Finite Group Databasebeta

← #10595986⋊C3 #13152446

Subgroups — #10595986⋊C3 #13152446

Order 1728.

855 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#64138⋊C3C212yesmaximal, complemented
4C4×#638⋊C3C212yesmaximal, complemented
5#64133⋊C3C211yesmaximal, characteristic, complemented
6#64107⋊C3C212yesmaximal, complemented
7#64105⋊C3C212yesmaximal, complemented
8#64103⋊C3C212yesmaximal, complemented
9#64098⋊C3C212yesmaximal, complemented
10#10595986C311yesmaximal, characteristic, complemented
11C3×#5515—31nomaximal
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#638⋊C3C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2212yes—
21#3757⋊C3C2212yes—
22#3752⋊C3C2212yes—
23#3752⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32C4×(C3⋊Dic3)⋊C3C2212yes—
33C2×C32×C4⋊C4—21no—
34C2×C32×C4⋊C4D311yescharacteristic, complemented
35C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
36#64139C2×C311yescharacteristic
37#64138C2×C312yescomplemented
38C4×#638C2×C312yescomplemented
39#64133C2×C311yescharacteristic, complemented
40#64107C2×C312yescomplemented
41#64105C2×C312yescomplemented
42#64103C2×C312yescomplemented
43#64098C2×C312yescomplemented
44C3×#930—31no—
45C3×#929—32no—
46C3×C4×#67—32no—
47C3×#924—31no—
48C3×#898—32no—
49C3×#896—32no—
50C3×#894—32no—
51C3×#889—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×(C3⋊D3)⋊C3C2312yes—
54C2×C4×H(3,𝔽3)—22no—
55C2×C4×H(3,𝔽3)—41no—
56C2×C4×H(3,𝔽3)—41no—
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×C4×H(3,𝔽3)C2312yes—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2312yes—
61#638⋊C3—24no—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C4⋊((C3⋊D3)⋊C3)—41no—
66C4×(C3⋊D3)⋊C3—42no—
67#634⋊C3—41no—
68C3×#539—91no—
69#5515—31no—
70#5515—91no—
71C22×C32×C4—21no—
72C22×C32×C4—21no—
73C22×C32×C4—21no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C22×C311yescharacteristic
78C22×C32×C4C22×C311yescharacteristic
79C22×C32×C4C22×C311yescharacteristic
80C2×#638C22×C312yes—
81C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
82C2×C4×C3⋊D3C22×C312yes—
83C2×#634C22×C311yescharacteristic
84C2×C3×#67—32no—
85C2×C3×D12—31no—
86C2×C3×C4×D3—32no—
87C2×C3×Dic6—31no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4—22no—
90C32×C4⋊C4C2×D312yes—
91C32×C4⋊C4C2×D312yes—
92C32×C4⋊C4C22×C312yes—
93C32×C4⋊C4C22×C312yes—
94#3757C22×C312yes—
95#3752C22×C312yes—
96#3752C22×C312yes—
97#3752C22×C312yes—
98#3752C22×C312yes—
99C4⋊(C3⋊Dic3)C22×C311yescharacteristic
100C4⋊(C3⋊Dic3)C22×C311yescharacteristic
101#3750C22×C312yes—
102#3750C22×C312yes—
103#3750C22×C312yes—
104#3750C22×C312yes—
105C4×C3⋊Dic3C22×C312yes—
106C3×#217—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—31no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×C4×Dic3—32no—
118C22×H(3,𝔽3)—21no—
119C22×H(3,𝔽3)—21no—
120C22×H(3,𝔽3)#4112yes—
121C22×H(3,𝔽3)C2411yescharacteristic
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—22no—
124C2×(C3⋊D3)⋊C3—22no—
125C4×H(3,𝔽3)—22no—
126C4×H(3,𝔽3)—21no—
127C4×H(3,𝔽3)—22no—
128C4×H(3,𝔽3)—21no—
129(C3⋊Dic3)⋊C3—22no—
130(C3⋊Dic3)⋊C3—22no—
131(C3⋊Dic3)⋊C3—22no—
132C3×#126—92no—
133C3×#125—92no—
134C3×#123—92no—
135C3×#123—92no—
136C3×C22⋊Q8—92no—
137C3×C22⋊Q8—91no—
138C3×#121—91no—
139C3×C4×D4—92no—
140C2×C3×C4⋊C4—31no—
141C2×C3×C4⋊C4—31no—
142C2×C3×C4⋊C4—61no—
143C2×C3×C4⋊C4C3×D311yescharacteristic
144#930—31no—
145#930—91no—
146#929—32no—
147#929—92no—
148C4×#67—32no—
149C4×#67—92no—
150#924—31no—
151#924—91no—
152#898—32no—
153#898—92no—
154#896—32no—
155#896—92no—
156#894—32no—
157#894—92no—
158#889—32no—
159#889—92no—
160C23×C32—21no—
161C23×C32C22×D311yescharacteristic
162C23×C32C23×C311yescharacteristic
163C22×C3⋊D3C23×C312yes—
164C22×C3×D3—32no—
165C2×C32×C4—24no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—22no—
169C2×C32×C4—22no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4—42no—
177C2×C32×C4—42no—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D312yes—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C23×C312yes—
183C2×C32×C4C23×C311yescharacteristic
184C2×C32×C4C23×C312yes—
185C2×C32×C4C23×C311yescharacteristic
186#638—24no—
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—64no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C3×D12—121no—
198C3×C4×D3—122no—
199C3×Dic6—121no—
200#539—91noSylow
201C2×H(3,𝔽3)—22no—
202C2×H(3,𝔽3)C2×#4111yescharacteristic
203C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
204C2×H(3,𝔽3)2−1+411yescharacteristic
205(C3⋊D3)⋊C3—42no—
206C2×C3×Q8—91no—
207C2×C3×D4—91no—
208C2×C3×D4—92no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—61no—
216C22×C3×C4—61no—
217C22×C3×C4—61no—
218C22×C3×C4—92no—
219C22×C3×C4C2×C3×D311yescharacteristic
220C22×C3×C4C2×C3×D311yescharacteristic
221C22×C3×C4C2×C3×D311yescharacteristic
222C2×#67—32no—
223C2×#67—92no—
224C2×D12—31no—
225C2×D12—91no—
226C2×C4×D3—32no—
227C2×C4×D3—92no—
228C2×Dic6—31no—
229C2×Dic6—91no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—32no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—62no—
235C3×C4⋊C4—62no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4C2×C3×D312yes—
243C3×C4⋊C4C2×C3×D312yes—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249C3×C42—92no—
250#217—32no—
251#217—92no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—32no—
255(C2×D3)⋊C4—32no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260C4⋊Dic3—31no—
261C4⋊Dic3—31no—
262C4⋊Dic3—91no—
263C4⋊Dic3—91no—
264Dic3⋊C4—32no—
265Dic3⋊C4—32no—
266Dic3⋊C4—32no—
267Dic3⋊C4—32no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272C4×Dic3—32no—
273C4×Dic3—92no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—22no—
282C22×C32#23512yes—
283C22×C32C3×#4112yes—
284C22×C32C23×D311yescharacteristic
285C22×C32C24×C311yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3⋊D3—22no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C32×C4—24no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—22no—
296C32×C4—21no—
297C32×C4—22no—
298C32×C4—21no—
299C32×C4—22no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4—42no—
303C32×C4—41no—
304C3⋊Dic3—22no—
305C3⋊Dic3—22no—
306C3⋊Dic3—22no—
307C3×Dic3—62no—
308C3×Dic3—62no—
309C3×Dic3—62no—
310#126—92no—
311#125—92no—
312#123—92no—
313#123—92no—
314C22⋊Q8—91no—
315C22⋊Q8—92no—
316#121—91no—
317C4×D4—92no—
318C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
319H(3,𝔽3)#53911yescharacteristic, complemented, Sylow
320C23×C3—31no—
321C23×C3—31no—
322C23×C3—61no—
323C23×C3—92no—
324C23×C3C22×C3×D311yescharacteristic
325C22×D3—32no—
326C22×D3—92no—
327C3×Q8—361no—
328C3×D4—184no—
329C3×D4—361no—
330C2×C3×C4—22no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—32no—
336C2×C3×C4—31no—
337C2×C3×C4—32no—
338C2×C3×C4—31no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—62no—
342C2×C3×C4—62no—
343C2×C3×C4—64no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—92no—
351C2×C3×C4—121no—
352C2×C3×C4—121no—
353C2×C3×C4—121no—
354C2×C3×C4—121no—
355C2×C3×C4—122no—
356C2×C3×C4—122no—
357C2×C3×C4—362no—
358C2×C3×C4C22×C3×D312yes—
359C2×C3×C4C22×C3×D311yescharacteristic
360C2×C3×C4C22×C3×D312yes—
361C2×C3×C4C22×C3×D311yescharacteristic
362#67—64no—
363#67—184no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—92no—
368C2×Dic3—92no—
369C2×Dic3—92no—
370D12—121no—
371D12—361no—
372C4×D3—122no—
373C4×D3—362no—
374Dic6—121no—
375Dic6—361no—
376C2×C32—24no—
377C2×C32—22no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32C2×#23511yescharacteristic
383C2×C32#100411yescharacteristic
384C2×C32#100711yescharacteristic
385C2×C32C2×C3×#4111yescharacteristic
386C2×C32C3×H(5,𝔽2)11yescharacteristic
387C2×C32C3×2−1+411yescharacteristic
388C3⋊D3—42no—
389C3×D3—122no—
390C2×Q8—91no—
391C2×D4—92no—
392C2×D4—91no—
393C22×C4—92no—
394C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
395C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
396C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
397C4⋊C4—92no—
398C4⋊C4—91no—
399C4⋊C4—92no—
400C4⋊C4—92no—
401C4⋊C4—91no—
402C4⋊C4—92no—
403C4⋊C4C2×(C3⋊D3)⋊C312yes—
404C4⋊C4C2×(C3⋊D3)⋊C312yes—
405C22⋊C4—92no—
406C22⋊C4—92no—
407C22⋊C4—92no—
408C22⋊C4—92no—
409C22⋊C4—92no—
410C42—92no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—31no—
414C22×C3—31no—
415C22×C3—32no—
416C22×C3—32no—
417C22×C3—61no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3—61no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—61no—
424C22×C3—62no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3—182no—
428C22×C3C3×#23512yes—
429C22×C3C23×C3×D311yescharacteristic
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—182no—
434C2×D3—182no—
435C2×D3—182no—
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—61no—
441C3×C4—62no—
442C3×C4—61no—
443C3×C4—62no—
444C3×C4—62no—
445C3×C4—62no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—64no—
449C3×C4—122no—
450C3×C4—121no—
451C3×C4—121no—
452C3×C4—182no—
453C3×C4—182no—
454C3×C4—182no—
455Dic3—62no—
456Dic3—62no—
457Dic3—62no—
458Dic3—182no—
459Dic3—182no—
460Dic3—182no—
461C32—21no—
462C32#551511yescharacteristic, complemented
463C32C3×#53911yescharacteristic, complemented
464C23—92no—
465C23C22×(C3⋊D3)⋊C311yescharacteristic
466Q8—361no—
467D4—184no—
468D4—361no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—92no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—362no—
476C2×C4C22×(C3⋊D3)⋊C312yes—
477C2×C4C22×(C3⋊D3)⋊C312yes—
478C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
479C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
480C2×C3—22no—
481C2×C3—31no—
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—62no—
488C2×C3—62no—
489C2×C3—64no—
490C2×C3—61no—
491C2×C3—61no—
492C2×C3—61no—
493C2×C3—362no—
494C2×C3C2×C3×#23511yescharacteristic
495C2×C3C3×#100411yescharacteristic
496C2×C3C3×#100711yescharacteristic
497D3—122no—
498D3—362no—
499C22—21no—
500C22—21no—
501C22—182no—
502C22—182no—
503C22—182no—
504C22#3828⋊C312yes—
505C22C23×(C3⋊D3)⋊C311yescharacteristic
506C4—22no—
507C4—22no—
508C4—21no—
509C4—21no—
510C4—182no—
511C4—182no—
512C4—182no—
513C3—31no—
514C3—31no—
515C3—61no—
516C3C3×#551511yescharacteristic
517C2—22no—
518C2—362no—
519C2C2×#3828⋊C311yescharacteristic
520C2#64361⋊C311yescharacteristic
521C2#1172262711yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radical#64139⋊C3C2
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. #64138⋊C3C2semidirect
4. C4×#638⋊C3C2semidirect
5. #64133⋊C3C2semidirect
6. #64107⋊C3C2semidirect
7. #64105⋊C3C2semidirect
8. #64103⋊C3C2semidirect
9. #64098⋊C3C2semidirect
10. #10595986C3semidirect
34. C2×C32×C4⋊C4D3semidirect
35. C2×C32×C4⋊C4C2×C3semidirect
37. #64138C2×C3semidirect
38. C4×#638C2×C3semidirect
39. #64133C2×C3semidirect
40. #64107C2×C3semidirect
41. #64105C2×C3semidirect
42. #64103C2×C3semidirect
43. #64098C2×C3semidirect
318. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
319. H(3,𝔽3)#539semidirect
462. C32#5515semidirect
463. C32C3×#539semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
#64139⋊C32
#64138⋊C32
C4×#638⋊C32
#64133⋊C32
#64107⋊C32
#64105⋊C32
#64103⋊C32
#64098⋊C32
#105959863
C3×#55153