Finite Group Databasebeta

← #10596170⋊C3 #13153189

Subgroups — #10596170⋊C3 #13153189

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#64107⋊C3C212yesmaximal, complemented
3#64107⋊C3C212yesmaximal, complemented
4#64106⋊C3C212yesmaximal, complemented
5#64105⋊C3C212yesmaximal, complemented
6#64103⋊C3C212yesmaximal
7#64097⋊C3C212yesmaximal, complemented
8#64084⋊C3C211yesmaximal, characteristic
9#64079⋊C3C211yesmaximal, characteristic, complemented
10#10596170C311yesmaximal, characteristic, complemented
11C3×#5699—31nomaximal
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×#634⋊C3C2211yescharacteristic
15C2×#634⋊C3C2212yes—
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2212yes—
18C4⋊C4×H(3,𝔽3)C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C42×H(3,𝔽3)C2211yescharacteristic
21#3752⋊C3C2212yes—
22#3752⋊C3C2212yes—
23#3752⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2212yes—
33C32×#125—21no—
34C32×#125D311yescharacteristic, complemented
35C32×#125C2×C311yescharacteristic, complemented
36#64107C2×C312yescomplemented
37#64107C2×C312yescomplemented
38#64106C2×C312yescomplemented
39#64105C2×C312yescomplemented
40#64103C2×C312yes—
41#64097C2×C312yescomplemented
42#64084C2×C311yescharacteristic
43#64079C2×C311yescharacteristic, complemented
44C3×#898—32no—
45C3×#898—32no—
46C3×#897—32no—
47C3×#896—32no—
48C3×#894—32no—
49C3×#888—32no—
50C3×#875—31no—
51C3×#870—31no—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)C2312yes—
54C2×C4×H(3,𝔽3)C2311yescharacteristic
55C2×C4×H(3,𝔽3)C2312yes—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C4⋊((C3⋊D3)⋊C3)—41no—
62C4×(C3⋊D3)⋊C3—42no—
63#634⋊C3—41no—
64#634⋊C3—42no—
65C3×#563—91no—
66#5699—31no—
67#5699—91no—
68C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
69C2×C4×C3⋊D3C22×C312yes—
70C2×#634C22×C311yescharacteristic
71C2×#634C22×C312yes—
72C2×C3×D12—31no—
73C2×C3×C4×D3—32no—
74C2×C3×Dic6—31no—
75C2×C3×Dic6—32no—
76C32×C4⋊C4—21no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—22no—
79C32×C4⋊C4—22no—
80C32×C4⋊C4C2×D312yes—
81C32×C4⋊C4C2×D311yescharacteristic
82C32×C4⋊C4C2×D312yes—
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C312yes—
85C32×C4⋊C4C22×C311yescharacteristic
86C32×C4⋊C4C22×C312yes—
87C32×C4⋊C4C22×C311yescharacteristic
88C32×C42—21no—
89C32×C42C2×D311yescharacteristic
90C32×C42C22×C311yescharacteristic
91#3752C22×C312yes—
92#3752C22×C312yes—
93#3752C22×C312yes—
94#3752C22×C312yes—
95#3752C22×C312yes—
96C4⋊(C3⋊Dic3)C22×C311yescharacteristic
97C4⋊(C3⋊Dic3)C22×C311yescharacteristic
98C4⋊(C3⋊Dic3)C22×C312yes—
99#3750C22×C312yes—
100#3750C22×C312yes—
101#3750C22×C312yes—
102C4×C3⋊Dic3C22×C312yes—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×C4⋊Dic3—32no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×C4×Dic3—32no—
115C22×H(3,𝔽3)C2411yescharacteristic
116C2×(C3⋊D3)⋊C3—22no—
117C2×(C3⋊D3)⋊C3—22no—
118C2×(C3⋊D3)⋊C3—22no—
119C4×H(3,𝔽3)—22no—
120C4×H(3,𝔽3)—21no—
121C4×H(3,𝔽3)—21no—
122C4×H(3,𝔽3)—22no—
123C4×H(3,𝔽3)—21no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127C3×C4⋊Q8—92no—
128C3×#126—92no—
129C3×#126—92no—
130C3×#125—31no—
131C3×#125—31no—
132C3×#125—61no—
133C3×#125—91no—
134C3×#125C3×D311yescharacteristic
135C3×#124—91no—
136C3×#123—92no—
137C3×C22⋊Q8—92no—
138C3×C22⋊Q8—92no—
139#898—32no—
140#898—32no—
141#898—92no—
142#898—92no—
143#897—32no—
144#897—92no—
145#896—32no—
146#896—92no—
147#894—32no—
148#894—92no—
149#888—32no—
150#888—92no—
151#875—31no—
152#875—91no—
153#870—31no—
154#870—91no—
155C22×C3⋊D3C23×C312yes—
156C22×C3×D3—32no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—22no—
162C2×C32×C4C22×D312yes—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D312yes—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C23×C312yes—
168C2×C32×C4C23×C311yescharacteristic
169C2×C32×C4C23×C311yescharacteristic
170C2×C32×C4C23×C312yes—
171C2×C32×C4C23×C311yescharacteristic
172C2×C3⋊Dic3C23×C312yes—
173C2×C3⋊Dic3C23×C312yes—
174C2×C3⋊Dic3C23×C312yes—
175C4⋊(C3⋊D3)—41no—
176C4×C3⋊D3—42no—
177#634—41no—
178#634—42no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C3×D12—121no—
183C3×C4×D3—122no—
184C3×Dic6—121no—
185C3×Dic6—122no—
186#563—91noSylow
187C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
188C2×H(3,𝔽3)2−1+411yescharacteristic
189C2×H(3,𝔽3)2−1+411yescharacteristic
190(C3⋊D3)⋊C3—42no—
191C2×C3×Q8—92no—
192C2×C3×Q8—91no—
193C2×C3×D4—91no—
194C22×C3×C4—92no—
195C2×D12—31no—
196C2×D12—91no—
197C2×C4×D3—32no—
198C2×C4×D3—92no—
199C2×Dic6—31no—
200C2×Dic6—32no—
201C2×Dic6—91no—
202C2×Dic6—92no—
203C3×C4⋊C4—32no—
204C3×C4⋊C4—31no—
205C3×C4⋊C4—32no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—32no—
209C3×C4⋊C4—32no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—62no—
213C3×C4⋊C4—62no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4C2×C3×D312yes—
222C3×C4⋊C4C2×C3×D311yescharacteristic
223C3×C4⋊C4C2×C3×D312yes—
224C3×C4⋊C4C2×C3×D311yescharacteristic
225C3×C22⋊C4—92no—
226C3×C22⋊C4—92no—
227C3×C22⋊C4—92no—
228C3×C22⋊C4—92no—
229C3×C22⋊C4—92no—
230C3×C42—31no—
231C3×C42—31no—
232C3×C42—61no—
233C3×C42—92no—
234C3×C42C2×C3×D311yescharacteristic
235(C2×D3)⋊C4—32no—
236(C2×D3)⋊C4—32no—
237(C2×D3)⋊C4—32no—
238(C2×D3)⋊C4—32no—
239(C2×D3)⋊C4—32no—
240(C2×D3)⋊C4—92no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245C4⋊Dic3—31no—
246C4⋊Dic3—31no—
247C4⋊Dic3—32no—
248C4⋊Dic3—91no—
249C4⋊Dic3—91no—
250C4⋊Dic3—92no—
251Dic3⋊C4—32no—
252Dic3⋊C4—32no—
253Dic3⋊C4—32no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257C4×Dic3—32no—
258C4×Dic3—92no—
259C22×C32—21no—
260C22×C32C23×D311yescharacteristic
261C22×C32C24×C311yescharacteristic
262C2×C3⋊D3—22no—
263C2×C3⋊D3—22no—
264C2×C3⋊D3—22no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C32×C4—22no—
269C32×C4—21no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4—22no—
273C32×C4—21no—
274C32×C4—22no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—22no—
278C32×C4—21no—
279C32×C4—42no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—42no—
283C3⋊Dic3—22no—
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C3×Dic3—62no—
289C4⋊Q8—92no—
290#126—92no—
291#126—92no—
292#125—91no—
293#125(C3⋊D3)⋊C311yescharacteristic, complemented
294#124—91no—
295#123—92no—
296C22⋊Q8—92no—
297C22⋊Q8—92no—
298H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
299C23×C3—92no—
300C22×D3—32no—
301C22×D3—92no—
302C3×Q8—362no—
303C3×Q8—361no—
304C3×D4—361no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—32no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—32no—
314C2×C3×C4—31no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—92no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4—362no—
324C2×C3×C4C22×C3×D312yes—
325C2×C3×C4C22×C3×D311yescharacteristic
326C2×C3×C4C22×C3×D311yescharacteristic
327C2×C3×C4C22×C3×D312yes—
328C2×C3×C4C22×C3×D311yescharacteristic
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—92no—
333C2×Dic3—92no—
334C2×Dic3—92no—
335D12—121no—
336D12—361no—
337C4×D3—122no—
338C4×D3—362no—
339Dic6—121no—
340Dic6—122no—
341Dic6—361no—
342Dic6—362no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32#100711yescharacteristic
347C2×C32#100911yescharacteristic
348C2×C32#101011yescharacteristic
349C2×C32C3×H(5,𝔽2)11yescharacteristic
350C2×C32C3×2−1+411yescharacteristic
351C2×C32C3×2−1+411yescharacteristic
352C3⋊D3—42no—
353C3×D3—122no—
354C2×Q8—92no—
355C2×Q8—91no—
356C2×D4—91no—
357C22×C4—92no—
358C4⋊C4—91no—
359C4⋊C4—92no—
360C4⋊C4—92no—
361C4⋊C4—92no—
362C4⋊C4—92no—
363C4⋊C4—91no—
364C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
365C4⋊C4C2×(C3⋊D3)⋊C312yes—
366C4⋊C4C2×(C3⋊D3)⋊C312yes—
367C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
368C22⋊C4—92no—
369C22⋊C4—92no—
370C22⋊C4—92no—
371C22⋊C4—92no—
372C22⋊C4—92no—
373C42—92no—
374C42C2×(C3⋊D3)⋊C311yescharacteristic
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—61no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3—182no—
381C22×C3C23×C3×D311yescharacteristic
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—182no—
386C2×D3—182no—
387C2×D3—182no—
388C3×C4—21no—
389C3×C4—22no—
390C3×C4—21no—
391C3×C4—22no—
392C3×C4—21no—
393C3×C4—62no—
394C3×C4—61no—
395C3×C4—62no—
396C3×C4—61no—
397C3×C4—61no—
398C3×C4—61no—
399C3×C4—62no—
400C3×C4—61no—
401C3×C4—61no—
402C3×C4—62no—
403C3×C4—62no—
404C3×C4—122no—
405C3×C4—121no—
406C3×C4—121no—
407C3×C4—122no—
408C3×C4—182no—
409C3×C4—182no—
410C3×C4—182no—
411Dic3—62no—
412Dic3—62no—
413Dic3—62no—
414Dic3—182no—
415Dic3—182no—
416Dic3—182no—
417C32—21no—
418C32#569911yescharacteristic, complemented
419C32C3×#56311yescharacteristic, complemented
420C23—92no—
421Q8—362no—
422Q8—361no—
423D4—361no—
424C2×C4—92no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—362no—
428C2×C4C22×(C3⋊D3)⋊C312yes—
429C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
430C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
431C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
432C2×C4C22×(C3⋊D3)⋊C312yes—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—61no—
440C2×C3—61no—
441C2×C3—61no—
442C2×C3—362no—
443C2×C3C3×#100711yescharacteristic
444C2×C3C3×#100911yescharacteristic
445C2×C3C3×#101011yescharacteristic
446D3—122no—
447D3—362no—
448C22—182no—
449C22—182no—
450C22—182no—
451C22C23×(C3⋊D3)⋊C311yescharacteristic
452C4—21no—
453C4—22no—
454C4—21no—
455C4—22no—
456C4—21no—
457C4—182no—
458C4—182no—
459C4—182no—
460C3—31no—
461C3—31no—
462C3—61no—
463C3C3×#569911yescharacteristic
464C2—362no—
465C2#1172262711yescharacteristic
466C2#1172263811yescharacteristic
467C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #64107⋊C3C2semidirect
3. #64107⋊C3C2semidirect
4. #64106⋊C3C2semidirect
5. #64105⋊C3C2semidirect
7. #64097⋊C3C2semidirect
9. #64079⋊C3C2semidirect
10. #10596170C3semidirect
34. C32×#125D3semidirect
35. C32×#125C2×C3semidirect
36. #64107C2×C3semidirect
37. #64107C2×C3semidirect
38. #64106C2×C3semidirect
39. #64105C2×C3semidirect
41. #64097C2×C3semidirect
43. #64079C2×C3semidirect
293. #125(C3⋊D3)⋊C3semidirect
298. H(3,𝔽3)#563semidirect
418. C32#5699semidirect
419. C32C3×#563semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64107⋊C32
#64106⋊C32
#64105⋊C32
#64103⋊C32
#64097⋊C32
#64084⋊C32
#64079⋊C32
#105961703
C3×#56993