Finite Group Databasebeta

← #10596262⋊C3 #13153558

Subgroups — #10596262⋊C3 #13153558

Order 1728.

1391 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3#11721702C212yesmaximal, complemented
4#64139⋊C3C212yesmaximal, complemented
5#64138⋊C3C212yesmaximal, complemented
6C4×#638⋊C3C214yesmaximal, complemented
7#64135⋊C3C211yesmaximal, characteristic, complemented
8#64133⋊C3C212yesmaximal, complemented
9#10596262C311yesmaximal, characteristic, complemented
10C3×#5791—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)—24no—
13C22×C4×H(3,𝔽3)C2212yes—
14C22×C4×H(3,𝔽3)C2212yes—
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C22×C4×H(3,𝔽3)C2211yescharacteristic
17C2×#638⋊C3C2212yes—
18C2×#638⋊C3C2214yes—
19#3828⋊C3—24no—
20C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
21C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
22C2×#634⋊C3C2211yescharacteristic, complemented
23#3757⋊C3C2214yes—
24#3757⋊C3C2212yes—
25#3752⋊C3C2214yescomplemented
26C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
27#3750⋊C3C2214yescomplemented
28C4×(C3⋊Dic3)⋊C3C2212yescomplemented
29C23×C32×C4—21no—
30C23×C32×C4D311yescharacteristic, complemented
31C23×C32×C4C2×C311yescharacteristic, complemented
32C2×#3828C2×C311yescharacteristic, complemented
33C2×C3×#235—31no—
34#64162C2×C312yescomplemented
35#64139C2×C312yescomplemented
36#64138C2×C312yescomplemented
37C4×#638C2×C314yescomplemented
38#64135C2×C311yescharacteristic, complemented
39#64133C2×C312yescomplemented
40C3×#953—32no—
41C3×#930—32no—
42C3×#929—32no—
43C3×C4×#67—34no—
44C3×#926—31no—
45C3×#924—32no—
46C23×H(3,𝔽3)—22no—
47C23×H(3,𝔽3)—24no—
48C23×H(3,𝔽3)C2311yescharacteristic
49C23×H(3,𝔽3)C2312yes—
50C22×(C3⋊D3)⋊C3C2312yes—
51C2×C4×H(3,𝔽3)—22no—
52C2×C4×H(3,𝔽3)—24no—
53C2×C4×H(3,𝔽3)—24no—
54C2×C4×H(3,𝔽3)D414yescomplemented
55C2×C4×H(3,𝔽3)C2312yes—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58#638⋊C3—24no—
59#638⋊C3—28no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2314yes—
62C4⋊((C3⋊D3)⋊C3)—22no—
63C4×(C3⋊D3)⋊C3—24no—
64#634⋊C3—22no—
65C3×#525—91no—
66#5791—31no—
67#5791—91no—
68C24×C32—21no—
69C24×C32C2×D311yescharacteristic
70C24×C32C22×C311yescharacteristic
71C22×C32×C4—28no—
72C22×C32×C4—22no—
73C22×C32×C4—22no—
74C22×C32×C4—21no—
75C22×C32×C4—21no—
76C22×C32×C4—24no—
77C22×C32×C4—24no—
78C22×C32×C4C2×D312yes—
79C22×C32×C4C2×D312yes—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C22×C312yes—
83C22×C32×C4C22×C312yes—
84C22×C32×C4C22×C311yescharacteristic
85C22×C32×C4C22×C311yescharacteristic
86C2×#638C22×C312yes—
87C2×#638C22×C314yes—
88#3828—24no—
89C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
90C2×C4×C3⋊D3C22×C312yescomplemented
91C2×#634C22×C311yescharacteristic, complemented
92C2×C3×#67—34no—
93C2×C3×#67—32no—
94C3×#235—64no—
95C2×C3×D12—31no—
96C2×C3×C4×D3—32no—
97C2×C3×Dic6—31no—
98#3757C22×C314yes—
99#3757C22×C312yes—
100#3752C22×C314yescomplemented
101C4⋊(C3⋊Dic3)C22×C312yescomplemented
102#3750C22×C314yescomplemented
103C4×C3⋊Dic3C22×C312yescomplemented
104C3×#217—34no—
105C3×#217—32no—
106C3×(C2×D3)⋊C4—34no—
107C3×C4⋊Dic3—32no—
108C3×Dic3⋊C4—34no—
109C3×C4×Dic3—32no—
110C22×H(3,𝔽3)—28no—
111C22×H(3,𝔽3)—22no—
112C22×H(3,𝔽3)—22no—
113C22×H(3,𝔽3)—22no—
114C22×H(3,𝔽3)C2×D412yes—
115C22×H(3,𝔽3)#4114yescomplemented
116C22×H(3,𝔽3)C2411yescharacteristic
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—24no—
119C4×H(3,𝔽3)—22no—
120C4×H(3,𝔽3)C2×D412yes—
121C4×H(3,𝔽3)C2×D412yes—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—24no—
124C2×C3×#41—91no—
125C23×C3×C4—31no—
126C23×C3×C4—31no—
127C23×C3×C4—61no—
128C23×C3×C4C3×D311yescharacteristic
129C2×#235—31no—
130C2×#235—91no—
131C3×#123—92no—
132C3×C22⋊Q8—92no—
133C3×#121—92no—
134C3×#120—92no—
135C3×C4×D4—94no—
136C3×#117—91no—
137#953—32no—
138#953—92no—
139#930—32no—
140#930—92no—
141#929—32no—
142#929—92no—
143C4×#67—34no—
144C4×#67—94no—
145#926—31no—
146#926—91no—
147#924—32no—
148#924—92no—
149C23×C32—28no—
150C23×C32—24no—
151C23×C32—24no—
152C23×C32—22no—
153C23×C32—22no—
154C23×C32—24no—
155C23×C32—22no—
156C23×C32—21no—
157C23×C32C22×D312yes—
158C23×C32C22×D311yescharacteristic
159C23×C32C23×C311yescharacteristic
160C23×C32C23×C312yes—
161C22×C3⋊D3C23×C312yes—
162C22×C3×D3—32no—
163C2×C32×C4—24no—
164C2×C32×C4—28no—
165C2×C32×C4—28no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—24no—
169C2×C32×C4—24no—
170C2×C32×C4—24no—
171C2×C32×C4—24no—
172C2×C32×C4—24no—
173C2×C32×C4—22no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4#6714yescomplemented
177C2×C32×C4C3×D414yescomplemented
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C23×C312yes—
182C2×C32×C4C23×C311yescharacteristic
183C2×C32×C4C23×C311yescharacteristic
184#638—24no—
185#638—28no—
186C2×C3⋊Dic3C23×C314yes—
187C2×C3⋊Dic3C23×C312yes—
188C4⋊(C3⋊D3)—22no—
189C4×C3⋊D3—24no—
190#634—22no—
191C3×#67—68no—
192C3×#67—64no—
193C2×C3×Dic3—34no—
194C2×C3×Dic3—32no—
195C3×D12—62no—
196C3×C4×D3—64no—
197C3×Dic6—62no—
198#525—91noSylow
199C2×H(3,𝔽3)—22no—
200C2×H(3,𝔽3)—24no—
201C2×H(3,𝔽3)C22×D411yescharacteristic
202C2×H(3,𝔽3)C2×#4112yes—
203(C3⋊D3)⋊C3—42no—
204C24×C3—31no—
205C24×C3—31no—
206C24×C3—61no—
207C24×C3C2×C3×D311yescharacteristic
208C3×#41—184no—
209C2×C3×Q8—91no—
210C2×C3×D4—91no—
211C2×C3×D4—94no—
212C2×C3×D4—92no—
213C22×C3×C4—24no—
214C22×C3×C4—31no—
215C22×C3×C4—31no—
216C22×C3×C4—32no—
217C22×C3×C4—32no—
218C22×C3×C4—31no—
219C22×C3×C4—32no—
220C22×C3×C4—32no—
221C22×C3×C4—31no—
222C22×C3×C4—64no—
223C22×C3×C4—64no—
224C22×C3×C4—68no—
225C22×C3×C4—62no—
226C22×C3×C4—61no—
227C22×C3×C4—62no—
228C22×C3×C4—61no—
229C22×C3×C4—92no—
230C22×C3×C4C2×C3×D312yes—
231C22×C3×C4C2×C3×D312yes—
232C22×C3×C4C2×C3×D311yescharacteristic
233C22×C3×C4C2×C3×D311yescharacteristic
234C2×#67—34no—
235C2×#67—32no—
236C2×#67—94no—
237C2×#67—92no—
238#235—64no—
239#235—184no—
240C2×D12—31no—
241C2×D12—91no—
242C2×C4×D3—32no—
243C2×C4×D3—92no—
244C2×Dic6—31no—
245C2×Dic6—91no—
246C3×C4⋊C4—94no—
247C3×C4⋊C4—92no—
248C3×C22⋊C4—94no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4—94no—
251C3×C42—92no—
252#217—34no—
253#217—32no—
254#217—94no—
255#217—92no—
256(C2×D3)⋊C4—34no—
257(C2×D3)⋊C4—94no—
258C4⋊Dic3—32no—
259C4⋊Dic3—92no—
260Dic3⋊C4—34no—
261Dic3⋊C4—94no—
262C4×Dic3—32no—
263C4×Dic3—92no—
264C22×C32—24no—
265C22×C32—216no—
266C22×C32—28no—
267C22×C32—28no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—24no—
278C22×C32—22no—
279C22×C32#23514yescomplemented
280C22×C32C2×#6712yes—
281C22×C32C2×C3×D412yes—
282C22×C32C3×#4114yescomplemented
283C22×C32C23×D311yescharacteristic
284C22×C32C24×C311yescharacteristic
285C2×C3⋊D3—22no—
286C2×C3⋊D3—24no—
287C2×C3×D3—64no—
288C2×C3×D3—62no—
289C32×C4—24no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4C2×#6712yes—
295C32×C4C2×#6712yes—
296C32×C4C2×C3×D412yes—
297C32×C4C2×C3×D412yes—
298C3⋊Dic3—22no—
299C3⋊Dic3—24no—
300C3×Dic3—64no—
301C3×Dic3—62no—
302C2×#41—91no—
303C23×C4(C3⋊D3)⋊C311yescharacteristic, complemented
304#123—92no—
305C22⋊Q8—92no—
306#121—92no—
307#120—92no—
308C4×D4—94no—
309#117—91no—
310H(3,𝔽3)#52511yescharacteristic, complemented, Sylow
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—32no—
314C23×C3—31no—
315C23×C3—32no—
316C23×C3—31no—
317C23×C3—62no—
318C23×C3—62no—
319C23×C3—64no—
320C23×C3—64no—
321C23×C3—64no—
322C23×C3—68no—
323C23×C3—62no—
324C23×C3—61no—
325C23×C3—92no—
326C23×C3C22×C3×D312yes—
327C23×C3C22×C3×D311yescharacteristic
328C22×D3—32no—
329C22×D3—92no—
330C3×Q8—182no—
331C3×D4—182no—
332C3×D4—188no—
333C3×D4—184no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—24no—
337C2×C3×C4—34no—
338C2×C3×C4—34no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—64no—
346C2×C3×C4—64no—
347C2×C3×C4—64no—
348C2×C3×C4—64no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—64no—
352C2×C3×C4—68no—
353C2×C3×C4—68no—
354C2×C3×C4—64no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—94no—
359C2×C3×C4—92no—
360C2×C3×C4—184no—
361C2×C3×C4C3×#6714yes—
362C2×C3×C4C22×C3×D312yes—
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×C3×C4C22×C3×D311yescharacteristic
365#67—68no—
366#67—64no—
367#67—184no—
368#67—188no—
369C2×Dic3—34no—
370C2×Dic3—32no—
371C2×Dic3—94no—
372C2×Dic3—92no—
373D12—62no—
374D12—182no—
375C4×D3—64no—
376C4×D3—184no—
377Dic6—62no—
378Dic6—182no—
379C2×C32—28no—
380C2×C32—24no—
381C2×C32—24no—
382C2×C32—22no—
383C2×C32—22no—
384C2×C32—24no—
385C2×C32—22no—
386C2×C32—21no—
387C2×C32C2×#23512yes—
388C2×C32C22×#6711yescharacteristic
389C2×C32C22×C3×D411yescharacteristic
390C2×C32C2×C3×#4112yes—
391C3⋊D3—42no—
392C3×D3—122no—
393C24C2×(C3⋊D3)⋊C311yescharacteristic
394#41—184no—
395C2×Q8—91no—
396C2×D4—94no—
397C2×D4—92no—
398C2×D4—91no—
399C22×C4—24no—
400C22×C4—92no—
401C22×C4C2×(C3⋊D3)⋊C312yes—
402C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
403C22×C4C2×(C3⋊D3)⋊C312yes—
404C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
405C4⋊C4—92no—
406C4⋊C4—94no—
407C22⋊C4—92no—
408C22⋊C4—94no—
409C22⋊C4—94no—
410C42—92no—
411C22×C3—28no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—31no—
416C22×C3—31no—
417C22×C3—34no—
418C22×C3—34no—
419C22×C3—32no—
420C22×C3—32no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3—64no—
428C22×C3—68no—
429C22×C3—68no—
430C22×C3—616no—
431C22×C3—64no—
432C22×C3—64no—
433C22×C3—61no—
434C22×C3—64no—
435C22×C3—62no—
436C22×C3—182no—
437C22×C3—184no—
438C22×C3C3×#23514yes—
439C22×C3C2×C3×#6712yes—
440C22×C3C23×C3×D311yescharacteristic
441C2×D3—64no—
442C2×D3—62no—
443C2×D3—182no—
444C2×D3—184no—
445C3×C4—22no—
446C3×C4—32no—
447C3×C4—32no—
448C3×C4—32no—
449C3×C4—32no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—64no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—182no—
456C3×C4—184no—
457C3×C4C2×C3×#6712yes—
458C3×C4C2×C3×#6712yes—
459Dic3—64no—
460Dic3—62no—
461Dic3—182no—
462Dic3—184no—
463C32—21no—
464C32#579111yescharacteristic, complemented
465C32C3×#52511yescharacteristic, complemented
466C23—24no—
467C23—22no—
468C23—92no—
469C23C22×(C3⋊D3)⋊C312yes—
470C23C22×(C3⋊D3)⋊C311yescharacteristic
471Q8—182no—
472D4—188no—
473D4—184no—
474D4—182no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—94no—
479C2×C4—92no—
480C2×C4—184no—
481C2×C4#638⋊C314yescomplemented
482C2×C4C22×(C3⋊D3)⋊C312yes—
483C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
484C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
485C2×C3—24no—
486C2×C3—22no—
487C2×C3—32no—
488C2×C3—32no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—64no—
492C2×C3—64no—
493C2×C3—62no—
494C2×C3—62no—
495C2×C3—68no—
496C2×C3—64no—
497C2×C3—62no—
498C2×C3—61no—
499C2×C3—362no—
500C2×C3C2×C3×#23512yes—
501C2×C3C22×C3×#6711yescharacteristic
502D3—122no—
503D3—362no—
504C22—28no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—184no—
509C22—182no—
510C22#3828⋊C314yescomplemented
511C22C2×#638⋊C312yes—
512C22C23×(C3⋊D3)⋊C311yescharacteristic
513C4—22no—
514C4—184no—
515C4—182no—
516C4C2×#638⋊C312yes—
517C4C2×#638⋊C312yes—
518C3—31no—
519C3—31no—
520C3—61no—
521C3C3×#579111yescharacteristic
522C2—24no—
523C2—22no—
524C2—362no—
525C2C2×#3828⋊C312yes—
526C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC23×C4×H(3,𝔽3)C2
hypercenterC23×C4(C3⋊D3)⋊C3
hypocenterC32C3×#525
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C23×C4×H(3,𝔽3)C2semidirect
2. C2×#3828⋊C3C2semidirect
3. #11721702C2semidirect
4. #64139⋊C3C2semidirect
5. #64138⋊C3C2semidirect
6. C4×#638⋊C3C2semidirect
7. #64135⋊C3C2semidirect
8. #64133⋊C3C2semidirect
9. #10596262C3semidirect
20. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
21. C2×C4×(C3⋊D3)⋊C3C22semidirect
22. C2×#634⋊C3C22semidirect
25. #3752⋊C3C22semidirect
26. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
27. #3750⋊C3C22semidirect
28. C4×(C3⋊Dic3)⋊C3C22semidirect
30. C23×C32×C4D3semidirect
31. C23×C32×C4C2×C3semidirect
32. C2×#3828C2×C3semidirect
34. #64162C2×C3semidirect
35. #64139C2×C3semidirect
36. #64138C2×C3semidirect
37. C4×#638C2×C3semidirect
38. #64135C2×C3semidirect
39. #64133C2×C3semidirect
54. C2×C4×H(3,𝔽3)D4semidirect
89. C2×C4⋊(C3⋊D3)C22×C3semidirect
90. C2×C4×C3⋊D3C22×C3semidirect
91. C2×#634C22×C3semidirect
100. #3752C22×C3semidirect
101. C4⋊(C3⋊Dic3)C22×C3semidirect
102. #3750C22×C3semidirect
103. C4×C3⋊Dic3C22×C3semidirect
115. C22×H(3,𝔽3)#41semidirect
176. C2×C32×C4#67semidirect
177. C2×C32×C4C3×D4semidirect
279. C22×C32#235semidirect
282. C22×C32C3×#41semidirect
303. C23×C4(C3⋊D3)⋊C3semidirect
310. H(3,𝔽3)#525semidirect
464. C32#5791semidirect
465. C32C3×#525semidirect
481. C2×C4#638⋊C3semidirect
510. C22#3828⋊C3semidirect

Maximal subgroups

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

subgroupindex
C23×C4×H(3,𝔽3)2
C2×#3828⋊C32
#117217022
#64139⋊C32
#64138⋊C32
C4×#638⋊C32
#64135⋊C32
#64133⋊C32
#105962623
C3×#57913