Finite Group Databasebeta

← #10596199⋊C3 #13153324

Subgroups — #10596199⋊C3 #13153324

Order 1728.

845 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64155⋊C3C212yesmaximal
3Q8×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
4#64106⋊C3C212yesmaximal
5#64101⋊C3C212yesmaximal
6#64099⋊C3C212yesmaximal, complemented
7#64096⋊C3C212yesmaximal, complemented
8C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
9#64078⋊C3C211yesmaximal, characteristic, complemented
10#10596199C311yesmaximal, characteristic, complemented
11C3×#5728—31nomaximal
12C2×Q8×H(3,𝔽3)C2212yes—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2212yes—
17C4⋊C4×H(3,𝔽3)C2212yes—
18C4⋊C4×H(3,𝔽3)C2212yes—
19C42×H(3,𝔽3)C2211yescharacteristic
20#3752⋊C3C2214yes—
21C4⋊((C3⋊Dic3)⋊C3)C2212yes—
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2214yes—
24#3750⋊C3C2214yes—
25C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2214yes—
29C32×C4⋊Q8—21no—
30C32×C4⋊Q8D311yescharacteristic, complemented
31C32×C4⋊Q8C2×C311yescharacteristic, complemented
32#64155C2×C312yes—
33Q8×C3⋊Dic3C2×C312yescomplemented
34#64106C2×C312yes—
35#64101C2×C312yes—
36#64099C2×C312yescomplemented
37#64096C2×C312yescomplemented
38C42×C3⋊D3C2×C311yescharacteristic
39#64078C2×C311yescharacteristic, complemented
40C3×#946—32no—
41C3×Q8×Dic3—32no—
42C3×#897—32no—
43C3×#892—32no—
44C3×#890—32no—
45C3×#887—32no—
46C3×C42×D3—31no—
47C3×#869—31no—
48C22×(C3⋊D3)⋊C3C2311yescharacteristic
49Q8×H(3,𝔽3)—24no—
50C2×C4×H(3,𝔽3)C2314yes—
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2311yescharacteristic
54C2×(C3⋊Dic3)⋊C3C2314yes—
55C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
56C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
57C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
58C4×(C3⋊D3)⋊C3—22no—
59C4×(C3⋊D3)⋊C3—22no—
60C4×(C3⋊D3)⋊C3Q814yescomplemented
61#634⋊C3—24no—
62C3×#533—91no—
63#5728—31no—
64#5728—91no—
65C2×C32×Q8—22no—
66C2×C32×Q8C2×D312yes—
67C2×C32×Q8C22×C312yes—
68C2×C4×C3⋊D3C22×C311yescharacteristic
69C2×C4×C3⋊D3C22×C311yescharacteristic
70C2×C4×C3⋊D3C22×C311yescharacteristic
71C2×#634C22×C312yes—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—31no—
75C2×C3×Dic6—32no—
76C32×C4⋊C4—22no—
77C32×C4⋊C4—22no—
78C32×C4⋊C4C2×D312yes—
79C32×C4⋊C4C2×D312yes—
80C32×C4⋊C4C22×C312yes—
81C32×C4⋊C4C22×C312yes—
82C32×C42—21no—
83C32×C42C2×D311yescharacteristic
84C32×C42C22×C311yescharacteristic
85#3752C22×C314yes—
86C4⋊(C3⋊Dic3)C22×C312yes—
87C4⋊(C3⋊Dic3)C22×C312yes—
88C4⋊(C3⋊Dic3)C22×C314yes—
89#3750C22×C314yes—
90C4×C3⋊Dic3C22×C311yescharacteristic
91C4×C3⋊Dic3C22×C311yescharacteristic
92C4×C3⋊Dic3C22×C311yescharacteristic
93C4×C3⋊Dic3C22×C314yes—
94C3×(C2×D3)⋊C4—34no—
95C3×C4⋊Dic3—32no—
96C3×C4⋊Dic3—34no—
97C3×C4⋊Dic3—32no—
98C3×Dic3⋊C4—34no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—34no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C22×H(3,𝔽3)C2411yescharacteristic
104C2×(C3⋊D3)⋊C3—21no—
105C2×(C3⋊D3)⋊C3—21no—
106C2×(C3⋊D3)⋊C3C2×Q812yes—
107C4×H(3,𝔽3)—24no—
108C4×H(3,𝔽3)C2×Q812yes—
109C4×H(3,𝔽3)#4112yes—
110C4×H(3,𝔽3)#4112yes—
111(C3⋊Dic3)⋊C3—24no—
112(C3⋊Dic3)⋊C3—21no—
113(C3⋊Dic3)⋊C3—21no—
114(C3⋊Dic3)⋊C3C2×Q812yes—
115C3×C4⋊Q8—31no—
116C3×C4⋊Q8—31no—
117C3×C4⋊Q8—61no—
118C3×C4⋊Q8—91no—
119C3×C4⋊Q8C3×D311yescharacteristic
120C3×#125—92no—
121C3×C22⋊Q8—92no—
122C3×C22⋊Q8—92no—
123C3×C4×Q8—92no—
124C3×C4×Q8—92no—
125C3×#117—92no—
126C2×C3×C42—91no—
127#946—32no—
128#946—92no—
129Q8×Dic3—32no—
130Q8×Dic3—92no—
131#897—32no—
132#897—92no—
133#892—32no—
134#892—92no—
135#890—32no—
136#890—92no—
137#887—32no—
138#887—92no—
139C42×D3—31no—
140C42×D3—91no—
141#869—31no—
142#869—91no—
143C22×C3⋊D3C23×C311yescharacteristic
144C22×C3×D3—31no—
145C32×Q8—24no—
146C32×Q8—24no—
147C32×Q8—44no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—24no—
152C2×C32×C4C22×D314yes—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C23×C314yes—
157C2×C32×C4C23×C311yescharacteristic
158C2×C32×C4C23×C311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160C2×C3⋊Dic3C23×C311yescharacteristic
161C2×C3⋊Dic3C23×C311yescharacteristic
162C2×C3⋊Dic3C23×C314yes—
163C2×C3⋊Dic3C23×C311yescharacteristic
164C4×C3⋊D3—22no—
165C4×C3⋊D3—22no—
166C4×C3⋊D3C3×Q814yescomplemented
167#634—24no—
168C2×C3×Dic3—31no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—34no—
172C3×C4×D3—34no—
173C3×C4×D3—62no—
174C3×C4×D3—62no—
175C3×Dic6—64no—
176#533—91noSylow
177C2×H(3,𝔽3)C22×Q811yescharacteristic
178C2×H(3,𝔽3)C2×#4111yescharacteristic
179C2×H(3,𝔽3)C2×#4111yescharacteristic
180(C3⋊D3)⋊C3—22no—
181C2×C3×Q8—32no—
182C2×C3×Q8—32no—
183C2×C3×Q8—62no—
184C2×C3×Q8—92no—
185C2×C3×Q8C2×C3×D312yes—
186C22×C3×C4—91no—
187C22×C3×C4—91no—
188C22×C3×C4—91no—
189C2×C4×D3—31no—
190C2×C4×D3—31no—
191C2×C4×D3—31no—
192C2×C4×D3—91no—
193C2×C4×D3—91no—
194C2×C4×D3—91no—
195C2×Dic6—32no—
196C2×Dic6—92no—
197C3×C4⋊C4—32no—
198C3×C4⋊C4—32no—
199C3×C4⋊C4—32no—
200C3×C4⋊C4—32no—
201C3×C4⋊C4—62no—
202C3×C4⋊C4—62no—
203C3×C4⋊C4—94no—
204C3×C4⋊C4—92no—
205C3×C4⋊C4—94no—
206C3×C4⋊C4—92no—
207C3×C4⋊C4C2×C3×D312yes—
208C3×C4⋊C4C2×C3×D312yes—
209C3×C22⋊C4—94no—
210C3×C42—31no—
211C3×C42—31no—
212C3×C42—61no—
213C3×C42—91no—
214C3×C42—91no—
215C3×C42—91no—
216C3×C42—94no—
217C3×C42C2×C3×D311yescharacteristic
218(C2×D3)⋊C4—34no—
219(C2×D3)⋊C4—94no—
220C4⋊Dic3—32no—
221C4⋊Dic3—32no—
222C4⋊Dic3—34no—
223C4⋊Dic3—92no—
224C4⋊Dic3—92no—
225C4⋊Dic3—94no—
226Dic3⋊C4—34no—
227Dic3⋊C4—94no—
228C4×Dic3—31no—
229C4×Dic3—31no—
230C4×Dic3—31no—
231C4×Dic3—34no—
232C4×Dic3—91no—
233C4×Dic3—91no—
234C4×Dic3—91no—
235C4×Dic3—94no—
236C22×C32—21no—
237C22×C32C23×D311yescharacteristic
238C22×C32C24×C311yescharacteristic
239C2×C3⋊D3—21no—
240C2×C3⋊D3—21no—
241C2×C3⋊D3C2×C3×Q812yes—
242C2×C3×D3—32no—
243C2×C3×D3—61no—
244C2×C3×D3—61no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—22no—
250C32×C4—44no—
251C32×C4#23712yes—
252C32×C4D3×Q812yes—
253C32×C4#23912yes—
254C32×C4C2×C3×Q812yes—
255C32×C4C3×#4112yes—
256C32×C4C3×#4112yes—
257C3⋊Dic3—24no—
258C3⋊Dic3—21no—
259C3⋊Dic3—21no—
260C3⋊Dic3C2×C3×Q812yes—
261C3×Dic3—32no—
262C3×Dic3—61no—
263C3×Dic3—64no—
264C3×Dic3—61no—
265C4⋊Q8—91no—
266C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
267#125—92no—
268C22⋊Q8—92no—
269C22⋊Q8—92no—
270C4×Q8—92no—
271C4×Q8—92no—
272#117—92no—
273C2×C42—91no—
274H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
275C23×C3—91no—
276C22×D3—31no—
277C22×D3—91no—
278C3×Q8—24no—
279C3×Q8—64no—
280C3×Q8—64no—
281C3×Q8—124no—
282C3×Q8—184no—
283C2×C3×C4—31no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—31no—
289C2×C3×C4—34no—
290C2×C3×C4—34no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—64no—
295C2×C3×C4—94no—
296C2×C3×C4—91no—
297C2×C3×C4—91no—
298C2×C3×C4—91no—
299C2×C3×C4—94no—
300C2×C3×C4—182no—
301C2×C3×C4—182no—
302C2×C3×C4C22×C3×D314yes—
303C2×C3×C4C22×C3×D311yescharacteristic
304C2×C3×C4C22×C3×D311yescharacteristic
305C2×C3×C4C22×C3×D311yescharacteristic
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—34no—
310C2×Dic3—91no—
311C2×Dic3—91no—
312C2×Dic3—91no—
313C2×Dic3—94no—
314C4×D3—34no—
315C4×D3—62no—
316C4×D3—62no—
317C4×D3—94no—
318C4×D3—182no—
319C4×D3—182no—
320Dic6—64no—
321Dic6—184no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32C2×#23711yescharacteristic
326C2×C32C2×D3×Q811yescharacteristic
327C2×C32C2×#23911yescharacteristic
328C2×C32C22×C3×Q811yescharacteristic
329C2×C32C2×C3×#4111yescharacteristic
330C2×C32C2×C3×#4111yescharacteristic
331C3⋊D3—22no—
332C3×D3—62no—
333C2×Q8—92no—
334C2×Q8C2×(C3⋊D3)⋊C312yes—
335C22×C4—91no—
336C22×C4—91no—
337C22×C4—91no—
338C4⋊C4—92no—
339C4⋊C4—94no—
340C4⋊C4—94no—
341C4⋊C4—92no—
342C4⋊C4C2×(C3⋊D3)⋊C312yes—
343C4⋊C4C2×(C3⋊D3)⋊C312yes—
344C22⋊C4—94no—
345C42—91no—
346C42—91no—
347C42—91no—
348C42—94no—
349C42C2×(C3⋊D3)⋊C311yescharacteristic
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—61no—
353C22×C3—92no—
354C22×C3—181no—
355C22×C3—181no—
356C22×C3C23×C3×D311yescharacteristic
357C2×D3—32no—
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—92no—
361C2×D3—181no—
362C2×D3—181no—
363C3×C4—24no—
364C3×C4—32no—
365C3×C4—32no—
366C3×C4—32no—
367C3×C4—32no—
368C3×C4—32no—
369C3×C4—32no—
370C3×C4—64no—
371C3×C4—64no—
372C3×C4—62no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—92no—
376C3×C4—124no—
377C3×C4—181no—
378C3×C4—181no—
379C3×C4—184no—
380C3×C4C3×#23712yes—
381C3×C4C3×D3×Q812yes—
382C3×C4C3×#23912yes—
383Dic3—32no—
384Dic3—61no—
385Dic3—64no—
386Dic3—61no—
387Dic3—92no—
388Dic3—184no—
389Dic3—181no—
390Dic3—181no—
391C32—21no—
392C32#572811yescharacteristic, complemented
393C32C3×#53311yescharacteristic, complemented
394C23—91no—
395Q8—24no—
396Q8—184no—
397C2×C4—91no—
398C2×C4—94no—
399C2×C4—91no—
400C2×C4—91no—
401C2×C4—94no—
402C2×C4—182no—
403C2×C4—182no—
404C2×C4C22×(C3⋊D3)⋊C314yes—
405C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
406C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
407C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
408C2×C3—31no—
409C2×C3—31no—
410C2×C3—31no—
411C2×C3—31no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—61no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—182no—
418C2×C3C2×C3×#23711yescharacteristic
419C2×C3C2×C3×D3×Q811yescharacteristic
420C2×C3C2×C3×#23911yescharacteristic
421D3—62no—
422D3—182no—
423C22—92no—
424C22—181no—
425C22—181no—
426C22C23×(C3⋊D3)⋊C311yescharacteristic
427C4—24no—
428C4—92no—
429C4—181no—
430C4—181no—
431C4—184no—
432C4#3830⋊C312yes—
433C4Q8×(C3⋊D3)⋊C312yes—
434C4#8872212yes—
435C3—31no—
436C3—31no—
437C3—61no—
438C3C3×#572811yescharacteristic
439C2—182no—
440C2C2×#3830⋊C311yescharacteristic
441C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
442C2C2×#8872211yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#533
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×C4⋊Q8C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
6. #64099⋊C3C2semidirect
7. #64096⋊C3C2semidirect
9. #64078⋊C3C2semidirect
10. #10596199C3semidirect
30. C32×C4⋊Q8D3semidirect
31. C32×C4⋊Q8C2×C3semidirect
33. Q8×C3⋊Dic3C2×C3semidirect
36. #64099C2×C3semidirect
37. #64096C2×C3semidirect
39. #64078C2×C3semidirect
60. C4×(C3⋊D3)⋊C3Q8semidirect
166. C4×C3⋊D3C3×Q8semidirect
266. C4⋊Q8(C3⋊D3)⋊C3semidirect
274. H(3,𝔽3)#533semidirect
392. C32#5728semidirect
393. C32C3×#533semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#64155⋊C32
Q8×(C3⋊Dic3)⋊C32
#64106⋊C32
#64101⋊C32
#64099⋊C32
#64096⋊C32
C42×(C3⋊D3)⋊C32
#64078⋊C32
#105961993
C3×#57283