Finite Group Databasebeta

← #13166030

Subgroups — #13166030

Order 1728.

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

normal subgroups only

This group has 569 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⋊GL(2,𝔽3))C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593629—31nomaximal
5#10593208—33nomaximal
6#10591607—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊GL(2,𝔽3))—21no—
9C3×C3⋊#226—21no—
10#88804—41nomaximal
11C2×C3⋊GL(2,𝔽3)—31no—
12C2×C3⋊GL(2,𝔽3)—33no—
13C2×C3⋊GL(2,𝔽3)—61no—
14C2×C3⋊GL(2,𝔽3)—63no—
15C2×C3⋊#226D311yescharacteristic, complemented
16C2×C3×#226—33no—
17C2×#3755—31no—
18C3×Q8×Dic3D311yescharacteristic, complemented
19C2×C3×#216—31no—
20C3×C4×SL(2,𝔽3)—31no—
21#63864—31no—
22Dic3×SL(2,𝔽3)D313yescomplemented
23Dic3⋊(C3⋊C8)—31no—
24#63565—31no—
25#63564—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30#5395—93no—
31#5185—91no—
32#5028—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C4⋊(C3⋊D3)—31no—
35C2×C3×Dic6—31no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—23no—
38C2×C3×SL(2,𝔽3)C2×D313yes—
39C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
40C3⋊GL(2,𝔽3)—62no—
41C3⋊GL(2,𝔽3)—66no—
42C3⋊GL(2,𝔽3)—63no—
43C3⋊GL(2,𝔽3)—61no—
44C3⋊#226—21no—
45C3×#226—63no—
46#3755—61no—
47#3755—61no—
48#3752—121no—
49C2×C3⋊(C3⋊C8)—31no—
50C3×#216—61no—
51C3×#216—61no—
52C3×C4⋊Dic3—31no—
53C3×C4×Dic3—31no—
54C2×C3×C3⋊C8—31no—
55#3722—61no—
56#3722—61no—
57#3722—123no—
58C22×C33—41no—
59C2×C3⋊(C3⋊D3)—42no—
60C2×C3⋊(C3⋊D3)—81no—
61C2×C3⋊(C3⋊D3)—81no—
62C3×C3⋊Dic3—81no—
63C32×Dic3—81no—
64C2×GL(2,𝔽3)—93no—
65C2×GL(2,𝔽3)—183no—
66C2×#226—33no—
67C2×C3×Q16—91no—
68C3×C4×Q8—31no—
69#947—91no—
70Q8×Dic3C3⋊D311yescharacteristic, complemented
71C2×#216—31no—
72C2×#215—91no—
73C2×#215—91no—
74C2×#215—181no—
75#875—91no—
76C4×SL(2,𝔽3)—33no—
77#835—91no—
78#821—91no—
79Dic3⋊C8—91no—
80#810—91no—
81#809—91no—
82C4⋊(C3⋊C8)—91no—
83C22×C3⋊D3—61no—
84C22×C3⋊D3—121no—
85C22×C3⋊D3—123no—
86C22×C3⋊D3—241no—
87C22×C3⋊D3—243no—
88C32×Q8—31no—
89C32×Q8D1211yescharacteristic
90C2×C32×C4—31no—
91C2×C32×C4—121no—
92C2×C3⋊Dic3—41no—
93C4⋊(C3⋊D3)—61no—
94C4⋊(C3⋊D3)—61no—
95C2×C3×Dic3—31no—
96C2×C3×Dic3—43no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—123no—
99C2×C3×Dic3S411yescharacteristic
100C3×Dic6—61no—
101C3×Dic6—61no—
102C3×SL(2,𝔽3)—21no—
103C3×SL(2,𝔽3)—23no—
104C3×SL(2,𝔽3)D1213yes—
105C3×SL(2,𝔽3)#6711yescharacteristic
106C3⋊(C3⋊C8)—61no—
107C3×C3⋊C8—61no—
108#464—271noSylow
109C2×C33—41no—
110C2×C33—41no—
111C2×C33—41no—
112C3⋊(C3⋊D3)—82no—
113C2×C3×Q8—21no—
114C2×C3×Q8—91no—
115C2×C3×Q8D3211yescharacteristic
116C2×C3×Q8C2×C3⋊D311yescharacteristic
117C2×D12—91no—
118C2×D12—91no—
119C2×D12—181no—
120C2×Dic6—31no—
121C2×SL(2,𝔽3)—23no—
122C2×SL(2,𝔽3)D3213yes—
123GL(2,𝔽3)—186no—
124GL(2,𝔽3)—183no—
125#226—63no—
126C3×Q16—181no—
127C3×Q16—181no—
128C2×C3×C8—91no—
129C3×C4⋊C4—91no—
130C3×C42—91no—
131#216—61no—
132#216—61no—
133#215—182no—
134#215—181no—
135#215—181no—
136#215—182no—
137#215—181no—
138#215—181no—
139(C2×D3)⋊C4—181no—
140(C2×D3)⋊C4—181no—
141(C2×D3)⋊C4—181no—
142(C2×D3)⋊C4—181no—
143(C2×D3)⋊C4—363no—
144C4⋊Dic3—31no—
145C4×Dic3—31no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—91no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—181no—
150C22×C32—41no—
151C22×C32—43no—
152C22×C32—83no—
153C22×C32—81no—
154C22×C32C2×S411yescharacteristic
155C2×C3⋊D3—122no—
156C2×C3⋊D3—121no—
157C2×C3⋊D3—126no—
158C2×C3⋊D3—121no—
159C2×C3⋊D3—121no—
160C2×C3⋊D3—243no—
161C2×C3⋊D3—241no—
162C2×C3⋊D3—243no—
163C2×C3⋊D3—241no—
164C2×C3⋊D3—246no—
165C2×C3⋊D3—242no—
166C2×C3⋊D3—242no—
167C2×C3⋊D3—246no—
168C2×C3⋊D3—242no—
169C2×C3⋊D3—246no—
170C32×C4—31no—
171C32×C4—31no—
172C32×C4—241no—
173C3⋊Dic3—81no—
174C3×Dic3—21no—
175C3×Dic3—61no—
176C3×Dic3—83no—
177C3×Dic3—121no—
178C3×Dic3—243no—
179C2×Q16—91no—
180C2×SD16—271no—
181#124—271no—
182C4×Q8—31no—
183C4⋊C8—271no—
184#103—271no—
185#102—271no—
186C33—41noSylow
187C22×D3—181no—
188C22×D3—181no—
189C22×D3—361no—
190C22×D3—363no—
191C22×D3—723no—
192C3×Q8—21no—
193C3×Q8—31no—
194C3×Q8—31no—
195C3×Q8—61no—
196C3×Q8—181no—
197C3×Q8—181no—
198C3×Q8#62611yescharacteristic
199C3×Q8C4⋊(C3⋊D3)11yescharacteristic
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—61no—
204C2×C3×C4—91no—
205C2×C3×C4—123no—
206C2×C3×C4—181no—
207C2×Dic3—31no—
208C2×Dic3—61no—
209C2×Dic3—123no—
210C2×Dic3C3⋊S411yescharacteristic
211D12—181no—
212D12—181no—
213D12—181no—
214D12—181no—
215D12—182no—
216D12—361no—
217Dic6—61no—
218Dic6—61no—
219SL(2,𝔽3)—23no—
220SL(2,𝔽3)#62613yes—
221C3×C8—181no—
222C3⋊C8—61no—
223C3⋊C8—181no—
224C3⋊C8—181no—
225C3⋊C8—182no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—43no—
230C2×C32—43no—
231C2×C32—43no—
232C2×C32—83no—
233C2×C32—83no—
234C2×C32—83no—
235C2×C32—81no—
236C2×C32—81no—
237C2×C32—81no—
238C2×C32C4⋊S411yescharacteristic
239C2×C32#98311yescharacteristic
240C2×C32#98511yescharacteristic
241C3⋊D3—242no—
242C3⋊D3—246no—
243C3⋊D3—2412no—
244C3⋊D3—244no—
245C3⋊D3—241no—
246C2×Q8—91no—
247C2×Q8D3×C3⋊D311yescharacteristic
248C2×D4—271no—
249Q16—181no—
250Q16—181no—
251SD16—541no—
252SD16—541no—
253C2×C8—91no—
254C2×C8—271no—
255C4⋊C4—91no—
256C22⋊C4—541no—
257C22⋊C4—541no—
258C42—91no—
259C22×C3—21no—
260C22×C3—43no—
261C22×C3—83no—
262C22×C3D3×S411yescharacteristic
263C22×C3C2×C3⋊S411yescharacteristic
264C2×D3—362no—
265C2×D3—366no—
266C2×D3—361no—
267C2×D3—361no—
268C2×D3—361no—
269C2×D3—361no—
270C2×D3—361no—
271C2×D3—361no—
272C2×D3—721no—
273C2×D3—721no—
274C2×D3—723no—
275C2×D3—723no—
276C2×D3—726no—
277C2×D3—726no—
278C2×D3—726no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—61no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—181no—
287C3×C4—243no—
288C3×C4—361no—
289Dic3—21no—
290Dic3—61no—
291Dic3—121no—
292Dic3—243no—
293C32—41no—
294C32—43no—
295C32—83no—
296C32—81no—
297C32#539511yescharacteristic, complemented
298C23—541no—
299Q8—31no—
300Q8—181no—
301Q8—181no—
302Q8#629211yescharacteristic
303D4—541no—
304D4—541no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—91no—
308C2×C4—181no—
309C8—181no—
310C8—541no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—43no—
315C2×C3—43no—
316C2×C3—43no—
317C2×C3—83no—
318C2×C3—83no—
319C2×C3—83no—
320C2×C3#6419811yescharacteristic
321C2×C3#6420211yescharacteristic
322C2×C3#6420711yescharacteristic
323C2×C3C4⋊(C3⋊S4)11yescharacteristic
324C2×C3#6426411yescharacteristic
325C2×C3#6426611yescharacteristic
326D3—722no—
327D3—726no—
328D3—7212no—
329D3—721no—
330D3—721no—
331C22—1081no—
332C22—1081no—
333C22—1081no—
334C22D3×C3⋊S411yescharacteristic
335C4—31no—
336C4—31no—
337C4—61no—
338C4—181no—
339C4—361no—
340C3—21no—
341C3—43no—
342C3—83no—
343C3#1059320811yescharacteristic, complemented
344C3#1059362911yescharacteristic, complemented
345C2—2161no—
346C2#1172419311yescharacteristic
347C2#1172419711yescharacteristic
348C2#1172420211yescharacteristic
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⋊GL(2,𝔽3))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×Dic3×SL(2,𝔽3)C2semidirect
15. C2×C3⋊#226D3semidirect
18. C3×Q8×Dic3D3semidirect
22. Dic3×SL(2,𝔽3)D3semidirect
70. Q8×Dic3C3⋊D3semidirect
297. C32#5395semidirect
343. C3#10593208semidirect
344. C3#10593629semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C3×C3⋊#2262
C3×Dic3×SL(2,𝔽3)2
#105936293
#105932083
#105916073
#888044