Finite Group Databasebeta

← #13166033

Subgroups — #13166033

Order 1728.

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

normal subgroups only

This group has 514 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⋊#226C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593653—31nomaximal
5#10593210—33nomaximal
6#10591594—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⋊#226D311yescharacteristic, complemented
15C2×C3×#226—33no—
16C2×C3×#216—31no—
17#63856—31no—
18C2×#3718—31no—
19C3⋊#859—31no—
20C3⋊#859—33no—
21C3⋊#859—61no—
22C3⋊#859—63no—
23#63662—31no—
24#63566—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#5186—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C3×D3×Q8—21no—
35C3×D3×Q8—61no—
36C2×C3×C4×D3—31no—
37C2×C3×Dic6—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×D313yes—
43C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
44D3×SL(2,𝔽3)—23no—
45C3⋊#226—21no—
46C3×#226—63no—
47#3757—121no—
48C4⋊(C3⋊Dic3)—31no—
49C2×C3⋊(C3⋊C8)—31no—
50C3×#216—61no—
51C3×#216—61no—
52C2×C3×C3⋊C8—31no—
53#3724—61no—
54(C2×D3)⋊Dic3—61no—
55(C2×D3)⋊Dic3—123no—
56#3718—62no—
57C22×C33—41no—
58C2×C32×D3—42no—
59C2×C32×D3—81no—
60C2×C32×D3—81no—
61C3⋊(C3⋊Dic3)—81no—
62C3×C3⋊Dic3—81no—
63C22×C3×Q8—31no—
64C2×D3×Q8C3⋊D311yescharacteristic
65C22×SL(2,𝔽3)—33no—
66C2×#226—33no—
67C2×C3×Q16—91no—
68#946—91no—
69C2×#216—31no—
70C2×#216—91no—
71C2×#216—91no—
72#924—91no—
73#859—93no—
74#859—183no—
75#835—91no—
76#835—91no—
77#835—91no—
78#835—181no—
79#830—91no—
80(C2×D3)⋊C8—91no—
81#816—91no—
82C23×C32—121no—
83C22×C3×D3—43no—
84C22×C3×D3S411yescharacteristic
85C32×Q8—31no—
86C32×Q8#6711yescharacteristic
87C2×C32×C4—31no—
88C2×C3⋊Dic3—41no—
89C2×C3⋊Dic3—61no—
90C2×C3⋊Dic3—121no—
91C2×C3⋊Dic3—123no—
92C2×C3⋊Dic3—241no—
93C2×C3⋊Dic3—243no—
94C2×C3×Dic3—31no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—123no—
97C3×C4×D3—61no—
98C3×C4×D3—61no—
99C3×Dic6—32no—
100C3×Dic6—61no—
101C3×Dic6—61no—
102C3×Dic6—61no—
103C3×SL(2,𝔽3)—21no—
104C3×SL(2,𝔽3)—23no—
105C3×SL(2,𝔽3)#6713yes—
106C3×SL(2,𝔽3)#6711yescharacteristic
107C3⋊(C3⋊C8)—61no—
108C3×C3⋊C8—61no—
109#451—271noSylow
110C2×C33—41no—
111C2×C33—41no—
112C2×C33—41no—
113C32×D3—82no—
114C2×C3×Q8—21no—
115C2×C3×Q8—61no—
116C2×C3×Q8—91no—
117C2×C3×Q8—91no—
118C2×C3×Q8—181no—
119C2×C3×Q8D3211yescharacteristic
120C2×C3×Q8C2×C3⋊D311yescharacteristic
121C22×C3×C4—91no—
122D3×Q8—21no—
123D3×Q8—61no—
124C2×C4×D3—31no—
125C2×Dic6—31no—
126C2×Dic6—31no—
127C2×SL(2,𝔽3)—23no—
128C2×SL(2,𝔽3)—63no—
129C2×SL(2,𝔽3)D3213yes—
130#226—63no—
131C3×Q16—181no—
132C3×Q16—181no—
133C2×C3×C8—91no—
134#217—181no—
135#217—363no—
136#216—61no—
137#216—61no—
138#216—182no—
139#216—182no—
140(C2×D3)⋊C4—181no—
141C4⋊Dic3—91no—
142C4⋊Dic3—91no—
143C4⋊Dic3—181no—
144Dic3⋊C4—181no—
145Dic3⋊C4—181no—
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×C32—122no—
155C22×C32—241no—
156C22×C32—241no—
157C22×C32C2×S411yescharacteristic
158C2×C3×D3—21no—
159C2×C3×D3—21no—
160C2×C3×D3—46no—
161C2×C3×D3—83no—
162C2×C3×D3—83no—
163C2×C3×D3#22612yescomplemented
164C32×C4—31no—
165C32×C4—31no—
166C3⋊Dic3—81no—
167C3⋊Dic3—121no—
168C3⋊Dic3—243no—
169C3⋊Dic3—241no—
170C3⋊Dic3—242no—
171C3⋊Dic3—246no—
172C3×Dic3—61no—
173C3×Dic3—121no—
174C3×Dic3—243no—
175C22×Q8—31no—
176C2×Q16—91no—
177C2×Q16—271no—
178C22⋊Q8—271no—
179#103—271no—
180#103—271no—
181C22⋊C8—271no—
182C33—41noSylow
183C23×C3—31no—
184C23×C3—123no—
185C22×D3C3⋊S411yescharacteristic
186C3×Q8—21no—
187C3×Q8—31no—
188C3×Q8—31no—
189C3×Q8—61no—
190C3×Q8—92no—
191C3×Q8—181no—
192C3×Q8—181no—
193C3×Q8—181no—
194C3×Q8#62511yescharacteristic
195C3×Q8#63811yescharacteristic
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—61no—
199C2×C3×C4—91no—
200C2×C3×C4—181no—
201C2×C3×C4—181no—
202C2×C3×C4—181no—
203C2×Dic3—31no—
204C2×Dic3—61no—
205C2×Dic3—123no—
206C2×Dic3—181no—
207C2×Dic3—181no—
208C2×Dic3—361no—
209C2×Dic3—363no—
210C2×Dic3—723no—
211C4×D3—61no—
212C4×D3—61no—
213Dic6—32no—
214Dic6—61no—
215Dic6—61no—
216Dic6—61no—
217SL(2,𝔽3)—23no—
218SL(2,𝔽3)#62513yes—
219C3×C8—181no—
220C3⋊C8—61no—
221C3⋊C8—181no—
222C3⋊C8—181no—
223C3⋊C8—361no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—41no—
227C2×C32—43no—
228C2×C32—43no—
229C2×C32—43no—
230C2×C32—83no—
231C2×C32—83no—
232C2×C32—83no—
233C2×C32—81no—
234C2×C32—81no—
235C2×C32—81no—
236C2×C32—242no—
237C2×C32C2×#22611yescharacteristic
238C2×C32#98311yescharacteristic
239C2×C32GL(2,ℤ/4)11yescharacteristic
240C3×D3—22no—
241C3×D3—86no—
242C2×Q8—61no—
243C2×Q8—91no—
244C2×Q8—91no—
245C2×Q8—181no—
246C2×Q8D3×C3⋊D311yescharacteristic
247C22×C4—91no—
248Q16—181no—
249Q16—181no—
250Q16—542no—
251C2×C8—91no—
252C2×C8—271no—
253C4⋊C4—271no—
254C4⋊C4—541no—
255C22⋊C4—541no—
256C22×C3—21no—
257C22×C3—32no—
258C22×C3—43no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—83no—
262C22×C3—126no—
263C22×C3—243no—
264C22×C3—243no—
265C22×C3D3×S411yescharacteristic
266C22×C3C2×C3⋊S411yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3C3⋊#22612yescomplemented
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—181no—
277C3×C4—361no—
278Dic3—61no—
279Dic3—121no—
280Dic3—243no—
281Dic3—361no—
282Dic3—361no—
283Dic3—721no—
284Dic3—723no—
285Dic3—726no—
286C32—41no—
287C32—43no—
288C32—83no—
289C32—81no—
290C32#541911yescharacteristic, complemented
291C23—31no—
292Q8—31no—
293Q8—92no—
294Q8—181no—
295Q8—181no—
296Q8—181no—
297Q8#629011yescharacteristic
298C2×C4—31no—
299C2×C4—91no—
300C2×C4—181no—
301C2×C4—181no—
302C2×C4—181no—
303C2×C4—541no—
304C8—181no—
305C8—541no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—43no—
310C2×C3—43no—
311C2×C3—43no—
312C2×C3—62no—
313C2×C3—83no—
314C2×C3—83no—
315C2×C3—83no—
316C2×C3—246no—
317C2×C3D3×#22611yescharacteristic
318C2×C3#6420211yescharacteristic
319C2×C3#6420911yescharacteristic
320C2×C3C2×C3⋊#22611yescharacteristic
321C2×C3#6426411yescharacteristic
322C2×C3#6426911yescharacteristic
323D3—22no—
324C22—32no—
325C22—61no—
326C22—61no—
327C22D3×C3⋊S411yescharacteristic
328C4—31no—
329C4—31no—
330C4—181no—
331C4—361no—
332C4—1081no—
333C3—21no—
334C3—43no—
335C3—83no—
336C3#1059321011yescharacteristic, complemented
337C3#1059365311yescharacteristic, complemented
338C2—62no—
339C2D3×C3⋊#22611yescharacteristic
340C2#1172419711yescharacteristic
341C2#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×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⋊#226C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
14. C2×C3⋊#226D3semidirect
163. C2×C3×D3#226semidirect
269. C2×D3C3⋊#226semidirect
290. C32#5419semidirect
336. C3#10593210semidirect
337. 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⋊#2262
C3⋊(C3⋊#859)2
#105936533
#105932103
#105915943
(C2×D3)⋊(C3⋊Dic3)4