Finite Group Databasebeta

← C3×#10593655 #13173023

Subgroups — C3×#10593655 #13173023

Order 1728.

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

normal subgroups only

This group has 707 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, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4#10593655C311yesmaximal, characteristic, direct product factor
5C3×#5421—33nomaximal
6C3×#5227—31nomaximal
7C2×C3×#638—41nomaximal
8C2×C32×SL(2,𝔽3)—21no—
9C2×C32×SL(2,𝔽3)C2211yescharacteristic
10C3×C3⋊GL(2,𝔽3)—21no—
11C22×C32×Q8D311yescharacteristic, complemented
12C22×C3×SL(2,𝔽3)—21no—
13C22×C3×SL(2,𝔽3)—23no—
14C22×C3×SL(2,𝔽3)D313yescomplemented
15C22×C3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
16C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
17C2×C3×GL(2,𝔽3)—33no—
18C2×C3×#215—31no—
19C2×C3×#215—31no—
20C3×#930—31no—
21C3⋊#859C2×C311yescharacteristic, complemented
22C3×#859—33no—
23C3×#835—31no—
24C3×#835—31no—
25C3×#830—31no—
26C23×C33—41no—
27C22×C3×C3⋊D3—41no—
28C3×#638—44no—
29C2×C3×C3⋊Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic, complemented
31#5421—33no—
32C3×#448—91no—
33#5227—31no—
34C2×C32×Q8—21no—
35C2×C32×Q8—31no—
36C2×C32×Q8—61no—
37C2×C32×Q8C2×D311yescharacteristic
38C22×C32×C4—31no—
39C2×#638—41no—
40C2×C3×#67—61no—
41C2×C3×#67—123no—
42C2×C3×D12—31no—
43C2×C3×SL(2,𝔽3)—22no—
44C2×C3×SL(2,𝔽3)—26no—
45C2×C3×SL(2,𝔽3)—21no—
46C2×C3×SL(2,𝔽3)—23no—
47C2×C3×SL(2,𝔽3)—23no—
48C2×C3×SL(2,𝔽3)—21no—
49C2×C3×SL(2,𝔽3)C2×D313yes—
50C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
51C3⋊GL(2,𝔽3)—21no—
52C3×GL(2,𝔽3)—63no—
53C3×#215—61no—
54C3×#215—62no—
55C3×#215—61no—
56C3×(C2×D3)⋊C4—61no—
57C3×C4⋊Dic3—31no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8—31no—
60C22×C33—41no—
61C22×C33—42no—
62C22×C33—81no—
63C22×C33—81no—
64C2×C3×C3⋊D3—42no—
65C2×C3×C3⋊D3—81no—
66C2×C3×C3⋊D3—81no—
67C3×C3⋊Dic3—42no—
68C22×C3×Q8—21no—
69C22×C3×Q8C3×D311yescharacteristic, complemented
70C22×C3×Q8C3⋊D311yescharacteristic, complemented
71C22×SL(2,𝔽3)—23no—
72C22×SL(2,𝔽3)C3×D313yescomplemented
73C2×GL(2,𝔽3)—33no—
74C2×C3×SD16—91no—
75C2×C3×SD16—91no—
76C3×#121—91no—
77C2×#215—31no—
78C2×#215—31no—
79#930—31no—
80#859—33no—
81C3×#103—91no—
82C3×#103—91no—
83C3×C22⋊C8—91no—
84#835—31no—
85#835—31no—
86#830—31no—
87C23×C32—41no—
88C23×C32—43no—
89C23×C32—83no—
90C23×C32—81no—
91C23×C32S411yescharacteristic
92C22×C3⋊D3—41no—
93C22×C3×D3—61no—
94C22×C3×D3—123no—
95C32×Q8—31no—
96C32×Q8—32no—
97C32×Q8—61no—
98C32×Q8#6711yescharacteristic, complemented
99C2×C32×C4—31no—
100C2×C32×C4—31no—
101C2×C32×C4—61no—
102C2×C32×C4—61no—
103#638—44no—
104C2×C3⋊Dic3—41no—
105C3×#67—1212no—
106C3×#67—122no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—123no—
109C3×D12—61no—
110C3×D12—61no—
111C3×SL(2,𝔽3)—21no—
112C3×SL(2,𝔽3)—23no—
113C3×SL(2,𝔽3)#6713yescomplemented
114C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
115C3×C3⋊C8—61no—
116C3×C3⋊C8—61no—
117#448—91noSylow
118C2×C33—41no—
119C2×C33—41no—
120C2×C33—41no—
121C2×C33—82no—
122C3×C3⋊D3—82no—
123C2×C3×Q8—22no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8—31no—
128C2×C3×Q8—31no—
129C2×C3×Q8—61no—
130C2×C3×Q8—61no—
131C2×C3×Q8—61no—
132C2×C3×Q8—62no—
133C2×C3×Q8C2×C3×D311yescharacteristic
134C2×C3×Q8C2×C3⋊D311yescharacteristic
135C2×C3×D4—91no—
136C2×C3×D4—181no—
137C22×C3×C4—31no—
138C22×C3×C4—31no—
139C22×C3×C4—61no—
140C2×#67—61no—
141C2×#67—123no—
142C2×D12—31no—
143C2×SL(2,𝔽3)—26no—
144C2×SL(2,𝔽3)—23no—
145C2×SL(2,𝔽3)—23no—
146C2×SL(2,𝔽3)C2×C3×D313yes—
147GL(2,𝔽3)—63no—
148C3×SD16—181no—
149C3×SD16—182no—
150C3×SD16—181no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—91no—
154C3×C22⋊C4—181no—
155#215—61no—
156#215—62no—
157#215—61no—
158(C2×D3)⋊C4—61no—
159C4⋊Dic3—31no—
160C2×C3⋊C8—31no—
161C2×C3⋊C8—31no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—41no—
165C22×C32—42no—
166C22×C32—43no—
167C22×C32—46no—
168C22×C32—86no—
169C22×C32—83no—
170C22×C32—83no—
171C22×C32—81no—
172C22×C32—81no—
173C22×C32—83no—
174C22×C32—86no—
175C22×C32—82no—
176C22×C32—86no—
177C22×C32—82no—
178C22×C32—81no—
179C22×C32—82no—
180C22×C32GL(2,𝔽3)12yescomplemented
181C22×C32C2×S411yescharacteristic
182C2×C3⋊D3—42no—
183C2×C3⋊D3—81no—
184C2×C3⋊D3—81no—
185C2×C3×D3—126no—
186C2×C3×D3—121no—
187C2×C3×D3—121no—
188C2×C3×D3—121no—
189C2×C3×D3—243no—
190C2×C3×D3—243no—
191C32×C4—31no—
192C32×C4—31no—
193C32×C4—61no—
194C3⋊Dic3—42no—
195C3×Dic3—126no—
196C3×Dic3—121no—
197C22×Q8C3×C3⋊D311yescharacteristic, complemented
198C2×SD16—91no—
199C2×SD16—91no—
200#121—91no—
201#103—91no—
202#103—91no—
203C22⋊C8—91no—
204C33—41noSylow
205C23×C3—21no—
206C23×C3—43no—
207C23×C3—83no—
208C23×C3—181no—
209C23×C3C3×S411yescharacteristic
210C23×C3C3⋊S411yescharacteristic
211C22×D3—61no—
212C22×D3—123no—
213C3×Q8—21no—
214C3×Q8—31no—
215C3×Q8—31no—
216C3×Q8—32no—
217C3×Q8—32no—
218C3×Q8—61no—
219C3×Q8—62no—
220C3×Q8—61no—
221C3×Q8—61no—
222C3×Q8—62no—
223C3×Q8C3×#6711yescharacteristic, complemented
224C3×Q8#63811yescharacteristic, complemented
225C3×D4—181no—
226C3×D4—181no—
227C3×D4—362no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—61no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—181no—
241#67—1212no—
242#67—122no—
243C2×Dic3—61no—
244C2×Dic3—123no—
245D12—61no—
246D12—61no—
247SL(2,𝔽3)—23no—
248SL(2,𝔽3)C3×#6713yescomplemented
249C3×C8—181no—
250C3×C8—181no—
251C3⋊C8—61no—
252C3⋊C8—61no—
253C2×C32—22no—
254C2×C32—41no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—43no—
258C2×C32—43no—
259C2×C32—43no—
260C2×C32—83no—
261C2×C32—83no—
262C2×C32—83no—
263C2×C32—82no—
264C2×C32—86no—
265C2×C32—812no—
266C2×C32—84no—
267C2×C32—81no—
268C2×C32—81no—
269C2×C32—81no—
270C2×C32C2×GL(2,𝔽3)11yescharacteristic
271C2×C32#98311yescharacteristic
272C2×C32GL(2,ℤ/4)11yescharacteristic
273C3⋊D3—82no—
274C3×D3—246no—
275C3×D3—241no—
276C2×Q8—21no—
277C2×Q8—31no—
278C2×Q8—61no—
279C2×Q8C2×C3×C3⋊D311yescharacteristic
280C2×D4—91no—
281C2×D4—181no—
282C22×C4—31no—
283SD16—181no—
284SD16—182no—
285SD16—181no—
286C2×C8—91no—
287C2×C8—91no—
288C4⋊C4—91no—
289C22⋊C4—181no—
290C22×C3—22no—
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—43no—
299C22×C3—46no—
300C22×C3—83no—
301C22×C3—83no—
302C22×C3—86no—
303C22×C3—86no—
304C22×C3—83no—
305C22×C3—86no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3C3×GL(2,𝔽3)12yescomplemented
310C22×C3C3⋊GL(2,𝔽3)12yescomplemented
311C22×C3C2×C3×S411yescharacteristic
312C22×C3C2×C3⋊S411yescharacteristic
313C2×D3—126no—
314C2×D3—121no—
315C2×D3—121no—
316C2×D3—121no—
317C2×D3—243no—
318C2×D3—243no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—31no—
323C3×C4—61no—
324C3×C4—61no—
325C3×C4—62no—
326C3×C4—61no—
327C3×C4—61no—
328C3×C4—361no—
329Dic3—126no—
330Dic3—121no—
331C32—41no—
332C32—43no—
333C32—83no—
334C32—81no—
335C32#542111yescharacteristic, complemented
336C23—181no—
337C23C3×C3⋊S411yescharacteristic
338Q8—31no—
339Q8—32no—
340Q8—61no—
341Q8C3×#63811yescharacteristic, complemented
342D4—181no—
343D4—181no—
344D4—362no—
345C2×C4—31no—
346C2×C4—31no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—181no—
350C8—181no—
351C8—181no—
352C2×C3—24no—
353C2×C3—22no—
354C2×C3—22no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—43no—
359C2×C3—43no—
360C2×C3—43no—
361C2×C3—86no—
362C2×C3—812no—
363C2×C3—83no—
364C2×C3—83no—
365C2×C3—83no—
366C2×C3—721no—
367C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
368C2×C3C3×#98311yescharacteristic
369C2×C3C3×GL(2,ℤ/4)11yescharacteristic
370C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
371C2×C3#6426411yescharacteristic
372C2×C3#6426911yescharacteristic
373D3—246no—
374D3—241no—
375C22—21no—
376C22—21no—
377C22—361no—
378C22—361no—
379C22—361no—
380C22C3×C3⋊GL(2,𝔽3)12yescomplemented
381C22C2×C3×C3⋊S411yescharacteristic
382C4—31no—
383C4—31no—
384C4—61no—
385C4—361no—
386C3—21no—
387C3—43no—
388C3—83no—
389C3C3×#542111yescharacteristic, complemented
390C3#1059365511yescharacteristic, direct product factor
391C2—22no—
392C2—721no—
393C2C2×C3×C3⋊GL(2,𝔽3)11yescharacteristic
394C2C3×#6426411yescharacteristic
395C2C3×#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
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
1. C22×C32×SL(2,𝔽3)C2semidirect
2. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
3. C3×C3⋊#859C2semidirect
4. #10593655C3direct (N × H)
11. C22×C32×Q8D3semidirect
14. C22×C3×SL(2,𝔽3)D3semidirect
15. C22×C3×SL(2,𝔽3)C2×C3semidirect
16. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
21. C3⋊#859C2×C3semidirect
30. C32×SL(2,𝔽3)D4semidirect
69. C22×C3×Q8C3×D3semidirect
70. C22×C3×Q8C3⋊D3semidirect
72. C22×SL(2,𝔽3)C3×D3semidirect
98. C32×Q8#67semidirect
113. C3×SL(2,𝔽3)#67semidirect
114. C3×SL(2,𝔽3)C3×D4semidirect
180. C22×C32GL(2,𝔽3)semidirect
197. C22×Q8C3×C3⋊D3semidirect
223. C3×Q8C3×#67semidirect
224. C3×Q8#638semidirect
248. SL(2,𝔽3)C3×#67semidirect
309. C22×C3C3×GL(2,𝔽3)semidirect
310. C22×C3C3⋊GL(2,𝔽3)semidirect
335. C32#5421semidirect
341. Q8C3×#638semidirect
380. C22C3×C3⋊GL(2,𝔽3)semidirect
389. C3C3×#5421semidirect
390. C3#10593655direct (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⋊GL(2,𝔽3)2
C3×C3⋊#8592
#105936553
C3×#54213
C3×#52273
C2×C3×#6384