Finite Group Databasebeta

← #10596155⋊C3 #13153125

Subgroups — #10596155⋊C3 #13153125

Order 1728.

846 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2#64155⋊C3C212yesmaximal, complemented
3#64148⋊C3C212yesmaximal, complemented
4#64095⋊C3C212yesmaximal, complemented
5#64093⋊C3C212yesmaximal, complemented
6#64092⋊C3C212yesmaximal, complemented
7#64087⋊C3C212yesmaximal, complemented
8#64084⋊C3C211yesmaximal, characteristic, complemented
9#64079⋊C3C211yesmaximal, characteristic, complemented
10#10596155C311yesmaximal, characteristic, complemented
11C3×#5684—31nomaximal
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C2×#638⋊C3C2212yes—
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×#634⋊C3C2211yescharacteristic
20C22⋊C4×H(3,𝔽3)C2212yes—
21C22⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2211yescharacteristic
23#3757⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29C4⋊((C3⋊Dic3)⋊C3)C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32#3750⋊C3C2212yes—
33C32×#124—21no—
34C32×#124D311yescharacteristic, complemented
35C32×#124C2×C311yescharacteristic, complemented
36#64155C2×C312yescomplemented
37#64148C2×C312yescomplemented
38#64095C2×C312yescomplemented
39#64093C2×C312yescomplemented
40#64092C2×C312yescomplemented
41#64087C2×C312yescomplemented
42#64084C2×C311yescharacteristic, complemented
43#64079C2×C311yescharacteristic, complemented
44C3×#946—32no—
45C3×#939—32no—
46C3×#886—32no—
47C3×#884—32no—
48C3×#883—32no—
49C3×#878—32no—
50C3×#875—31no—
51C3×#870—31no—
52C23×H(3,𝔽3)C2312yes—
53C22×(C3⋊D3)⋊C3C2312yes—
54Q8×H(3,𝔽3)—41no—
55D4×H(3,𝔽3)—41no—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×C4×H(3,𝔽3)C2312yes—
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60#638⋊C3—42no—
61#638⋊C3—42no—
62C2×(C3⋊Dic3)⋊C3—42no—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C4⋊((C3⋊D3)⋊C3)—41no—
67C4×(C3⋊D3)⋊C3—42no—
68#634⋊C3—41no—
69C3×#562—91no—
70#5684—31no—
71#5684—91no—
72C2×C32×Q8—21no—
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C22×C311yescharacteristic
75C2×C32×D4—21no—
76C2×C32×D4C2×D311yescharacteristic
77C2×C32×D4C22×C311yescharacteristic
78C2×#638C22×C312yes—
79C2×#638C22×C312yes—
80C22×C3⋊Dic3C22×C312yes—
81C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
82C2×C4×C3⋊D3C22×C312yes—
83C2×#634C22×C311yescharacteristic
84C2×C3×#67—32no—
85C2×C3×#67—32no—
86C22×C3×Dic3—32no—
87C2×C3×D12—31no—
88C2×C3×C4×D3—32no—
89C2×C3×Dic6—31no—
90C32×C22⋊C4—22no—
91C32×C22⋊C4—22no—
92C32×C22⋊C4C2×D312yes—
93C32×C22⋊C4C2×D312yes—
94C32×C22⋊C4C22×C312yes—
95C32×C22⋊C4C22×C312yes—
96C32×C42—21no—
97C32×C42C2×D311yescharacteristic
98C32×C42C22×C311yescharacteristic
99#3757C22×C312yes—
100#3752C22×C312yes—
101#3752C22×C312yes—
102#3752C22×C312yes—
103C4⋊(C3⋊Dic3)C22×C311yescharacteristic
104C4⋊(C3⋊Dic3)C22×C311yescharacteristic
105C4⋊(C3⋊Dic3)C22×C312yes—
106#3750C22×C312yes—
107#3750C22×C312yes—
108#3750C22×C312yes—
109C3×#217—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4—32no—
112C3×(C2×D3)⋊C4—32no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—32no—
119C22×H(3,𝔽3)—22no—
120C22×H(3,𝔽3)—22no—
121C22×H(3,𝔽3)—22no—
122C22×H(3,𝔽3)C2411yescharacteristic
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C2×(C3⋊D3)⋊C3—22no—
126C4×H(3,𝔽3)—21no—
127C4×H(3,𝔽3)—21no—
128C4×H(3,𝔽3)—22no—
129C4×H(3,𝔽3)—21no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132(C3⋊Dic3)⋊C3—22no—
133C3×#125—91no—
134C3×#124—31no—
135C3×#124—31no—
136C3×#124—61no—
137C3×#124—91no—
138C3×#124C3×D311yescharacteristic
139C3×#123—92no—
140C3×#123—92no—
141C3×C22⋊Q8—92no—
142C3×C22⋊Q8—92no—
143C3×#121—92no—
144C3×#121—92no—
145#946—32no—
146#946—92no—
147#939—32no—
148#939—92no—
149#886—32no—
150#886—92no—
151#884—32no—
152#884—92no—
153#883—32no—
154#883—92no—
155#878—32no—
156#878—92no—
157#875—31no—
158#875—91no—
159#870—31no—
160#870—91no—
161C23×C32—22no—
162C23×C32C22×D312yes—
163C23×C32C23×C312yes—
164C22×C3⋊D3C23×C312yes—
165C22×C3×D3—32no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—42no—
169C32×D4—41no—
170C32×D4—41no—
171C32×D4—42no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C23×C311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182C2×C32×C4C23×C312yes—
183C2×C32×C4C23×C311yescharacteristic
184#638—42no—
185#638—42no—
186C2×C3⋊Dic3—42no—
187C2×C3⋊Dic3C23×C312yes—
188C2×C3⋊Dic3C23×C312yes—
189C2×C3⋊Dic3C23×C312yes—
190C4⋊(C3⋊D3)—41no—
191C4×C3⋊D3—42no—
192#634—41no—
193C3×#67—122no—
194C3×#67—122no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—122no—
199C3×D12—121no—
200C3×C4×D3—122no—
201C3×Dic6—121no—
202#562—91noSylow
203C2×H(3,𝔽3)—42no—
204C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
205C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
206C2×H(3,𝔽3)2−1+411yescharacteristic
207(C3⋊D3)⋊C3—42no—
208C2×C3×Q8—31no—
209C2×C3×Q8—31no—
210C2×C3×Q8—61no—
211C2×C3×Q8—91no—
212C2×C3×Q8C2×C3×D311yescharacteristic
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C2×C3×D4—61no—
216C2×C3×D4—91no—
217C2×C3×D4—92no—
218C2×C3×D4—92no—
219C2×C3×D4C2×C3×D311yescharacteristic
220C22×C3×C4—92no—
221C22×C3×C4—92no—
222C2×#67—32no—
223C2×#67—32no—
224C2×#67—92no—
225C2×#67—92no—
226C22×Dic3—32no—
227C22×Dic3—92no—
228C2×D12—31no—
229C2×D12—91no—
230C2×C4×D3—32no—
231C2×C4×D3—92no—
232C2×Dic6—31no—
233C2×Dic6—91no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4—91no—
240C3×C22⋊C4—32no—
241C3×C22⋊C4—32no—
242C3×C22⋊C4—32no—
243C3×C22⋊C4—32no—
244C3×C22⋊C4—62no—
245C3×C22⋊C4—62no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C22⋊C4—92no—
250C3×C22⋊C4C2×C3×D312yes—
251C3×C22⋊C4C2×C3×D312yes—
252C3×C42—31no—
253C3×C42—31no—
254C3×C42—61no—
255C3×C42C2×C3×D311yescharacteristic
256#217—32no—
257#217—92no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266C4⋊Dic3—32no—
267C4⋊Dic3—91no—
268C4⋊Dic3—91no—
269C4⋊Dic3—92no—
270Dic3⋊C4—32no—
271Dic3⋊C4—32no—
272Dic3⋊C4—32no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—21no—
283C22×C32—42no—
284C22×C32—42no—
285C22×C32—42no—
286C22×C32C23×D311yescharacteristic
287C22×C32C24×C311yescharacteristic
288C2×C3⋊D3—22no—
289C2×C3⋊D3—22no—
290C2×C3⋊D3—22no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C32×C4—22no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—22no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—42no—
306C3⋊Dic3—22no—
307C3⋊Dic3—22no—
308C3⋊Dic3—22no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—62no—
312#125—91no—
313#124—91no—
314#124(C3⋊D3)⋊C311yescharacteristic, complemented
315#123—92no—
316#123—92no—
317C22⋊Q8—92no—
318C22⋊Q8—92no—
319#121—92no—
320#121—92no—
321H(3,𝔽3)#56211yescharacteristic, complemented, Sylow
322C23×C3—32no—
323C23×C3—32no—
324C23×C3—62no—
325C23×C3—92no—
326C23×C3C22×C3×D312yes—
327C22×D3—32no—
328C22×D3—92no—
329C3×Q8—41no—
330C3×Q8—121no—
331C3×Q8—121no—
332C3×Q8—122no—
333C3×Q8—361no—
334C3×D4—41no—
335C3×D4—121no—
336C3×D4—121no—
337C3×D4—122no—
338C3×D4—361no—
339C3×D4—362no—
340C3×D4—362no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—32no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—32no—
348C2×C3×C4—31no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—62no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4—92no—
356C2×C3×C4—362no—
357C2×C3×C4—362no—
358C2×C3×C4C22×C3×D311yescharacteristic
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×C3×C4C22×C3×D312yes—
361C2×C3×C4C22×C3×D311yescharacteristic
362#67—122no—
363#67—122no—
364#67—362no—
365#67—362no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—92no—
370C2×Dic3—92no—
371C2×Dic3—92no—
372C2×Dic3—122no—
373C2×Dic3—362no—
374D12—121no—
375D12—361no—
376C4×D3—122no—
377C4×D3—362no—
378Dic6—121no—
379Dic6—361no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—42no—
384C2×C32—42no—
385C2×C32—82no—
386C2×C32#100411yescharacteristic
387C2×C32#100911yescharacteristic
388C2×C32#101011yescharacteristic
389C2×C32C3×H(5,𝔽2)11yescharacteristic
390C2×C32C3×H(5,𝔽2)11yescharacteristic
391C2×C32C3×2−1+411yescharacteristic
392C3⋊D3—42no—
393C3×D3—122no—
394C2×Q8—91no—
395C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
396C2×D4—92no—
397C2×D4—92no—
398C2×D4—91no—
399C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
400C22×C4—92no—
401C22×C4—92no—
402C4⋊C4—92no—
403C4⋊C4—92no—
404C4⋊C4—92no—
405C4⋊C4—92no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C22⋊C4—92no—
409C22⋊C4—92no—
410C22⋊C4—92no—
411C22⋊C4—92no—
412C22⋊C4C2×(C3⋊D3)⋊C312yes—
413C22⋊C4C2×(C3⋊D3)⋊C312yes—
414C42C2×(C3⋊D3)⋊C311yescharacteristic
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—31no—
419C22×C3—31no—
420C22×C3—62no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—62no—
425C22×C3—62no—
426C22×C3—61no—
427C22×C3—122no—
428C22×C3—122no—
429C22×C3—122no—
430C22×C3—182no—
431C22×C3—182no—
432C22×C3—182no—
433C22×C3C23×C3×D311yescharacteristic
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—182no—
438C2×D3—182no—
439C2×D3—182no—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—61no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—61no—
448C3×C4—61no—
449C3×C4—61no—
450C3×C4—61no—
451C3×C4—61no—
452C3×C4—62no—
453C3×C4—122no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—182no—
457C3×C4—182no—
458C3×C4—182no—
459Dic3—62no—
460Dic3—62no—
461Dic3—62no—
462Dic3—182no—
463Dic3—182no—
464Dic3—182no—
465C32—21no—
466C32#568411yescharacteristic, complemented
467C32C3×#56211yescharacteristic, complemented
468C23—92no—
469C23C22×(C3⋊D3)⋊C312yes—
470Q8—41no—
471Q8—361no—
472D4—41no—
473D4—361no—
474D4—362no—
475D4—362no—
476C2×C4—92no—
477C2×C4—92no—
478C2×C4—92no—
479C2×C4—362no—
480C2×C4—362no—
481C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
482C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
483C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
484C2×C4C22×(C3⋊D3)⋊C312yes—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—42no—
492C2×C3—61no—
493C2×C3—61no—
494C2×C3—61no—
495C2×C3—122no—
496C2×C3—122no—
497C2×C3—242no—
498C2×C3—362no—
499C2×C3C3×#100411yescharacteristic
500C2×C3C3×#100911yescharacteristic
501C2×C3C3×#101011yescharacteristic
502D3—122no—
503D3—362no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—182no—
508C22—182no—
509C22—182no—
510C22C23×(C3⋊D3)⋊C311yescharacteristic
511C4—22no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—182no—
516C4—182no—
517C4—182no—
518C3—31no—
519C3—31no—
520C3—61no—
521C3C3×#568411yescharacteristic
522C2—42no—
523C2—362no—
524C2#64361⋊C311yescharacteristic
525C2#1172263811yescharacteristic
526C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#124C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. #64155⋊C3C2semidirect
3. #64148⋊C3C2semidirect
4. #64095⋊C3C2semidirect
5. #64093⋊C3C2semidirect
6. #64092⋊C3C2semidirect
7. #64087⋊C3C2semidirect
8. #64084⋊C3C2semidirect
9. #64079⋊C3C2semidirect
10. #10596155C3semidirect
34. C32×#124D3semidirect
35. C32×#124C2×C3semidirect
36. #64155C2×C3semidirect
37. #64148C2×C3semidirect
38. #64095C2×C3semidirect
39. #64093C2×C3semidirect
40. #64092C2×C3semidirect
41. #64087C2×C3semidirect
42. #64084C2×C3semidirect
43. #64079C2×C3semidirect
314. #124(C3⋊D3)⋊C3semidirect
321. H(3,𝔽3)#562semidirect
466. C32#5684semidirect
467. C32C3×#562semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
#64155⋊C32
#64148⋊C32
#64095⋊C32
#64093⋊C32
#64092⋊C32
#64087⋊C32
#64084⋊C32
#64079⋊C32
#105961553
C3×#56843