Finite Group Databasebeta

← C3×#10593653 #13173021

Subgroups — C3×#10593653 #13173021

Order 1728.

607 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 607 conjugacy classes of subgroups — too many to draw. Only its 74 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×C32×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3×C3⋊#226C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593653C311yesmaximal, characteristic, direct product factor
5C3×#5419—33nomaximal
6C3×#5228—31nomaximal
7C2×C32×SL(2,𝔽3)—21no—
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C3×C3⋊#226—21no—
10C3×#3757—41nomaximal
11C22×C32×Q8D311yescharacteristic
12C22×C3×SL(2,𝔽3)—21no—
13C22×C3×SL(2,𝔽3)—23no—
14C22×C3×SL(2,𝔽3)D313yes—
15C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
16C2×C3⋊#226C2×C311yescharacteristic, complemented
17C2×C3×#226—33no—
18C2×C3×#216—31no—
19C2×C3×#216—31no—
20C3×#924—31no—
21C3⋊#859C2×C311yescharacteristic, complemented
22C3×#859—33no—
23C3×#835—31no—
24C3×#835—31no—
25C3×#830—31no—
26C23×C33—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C3×C3⋊Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic
30#5419—33no—
31C3×#451—91no—
32#5228—31no—
33C2×C32×Q8—21no—
34C2×C32×Q8—31no—
35C2×C32×Q8—61no—
36C2×C32×Q8C2×D311yescharacteristic
37C22×C32×C4—31no—
38C2×C3×Dic6—31no—
39C2×C3×SL(2,𝔽3)—22no—
40C2×C3×SL(2,𝔽3)—26no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—23no—
43C2×C3×SL(2,𝔽3)—23no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)C2×D313yes—
46C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
47C3⋊#226—21no—
48C3×#226—63no—
49#3757—41no—
50C3×#217—61no—
51C3×#217—123no—
52C3×#216—61no—
53C3×#216—62no—
54C3×#216—61no—
55C3×C4⋊Dic3—31no—
56C3×Dic3⋊C4—61no—
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8—31no—
59C22×C33—41no—
60C22×C33—42no—
61C22×C33—81no—
62C22×C33—81no—
63C3×C3⋊Dic3—81no—
64C3×C3⋊Dic3—81no—
65C22×C3×Q8—21no—
66C22×C3×Q8C3×D311yescharacteristic
67C22×C3×Q8C3⋊D311yescharacteristic
68C22×SL(2,𝔽3)—23no—
69C22×SL(2,𝔽3)C3×D313yes—
70C2×#226—33no—
71C2×C3×Q16—91no—
72C2×C3×Q16—91no—
73C3×C22⋊Q8—91no—
74C2×#216—31no—
75C2×#216—31no—
76#924—31no—
77#859—33no—
78C3×#103—91no—
79C3×#103—91no—
80C3×C22⋊C8—91no—
81#835—31no—
82#835—31no—
83#830—31no—
84C23×C32—41no—
85C23×C32—43no—
86C23×C32—83no—
87C23×C32—81no—
88C23×C32S411yescharacteristic
89C32×Q8—31no—
90C32×Q8—32no—
91C32×Q8—61no—
92C32×Q8#6711yescharacteristic
93C2×C32×C4—31no—
94C2×C32×C4—31no—
95C2×C32×C4—61no—
96C2×C32×C4—61no—
97C2×C3⋊Dic3—41no—
98C2×C3⋊Dic3—41no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C2×C3×Dic3—123no—
102C2×C3×Dic3—123no—
103C3×Dic6—61no—
104C3×Dic6—61no—
105C3×SL(2,𝔽3)—21no—
106C3×SL(2,𝔽3)—23no—
107C3×SL(2,𝔽3)#6713yes—
108C3×SL(2,𝔽3)C3×D411yescharacteristic
109C3×C3⋊C8—61no—
110C3×C3⋊C8—61no—
111#451—91noSylow
112C2×C33—41no—
113C2×C33—41no—
114C2×C33—41no—
115C2×C33—82no—
116C2×C3×Q8—22no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—21no—
120C2×C3×Q8—31no—
121C2×C3×Q8—31no—
122C2×C3×Q8—61no—
123C2×C3×Q8—61no—
124C2×C3×Q8—61no—
125C2×C3×Q8—62no—
126C2×C3×Q8—91no—
127C2×C3×Q8C2×C3×D311yescharacteristic
128C2×C3×Q8C2×C3⋊D311yescharacteristic
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—61no—
132C2×Dic6—31no—
133C2×SL(2,𝔽3)—26no—
134C2×SL(2,𝔽3)—23no—
135C2×SL(2,𝔽3)—23no—
136C2×SL(2,𝔽3)C2×C3×D313yes—
137#226—63no—
138C3×Q16—181no—
139C3×Q16—182no—
140C3×Q16—181no—
141C2×C3×C8—91no—
142C2×C3×C8—91no—
143C3×C4⋊C4—91no—
144C3×C4⋊C4—181no—
145C3×C22⋊C4—181no—
146#217—61no—
147#217—123no—
148#216—61no—
149#216—62no—
150#216—61no—
151C4⋊Dic3—31no—
152Dic3⋊C4—61no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—31no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—41no—
158C22×C32—42no—
159C22×C32—43no—
160C22×C32—46no—
161C22×C32—86no—
162C22×C32—83no—
163C22×C32—83no—
164C22×C32—81no—
165C22×C32—81no—
166C22×C32—83no—
167C22×C32—86no—
168C22×C32—82no—
169C22×C32—86no—
170C22×C32—82no—
171C22×C32—81no—
172C22×C32—82no—
173C22×C32#22612yescomplemented
174C22×C32C2×S411yescharacteristic
175C32×C4—31no—
176C32×C4—31no—
177C32×C4—61no—
178C3⋊Dic3—81no—
179C3⋊Dic3—81no—
180C3×Dic3—121no—
181C3×Dic3—121no—
182C3×Dic3—243no—
183C3×Dic3—243no—
184C22×Q8C3×C3⋊D311yescharacteristic
185C2×Q16—91no—
186C2×Q16—91no—
187C22⋊Q8—91no—
188#103—91no—
189#103—91no—
190C22⋊C8—91no—
191C33—41noSylow
192C23×C3—21no—
193C23×C3—43no—
194C23×C3—83no—
195C23×C3C3×S411yescharacteristic
196C23×C3C3⋊S411yescharacteristic
197C3×Q8—21no—
198C3×Q8—31no—
199C3×Q8—31no—
200C3×Q8—32no—
201C3×Q8—32no—
202C3×Q8—61no—
203C3×Q8—62no—
204C3×Q8—61no—
205C3×Q8—61no—
206C3×Q8—62no—
207C3×Q8—181no—
208C3×Q8—181no—
209C3×Q8C3×#6711yescharacteristic
210C3×Q8#63811yescharacteristic
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—181no—
224C2×C3×C4—181no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—123no—
228C2×Dic3—123no—
229Dic6—61no—
230Dic6—61no—
231SL(2,𝔽3)—23no—
232SL(2,𝔽3)C3×#6713yes—
233C3×C8—181no—
234C3×C8—181no—
235C3⋊C8—61no—
236C3⋊C8—61no—
237C2×C32—22no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—43no—
242C2×C32—43no—
243C2×C32—43no—
244C2×C32—83no—
245C2×C32—83no—
246C2×C32—83no—
247C2×C32—82no—
248C2×C32—86no—
249C2×C32—812no—
250C2×C32—84no—
251C2×C32—81no—
252C2×C32—81no—
253C2×C32—81no—
254C2×C32C2×#22611yescharacteristic
255C2×C32#98311yescharacteristic
256C2×C32GL(2,ℤ/4)11yescharacteristic
257C2×Q8—21no—
258C2×Q8—31no—
259C2×Q8—61no—
260C2×Q8—91no—
261C2×Q8C2×C3×C3⋊D311yescharacteristic
262C22×C4—31no—
263Q16—181no—
264Q16—182no—
265Q16—181no—
266C2×C8—91no—
267C2×C8—91no—
268C4⋊C4—91no—
269C4⋊C4—181no—
270C22⋊C4—181no—
271C22×C3—22no—
272C22×C3—22no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—22no—
279C22×C3—43no—
280C22×C3—46no—
281C22×C3—83no—
282C22×C3—83no—
283C22×C3—86no—
284C22×C3—86no—
285C22×C3—83no—
286C22×C3—86no—
287C22×C3C3×#22612yescomplemented
288C22×C3C3⋊#22612yescomplemented
289C22×C3C2×C3×S411yescharacteristic
290C22×C3C2×C3⋊S411yescharacteristic
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—31no—
294C3×C4—31no—
295C3×C4—61no—
296C3×C4—61no—
297C3×C4—62no—
298C3×C4—61no—
299C3×C4—61no—
300C3×C4—361no—
301C3×C4—361no—
302Dic3—121no—
303Dic3—121no—
304Dic3—243no—
305Dic3—243no—
306C32—41no—
307C32—43no—
308C32—83no—
309C32—81no—
310C32#541911yescharacteristic, complemented
311C23C3×C3⋊S411yescharacteristic
312Q8—31no—
313Q8—32no—
314Q8—61no—
315Q8—181no—
316Q8—181no—
317Q8C3×#63811yescharacteristic
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—181no—
323C2×C4—181no—
324C8—181no—
325C8—181no—
326C2×C3—24no—
327C2×C3—22no—
328C2×C3—22no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—43no—
333C2×C3—43no—
334C2×C3—43no—
335C2×C3—86no—
336C2×C3—812no—
337C2×C3—83no—
338C2×C3—83no—
339C2×C3—83no—
340C2×C3C2×C3×#22611yescharacteristic
341C2×C3C3×#98311yescharacteristic
342C2×C3C3×GL(2,ℤ/4)11yescharacteristic
343C2×C3C2×C3⋊#22611yescharacteristic
344C2×C3#6426411yescharacteristic
345C2×C3#6426911yescharacteristic
346C22—21no—
347C22—21no—
348C22C3×C3⋊#22612yescomplemented
349C22C2×C3×C3⋊S411yescharacteristic
350C4—31no—
351C4—31no—
352C4—61no—
353C4—361no—
354C4—361no—
355C3—21no—
356C3—43no—
357C3—83no—
358C3C3×#541911yescharacteristic, complemented
359C3#1059365311yescharacteristic, direct product factor
360C2—22no—
361C2C2×C3×C3⋊#22611yescharacteristic
362C2C3×#6426411yescharacteristic
363C2C3×#6426911yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C2×C3×C3⋊S4
FittingC22×C32×Q8D3
socleC22×C32C2×S4
prime radicalC22×C32×SL(2,𝔽3)C2
hypercenterC23×C3C3⋊S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊#226C2semidirect
3. C3×C3⋊#859C2semidirect
4. #10593653C3direct (N × H)
16. C2×C3⋊#226C2×C3semidirect
21. C3⋊#859C2×C3semidirect
173. C22×C32#226semidirect
287. C22×C3C3×#226semidirect
288. C22×C3C3⋊#226semidirect
310. C32#5419semidirect
348. C22C3×C3⋊#226semidirect
358. C3C3×#5419semidirect
359. C3#10593653direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C2×C3×C3⋊#2262
C3×C3⋊#8592
#105936533
C3×#54193
C3×#52283
C3×#37574