Finite Group Databasebeta

← C3×#10593629 #13172997

Subgroups — C3×#10593629 #13172997

Order 1728.

488 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 66 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
3C32×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593629C311yesmaximal, characteristic, direct product factor
5C3×#5395—33nomaximal
6C3×#5028—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×C3⋊GL(2,𝔽3)—21no—
9C3×C3⋊#226—21no—
10C3×#3752—41nomaximal
11C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic
12C2×C3⋊#226C2×C311yescharacteristic, complemented
13C2×C3×GL(2,𝔽3)—33no—
14C2×C3×#226—33no—
15C32×C4×Q8D311yescharacteristic, complemented
16C2×C3×#216—31no—
17C2×C3×#215—31no—
18C3×#875—31no—
19C3×C4×SL(2,𝔽3)—21no—
20C3×C4×SL(2,𝔽3)—23no—
21C3×C4×SL(2,𝔽3)D313yescomplemented
22C3×C4×SL(2,𝔽3)C2×C311yescharacteristic, complemented
23C3×#810—31no—
24C3×#809—31no—
25C3×C4⋊(C3⋊C8)—31no—
26C22×C3×C3⋊D3—41no—
27C2×C33×C4—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30#5395—33no—
31C3×#464—91no—
32#5028—31no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C3×D12—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)C22×C311yescharacteristic
40C3⋊GL(2,𝔽3)—21no—
41C3⋊#226—21no—
42C3×GL(2,𝔽3)—63no—
43C3×#226—63no—
44C32×C4⋊C4—31no—
45C32×C42—31no—
46#3752—41no—
47C3×#216—61no—
48C3×#216—61no—
49C3×#215—61no—
50C3×#215—61no—
51C3×(C2×D3)⋊C4—61no—
52C3×(C2×D3)⋊C4—61no—
53C3×(C2×D3)⋊C4—123no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—31no—
56C22×C33—41no—
57C2×C3×C3⋊D3—42no—
58C2×C3×C3⋊D3—81no—
59C2×C3×C3⋊D3—81no—
60C33×C4—81no—
61C3×C3⋊Dic3—81no—
62C2×GL(2,𝔽3)—33no—
63C2×#226—33no—
64C2×C3×Q16—91no—
65C2×C3×SD16—91no—
66C3×#124—91no—
67C3×C4×Q8—21no—
68C3×C4×Q8C3×D311yescharacteristic, complemented
69C3×C4×Q8C3⋊D311yescharacteristic, complemented
70C2×#216—31no—
71C2×#215—31no—
72#875—31no—
73C4×SL(2,𝔽3)—23no—
74C4×SL(2,𝔽3)C3×D313yescomplemented
75C3×C4⋊C8—91no—
76C3×#103—91no—
77C3×#102—91no—
78#810—31no—
79#809—31no—
80C4⋊(C3⋊C8)—31no—
81C22×C3⋊D3—41no—
82C22×C3×D3—61no—
83C22×C3×D3—123no—
84C32×Q8—31no—
85C32×Q8D1211yescharacteristic
86C2×C32×C4—31no—
87C2×C32×C4—31no—
88C2×C32×C4—41no—
89C2×C32×C4—43no—
90C2×C32×C4—83no—
91C2×C32×C4—81no—
92C2×C32×C4S411yescharacteristic
93C2×C3⋊Dic3—41no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—123no—
96C3×D12—61no—
97C3×D12—61no—
98C3×Dic6—61no—
99C3×Dic6—61no—
100C3×SL(2,𝔽3)—21no—
101C3×SL(2,𝔽3)—23no—
102C3×SL(2,𝔽3)D1213yes—
103C3×SL(2,𝔽3)C3×D411yescharacteristic
104C3×C3⋊C8—61no—
105C3×C3⋊C8—61no—
106#464—91noSylow
107C2×C33—41no—
108C2×C33—41no—
109C2×C33—41no—
110C3×C3⋊D3—82no—
111C2×C3×Q8—21no—
112C2×C3×Q8—91no—
113C2×C3×Q8C2×C3×D311yescharacteristic
114C2×C3×Q8C2×C3⋊D311yescharacteristic
115C2×C3×D4—91no—
116C2×D12—31no—
117C2×Dic6—31no—
118C2×SL(2,𝔽3)—23no—
119C2×SL(2,𝔽3)C2×C3×D313yes—
120GL(2,𝔽3)—63no—
121#226—63no—
122C3×Q16—181no—
123C3×Q16—181no—
124C3×SD16—181no—
125C3×SD16—181no—
126C2×C3×C8—91no—
127C2×C3×C8—91no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—31no—
130C3×C4⋊C4—61no—
131C3×C22⋊C4—181no—
132C3×C22⋊C4—181no—
133C3×C42—31no—
134C3×C42—31no—
135C3×C42—61no—
136#216—61no—
137#216—61no—
138#215—61no—
139#215—61no—
140(C2×D3)⋊C4—61no—
141(C2×D3)⋊C4—61no—
142(C2×D3)⋊C4—123no—
143C2×C3⋊C8—31no—
144C2×C3⋊C8—31no—
145C22×C32—41no—
146C22×C32—43no—
147C22×C32—83no—
148C22×C32—81no—
149C22×C32C2×S411yescharacteristic
150C2×C3⋊D3—42no—
151C2×C3⋊D3—81no—
152C2×C3⋊D3—81no—
153C2×C3×D3—126no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—243no—
158C2×C3×D3—243no—
159C32×C4—21no—
160C32×C4—31no—
161C32×C4—31no—
162C32×C4—61no—
163C32×C4—83no—
164C32×C4—81no—
165C32×C4—86no—
166C32×C4—82no—
167C3⋊Dic3—81no—
168C3×Dic3—121no—
169C3×Dic3—243no—
170C2×Q16—91no—
171C2×SD16—91no—
172#124—91no—
173C4×Q8C3×C3⋊D311yescharacteristic, complemented
174C4⋊C8—91no—
175#103—91no—
176#102—91no—
177C33—41noSylow
178C23×C3—181no—
179C22×D3—61no—
180C22×D3—123no—
181C3×Q8—21no—
182C3×Q8—31no—
183C3×Q8—31no—
184C3×Q8—61no—
185C3×Q8—181no—
186C3×Q8—181no—
187C3×Q8C3×D1211yescharacteristic
188C3×Q8C4⋊(C3⋊D3)11yescharacteristic
189C3×D4—181no—
190C3×D4—181no—
191C2×C3×C4—21no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—43no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—83no—
200C2×C3×C4—181no—
201C2×C3×C4C3×S411yescharacteristic
202C2×C3×C4C3⋊S411yescharacteristic
203C2×Dic3—61no—
204C2×Dic3—123no—
205D12—61no—
206D12—61no—
207Dic6—61no—
208Dic6—61no—
209SL(2,𝔽3)—23no—
210SL(2,𝔽3)C3×D1213yes—
211C3×C8—181no—
212C3×C8—181no—
213C3⋊C8—61no—
214C3⋊C8—61no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—43no—
219C2×C32—43no—
220C2×C32—43no—
221C2×C32—83no—
222C2×C32—83no—
223C2×C32—83no—
224C2×C32—81no—
225C2×C32—81no—
226C2×C32—81no—
227C2×C32C4⋊S411yescharacteristic
228C2×C32#98311yescharacteristic
229C2×C32#98511yescharacteristic
230C3⋊D3—82no—
231C3×D3—246no—
232C3×D3—241no—
233C2×Q8—91no—
234C2×Q8C2×C3×C3⋊D311yescharacteristic
235C2×D4—91no—
236Q16—181no—
237Q16—181no—
238SD16—181no—
239SD16—181no—
240C2×C8—91no—
241C2×C8—91no—
242C4⋊C4—31no—
243C22⋊C4—181no—
244C22⋊C4—181no—
245C42—31no—
246C22×C3—21no—
247C22×C3—43no—
248C22×C3—83no—
249C22×C3—361no—
250C22×C3—361no—
251C22×C3—361no—
252C22×C3C2×C3×S411yescharacteristic
253C22×C3C2×C3⋊S411yescharacteristic
254C2×D3—126no—
255C2×D3—121no—
256C2×D3—121no—
257C2×D3—121no—
258C2×D3—243no—
259C2×D3—243no—
260C3×C4—22no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—83no—
272C3×C4—86no—
273C3×C4—121no—
274C3×C4—361no—
275Dic3—121no—
276Dic3—243no—
277C32—41no—
278C32—43no—
279C32—83no—
280C32—81no—
281C32#539511yescharacteristic, complemented
282C23—181no—
283Q8—31no—
284Q8—181no—
285Q8—181no—
286Q8C3×C4⋊(C3⋊D3)11yescharacteristic
287D4—181no—
288D4—181no—
289C2×C4—31no—
290C2×C4—31no—
291C2×C4—181no—
292C2×C4C3×C3⋊S411yescharacteristic
293C8—181no—
294C8—181no—
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—721no—
305C2×C3C3×C4⋊S411yescharacteristic
306C2×C3C3×#98311yescharacteristic
307C2×C3C3×#98511yescharacteristic
308C2×C3C4⋊(C3⋊S4)11yescharacteristic
309C2×C3#6426411yescharacteristic
310C2×C3#6426611yescharacteristic
311D3—246no—
312D3—241no—
313C22—361no—
314C22—361no—
315C22—361no—
316C22C2×C3×C3⋊S411yescharacteristic
317C4—21no—
318C4—31no—
319C4—31no—
320C4—61no—
321C4—361no—
322C3—21no—
323C3—43no—
324C3—83no—
325C3C3×#539511yescharacteristic, complemented
326C3#1059362911yescharacteristic, direct product factor
327C2—721no—
328C2C3×C4⋊(C3⋊S4)11yescharacteristic
329C2C3×#6426411yescharacteristic
330C2C3×#6426611yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C2×C3×C3⋊S4
FittingC32×C4×Q8D3
socleC22×C32C2×S4
prime radicalC2×C3×C3⋊GL(2,𝔽3)C2
hypercenterC2×C3×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×C3⋊#226C2semidirect
3. C32×C4×SL(2,𝔽3)C2semidirect
4. #10593629C3direct (N × H)
12. C2×C3⋊#226C2×C3semidirect
15. C32×C4×Q8D3semidirect
21. C3×C4×SL(2,𝔽3)D3semidirect
22. C3×C4×SL(2,𝔽3)C2×C3semidirect
68. C3×C4×Q8C3×D3semidirect
69. C3×C4×Q8C3⋊D3semidirect
74. C4×SL(2,𝔽3)C3×D3semidirect
173. C4×Q8C3×C3⋊D3semidirect
281. C32#5395semidirect
325. C3C3×#5395semidirect
326. C3#10593629direct (N × H)

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
C32×C4×SL(2,𝔽3)2
#105936293
C3×#53953
C3×#50283
C3×#37524