Finite Group Databasebeta

← #10593655⋊C3 #13146907

Subgroups — #10593655⋊C3 #13146907

Order 1728.

406 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×(C3⋊GL(2,𝔽3))⋊C3C211yesmaximal, characteristic, complemented
3(C3⋊#859)⋊C3C211yesmaximal, characteristic, complemented
4#10593655C311yesmaximal, characteristic, complemented
5C3×#5227—31nomaximal
6C2×#638⋊C3—41nomaximal
7C2×Q8⋊H(3,𝔽3)—21no—
8C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
9(C3⋊GL(2,𝔽3))⋊C3—21no—
10C22×C32×Q8D311yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)—21no—
12C22×C3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
13C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
14C2×C3×#215—31no—
15C2×C3×#215—31no—
16C3×#930—31no—
17C3⋊#859C2×C311yescharacteristic, complemented
18C3×#835—31no—
19C3×#835—31no—
20C3×#830—31no—
21C23×H(3,𝔽3)—41no—
22C22×(C3⋊D3)⋊C3—41no—
23#638⋊C3—44no—
24C2×(C3⋊Dic3)⋊C3—41no—
25Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
26#5421—91no—
27C3×#448—91no—
28#5227—31no—
29C2×C32×Q8—21no—
30C2×C32×Q8—31no—
31C2×C32×Q8—61no—
32C2×C32×Q8C2×D311yescharacteristic
33C22×C32×C4—31no—
34C2×#638—41no—
35C2×C3×#67—61no—
36C2×C3×D12—31no—
37C2×C3×SL(2,𝔽3)—22no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
41C3⋊GL(2,𝔽3)—21no—
42C3×#215—61no—
43C3×#215—62no—
44C3×#215—61no—
45C3×(C2×D3)⋊C4—61no—
46C3×C4⋊Dic3—31no—
47C2×C3×C3⋊C8—31no—
48C2×C3×C3⋊C8—31no—
49C22×H(3,𝔽3)—41no—
50C22×H(3,𝔽3)—42no—
51C22×H(3,𝔽3)—81no—
52C22×H(3,𝔽3)—81no—
53C2×(C3⋊D3)⋊C3—42no—
54C2×(C3⋊D3)⋊C3—81no—
55C2×(C3⋊D3)⋊C3—81no—
56(C3⋊Dic3)⋊C3—42no—
57C22×C3×Q8—31no—
58C22×C3×Q8C3×D311yescharacteristic
59C22×SL(2,𝔽3)—31no—
60C22×SL(2,𝔽3)—61no—
61C2×GL(2,𝔽3)—91no—
62C2×C3×SD16—91no—
63C2×C3×SD16—91no—
64C3×#121—91no—
65C2×#215—31no—
66C2×#215—31no—
67#930—31no—
68#859—91no—
69C3×#103—91no—
70C3×#103—91no—
71C3×C22⋊C8—91no—
72#835—31no—
73#835—31no—
74#830—31no—
75C23×C32—41no—
76C23×C32—81no—
77C23×C32S411yescharacteristic
78C22×C3⋊D3—41no—
79C22×C3×D3—61no—
80C32×Q8—31no—
81C32×Q8—32no—
82C32×Q8—61no—
83C32×Q8#6711yescharacteristic, complemented
84C2×C32×C4—31no—
85C2×C32×C4—31no—
86C2×C32×C4—61no—
87C2×C32×C4—61no—
88#638—44no—
89C2×C3⋊Dic3—41no—
90C3×#67—122no—
91C2×C3×Dic3—61no—
92C3×D12—61no—
93C3×D12—61no—
94C3×SL(2,𝔽3)—21no—
95C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
96C3×C3⋊C8—61no—
97C3×C3⋊C8—61no—
98#448—91noSylow
99C2×H(3,𝔽3)—41no—
100C2×H(3,𝔽3)—41no—
101C2×H(3,𝔽3)—41no—
102C2×H(3,𝔽3)—82no—
103(C3⋊D3)⋊C3—82no—
104C2×C3×Q8—21no—
105C2×C3×Q8—31no—
106C2×C3×Q8—31no—
107C2×C3×Q8—31no—
108C2×C3×Q8—61no—
109C2×C3×Q8—61no—
110C2×C3×Q8—62no—
111C2×C3×Q8—61no—
112C2×C3×Q8—61no—
113C2×C3×Q8C2×C3×D311yescharacteristic
114C2×C3×D4—91no—
115C2×C3×D4—181no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—61no—
119C2×#67—61no—
120C2×#67—361no—
121C2×D12—31no—
122C2×SL(2,𝔽3)—31no—
123C2×SL(2,𝔽3)—61no—
124C2×SL(2,𝔽3)—62no—
125C2×SL(2,𝔽3)—61no—
126GL(2,𝔽3)—181no—
127C3×SD16—181no—
128C3×SD16—182no—
129C3×SD16—181no—
130C2×C3×C8—91no—
131C2×C3×C8—91no—
132C3×C4⋊C4—91no—
133C3×C22⋊C4—181no—
134#215—61no—
135#215—62no—
136#215—61no—
137(C2×D3)⋊C4—61no—
138C4⋊Dic3—31no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—31no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—41no—
144C22×C32—42no—
145C22×C32—81no—
146C22×C32—81no—
147C22×C32—82no—
148C22×C32—82no—
149C22×C32—81no—
150C22×C32—82no—
151C22×C32GL(2,𝔽3)12yescomplemented
152C22×C32C2×S411yescharacteristic
153C2×C3⋊D3—42no—
154C2×C3⋊D3—81no—
155C2×C3⋊D3—81no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C2×C3×D3—121no—
159C32×C4—31no—
160C32×C4—31no—
161C32×C4—61no—
162C3⋊Dic3—42no—
163C3×Dic3—121no—
164C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
165C2×SD16—91no—
166C2×SD16—91no—
167#121—91no—
168#103—91no—
169#103—91no—
170C22⋊C8—91no—
171H(3,𝔽3)—41noSylow
172C23×C3—31no—
173C23×C3—121no—
174C23×C3—181no—
175C23×C3—241no—
176C23×C3C3×S411yescharacteristic
177C22×D3—61no—
178C22×D3—361no—
179C3×Q8—31no—
180C3×Q8—32no—
181C3×Q8—31no—
182C3×Q8—32no—
183C3×Q8—31no—
184C3×Q8—61no—
185C3×Q8—62no—
186C3×Q8—61no—
187C3×Q8—61no—
188C3×Q8—62no—
189C3×Q8C3×#6711yescharacteristic
190C3×D4—181no—
191C3×D4—181no—
192C3×D4—362no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—181no—
206#67—122no—
207#67—364no—
208C2×Dic3—61no—
209C2×Dic3—361no—
210D12—61no—
211D12—61no—
212SL(2,𝔽3)—31no—
213SL(2,𝔽3)—61no—
214C3×C8—181no—
215C3×C8—181no—
216C3⋊C8—61no—
217C3⋊C8—61no—
218C2×C32—22no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—82no—
223C2×C32—84no—
224C2×C32—81no—
225C2×C32—81no—
226C2×C32—81no—
227C2×C32C2×GL(2,𝔽3)11yescharacteristic
228C2×C32#98311yescharacteristic
229C2×C32GL(2,ℤ/4)11yescharacteristic
230C3⋊D3—82no—
231C3×D3—241no—
232C2×Q8—21no—
233C2×Q8—31no—
234C2×Q8—61no—
235C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
236C2×D4—91no—
237C2×D4—181no—
238C22×C4—31no—
239SD16—181no—
240SD16—182no—
241SD16—181no—
242C2×C8—91no—
243C2×C8—91no—
244C4⋊C4—91no—
245C22⋊C4—181no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—32no—
249C22×C3—31no—
250C22×C3—61no—
251C22×C3—61no—
252C22×C3—122no—
253C22×C3—121no—
254C22×C3—241no—
255C22×C3—241no—
256C22×C3—242no—
257C22×C3—242no—
258C22×C3—242no—
259C22×C3—241no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3—361no—
263C22×C3C3×GL(2,𝔽3)12yes—
264C22×C3C2×C3×S411yescharacteristic
265C2×D3—121no—
266C2×D3—121no—
267C2×D3—121no—
268C2×D3—362no—
269C2×D3—721no—
270C2×D3—721no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—61no—
276C3×C4—61no—
277C3×C4—61no—
278C3×C4—62no—
279C3×C4—61no—
280C3×C4—361no—
281Dic3—121no—
282Dic3—362no—
283C32—41no—
284C32—81no—
285C32#542111yescharacteristic, complemented
286C23—181no—
287C23(C3⋊S4)⋊C311yescharacteristic
288Q8—31no—
289Q8—32no—
290Q8—61no—
291Q8#638⋊C311yescharacteristic, complemented
292D4—181no—
293D4—181no—
294D4—362no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—61no—
298C2×C4—61no—
299C2×C4—181no—
300C8—181no—
301C8—181no—
302C2×C3—22no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—62no—
307C2×C3—121no—
308C2×C3—121no—
309C2×C3—121no—
310C2×C3—242no—
311C2×C3—244no—
312C2×C3—241no—
313C2×C3—241no—
314C2×C3—241no—
315C2×C3—721no—
316C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
317C2×C3C3×#98311yescharacteristic
318C2×C3C3×GL(2,ℤ/4)11yescharacteristic
319D3—241no—
320D3—722no—
321C22—21no—
322C22—21no—
323C22—361no—
324C22—361no—
325C22—361no—
326C22(C3⋊GL(2,𝔽3))⋊C312yescomplemented
327C22C2×(C3⋊S4)⋊C311yescharacteristic
328C4—31no—
329C4—31no—
330C4—61no—
331C4—361no—
332C3—31no—
333C3—121no—
334C3—241no—
335C3C3×#542111yescharacteristic
336C2—22no—
337C2—721no—
338C2C2×(C3⋊GL(2,𝔽3))⋊C311yescharacteristic
339C2#64264⋊C311yescharacteristic
340C2#64269⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
Frattini, socleC22×C3C2×C3×S4
FittingC22×C32×Q8D3
hypercenterC23(C3⋊S4)⋊C3
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. C2×(C3⋊GL(2,𝔽3))⋊C3C2semidirect
3. (C3⋊#859)⋊C3C2semidirect
4. #10593655C3semidirect
10. C22×C32×Q8D3semidirect
12. C22×C3×SL(2,𝔽3)C2×C3semidirect
13. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
17. C3⋊#859C2×C3semidirect
25. Q8⋊H(3,𝔽3)D4semidirect
83. C32×Q8#67semidirect
95. C3×SL(2,𝔽3)C3×D4semidirect
151. C22×C32GL(2,𝔽3)semidirect
164. C22×Q8(C3⋊D3)⋊C3semidirect
285. C32#5421semidirect
291. Q8#638⋊C3semidirect
326. C22(C3⋊GL(2,𝔽3))⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×(C3⋊GL(2,𝔽3))⋊C32
(C3⋊#859)⋊C32
#105936553
C3×#52273
C2×#638⋊C34