Finite Group Databasebeta

← C3×#10593215 #13165891

Subgroups — C3×#10593215 #13165891

Order 1728.

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

normal subgroups only

This group has 493 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×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C32×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C3×C4⋊GL(2,𝔽3)—31nomaximal
5#10593215C311yesmaximal, characteristic, direct product factor
6C3×#5163—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×#626—41nomaximal
9C3×C3⋊GL(2,𝔽3)—21no—
10C32×GL(2,𝔽3)—21no—
11C2×C3⋊GL(2,𝔽3)C2×C311yescharacteristic, complemented
12C2×C3×GL(2,𝔽3)—21no—
13C2×C3×GL(2,𝔽3)—31no—
14C2×C3×GL(2,𝔽3)—61no—
15C2×C3×GL(2,𝔽3)D311yescharacteristic, complemented
16C2×C3×GL(2,𝔽3)C2×C311yescharacteristic, complemented
17C2×C32×SD16—31no—
18C3×Q8×Dic3D311yescharacteristic, complemented
19C2×C3×#215—31no—
20C3×#940—31no—
21C3×C4×SL(2,𝔽3)—31no—
22Dic3×SL(2,𝔽3)—21no—
23Dic3×SL(2,𝔽3)C2×C311yescharacteristic, complemented
24C3×#832—31no—
25C3×#821—31no—
26C3×Dic3⋊C8—31no—
27C22×C3×C3⋊D3—41no—
28C22×C32×D3—41no—
29C2×C32×Dic3—41no—
30C32×SL(2,𝔽3)D411yescharacteristic, complemented
31C3×#626—44no—
32C4⋊GL(2,𝔽3)—31no—
33C3×#460—91no—
34#5163—31no—
35C2×C32×Q8C2×D311yescharacteristic
36C2×C32×D4—31no—
37C2×C3×#67—61no—
38C2×C3×#67—61no—
39C2×C3×D12—31no—
40C2×C3×D12—121no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)—41no—
45C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
46C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
47C2×#626—41no—
48C3⋊GL(2,𝔽3)—21no—
49C3×GL(2,𝔽3)—22no—
50C3×GL(2,𝔽3)—21no—
51C3×GL(2,𝔽3)—21no—
52C3×GL(2,𝔽3)—62no—
53C3×GL(2,𝔽3)—61no—
54C32×SD16—61no—
55C32×SD16—61no—
56C2×C32×C8—31no—
57C3×#215—61no—
58C3×#215—61no—
59C3×C4⋊Dic3—31no—
60C3×C4×Dic3—31no—
61C2×C3×C3⋊C8—31no—
62C22×C33—41no—
63C2×C3×C3⋊D3—42no—
64C2×C3×C3⋊D3—81no—
65C2×C3×C3⋊D3—81no—
66C2×C32×D3—42no—
67C2×C32×D3—81no—
68C2×C32×D3—81no—
69C32×Dic3—42no—
70C2×GL(2,𝔽3)—31no—
71C2×GL(2,𝔽3)—61no—
72C2×GL(2,𝔽3)C3×D311yescharacteristic, complemented
73C2×C3×SD16—31no—
74C2×C3×SD16—31no—
75C2×C3×SD16—61no—
76C2×C3×SD16—91no—
77C3×#127—91no—
78C3×C4×Q8—31no—
79Q8×Dic3C3×D311yescharacteristic, complemented
80C2×#215—31no—
81#940—31no—
82C4×SL(2,𝔽3)—31no—
83C4×SL(2,𝔽3)—61no—
84C3×C4⋊C8—91no—
85C3×#102—91no—
86C3×#102—91no—
87#832—31no—
88#821—31no—
89Dic3⋊C8—31no—
90C23×C32—61no—
91C22×C3⋊D3—41no—
92C22×C3×D3—41no—
93C22×C3×D3—41no—
94C22×C3×D3—61no—
95C22×C3×D3—81no—
96C22×C3×D3—121no—
97C22×C3×D3—241no—
98C32×Q8—31no—
99C32×Q8D1211yescharacteristic, complemented
100C32×D4—61no—
101C32×D4—61no—
102C2×C32×C4—31no—
103C2×C32×C4—121no—
104C3×#67—122no—
105C3×#67—122no—
106C2×C3×Dic3—31no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—81no—
109C2×C3×Dic3S411yescharacteristic
110C3×D12—61no—
111C3×D12—61no—
112C3×D12—124no—
113C3×SL(2,𝔽3)—21no—
114C3×SL(2,𝔽3)—21no—
115C3×SL(2,𝔽3)—21no—
116C3×SL(2,𝔽3)—41no—
117C3×SL(2,𝔽3)#6711yescharacteristic, complemented
118C3×SL(2,𝔽3)C3×D411yescharacteristic, complemented
119#626—44no—
120C32×C8—61no—
121C3×C3⋊C8—61no—
122#460—91noSylow
123C2×C33—41no—
124C2×C33—41no—
125C2×C33—41no—
126C3×C3⋊D3—82no—
127C32×D3—82no—
128C2×C3×Q8—21no—
129C2×C3×Q8D3211yescharacteristic
130C2×C3×Q8C2×C3×D311yescharacteristic
131C2×C3×D4—31no—
132C2×C3×D4—31no—
133C2×C3×D4—61no—
134C2×C3×D4—91no—
135C2×C3×D4—181no—
136C2×C3×D4—181no—
137C2×#67—61no—
138C2×#67—61no—
139C2×D12—31no—
140C2×D12—121no—
141C2×SL(2,𝔽3)—21no—
142C2×SL(2,𝔽3)—21no—
143C2×SL(2,𝔽3)—41no—
144C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
145GL(2,𝔽3)—21no—
146GL(2,𝔽3)—62no—
147GL(2,𝔽3)—61no—
148C3×SD16—62no—
149C3×SD16—61no—
150C3×SD16—61no—
151C3×SD16—62no—
152C3×SD16—61no—
153C3×SD16—61no—
154C3×SD16—181no—
155C3×SD16—181no—
156C2×C3×C8—31no—
157C2×C3×C8—31no—
158C2×C3×C8—61no—
159C2×C3×C8—91no—
160C3×C4⋊C4—91no—
161C3×C42—91no—
162#215—61no—
163#215—61no—
164C4⋊Dic3—31no—
165C4×Dic3—31no—
166C2×C3⋊C8—31no—
167C22×C32—41no—
168C22×C32—41no—
169C22×C32—81no—
170C22×C32—81no—
171C22×C32—81no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32—161no—
176C22×C32C2×S411yescharacteristic
177C2×C3⋊D3—42no—
178C2×C3⋊D3—81no—
179C2×C3⋊D3—81no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—82no—
183C2×C3×D3—81no—
184C2×C3×D3—81no—
185C2×C3×D3—81no—
186C2×C3×D3—81no—
187C2×C3×D3—82no—
188C2×C3×D3—82no—
189C2×C3×D3—122no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C2×C3×D3—241no—
194C2×C3×D3—241no—
195C2×C3×D3—242no—
196C2×C3×D3—242no—
197C2×C3×D3—242no—
198C32×C4—31no—
199C32×C4—31no—
200C32×C4—122no—
201C3×Dic3—42no—
202C3×Dic3—61no—
203C3×Dic3—82no—
204C3×Dic3GL(2,𝔽3)12yescomplemented
205C2×SD16—31no—
206C2×SD16—91no—
207#127—91no—
208C4×Q8—31no—
209C4⋊C8—91no—
210#102—91no—
211#102—91no—
212C33—41noSylow
213C23×C3—61no—
214C23×C3—61no—
215C23×C3—121no—
216C23×C3—181no—
217C22×D3—41no—
218C22×D3—61no—
219C22×D3—121no—
220C22×D3—241no—
221C3×Q8—21no—
222C3×Q8—31no—
223C3×Q8—31no—
224C3×Q8—61no—
225C3×Q8#62611yescharacteristic, complemented
226C3×Q8C3×D1211yescharacteristic, complemented
227C3×D4—61no—
228C3×D4—61no—
229C3×D4—61no—
230C3×D4—61no—
231C3×D4—62no—
232C3×D4—121no—
233C3×D4—181no—
234C3×D4—181no—
235C3×D4—362no—
236C3×D4—362no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—61no—
241C2×C3×C4—91no—
242C2×C3×C4—121no—
243C2×C3×C4—241no—
244#67—122no—
245#67—122no—
246C2×Dic3—31no—
247C2×Dic3C3×S411yescharacteristic
248D12—61no—
249D12—61no—
250D12—124no—
251SL(2,𝔽3)—21no—
252SL(2,𝔽3)—21no—
253SL(2,𝔽3)—41no—
254SL(2,𝔽3)C3×#6711yescharacteristic, complemented
255C3×C8—61no—
256C3×C8—61no—
257C3×C8—62no—
258C3×C8—181no—
259C3⋊C8—61no—
260C2×C32—41no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—81no—
267C2×C32—81no—
268C2×C32—81no—
269C2×C32—81no—
270C2×C32—81no—
271C2×C32—81no—
272C2×C32—81no—
273C2×C32—81no—
274C2×C32—81no—
275C2×C32—161no—
276C2×C32—161no—
277C2×C32—161no—
278C2×C32—241no—
279C2×C32C4⋊S411yescharacteristic
280C2×C32C2×GL(2,𝔽3)11yescharacteristic
281C2×C32#98611yescharacteristic
282C3⋊D3—82no—
283C3×D3—82no—
284C3×D3—82no—
285C3×D3—84no—
286C3×D3—242no—
287C3×D3—244no—
288C3×D3—241no—
289C2×Q8C3×D3211yescharacteristic
290C2×D4—31no—
291C2×D4—91no—
292C2×D4—181no—
293C2×D4—181no—
294SD16—61no—
295SD16—61no—
296SD16—181no—
297SD16—181no—
298C2×C8—31no—
299C2×C8—91no—
300C4⋊C4—91no—
301C42—91no—
302C22×C3—21no—
303C22×C3—41no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3—122no—
307C22×C3—121no—
308C22×C3—121no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—161no—
314C22×C3—241no—
315C22×C3—241no—
316C22×C3—361no—
317C22×C3—361no—
318C22×C3—361no—
319C22×C3D3×S411yescharacteristic
320C22×C3C2×C3×S411yescharacteristic
321C2×D3—42no—
322C2×D3—81no—
323C2×D3—81no—
324C2×D3—122no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—121no—
328C2×D3—241no—
329C2×D3—241no—
330C2×D3—242no—
331C2×D3—242no—
332C2×D3—242no—
333C3×C4—32no—
334C3×C4—31no—
335C3×C4—31no—
336C3×C4—31no—
337C3×C4—31no—
338C3×C4—61no—
339C3×C4—61no—
340C3×C4—122no—
341C3×C4—181no—
342C3×C4—242no—
343Dic3—61no—
344Dic3C3×GL(2,𝔽3)12yescomplemented
345C32—41no—
346C32—41no—
347C32—81no—
348C32—81no—
349C32—81no—
350C32—161no—
351C32C4⋊GL(2,𝔽3)11yescharacteristic, complemented
352C23—61no—
353C23—181no—
354Q8—31no—
355Q8C3×#62611yescharacteristic, complemented
356D4—61no—
357D4—61no—
358D4—181no—
359D4—181no—
360D4—362no—
361D4—362no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—91no—
365C8—61no—
366C8—181no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—81no—
374C2×C3—81no—
375C2×C3—81no—
376C2×C3—81no—
377C2×C3—81no—
378C2×C3—81no—
379C2×C3—161no—
380C2×C3—161no—
381C2×C3—161no—
382C2×C3—242no—
383C2×C3—241no—
384C2×C3—241no—
385C2×C3—721no—
386C2×C3#6419911yescharacteristic
387C2×C3D3×GL(2,𝔽3)11yescharacteristic
388C2×C3#6420711yescharacteristic
389C2×C3C3×C4⋊S411yescharacteristic
390C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
391C2×C3C3×#98611yescharacteristic
392D3—82no—
393D3—242no—
394D3—244no—
395D3—241no—
396C22—121no—
397C22—121no—
398C22—121no—
399C22—361no—
400C22—361no—
401C22—361no—
402C22C3×D3×S411yescharacteristic
403C4—32no—
404C4—31no—
405C4—31no—
406C4—181no—
407C3—21no—
408C3—41no—
409C3—81no—
410C3—81no—
411C3—161no—
412C3#1059321511yescharacteristic, direct product factor
413C3C3×C4⋊GL(2,𝔽3)11yescharacteristic, complemented
414C2—241no—
415C2—721no—
416C2C3×#6419911yescharacteristic
417C2C3×D3×GL(2,𝔽3)11yescharacteristic
418C2C3×#6420711yescharacteristic
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×C3⋊GL(2,𝔽3)C2semidirect
2. C2×C32×GL(2,𝔽3)C2semidirect
3. C3×Dic3×SL(2,𝔽3)C2semidirect
5. #10593215C3direct (N × H)
11. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
15. C2×C3×GL(2,𝔽3)D3semidirect
16. C2×C3×GL(2,𝔽3)C2×C3semidirect
18. C3×Q8×Dic3D3semidirect
23. Dic3×SL(2,𝔽3)C2×C3semidirect
30. C32×SL(2,𝔽3)D4semidirect
72. C2×GL(2,𝔽3)C3×D3semidirect
79. Q8×Dic3C3×D3semidirect
99. C32×Q8D12semidirect
117. C3×SL(2,𝔽3)#67semidirect
118. C3×SL(2,𝔽3)C3×D4semidirect
204. C3×Dic3GL(2,𝔽3)semidirect
225. C3×Q8#626semidirect
226. C3×Q8C3×D12semidirect
254. SL(2,𝔽3)C3×#67semidirect
344. Dic3C3×GL(2,𝔽3)semidirect
351. C32C4⋊GL(2,𝔽3)semidirect
355. Q8C3×#626semidirect
412. C3#10593215direct (N × H)
413. C3C3×C4⋊GL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C3⋊GL(2,𝔽3)2
C2×C32×GL(2,𝔽3)2
C3×Dic3×SL(2,𝔽3)2
C3×C4⋊GL(2,𝔽3)3
#105932153
C3×#51633
C2×C3×#6264