Finite Group Databasebeta

← #13166029

Subgroups — #13166029

Order 1728.

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

normal subgroups only

This group has 509 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⋊#226)C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593629—31nomaximal
5#10593205—33nomaximal
6#10591604—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3⋊(C3⋊#226)—21no—
9C3×C3⋊GL(2,𝔽3)—21no—
10#88803—41nomaximal
11C2×C3⋊GL(2,𝔽3)D311yescharacteristic
12C2×C3⋊#226—31no—
13C2×C3⋊#226—33no—
14C2×C3⋊#226—61no—
15C2×C3⋊#226—63no—
16C2×C3×GL(2,𝔽3)—33no—
17C2×#3756—31no—
18C3×Q8×Dic3D311yescharacteristic, complemented
19C2×C3×#215—31no—
20C3×C4×SL(2,𝔽3)—31no—
21#63860—31no—
22Dic3×SL(2,𝔽3)D313yescomplemented
23Dic3⋊(C3⋊C8)—31no—
24#63567—31no—
25#63562—31no—
26C22×C3×C3⋊D3—41no—
27C2×C3⋊(C3⋊Dic3)—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30#5395—93no—
31#5166—91no—
32#5028—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×#634—31no—
35C2×C3×D12—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)—21no—
41C3⋊#226—62no—
42C3⋊#226—66no—
43C3⋊#226—63no—
44C3⋊#226—61no—
45C3×GL(2,𝔽3)—63no—
46#3756—61no—
47#3756—61no—
48#3752—121no—
49C2×C3⋊(C3⋊C8)—31no—
50C3×#215—61no—
51C3×#215—61no—
52C3×C4⋊Dic3—31no—
53C3×C4×Dic3—31no—
54C2×C3×C3⋊C8—31no—
55(C2×D3)⋊Dic3—61no—
56(C2×D3)⋊Dic3—61no—
57(C2×D3)⋊Dic3—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)—33no—
65C2×#226—93no—
66C2×#226—183no—
67C2×C3×SD16—91no—
68C3×C4×Q8—31no—
69Q8×Dic3C3⋊D311yescharacteristic, complemented
70C2×#216—91no—
71C2×#216—91no—
72C2×#216—181no—
73C2×#215—31no—
74#936—91no—
75#875—91no—
76C4×SL(2,𝔽3)—33no—
77#832—91no—
78#816—91no—
79Dic3⋊C8—91no—
80#810—91no—
81#809—91no—
82C4⋊(C3⋊C8)—91no—
83C22×C3⋊D3—41no—
84C22×C3×D3—61no—
85C22×C3×D3—123no—
86C32×Q8—31no—
87C32×Q8D1211yescharacteristic
88C2×C32×C4—31no—
89C2×C32×C4—121no—
90C2×C3⋊Dic3—61no—
91C2×C3⋊Dic3—121no—
92C2×C3⋊Dic3—123no—
93C2×C3⋊Dic3—241no—
94C2×C3⋊Dic3—243no—
95#634—61no—
96#634—61no—
97C2×C3×Dic3—31no—
98C2×C3×Dic3—43no—
99C2×C3×Dic3S411yescharacteristic
100C3×D12—61no—
101C3×D12—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×Q8D3211yescharacteristic
115C2×C3×Q8C2×C3⋊D311yescharacteristic
116C2×C3×D4—91no—
117C2×D12—31no—
118C2×Dic6—91no—
119C2×Dic6—91no—
120C2×Dic6—181no—
121C2×SL(2,𝔽3)—23no—
122C2×SL(2,𝔽3)D3213yes—
123GL(2,𝔽3)—63no—
124#226—186no—
125#226—183no—
126C3×SD16—181no—
127C3×SD16—181no—
128C2×C3×C8—91no—
129C3×C4⋊C4—91no—
130C3×C42—91no—
131#217—181no—
132#217—181no—
133#216—182no—
134#216—181no—
135#216—181no—
136#216—182no—
137#216—181no—
138#216—181no—
139#215—61no—
140#215—61no—
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—42no—
156C2×C3⋊D3—81no—
157C2×C3⋊D3—81no—
158C2×C3×D3—126no—
159C2×C3×D3—121no—
160C2×C3×D3—121no—
161C2×C3×D3—121no—
162C2×C3×D3—243no—
163C2×C3×D3—243no—
164C32×C4—31no—
165C32×C4—31no—
166C32×C4—241no—
167C3⋊Dic3—121no—
168C3⋊Dic3—243no—
169C3⋊Dic3—241no—
170C3⋊Dic3—246no—
171C3⋊Dic3—242no—
172C3×Dic3—21no—
173C3×Dic3—61no—
174C3×Dic3—83no—
175C2×Q16—271no—
176C2×SD16—91no—
177#124—271no—
178C4×Q8—31no—
179C4⋊C8—271no—
180#103—271no—
181#102—271no—
182C33—41noSylow
183C23×C3—181no—
184C22×D3—61no—
185C22×D3—123no—
186C3×Q8—21no—
187C3×Q8—31no—
188C3×Q8—31no—
189C3×Q8—61no—
190C3×Q8#62611yescharacteristic
191C3×Q8C4⋊(C3⋊D3)11yescharacteristic
192C3×D4—181no—
193C3×D4—181no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—91no—
199C2×C3×C4—123no—
200C2×Dic3—31no—
201C2×Dic3—181no—
202C2×Dic3—181no—
203C2×Dic3—361no—
204C2×Dic3—363no—
205C2×Dic3—723no—
206C2×Dic3C3⋊S411yescharacteristic
207D12—61no—
208D12—61no—
209Dic6—181no—
210Dic6—181no—
211Dic6—181no—
212Dic6—181no—
213Dic6—182no—
214Dic6—361no—
215SL(2,𝔽3)—23no—
216SL(2,𝔽3)#62613yes—
217C3×C8—181no—
218C3⋊C8—61no—
219C3⋊C8—181no—
220C3⋊C8—181no—
221C3⋊C8—182no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—43no—
226C2×C32—43no—
227C2×C32—43no—
228C2×C32—83no—
229C2×C32—83no—
230C2×C32—83no—
231C2×C32—81no—
232C2×C32—81no—
233C2×C32—81no—
234C2×C32C4⋊S411yescharacteristic
235C2×C32#98311yescharacteristic
236C2×C32#98511yescharacteristic
237C3⋊D3—82no—
238C3×D3—246no—
239C3×D3—241no—
240C2×Q8—271no—
241C2×Q8D3×C3⋊D311yescharacteristic
242C2×D4—91no—
243Q16—541no—
244Q16—541no—
245SD16—181no—
246SD16—181no—
247C2×C8—91no—
248C2×C8—271no—
249C4⋊C4—91no—
250C22⋊C4—541no—
251C22⋊C4—541no—
252C42—91no—
253C22×C3—21no—
254C22×C3—43no—
255C22×C3—83no—
256C22×C3—361no—
257C22×C3—361no—
258C22×C3—361no—
259C22×C3D3×S411yescharacteristic
260C22×C3C2×C3⋊S411yescharacteristic
261C2×D3—126no—
262C2×D3—121no—
263C2×D3—121no—
264C2×D3—121no—
265C2×D3—243no—
266C2×D3—243no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—31no—
271C3×C4—61no—
272C3×C4—61no—
273C3×C4—61no—
274C3×C4—181no—
275C3×C4—243no—
276Dic3—21no—
277Dic3—61no—
278Dic3—362no—
279Dic3—361no—
280Dic3—361no—
281Dic3—723no—
282Dic3—726no—
283C32—41no—
284C32—43no—
285C32—83no—
286C32—81no—
287C32#539511yescharacteristic, complemented
288C23—181no—
289Q8—31no—
290Q8—541no—
291Q8—541no—
292Q8#629211yescharacteristic
293D4—181no—
294D4—181no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—91no—
298C2×C4—541no—
299C8—181no—
300C8—541no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—43no—
305C2×C3—43no—
306C2×C3—43no—
307C2×C3—83no—
308C2×C3—83no—
309C2×C3—83no—
310C2×C3—721no—
311C2×C3#6419711yescharacteristic
312C2×C3#6420111yescharacteristic
313C2×C3#6420711yescharacteristic
314C2×C3C4⋊(C3⋊S4)11yescharacteristic
315C2×C3#6426411yescharacteristic
316C2×C3#6426611yescharacteristic
317D3—246no—
318D3—241no—
319C22—361no—
320C22—361no—
321C22—361no—
322C22D3×C3⋊S411yescharacteristic
323C4—31no—
324C4—31no—
325C4—61no—
326C4—181no—
327C4—1081no—
328C3—21no—
329C3—43no—
330C3—83no—
331C3#1059320511yescharacteristic, complemented
332C3#1059362911yescharacteristic, complemented
333C2—721no—
334C2#1172419211yescharacteristic
335C2#1172419611yescharacteristic
336C2#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
1. C2×C3⋊(C3⋊#226)C2semidirect
3. C3×Dic3×SL(2,𝔽3)C2semidirect
18. C3×Q8×Dic3D3semidirect
22. Dic3×SL(2,𝔽3)D3semidirect
69. Q8×Dic3C3⋊D3semidirect
287. C32#5395semidirect
331. C3#10593205semidirect
332. 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⋊#226)2
C2×C3×C3⋊GL(2,𝔽3)2
C3×Dic3×SL(2,𝔽3)2
#105936293
#105932053
#105916043
#888034