Finite Group Databasebeta

← C3×#10593218 #13165894

Subgroups — C3×#10593218 #13165894

Order 1728.

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

normal subgroups only

This group has 506 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×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4C3×#5421—31nomaximal
5#10593218C311yesmaximal, characteristic, direct product factor
6C3×#4806—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#626—41nomaximal
9C3×D3×SL(2,𝔽3)—21no—
10C3×C3⋊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)C2×C311yescharacteristic, complemented
16C2×C3×GL(2,𝔽3)—31no—
17C2×C3×GL(2,𝔽3)—61no—
18C2×C3×#215—31no—
19C2×C3×#204—31no—
20C3×#895—31no—
21C3×#859—21no—
22C3×#859D311yescharacteristic, complemented
23C3×#859C2×C311yescharacteristic, complemented
24C32×#103—31no—
25C3×(C2×D3)⋊C8—31no—
26C3×#810—31no—
27C22×C3×C3⋊D3—41no—
28C22×C32×D3—41no—
29C2×C32×Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic, complemented
31C3×#626—44no—
32#5421—31no—
33C3×#448—91no—
34#4806—31no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C3×#67—121no—
37C3×D3×Q8—21no—
38C3×D3×Q8—61no—
39C2×C3×D12—31no—
40C2×C3×D12—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×#626—41no—
51D3×SL(2,𝔽3)—22no—
52D3×SL(2,𝔽3)—21no—
53C3⋊GL(2,𝔽3)—21no—
54C3×GL(2,𝔽3)—62no—
55C3×GL(2,𝔽3)—61no—
56C2×C32×C8—31no—
57C32×C4⋊C4—31no—
58C3×#215—61no—
59C3×#215—61no—
60C3×(C2×D3)⋊C4—61no—
61C2×C3×C3⋊C8—31no—
62C3×#204—62no—
63C22×C33—41no—
64C2×C3×C3⋊D3—42no—
65C2×C3×C3⋊D3—81no—
66C2×C3×C3⋊D3—81no—
67C2×C32×D3—42no—
68C2×C32×D3—81no—
69C2×C32×D3—81no—
70C32×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)—31no—
76C2×GL(2,𝔽3)—61no—
77C2×C3×SD16—91no—
78C2×C3×SD16—91no—
79C3×#121—91no—
80C2×#215—31no—
81C2×#204—31no—
82#895—31no—
83#859C3×D311yescharacteristic, complemented
84C3×#103—31no—
85C3×#103—31no—
86C3×#103—61no—
87C3×#103—91no—
88C3×C22⋊C8—91no—
89(C2×D3)⋊C8—31no—
90#810—31no—
91C23×C32—121no—
92C22×C3⋊D3—41no—
93C22×C3×D3—41no—
94C22×C3×D3—61no—
95C22×C3×D3—81no—
96C22×C3×D3—121no—
97C22×C3×D3—241no—
98C22×C3×D3S411yescharacteristic
99C32×Q8—31no—
100C32×Q8#6711yescharacteristic, complemented
101C2×C32×C4—31no—
102C2×C32×C4—61no—
103C3×#67—124no—
104C2×C3×Dic3—31no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—81no—
108C3×D12—61no—
109C3×D12—61no—
110C3×D12—122no—
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)D1211yescharacteristic, complemented
120C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
121#626—44no—
122C32×C8—61no—
123C3×C3⋊C8—61no—
124#448—91noSylow
125C2×C33—41no—
126C2×C33—41no—
127C2×C33—41no—
128C3×C3⋊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—91no—
137C2×C3×D4—181no—
138C22×C3×C4—91no—
139C2×#67—121no—
140D3×Q8—21no—
141D3×Q8—61no—
142C2×D12—31no—
143C2×D12—61no—
144C2×C4×D3—31no—
145C2×Dic6—31no—
146C2×SL(2,𝔽3)—21no—
147C2×SL(2,𝔽3)—21no—
148C2×SL(2,𝔽3)—41no—
149C2×SL(2,𝔽3)—62no—
150C2×SL(2,𝔽3)—61no—
151C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
152GL(2,𝔽3)—62no—
153GL(2,𝔽3)—61no—
154C3×SD16—182no—
155C3×SD16—181no—
156C3×SD16—181no—
157C2×C3×C8—31no—
158C2×C3×C8—31no—
159C2×C3×C8—61no—
160C2×C3×C8—91no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—31no—
163C3×C4⋊C4—61no—
164C3×C22⋊C4—181no—
165#215—61no—
166#215—61no—
167(C2×D3)⋊C4—61no—
168C2×C3⋊C8—31no—
169#204—62no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—81no—
173C22×C32—81no—
174C22×C32—81no—
175C22×C32—122no—
176C22×C32—161no—
177C22×C32—241no—
178C22×C32—241no—
179C22×C32C2×S411yescharacteristic
180C2×C3⋊D3—42no—
181C2×C3⋊D3—81no—
182C2×C3⋊D3—81no—
183C2×C3×D3—21no—
184C2×C3×D3—21no—
185C2×C3×D3—42no—
186C2×C3×D3—81no—
187C2×C3×D3—81no—
188C2×C3×D3—82no—
189C2×C3×D3—82no—
190C2×C3×D3—82no—
191C2×C3×D3—122no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C2×C3×D3—121no—
195C2×C3×D3—241no—
196C2×C3×D3—241no—
197C2×C3×D3—242no—
198C2×C3×D3—242no—
199C2×C3×D3—242no—
200C2×C3×D3GL(2,𝔽3)12yescomplemented
201C32×C4—31no—
202C32×C4—31no—
203C32×C4—121no—
204C3×Dic3—42no—
205C3×Dic3—42no—
206C3×Dic3—61no—
207C3×Dic3—82no—
208C22×Q8—31no—
209C2×SD16—91no—
210C2×SD16—91no—
211#121—91no—
212#103—31no—
213#103—91no—
214C22⋊C8—91no—
215C33—41noSylow
216C23×C3—31no—
217C23×C3—121no—
218C23×C3—181no—
219C23×C3—241no—
220C22×D3—61no—
221C22×D3—121no—
222C22×D3—241no—
223C22×D3C3×S411yescharacteristic
224C3×Q8—21no—
225C3×Q8—31no—
226C3×Q8—31no—
227C3×Q8—61no—
228C3×Q8—92no—
229C3×Q8—181no—
230C3×Q8#62611yescharacteristic, complemented
231C3×Q8C3×#6711yescharacteristic, complemented
232C3×D4—181no—
233C3×D4—181no—
234C3×D4—362no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—91no—
241C2×C3×C4—121no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244#67—124no—
245C2×Dic3—31no—
246C2×Dic3—41no—
247D12—61no—
248D12—61no—
249D12—122no—
250C4×D3—61no—
251C4×D3—61no—
252Dic6—32no—
253Dic6—61no—
254SL(2,𝔽3)—21no—
255SL(2,𝔽3)—21no—
256SL(2,𝔽3)—41no—
257SL(2,𝔽3)C3×D1211yescharacteristic, complemented
258C3×C8—61no—
259C3×C8—61no—
260C3×C8—121no—
261C3×C8—181no—
262C3⋊C8—61no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—81no—
270C2×C32—81no—
271C2×C32—81no—
272C2×C32—81no—
273C2×C32—81no—
274C2×C32—81no—
275C2×C32—81no—
276C2×C32—81no—
277C2×C32—81no—
278C2×C32—161no—
279C2×C32—161no—
280C2×C32—161no—
281C2×C32—242no—
282C2×C32C2×GL(2,𝔽3)11yescharacteristic
283C2×C32#98311yescharacteristic
284C2×C32GL(2,ℤ/4)11yescharacteristic
285C3⋊D3—82no—
286C3×D3—22no—
287C3×D3—82no—
288C3×D3—84no—
289C3×D3—242no—
290C3×D3—244no—
291C3×D3—241no—
292C2×Q8—61no—
293C2×Q8—91no—
294C2×Q8—181no—
295C2×Q8C3×D3211yescharacteristic
296C2×D4—91no—
297C2×D4—181no—
298C22×C4—91no—
299SD16—182no—
300SD16—181no—
301SD16—181no—
302C2×C8—31no—
303C2×C8—91no—
304C4⋊C4—31no—
305C22⋊C4—181no—
306C22×C3—21no—
307C22×C3—32no—
308C22×C3—41no—
309C22×C3—61no—
310C22×C3—61no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—122no—
314C22×C3—161no—
315C22×C3—242no—
316C22×C3—241no—
317C22×C3—241no—
318C22×C3—242no—
319C22×C3—242no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3D3×S411yescharacteristic
324C22×C3C2×C3×S411yescharacteristic
325C2×D3—21no—
326C2×D3—21no—
327C2×D3—122no—
328C2×D3—121no—
329C2×D3—121no—
330C2×D3—121no—
331C2×D3—241no—
332C2×D3—241no—
333C2×D3—242no—
334C2×D3—242no—
335C2×D3—242no—
336C2×D3C3×GL(2,𝔽3)12yescomplemented
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—31no—
340C3×C4—31no—
341C3×C4—61no—
342C3×C4—61no—
343C3×C4—121no—
344C3×C4—121no—
345C3×C4—181no—
346C3×C4—241no—
347Dic3—42no—
348Dic3—61no—
349C32—41no—
350C32—41no—
351C32—81no—
352C32—81no—
353C32—81no—
354C32—161no—
355C32#542111yescharacteristic, complemented
356C23—31no—
357C23—181no—
358Q8—31no—
359Q8—92no—
360Q8—181no—
361Q8C3×#62611yescharacteristic, complemented
362D4—181no—
363D4—181no—
364D4—362no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—91no—
368C2×C4—181no—
369C2×C4—181no—
370C8—61no—
371C8—181no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—62no—
379C2×C3—81no—
380C2×C3—81no—
381C2×C3—81no—
382C2×C3—81no—
383C2×C3—81no—
384C2×C3—81no—
385C2×C3—161no—
386C2×C3—161no—
387C2×C3—161no—
388C2×C3—242no—
389C2×C3—244no—
390C2×C3—721no—
391C2×C3#6420211yescharacteristic
392C2×C3D3×GL(2,𝔽3)11yescharacteristic
393C2×C3#6421011yescharacteristic
394C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
395C2×C3C3×#98311yescharacteristic
396C2×C3C3×GL(2,ℤ/4)11yescharacteristic
397D3—22no—
398D3—242no—
399D3—244no—
400D3—241no—
401C22—32no—
402C22—61no—
403C22—61no—
404C22—361no—
405C22—361no—
406C22—361no—
407C22C3×D3×S411yescharacteristic
408C4—31no—
409C4—31no—
410C4—121no—
411C4—181no—
412C3—21no—
413C3—41no—
414C3—81no—
415C3—81no—
416C3—161no—
417C3#1059321811yescharacteristic, direct product factor
418C3C3×#542111yescharacteristic, complemented
419C2—62no—
420C2—721no—
421C2C3×#6420211yescharacteristic
422C2C3×D3×GL(2,𝔽3)11yescharacteristic
423C2C3×#6421011yescharacteristic
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×C3×C3⋊GL(2,𝔽3)C2semidirect
3. C32×#859C2semidirect
5. #10593218C3direct (N × H)
11. C2×C3×D3×Q8D3semidirect
14. C2×D3×SL(2,𝔽3)C2×C3semidirect
15. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
22. C3×#859D3semidirect
23. C3×#859C2×C3semidirect
30. C32×SL(2,𝔽3)D4semidirect
72. C2×D3×Q8C3×D3semidirect
83. #859C3×D3semidirect
100. C32×Q8#67semidirect
119. C3×SL(2,𝔽3)D12semidirect
120. C3×SL(2,𝔽3)C3×D4semidirect
200. C2×C3×D3GL(2,𝔽3)semidirect
230. C3×Q8#626semidirect
231. C3×Q8C3×#67semidirect
257. SL(2,𝔽3)C3×D12semidirect
336. C2×D3C3×GL(2,𝔽3)semidirect
355. C32#5421semidirect
361. Q8C3×#626semidirect
417. C3#10593218direct (N × H)
418. 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×C3×C3⋊GL(2,𝔽3)2
C32×#8592
C3×#54213
#105932183
C3×#48063
C2×C3×#6264