Finite Group Databasebeta

← #13166035

Subgroups — #13166035

Order 1728.

537 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 60 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊(C3⋊#226)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593653—31nomaximal
5#10593211—33nomaximal
6#10591595—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×D3×SL(2,𝔽3)—21no—
9C3⋊(C3⋊#226)—21no—
10(C2×D3)⋊(C3⋊Dic3)—41nomaximal
11C2×C3×D3×Q8D311yescharacteristic
12C22×C3×SL(2,𝔽3)—31no—
13C2×D3×SL(2,𝔽3)D313yes—
14C2×C3⋊#226—31no—
15C2×C3⋊#226—33no—
16C2×C3⋊#226—61no—
17C2×C3⋊#226—63no—
18C2×#3756—31no—
19#63857—31no—
20C2×#3719—31no—
21C3⋊#859D311yescharacteristic, complemented
22C3×#859—33no—
23C3×#835—31no—
24#63565—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26C22×C32×D3—41no—
27C2×C3⋊(C3⋊Dic3)—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30#5419—93no—
31#5228—91no—
32#4809—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×#634—31no—
35C3×D3×Q8—21no—
36C3×D3×Q8—61no—
37C2×C3×C4×D3—31no—
38C2×C3×Dic6—31no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—23no—
41C2×C3×SL(2,𝔽3)—61no—
42C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
43C2×C3×SL(2,𝔽3)C2×D313yes—
44D3×SL(2,𝔽3)—23no—
45C3⋊#226—62no—
46C3⋊#226—66no—
47C3⋊#226—61no—
48C3⋊#226—63no—
49#3757—121no—
50#3756—61no—
51#3756—61no—
52C2×C3⋊(C3⋊C8)—31no—
53C3×C4⋊Dic3—31no—
54C2×C3×C3⋊C8—31no—
55Dic3⋊Dic3—61no—
56(C2×D3)⋊Dic3—61no—
57(C2×D3)⋊Dic3—123no—
58#3719—62no—
59C22×C33—41no—
60C2×C32×D3—42no—
61C2×C32×D3—81no—
62C2×C32×D3—81no—
63C3⋊(C3⋊Dic3)—81no—
64C3×C3⋊Dic3—81no—
65C22×C3×Q8—31no—
66C2×D3×Q8C3⋊D311yescharacteristic
67C22×SL(2,𝔽3)—33no—
68C2×#226—93no—
69C2×#226—183no—
70C2×#216—91no—
71C2×#216—91no—
72C2×#216—91no—
73C2×#216—181no—
74#924—91no—
75C2×Dic12—91no—
76#897—91no—
77#859—33no—
78C3×#103—91no—
79#835—31no—
80#835—91no—
81#830—91no—
82(C2×D3)⋊C8—91no—
83#810—91no—
84C23×C32—121no—
85C22×C3×D3—43no—
86C22×C3×D3S411yescharacteristic
87C32×Q8—31no—
88C32×Q8#6711yescharacteristic
89C2×C32×C4—31no—
90C2×C3⋊Dic3—41no—
91C2×C3⋊Dic3—61no—
92C2×C3⋊Dic3—121no—
93C2×C3⋊Dic3—123no—
94C2×C3⋊Dic3—241no—
95C2×C3⋊Dic3—243no—
96#634—61no—
97#634—61no—
98C2×C3×Dic3—31no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—123no—
101C3×C4×D3—61no—
102C3×C4×D3—61no—
103C3×Dic6—32no—
104C3×Dic6—61no—
105C3×SL(2,𝔽3)—21no—
106C3×SL(2,𝔽3)—23no—
107C3×SL(2,𝔽3)D1211yescharacteristic
108C3×SL(2,𝔽3)#6713yes—
109C3⋊(C3⋊C8)—61no—
110C3×C3⋊C8—61no—
111#451—271noSylow
112C2×C33—41no—
113C2×C33—41no—
114C2×C33—41no—
115C32×D3—82no—
116C2×C3×Q8—21no—
117C2×C3×Q8—61no—
118C2×C3×Q8—91no—
119C2×C3×Q8—181no—
120C2×C3×Q8D3211yescharacteristic
121C2×C3×Q8C2×C3⋊D311yescharacteristic
122C22×C3×C4—91no—
123D3×Q8—21no—
124D3×Q8—61no—
125C2×C4×D3—31no—
126C2×Dic6—31no—
127C2×Dic6—91no—
128C2×Dic6—91no—
129C2×Dic6—181no—
130C2×SL(2,𝔽3)—23no—
131C2×SL(2,𝔽3)—63no—
132C2×SL(2,𝔽3)D3213yes—
133#226—186no—
134#226—183no—
135C2×C3×C8—91no—
136C3×C4⋊C4—91no—
137#217—181no—
138#217—363no—
139#216—182no—
140#216—182no—
141#216—181no—
142#216—181no—
143#216—182no—
144#216—181no—
145#216—181no—
146(C2×D3)⋊C4—181no—
147C4⋊Dic3—31no—
148C4⋊Dic3—181no—
149Dic3⋊C4—181no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—91no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—181no—
154Dic12—182no—
155C22×C32—41no—
156C22×C32—43no—
157C22×C32—83no—
158C22×C32—81no—
159C22×C32—122no—
160C22×C32—241no—
161C22×C32—241no—
162C22×C32C2×S411yescharacteristic
163C2×C3×D3—21no—
164C2×C3×D3—21no—
165C2×C3×D3—46no—
166C2×C3×D3—83no—
167C2×C3×D3—83no—
168C2×C3×D3#22612yescomplemented
169C32×C4—31no—
170C32×C4—31no—
171C3⋊Dic3—81no—
172C3⋊Dic3—121no—
173C3⋊Dic3—243no—
174C3⋊Dic3—241no—
175C3⋊Dic3—246no—
176C3⋊Dic3—242no—
177C3×Dic3—61no—
178C3×Dic3—121no—
179C3×Dic3—243no—
180C22×Q8—31no—
181C2×Q16—271no—
182C2×Q16—271no—
183C22⋊Q8—271no—
184#103—91no—
185#103—271no—
186C22⋊C8—271no—
187C33—41noSylow
188C23×C3—31no—
189C23×C3—123no—
190C22×D3C3⋊S411yescharacteristic
191C3×Q8—21no—
192C3×Q8—31no—
193C3×Q8—31no—
194C3×Q8—61no—
195C3×Q8—92no—
196C3×Q8—181no—
197C3×Q8#62611yescharacteristic
198C3×Q8#63811yescharacteristic
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—91no—
203C2×C3×C4—181no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206C2×Dic3—31no—
207C2×Dic3—61no—
208C2×Dic3—123no—
209C2×Dic3—181no—
210C2×Dic3—181no—
211C2×Dic3—361no—
212C2×Dic3—363no—
213C2×Dic3—723no—
214C4×D3—61no—
215C4×D3—61no—
216Dic6—32no—
217Dic6—61no—
218Dic6—181no—
219Dic6—181no—
220Dic6—181no—
221Dic6—181no—
222Dic6—182no—
223Dic6—361no—
224SL(2,𝔽3)—23no—
225SL(2,𝔽3)#62613yes—
226C3×C8—181no—
227C3⋊C8—61no—
228C3⋊C8—181no—
229C3⋊C8—181no—
230C3⋊C8—182no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—43no—
235C2×C32—43no—
236C2×C32—43no—
237C2×C32—83no—
238C2×C32—83no—
239C2×C32—83no—
240C2×C32—81no—
241C2×C32—81no—
242C2×C32—81no—
243C2×C32—242no—
244C2×C32C2×#22611yescharacteristic
245C2×C32#98311yescharacteristic
246C2×C32GL(2,ℤ/4)11yescharacteristic
247C3×D3—22no—
248C3×D3—86no—
249C2×Q8—61no—
250C2×Q8—91no—
251C2×Q8—181no—
252C2×Q8—271no—
253C2×Q8D3×C3⋊D311yescharacteristic
254C22×C4—91no—
255Q16—542no—
256Q16—541no—
257Q16—541no—
258C2×C8—91no—
259C2×C8—271no—
260C4⋊C4—91no—
261C4⋊C4—541no—
262C22⋊C4—541no—
263C22×C3—21no—
264C22×C3—32no—
265C22×C3—43no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3—83no—
269C22×C3—126no—
270C22×C3—243no—
271C22×C3—243no—
272C22×C3D3×S411yescharacteristic
273C22×C3C2×C3⋊S411yescharacteristic
274C2×D3—21no—
275C2×D3—21no—
276C2×D3C3⋊#22612yescomplemented
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—181no—
284C3×C4—361no—
285Dic3—61no—
286Dic3—121no—
287Dic3—243no—
288Dic3—362no—
289Dic3—361no—
290Dic3—361no—
291Dic3—723no—
292Dic3—726no—
293C32—41no—
294C32—43no—
295C32—83no—
296C32—81no—
297C32#541911yescharacteristic, complemented
298C23—31no—
299Q8—31no—
300Q8—92no—
301Q8—181no—
302Q8—541no—
303Q8—541no—
304Q8#629111yescharacteristic
305C2×C4—31no—
306C2×C4—91no—
307C2×C4—181no—
308C2×C4—181no—
309C2×C4—181no—
310C2×C4—541no—
311C8—181no—
312C8—541no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—43no—
317C2×C3—43no—
318C2×C3—43no—
319C2×C3—62no—
320C2×C3—83no—
321C2×C3—83no—
322C2×C3—83no—
323C2×C3—246no—
324C2×C3D3×#22611yescharacteristic
325C2×C3#6420111yescharacteristic
326C2×C3#6421011yescharacteristic
327C2×C3C2×C3⋊#22611yescharacteristic
328C2×C3#6426411yescharacteristic
329C2×C3#6426911yescharacteristic
330D3—22no—
331C22—32no—
332C22—61no—
333C22—61no—
334C22D3×C3⋊S411yescharacteristic
335C4—31no—
336C4—31no—
337C4—181no—
338C4—361no—
339C4—1081no—
340C3—21no—
341C3—43no—
342C3—83no—
343C3#1059321111yescharacteristic, complemented
344C3#1059365311yescharacteristic, complemented
345C2—62no—
346C2D3×C3⋊#22611yescharacteristic
347C2#1172419611yescharacteristic
348C2#1172420511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×D3×SL(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⋊#226)C2semidirect
3. C3×C3⋊#859C2semidirect
21. C3⋊#859D3semidirect
168. C2×C3×D3#226semidirect
276. C2×D3C3⋊#226semidirect
297. C32#5419semidirect
343. C3#10593211semidirect
344. C3#10593653semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×SL(2,𝔽3)2
C2×C3⋊(C3⋊#226)2
C3×C3⋊#8592
#105936533
#105932113
#105915953
(C2×D3)⋊(C3⋊Dic3)4