Finite Group Databasebeta

← C3×#10593217 #13165893

Subgroups — C3×#10593217 #13165893

Order 1728.

509 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 54 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4C3×#5421—31nomaximal
5#10593217C311yesmaximal, characteristic, direct product factor
6C3×#5170—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#625—41nomaximal
9C3×D3×SL(2,𝔽3)—21no—
10C32×GL(2,𝔽3)—21no—
11C2×C3×D3×Q8D311yescharacteristic, complemented
12C22×C3×SL(2,𝔽3)—31no—
13C2×D3×SL(2,𝔽3)—21no—
14C2×D3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
15C2×C3×GL(2,𝔽3)—21no—
16C2×C3×GL(2,𝔽3)D311yescharacteristic, complemented
17C2×C3×GL(2,𝔽3)C2×C311yescharacteristic, complemented
18C2×C32×SD16—31no—
19C3×#938—31no—
20C2×C3×#214—31no—
21C3⋊#859C2×C311yescharacteristic, complemented
22C3×#859—31no—
23C3×#859—61no—
24C3×#835—31no—
25C3×(C2×D3)⋊C8—31no—
26C3×#816—31no—
27C22×C32×D3—41no—
28C22×C32×D3—41no—
29C2×C3×C3⋊Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic, complemented
31C3×#625—44no—
32#5421—31no—
33C3×#448—91no—
34#5170—31no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×D4—31no—
37C2×C3×#67—61no—
38C2×C3×#67—121no—
39C3×D3×Q8—21no—
40C3×D3×Q8—61no—
41C2×C3×C4×D3—31no—
42C2×C3×Dic6—31no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)—21no—
45C2×C3×SL(2,𝔽3)—21no—
46C2×C3×SL(2,𝔽3)—41no—
47C2×C3×SL(2,𝔽3)—61no—
48C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
49C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
50C2×#625—41no—
51D3×SL(2,𝔽3)—22no—
52D3×SL(2,𝔽3)—21no—
53C3×GL(2,𝔽3)—22no—
54C3×GL(2,𝔽3)—21no—
55C3×GL(2,𝔽3)—21no—
56C32×SD16—61no—
57C32×SD16—61no—
58C2×C32×C8—31no—
59C3×#217—61no—
60C3×#214—62no—
61C3×C4⋊Dic3—31no—
62C2×C3×C3⋊C8—31no—
63C22×C33—41no—
64C2×C32×D3—42no—
65C2×C32×D3—42no—
66C2×C32×D3—81no—
67C2×C32×D3—81no—
68C2×C32×D3—81no—
69C2×C32×D3—81no—
70C3×C3⋊Dic3—42no—
71C22×C3×Q8—31no—
72C2×D3×Q8C3×D311yescharacteristic, complemented
73C22×SL(2,𝔽3)—31no—
74C22×SL(2,𝔽3)—61no—
75C2×GL(2,𝔽3)C3×D311yescharacteristic, complemented
76C2×C3×SD16—31no—
77C2×C3×SD16—31no—
78C2×C3×SD16—61no—
79C2×C3×SD16—91no—
80C3×#121—91no—
81#938—31no—
82C2×#214—31no—
83#859—31no—
84#859—61no—
85C3×#103—91no—
86C3×#103—91no—
87C3×C22⋊C8—91no—
88#835—31no—
89(C2×D3)⋊C8—31no—
90#816—31no—
91C23×C32—61no—
92C23×C32—121no—
93C22×C3×D3—41no—
94C22×C3×D3—41no—
95C22×C3×D3—41no—
96C22×C3×D3—81no—
97C22×C3×D3—81no—
98C22×C3×D3S411yescharacteristic
99C32×Q8—31no—
100C32×Q8#6711yescharacteristic, complemented
101C32×D4—61no—
102C32×D4—61no—
103C2×C32×C4—31no—
104C2×C3⋊Dic3—41no—
105C3×#67—124no—
106C3×#67—122no—
107C2×C3×Dic3—31no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—121no—
110C2×C3×Dic3—241no—
111C3×C4×D3—61no—
112C3×C4×D3—61no—
113C3×Dic6—32no—
114C3×Dic6—61no—
115C3×SL(2,𝔽3)—21no—
116C3×SL(2,𝔽3)—21no—
117C3×SL(2,𝔽3)—21no—
118C3×SL(2,𝔽3)—41no—
119C3×SL(2,𝔽3)#6711yescharacteristic, complemented
120C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
121#625—44no—
122C32×C8—61no—
123C3×C3⋊C8—61no—
124#448—91noSylow
125C2×C33—41no—
126C2×C33—41no—
127C2×C33—41no—
128C32×D3—82no—
129C32×D3—82no—
130C2×C3×Q8—21no—
131C2×C3×Q8—61no—
132C2×C3×Q8—91no—
133C2×C3×Q8—181no—
134C2×C3×Q8D3211yescharacteristic
135C2×C3×Q8C2×C3×D311yescharacteristic
136C2×C3×D4—31no—
137C2×C3×D4—31no—
138C2×C3×D4—61no—
139C2×C3×D4—181no—
140C22×C3×C4—91no—
141C2×#67—61no—
142C2×#67—121no—
143D3×Q8—21no—
144D3×Q8—61no—
145C2×C4×D3—31no—
146C2×Dic6—31no—
147C2×SL(2,𝔽3)—21no—
148C2×SL(2,𝔽3)—21no—
149C2×SL(2,𝔽3)—41no—
150C2×SL(2,𝔽3)—62no—
151C2×SL(2,𝔽3)—61no—
152C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
153GL(2,𝔽3)—21no—
154C3×SD16—62no—
155C3×SD16—61no—
156C3×SD16—61no—
157C3×SD16—62no—
158C3×SD16—61no—
159C3×SD16—61no—
160C3×SD16—182no—
161C2×C3×C8—31no—
162C2×C3×C8—31no—
163C2×C3×C8—61no—
164C2×C3×C8—91no—
165C3×C4⋊C4—91no—
166C3×C22⋊C4—181no—
167#217—61no—
168#214—62no—
169C4⋊Dic3—31no—
170C2×C3⋊C8—31no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—81no—
174C22×C32—81no—
175C22×C32—81no—
176C22×C32—122no—
177C22×C32—121no—
178C22×C32—121no—
179C22×C32—121no—
180C22×C32—161no—
181C22×C32—241no—
182C22×C32—241no—
183C22×C32C2×S411yescharacteristic
184C2×C3×D3—21no—
185C2×C3×D3—21no—
186C2×C3×D3—42no—
187C2×C3×D3—42no—
188C2×C3×D3—42no—
189C2×C3×D3—81no—
190C2×C3×D3—81no—
191C2×C3×D3—82no—
192C2×C3×D3—82no—
193C2×C3×D3—82no—
194C2×C3×D3—81no—
195C2×C3×D3—81no—
196C2×C3×D3—81no—
197C2×C3×D3—81no—
198C2×C3×D3—82no—
199C2×C3×D3—82no—
200C2×C3×D3—82no—
201C2×C3×D3GL(2,𝔽3)12yescomplemented
202C32×C4—31no—
203C32×C4—31no—
204C3⋊Dic3—42no—
205C3×Dic3—61no—
206C3×Dic3—122no—
207C3×Dic3—121no—
208C3×Dic3—242no—
209C22×Q8—31no—
210C2×SD16—31no—
211C2×SD16—91no—
212#121—91no—
213#103—91no—
214#103—91no—
215C22⋊C8—91no—
216C33—41noSylow
217C23×C3—31no—
218C23×C3—61no—
219C23×C3—61no—
220C23×C3—121no—
221C23×C3—121no—
222C23×C3—241no—
223C22×D3—41no—
224C22×D3C3×S411yescharacteristic
225C3×Q8—21no—
226C3×Q8—31no—
227C3×Q8—31no—
228C3×Q8—61no—
229C3×Q8—92no—
230C3×Q8—181no—
231C3×Q8#62511yescharacteristic, complemented
232C3×Q8C3×#6711yescharacteristic, complemented
233C3×D4—61no—
234C3×D4—61no—
235C3×D4—61no—
236C3×D4—61no—
237C3×D4—62no—
238C3×D4—121no—
239C3×D4—362no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—61no—
243C2×C3×C4—91no—
244C2×C3×C4—181no—
245C2×C3×C4—181no—
246C2×C3×C4—181no—
247#67—124no—
248#67—122no—
249C2×Dic3—31no—
250C2×Dic3—61no—
251C2×Dic3—121no—
252C2×Dic3—241no—
253C4×D3—61no—
254C4×D3—61no—
255Dic6—32no—
256Dic6—61no—
257SL(2,𝔽3)—21no—
258SL(2,𝔽3)—21no—
259SL(2,𝔽3)—41no—
260SL(2,𝔽3)C3×#6711yescharacteristic, complemented
261C3×C8—61no—
262C3×C8—61no—
263C3×C8—62no—
264C3×C8—181no—
265C3⋊C8—61no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—81no—
273C2×C32—81no—
274C2×C32—81no—
275C2×C32—81no—
276C2×C32—81no—
277C2×C32—81no—
278C2×C32—81no—
279C2×C32—81no—
280C2×C32—81no—
281C2×C32—161no—
282C2×C32—161no—
283C2×C32—161no—
284C2×C32—242no—
285C2×C32—241no—
286C2×C32C2×GL(2,𝔽3)11yescharacteristic
287C2×C32#98311yescharacteristic
288C2×C32GL(2,ℤ/4)11yescharacteristic
289C3×D3—22no—
290C3×D3—82no—
291C3×D3—84no—
292C3×D3—82no—
293C3×D3—82no—
294C3×D3—84no—
295C2×Q8—61no—
296C2×Q8—91no—
297C2×Q8—181no—
298C2×Q8C3×D3211yescharacteristic
299C2×D4—31no—
300C2×D4—181no—
301C22×C4—91no—
302SD16—61no—
303SD16—61no—
304SD16—182no—
305C2×C8—31no—
306C2×C8—91no—
307C4⋊C4—91no—
308C22⋊C4—181no—
309C22×C3—21no—
310C22×C3—32no—
311C22×C3—41no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3—122no—
317C22×C3—122no—
318C22×C3—121no—
319C22×C3—121no—
320C22×C3—121no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—161no—
325C22×C3—242no—
326C22×C3—241no—
327C22×C3—241no—
328C22×C3—242no—
329C22×C3—242no—
330C22×C3—241no—
331C22×C3—241no—
332C22×C3D3×S411yescharacteristic
333C22×C3C2×C3×S411yescharacteristic
334C2×D3—21no—
335C2×D3—21no—
336C2×D3—42no—
337C2×D3—81no—
338C2×D3—81no—
339C2×D3C3×GL(2,𝔽3)12yescomplemented
340C3×C4—31no—
341C3×C4—31no—
342C3×C4—31no—
343C3×C4—31no—
344C3×C4—61no—
345C3×C4—61no—
346C3×C4—181no—
347C3×C4—361no—
348Dic3—61no—
349Dic3—122no—
350Dic3—121no—
351Dic3—242no—
352C32—41no—
353C32—41no—
354C32—81no—
355C32—81no—
356C32—81no—
357C32—161no—
358C32#542111yescharacteristic, complemented
359C23—31no—
360C23—61no—
361Q8—31no—
362Q8—92no—
363Q8—181no—
364Q8C3×#62511yescharacteristic, complemented
365D4—61no—
366D4—61no—
367D4—362no—
368C2×C4—31no—
369C2×C4—91no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C8—61no—
374C8—181no—
375C2×C3—21no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—62no—
382C2×C3—81no—
383C2×C3—81no—
384C2×C3—81no—
385C2×C3—81no—
386C2×C3—81no—
387C2×C3—81no—
388C2×C3—161no—
389C2×C3—161no—
390C2×C3—161no—
391C2×C3—242no—
392C2×C3—244no—
393C2×C3—242no—
394C2×C3—241no—
395C2×C3—241no—
396C2×C3#6420111yescharacteristic
397C2×C3D3×GL(2,𝔽3)11yescharacteristic
398C2×C3#6420911yescharacteristic
399C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
400C2×C3C3×#98311yescharacteristic
401C2×C3C3×GL(2,ℤ/4)11yescharacteristic
402D3—22no—
403D3—82no—
404C22—32no—
405C22—61no—
406C22—61no—
407C22—121no—
408C22—121no—
409C22—121no—
410C22C3×D3×S411yescharacteristic
411C4—31no—
412C4—31no—
413C4—181no—
414C4—361no—
415C3—21no—
416C3—41no—
417C3—81no—
418C3—81no—
419C3—161no—
420C3#1059321711yescharacteristic, direct product factor
421C3C3×#542111yescharacteristic, complemented
422C2—62no—
423C2—241no—
424C2C3×#6420111yescharacteristic
425C2C3×D3×GL(2,𝔽3)11yescharacteristic
426C2C3×#6420911yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C22×C3
FrattiniC22C3×D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC3×SL(2,𝔽3)C3×D4
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×SL(2,𝔽3)C2semidirect
2. C2×C32×GL(2,𝔽3)C2semidirect
3. C3×C3⋊#859C2semidirect
5. #10593217C3direct (N × H)
11. C2×C3×D3×Q8D3semidirect
14. C2×D3×SL(2,𝔽3)C2×C3semidirect
16. C2×C3×GL(2,𝔽3)D3semidirect
17. C2×C3×GL(2,𝔽3)C2×C3semidirect
21. C3⋊#859C2×C3semidirect
30. C32×SL(2,𝔽3)D4semidirect
72. C2×D3×Q8C3×D3semidirect
75. C2×GL(2,𝔽3)C3×D3semidirect
100. C32×Q8#67semidirect
119. C3×SL(2,𝔽3)#67semidirect
120. C3×SL(2,𝔽3)C3×D4semidirect
201. C2×C3×D3GL(2,𝔽3)semidirect
231. C3×Q8#625semidirect
232. C3×Q8C3×#67semidirect
260. SL(2,𝔽3)C3×#67semidirect
339. C2×D3C3×GL(2,𝔽3)semidirect
358. C32#5421semidirect
364. Q8C3×#625semidirect
420. C3#10593217direct (N × H)
421. C3C3×#5421semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×SL(2,𝔽3)2
C2×C32×GL(2,𝔽3)2
C3×C3⋊#8592
C3×#54213
#105932173
C3×#51703
C2×C3×#6254