#542511yescharacteristic, complemented
280C23—91no—
281Q8—31no—
282Q8—181no—
283Q8—181no—
284Q8#629011yescharacteristic
285D4—92no—
286D4—181no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—181no—
290C2×C4—181no—
291C2×C4—181no—
292C2×C4—541no—
293C8—181no—
294C8—541no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—43no—
299C2×C3—43no—
300C2×C3—43no—
301C2×C3—83no—
302C2×C3—83no—
303C2×C3—83no—
304C2×C3—361no—
305C2×C3#6419611yescharacteristic
306C2×C3#6419811yescharacteristic
307C2×C3#6420911yescharacteristic
308C2×C3#6426511yescharacteristic
309C2×C3#6426611yescharacteristic
310C2×C3#6426911yescharacteristic
311D3—121no—
312C22—181no—
313C22—181no—
314C22—181no—
315C22D3×C3⋊S411yescharacteristic
316C4—31no—
317C4—31no—
318C4—61no—
319C4—361no—
320C4—1081no—
321C3—21no—
322C3—43no—
323C3—83no—
324C3#1059319411yescharacteristic, complemented
325C3#1059365911yescharacteristic, complemented
326C2—361no—
327C2#1172419111yescharacteristic
328C2#1172419311yescharacteristic
329C2#1172420411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊(#41⋊C3)C2
hypocenterC32×SL(2,𝔽3)D4
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
2. C2×C3×C3⋊#226C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
14. C2×C3⋊#226D3semidirect
279. C32#5425semidirect
324. C3#10593194semidirect
325. C3#10593659semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊(#41⋊C3)2
C2×C3×C3⋊#2262
C3⋊(C3⋊#859)2
#105936593
#105931943
#105915973
#888064