Finite Group Databasebeta

← #10596197⋊C3 #13153323

Subgroups — #10596197⋊C3 #13153323

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C212yesmaximal, complemented
2H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
3#64155⋊C3C212yesmaximal, complemented
4#64104⋊C3C212yesmaximal, complemented
5#64103⋊C3C214yesmaximal, complemented
6C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
7C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
8C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
9#10596197C311yesmaximal, characteristic, complemented
10C3×#5726—31nomaximal
11C2×Q8×H(3,𝔽3)C2212yes—
12#88722—28no—
13C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2214yes—
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2212yes—
21C42×H(3,𝔽3)C2211yescharacteristic
22#3752⋊C3C2212yes—
23#3752⋊C3C2212yes—
24#3752⋊C3C2214yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27#3750⋊C3C2214yes—
28C2×#3832C2×C312yescomplemented
29C2×C3×#239—32no—
30C32×C4⋊Q8—21no—
31C32×C4⋊Q8D311yescharacteristic, complemented
32C32×C4⋊Q8C2×C311yescharacteristic, complemented
33#64155C2×C312yescomplemented
34#64104C2×C312yescomplemented
35#64103C2×C314yescomplemented
36C4⋊C4×C3⋊D3C2×C312yescomplemented
37C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
38C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
39C3×#946—32no—
40C3×#895—32no—
41C3×#894—34no—
42C3×D3×C4⋊C4—32no—
43C3×C4×D12—31no—
44C3×C4×D12—31no—
45C22×(C3⋊D3)⋊C3C2312yes—
46C22×(C3⋊D3)⋊C3C2312yes—
47Q8×H(3,𝔽3)—24no—
48C2×C4×H(3,𝔽3)C2314yes—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C4⋊((C3⋊D3)⋊C3)—28no—
55C4⋊((C3⋊D3)⋊C3)—22no—
56C4⋊((C3⋊D3)⋊C3)D414yescomplemented
57C4×(C3⋊D3)⋊C3—28no—
58C4×(C3⋊D3)⋊C3—24no—
59C4×(C3⋊D3)⋊C3—24no—
60C3×#547—91no—
61#5726—31no—
62#5726—91no—
63C2×C32×Q8—22no—
64C2×C32×Q8C2×D312yes—
65C2×C32×Q8C22×C312yes—
66#3832—28no—
67C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
68C2×C4⋊(C3⋊D3)C22×C314yes—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C4×C3⋊D3C22×C312yes—
71C2×C4×C3⋊D3C22×C312yes—
72C2×C4×C3⋊D3C22×C314yes—
73C3×#239—68no—
74C2×C3×D12—31no—
75C2×C3×D12—34no—
76C2×C3×D12—31no—
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—32no—
80C32×C4⋊C4—22no—
81C32×C4⋊C4—22no—
82C32×C4⋊C4C2×D312yes—
83C32×C4⋊C4C2×D312yes—
84C32×C4⋊C4C22×C312yes—
85C32×C4⋊C4C22×C312yes—
86C32×C42—21no—
87C32×C42C2×D311yescharacteristic
88C32×C42C22×C311yescharacteristic
89#3752C22×C312yes—
90#3752C22×C312yes—
91#3752C22×C314yes—
92C4⋊(C3⋊Dic3)C22×C311yescharacteristic
93C4⋊(C3⋊Dic3)C22×C311yescharacteristic
94#3750C22×C314yes—
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4—34no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×Dic3⋊C4—34no—
101C22×H(3,𝔽3)C2411yescharacteristic
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—22no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3C2×D414yes—
108C4×H(3,𝔽3)—24no—
109C4×H(3,𝔽3)—21no—
110C4×H(3,𝔽3)C2×D412yes—
111C4×H(3,𝔽3)#4112yescomplemented
112(C3⋊Dic3)⋊C3—22no—
113(C3⋊Dic3)⋊C3—22no—
114C2×C3×#41—92no—
115C2×#239—32no—
116C2×#239—92no—
117C3×C4⋊Q8—31no—
118C3×C4⋊Q8—31no—
119C3×C4⋊Q8—61no—
120C3×C4⋊Q8C3×D311yescharacteristic
121C3×#123—94no—
122C3×C22⋊Q8—92no—
123C3×#121—92no—
124C3×C4×D4—91no—
125C3×C4×D4—91no—
126C2×C3×C4⋊C4—92no—
127#946—32no—
128#946—92no—
129#895—32no—
130#895—92no—
131#894—34no—
132#894—94no—
133D3×C4⋊C4—32no—
134D3×C4⋊C4—92no—
135C4×D12—31no—
136C4×D12—31no—
137C4×D12—91no—
138C4×D12—91no—
139C22×C3⋊D3C23×C312yes—
140C22×C3⋊D3C23×C312yes—
141C22×C3×D3—32no—
142C22×C3×D3—32no—
143C32×Q8—24no—
144C32×Q8—24no—
145C32×Q8—44no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—24no—
150C2×C32×C4C22×D314yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C23×C314yes—
155C2×C32×C4C23×C311yescharacteristic
156C2×C32×C4C23×C311yescharacteristic
157C2×C32×C4C23×C311yescharacteristic
158C2×C3⋊Dic3C23×C312yes—
159C2×C3⋊Dic3C23×C312yes—
160C4⋊(C3⋊D3)—28no—
161C4⋊(C3⋊D3)—22no—
162C4⋊(C3⋊D3)C3×D414yescomplemented
163C4×C3⋊D3—28no—
164C4×C3⋊D3—24no—
165C4×C3⋊D3—24no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C3×D12—34no—
169C3×D12—68no—
170C3×D12—62no—
171C3×C4×D3—68no—
172C3×C4×D3—64no—
173C3×C4×D3—64no—
174#547—91noSylow
175C2×H(3,𝔽3)C22×D411yescharacteristic
176C2×H(3,𝔽3)C2×#4111yescharacteristic
177C2×H(3,𝔽3)2−1+411yescharacteristic
178(C3⋊D3)⋊C3—24no—
179(C3⋊D3)⋊C3—42no—
180C3×#41—188no—
181C2×C3×Q8—32no—
182C2×C3×Q8—32no—
183C2×C3×Q8—62no—
184C2×C3×Q8C2×C3×D312yes—
185C2×C3×D4—91no—
186C2×C3×D4—91no—
187C2×C3×D4—94no—
188C22×C3×C4—92no—
189C22×C3×C4—94no—
190C22×C3×C4—92no—
191#239—68no—
192#239—188no—
193C2×D12—31no—
194C2×D12—31no—
195C2×D12—34no—
196C2×D12—91no—
197C2×D12—91no—
198C2×D12—94no—
199C2×C4×D3—32no—
200C2×C4×D3—32no—
201C2×C4×D3—34no—
202C2×C4×D3—92no—
203C2×C4×D3—92no—
204C2×C4×D3—94no—
205C3×C4⋊C4—32no—
206C3×C4⋊C4—32no—
207C3×C4⋊C4—32no—
208C3×C4⋊C4—32no—
209C3×C4⋊C4—62no—
210C3×C4⋊C4—62no—
211C3×C4⋊C4—94no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—91no—
214C3×C4⋊C4C2×C3×D312yes—
215C3×C4⋊C4C2×C3×D312yes—
216C3×C22⋊C4—92no—
217C3×C22⋊C4—92no—
218C3×C22⋊C4—94no—
219C3×C42—31no—
220C3×C42—31no—
221C3×C42—61no—
222C3×C42C2×C3×D311yescharacteristic
223(C2×D3)⋊C4—32no—
224(C2×D3)⋊C4—32no—
225(C2×D3)⋊C4—34no—
226(C2×D3)⋊C4—92no—
227(C2×D3)⋊C4—92no—
228(C2×D3)⋊C4—94no—
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—91no—
232C4⋊Dic3—91no—
233Dic3⋊C4—34no—
234Dic3⋊C4—94no—
235C22×C32—21no—
236C22×C32C23×D311yescharacteristic
237C22×C32C24×C311yescharacteristic
238C2×C3⋊D3—22no—
239C2×C3⋊D3—22no—
240C2×C3⋊D3—22no—
241C2×C3⋊D3—22no—
242C2×C3⋊D3—22no—
243C2×C3⋊D3C2×C3×D414yes—
244C2×C3×D3—34no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—62no—
249C2×C3×D3—62no—
250C32×C4—22no—
251C32×C4—24no—
252C32×C4—21no—
253C32×C4—24no—
254C32×C4—21no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—44no—
258C32×C4D3×D412yes—
259C32×C4#23912yescomplemented
260C32×C4C2×C3×D412yes—
261C32×C4C3×#4112yescomplemented
262C3⋊Dic3—22no—
263C3⋊Dic3—22no—
264C3×Dic3—62no—
265C3×Dic3—62no—
266C2×#41—92no—
267C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
268#123—94no—
269C22⋊Q8—92no—
270#121—92no—
271C4×D4—91no—
272C4×D4—91no—
273C2×C4⋊C4—92no—
274H(3,𝔽3)#54711yescharacteristic, complemented, Sylow
275C23×C3—92no—
276C23×C3—92no—
277C22×D3—32no—
278C22×D3—32no—
279C22×D3—92no—
280C22×D3—92no—
281C3×Q8—24no—
282C3×Q8—64no—
283C3×Q8—64no—
284C3×Q8—124no—
285C3×D4—94no—
286C3×D4—182no—
287C3×D4—188no—
288C2×C3×C4—31no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—34no—
294C2×C3×C4—34no—
295C2×C3×C4—31no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—64no—
300C2×C3×C4—92no—
301C2×C3×C4—92no—
302C2×C3×C4—184no—
303C2×C3×C4—184no—
304C2×C3×C4—188no—
305C2×C3×C4C22×C3×D314yes—
306C2×C3×C4C22×C3×D311yescharacteristic
307C2×C3×C4C22×C3×D311yescharacteristic
308C2×C3×C4C22×C3×D311yescharacteristic
309C2×Dic3—32no—
310C2×Dic3—32no—
311C2×Dic3—92no—
312C2×Dic3—92no—
313D12—34no—
314D12—68no—
315D12—62no—
316D12—94no—
317D12—182no—
318D12—188no—
319C4×D3—68no—
320C4×D3—64no—
321C4×D3—64no—
322C4×D3—184no—
323C4×D3—184no—
324C4×D3—188no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32C2×D3×D411yescharacteristic
329C2×C32C2×#23911yescharacteristic
330C2×C32#100711yescharacteristic
331C2×C32C22×C3×D411yescharacteristic
332C2×C32C2×C3×#4111yescharacteristic
333C2×C32C3×2−1+411yescharacteristic
334C3⋊D3—24no—
335C3⋊D3—42no—
336C3×D3—64no—
337C3×D3—122no—
338#41—188no—
339C2×Q8C2×(C3⋊D3)⋊C312yes—
340C2×D4—91no—
341C2×D4—91no—
342C2×D4—94no—
343C22×C4—92no—
344C22×C4—94no—
345C22×C4—92no—
346C4⋊C4—91no—
347C4⋊C4—94no—
348C4⋊C4—91no—
349C4⋊C4C2×(C3⋊D3)⋊C312yes—
350C4⋊C4C2×(C3⋊D3)⋊C312yes—
351C22⋊C4—92no—
352C22⋊C4—92no—
353C22⋊C4—94no—
354C42C2×(C3⋊D3)⋊C311yescharacteristic
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—61no—
358C22×C3—94no—
359C22×C3—182no—
360C22×C3—182no—
361C22×C3—182no—
362C22×C3—182no—
363C22×C3—182no—
364C22×C3C23×C3×D311yescharacteristic
365C2×D3—34no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—94no—
372C2×D3—182no—
373C2×D3—182no—
374C2×D3—182no—
375C2×D3—182no—
376C2×D3—182no—
377C3×C4—24no—
378C3×C4—21no—
379C3×C4—32no—
380C3×C4—32no—
381C3×C4—32no—
382C3×C4—32no—
383C3×C4—64no—
384C3×C4—64no—
385C3×C4—61no—
386C3×C4—61no—
387C3×C4—62no—
388C3×C4—62no—
389C3×C4—62no—
390C3×C4—124no—
391C3×C4—182no—
392C3×C4—182no—
393C3×C4C3×D3×D412yes—
394C3×C4C3×#23912yes—
395Dic3—62no—
396Dic3—62no—
397Dic3—182no—
398Dic3—182no—
399C32—21no—
400C32#572611yescharacteristic, complemented
401C32C3×#54711yescharacteristic, complemented
402C23—92no—
403C23—92no—
404Q8—24no—
405D4—94no—
406D4—188no—
407D4—182no—
408C2×C4—92no—
409C2×C4—92no—
410C2×C4—188no—
411C2×C4—184no—
412C2×C4—184no—
413C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
414C2×C4C22×(C3⋊D3)⋊C314yes—
415C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
416C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—61no—
426C2×C3—184no—
427C2×C3—362no—
428C2×C3C2×C3×D3×D411yescharacteristic
429C2×C3C2×C3×#23911yescharacteristic
430C2×C3C3×#100711yescharacteristic
431D3—64no—
432D3—122no—
433D3—184no—
434D3—362no—
435C22—94no—
436C22—182no—
437C22—182no—
438C22—182no—
439C22—182no—
440C22—182no—
441C22C23×(C3⋊D3)⋊C311yescharacteristic
442C4—24no—
443C4—21no—
444C4—182no—
445C4—182no—
446C4D4×(C3⋊D3)⋊C312yes—
447C4#8872212yescomplemented
448C3—31no—
449C3—31no—
450C3—61no—
451C3C3×#572611yescharacteristic
452C2—184no—
453C2—362no—
454C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
455C2C2×#8872211yescharacteristic
456C2#1172262711yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88722C2semidirect
2. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #64155⋊C3C2semidirect
4. #64104⋊C3C2semidirect
5. #64103⋊C3C2semidirect
6. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
7. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
8. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
9. #10596197C3semidirect
28. C2×#3832C2×C3semidirect
31. C32×C4⋊Q8D3semidirect
32. C32×C4⋊Q8C2×C3semidirect
33. #64155C2×C3semidirect
34. #64104C2×C3semidirect
35. #64103C2×C3semidirect
36. C4⋊C4×C3⋊D3C2×C3semidirect
37. C4×C4⋊(C3⋊D3)C2×C3semidirect
38. C4×C4⋊(C3⋊D3)C2×C3semidirect
56. C4⋊((C3⋊D3)⋊C3)D4semidirect
111. C4×H(3,𝔽3)#41semidirect
162. C4⋊(C3⋊D3)C3×D4semidirect
259. C32×C4#239semidirect
261. C32×C4C3×#41semidirect
267. C4⋊Q8(C3⋊D3)⋊C3semidirect
274. H(3,𝔽3)#547semidirect
400. C32#5726semidirect
401. C32C3×#547semidirect
447. C4#88722semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
H(3,𝔽3)×C4⋊Q82
#64155⋊C32
#64104⋊C32
#64103⋊C32
C4⋊C4×(C3⋊D3)⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#105961973
C3×#57263