Finite Group Databasebeta

← #10596303⋊C3 #13153722

Subgroups — #10596303⋊C3 #13153722

Order 1728.

1249 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3828⋊C3C211yesmaximal, characteristic, complemented
3#64156⋊C3C211yesmaximal, characteristic, complemented
4#64153⋊C3C211yesmaximal, characteristic, complemented
5#64149⋊C3C211yesmaximal, characteristic, complemented
6#64148⋊C3C214yesmaximal, complemented
7#64145⋊C3C211yesmaximal, characteristic, complemented
8C4×#638⋊C3C214yesmaximal, complemented
9C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
10#10596303C311yesmaximal, characteristic, complemented
11C3×#5832—31nomaximal
12#41×H(3,𝔽3)—24no—
13C2×Q8×H(3,𝔽3)C2211yescharacteristic
14C2×D4×H(3,𝔽3)C2211yescharacteristic
15C2×D4×H(3,𝔽3)C2212yes—
16C22×C4×H(3,𝔽3)C2212yes—
17C22×C4×H(3,𝔽3)C2211yescharacteristic
18C2×#638⋊C3C2212yes—
19C2×#638⋊C3C2214yes—
20C22×(C3⋊Dic3)⋊C3C2212yes—
21#3828⋊C3—24no—
22C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
23C2×C4×(C3⋊D3)⋊C3C2212yes—
24C2×#634⋊C3C2211yescharacteristic
25#3757⋊C3C2214yes—
26#3752⋊C3C2214yes—
27#3750⋊C3C2214yes—
28C4×(C3⋊Dic3)⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31C2×C32×#41—21no—
32C2×C32×#41D311yescharacteristic, complemented
33C2×C32×#41C2×C311yescharacteristic, complemented
34C2×#3828C2×C311yescharacteristic, complemented
35C2×C3×#235—31no—
36#64156C2×C311yescharacteristic, complemented
37#64153C2×C311yescharacteristic, complemented
38#64149C2×C311yescharacteristic, complemented
39#64148C2×C314yescomplemented
40#64145C2×C311yescharacteristic, complemented
41C4×#638C2×C314yescomplemented
42C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
43C3×#947—31no—
44C3×#944—31no—
45C3×#940—31no—
46C3×#939—34no—
47C3×#936—31no—
48C3×C4×#67—34no—
49C2×C3×C4×Dic3—31no—
50C23×H(3,𝔽3)C2312yes—
51C23×H(3,𝔽3)C2311yescharacteristic
52C22×(C3⋊D3)⋊C3C2312yes—
53Q8×H(3,𝔽3)—22no—
54D4×H(3,𝔽3)—24no—
55D4×H(3,𝔽3)—22no—
56C2×C4×H(3,𝔽3)—24no—
57C2×C4×H(3,𝔽3)D414yescomplemented
58C2×C4×H(3,𝔽3)C2312yes—
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61#638⋊C3—24no—
62#638⋊C3—28no—
63C2×(C3⋊Dic3)⋊C3—24no—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67C4⋊((C3⋊D3)⋊C3)—22no—
68C4×(C3⋊D3)⋊C3—24no—
69#634⋊C3—22no—
70C3×#532—91no—
71#5832—31no—
72#5832—91no—
73C32×#41—28no—
74C32×#41—24no—
75C32×#41—24no—
76C2×C32×Q8—21no—
77C2×C32×Q8C2×D311yescharacteristic
78C2×C32×Q8C22×C311yescharacteristic
79C2×C32×D4—21no—
80C2×C32×D4—22no—
81C2×C32×D4C2×D311yescharacteristic
82C2×C32×D4C2×D312yes—
83C2×C32×D4C22×C311yescharacteristic
84C2×C32×D4C22×C312yes—
85C22×C32×C4—22no—
86C22×C32×C4—21no—
87C22×C32×C4C2×D312yes—
88C22×C32×C4C2×D311yescharacteristic
89C22×C32×C4C22×C312yes—
90C22×C32×C4C22×C311yescharacteristic
91C2×#638C22×C312yes—
92C2×#638C22×C314yes—
93C22×C3⋊Dic3C22×C312yes—
94#3828—24no—
95C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
96C2×C4×C3⋊D3C22×C312yes—
97C2×#634C22×C311yescharacteristic
98C2×C3×#67—34no—
99C2×C3×#67—32no—
100C22×C3×Dic3—32no—
101C3×#235—64no—
102C2×C3×D12—31no—
103C2×C3×C4×D3—32no—
104C2×C3×Dic6—31no—
105#3757C22×C314yes—
106#3752C22×C314yes—
107#3750C22×C314yes—
108C4×C3⋊Dic3C22×C312yes—
109C4×C3⋊Dic3C22×C311yescharacteristic
110C4×C3⋊Dic3C22×C311yescharacteristic
111C3×#217—34no—
112C3×(C2×D3)⋊C4—34no—
113C3×Dic3⋊C4—34no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—31no—
117C22×H(3,𝔽3)—22no—
118C22×H(3,𝔽3)—22no—
119C22×H(3,𝔽3)—24no—
120C22×H(3,𝔽3)C2×D412yes—
121C22×H(3,𝔽3)C2411yescharacteristic
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—24no—
124C4×H(3,𝔽3)—22no—
125C4×H(3,𝔽3)C2×D412yes—
126C4×H(3,𝔽3)C2×D412yes—
127(C3⋊Dic3)⋊C3—22no—
128(C3⋊Dic3)⋊C3—22no—
129(C3⋊Dic3)⋊C3#4114yescomplemented
130C2×C3×#41—31no—
131C2×C3×#41—31no—
132C2×C3×#41—61no—
133C2×C3×#41—91no—
134C2×C3×#41C3×D311yescharacteristic
135C2×#235—31no—
136C2×#235—91no—
137C3×C4⋊Q8—91no—
138C3×#127—91no—
139C3×#124—91no—
140C3×#124—91no—
141C3×#121—94no—
142C3×C4×D4—94no—
143C2×C3×C42—91no—
144#947—31no—
145#947—91no—
146#944—31no—
147#944—91no—
148#940—31no—
149#940—91no—
150#939—34no—
151#939—94no—
152#936—31no—
153#936—91no—
154C4×#67—34no—
155C4×#67—94no—
156C2×C4×Dic3—31no—
157C2×C4×Dic3—91no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32C22×D312yes—
161C23×C32C22×D311yescharacteristic
162C23×C32C23×C312yes—
163C23×C32C23×C311yescharacteristic
164C22×C3⋊D3C23×C312yes—
165C22×C3×D3—32no—
166C32×Q8—24no—
167C32×Q8—22no—
168C32×Q8—22no—
169C32×D4—28no—
170C32×D4—24no—
171C32×D4—24no—
172C32×D4—24no—
173C32×D4—22no—
174C32×D4—22no—
175C2×C32×C4—28no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—21no—
180C2×C32×C4—22no—
181C2×C32×C4—21no—
182C2×C32×C4#6714yescomplemented
183C2×C32×C4C3×D414yescomplemented
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C23×C312yes—
188C2×C32×C4C23×C311yescharacteristic
189C2×C32×C4C23×C311yescharacteristic
190#638—24no—
191#638—28no—
192C2×C3⋊Dic3—24no—
193C2×C3⋊Dic3C23×C312yes—
194C2×C3⋊Dic3C23×C312yes—
195C2×C3⋊Dic3C23×C312yes—
196C4⋊(C3⋊D3)—22no—
197C4×C3⋊D3—24no—
198#634—22no—
199C3×#67—68no—
200C3×#67—64no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—64no—
205C3×D12—62no—
206C3×C4×D3—64no—
207C3×Dic6—62no—
208#532—91noSylow
209C2×H(3,𝔽3)—22no—
210C2×H(3,𝔽3)—42no—
211C2×H(3,𝔽3)C22×D411yescharacteristic
212C2×H(3,𝔽3)C2×#4112yes—
213(C3⋊D3)⋊C3—42no—
214C3×#41—24no—
215C3×#41—64no—
216C3×#41—64no—
217C3×#41—68no—
218C3×#41—184no—
219C2×C3×Q8—31no—
220C2×C3×Q8—31no—
221C2×C3×Q8—61no—
222C2×C3×Q8—91no—
223C2×C3×Q8C2×C3×D311yescharacteristic
224C2×C3×D4—31no—
225C2×C3×D4—32no—
226C2×C3×D4—32no—
227C2×C3×D4—31no—
228C2×C3×D4—62no—
229C2×C3×D4—61no—
230C2×C3×D4—91no—
231C2×C3×D4—94no—
232C2×C3×D4—92no—
233C2×C3×D4C2×C3×D311yescharacteristic
234C2×C3×D4C2×C3×D312yes—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—32no—
238C22×C3×C4—32no—
239C22×C3×C4—62no—
240C22×C3×C4—61no—
241C22×C3×C4—92no—
242C22×C3×C4—92no—
243C22×C3×C4C2×C3×D312yes—
244C22×C3×C4C2×C3×D311yescharacteristic
245C2×#67—34no—
246C2×#67—32no—
247C2×#67—94no—
248C2×#67—92no—
249C22×Dic3—32no—
250C22×Dic3—92no—
251#235—64no—
252#235—184no—
253C2×D12—31no—
254C2×D12—91no—
255C2×C4×D3—32no—
256C2×C4×D3—92no—
257C2×Dic6—31no—
258C2×Dic6—91no—
259C3×C4⋊C4—94no—
260C3×C22⋊C4—94no—
261C3×C22⋊C4—94no—
262C3×C42—92no—
263C3×C42—91no—
264C3×C42—91no—
265#217—34no—
266#217—94no—
267(C2×D3)⋊C4—34no—
268(C2×D3)⋊C4—94no—
269Dic3⋊C4—34no—
270Dic3⋊C4—94no—
271C4×Dic3—32no—
272C4×Dic3—31no—
273C4×Dic3—31no—
274C4×Dic3—92no—
275C4×Dic3—91no—
276C4×Dic3—91no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—24no—
280C22×C32—24no—
281C22×C32—22no—
282C22×C32—22no—
283C22×C32—24no—
284C22×C32—21no—
285C22×C32—22no—
286C22×C32—42no—
287C22×C32—44no—
288C22×C32C2×#6712yes—
289C22×C32C2×C3×D412yes—
290C22×C32C23×D311yescharacteristic
291C22×C32C24×C311yescharacteristic
292C2×C3⋊D3—22no—
293C2×C3⋊D3—24no—
294C2×C3×D3—64no—
295C2×C3×D3—62no—
296C32×C4—24no—
297C32×C4—22no—
298C32×C4—22no—
299C32×C4—22no—
300C32×C4—22no—
301C32×C4C2×#6712yes—
302C32×C4C2×#6712yes—
303C32×C4C2×C3×D412yes—
304C32×C4C2×C3×D412yes—
305C3⋊Dic3—22no—
306C3⋊Dic3—22no—
307C3⋊Dic3C3×#4114yescomplemented
308C3×Dic3—34no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C2×#41—91no—
312C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
313C4⋊Q8—91no—
314#127—91no—
315#124—91no—
316#124—91no—
317#121—94no—
318C4×D4—94no—
319C2×C42—91no—
320H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
321C23×C3—31no—
322C23×C3—31no—
323C23×C3—32no—
324C23×C3—32no—
325C23×C3—62no—
326C23×C3—61no—
327C23×C3—92no—
328C23×C3C22×C3×D312yes—
329C23×C3C22×C3×D311yescharacteristic
330C22×D3—32no—
331C22×D3—92no—
332C3×Q8—22no—
333C3×Q8—62no—
334C3×Q8—62no—
335C3×Q8—64no—
336C3×Q8—182no—
337C3×D4—24no—
338C3×D4—22no—
339C3×D4—62no—
340C3×D4—62no—
341C3×D4—64no—
342C3×D4—64no—
343C3×D4—68no—
344C3×D4—64no—
345C3×D4—182no—
346C3×D4—188no—
347C3×D4—184no—
348C2×C3×C4—24no—
349C2×C3×C4—34no—
350C2×C3×C4—34no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—32no—
354C2×C3×C4—32no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—64no—
358C2×C3×C4—64no—
359C2×C3×C4—68no—
360C2×C3×C4—64no—
361C2×C3×C4—62no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—92no—
365C2×C3×C4—92no—
366C2×C3×C4—92no—
367C2×C3×C4—184no—
368C2×C3×C4—184no—
369C2×C3×C4C3×#6714yes—
370C2×C3×C4C22×C3×D312yes—
371C2×C3×C4C22×C3×D311yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373#67—68no—
374#67—64no—
375#67—188no—
376#67—184no—
377C2×Dic3—32no—
378C2×Dic3—32no—
379C2×Dic3—32no—
380C2×Dic3—64no—
381C2×Dic3—92no—
382C2×Dic3—92no—
383C2×Dic3—92no—
384C2×Dic3—184no—
385D12—62no—
386D12—182no—
387C4×D3—64no—
388C4×D3—184no—
389Dic6—62no—
390Dic6—182no—
391C2×C32—22no—
392C2×C32—22no—
393C2×C32—22no—
394C2×C32—21no—
395C2×C32—42no—
396C2×C32—44no—
397C2×C32—42no—
398C2×C32—42no—
399C2×C32D3×#4112yes—
400C2×C32C22×#6711yescharacteristic
401C2×C32C22×C3×D411yescharacteristic
402C2×C32C2×C3×#4112yes—
403C3⋊D3—42no—
404C3×D3—122no—
405#41—24no—
406#41—184no—
407C2×Q8—91no—
408C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
409C2×D4—94no—
410C2×D4—92no—
411C2×D4—91no—
412C2×D4C2×(C3⋊D3)⋊C312yes—
413C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
414C22×C4—92no—
415C22×C4—92no—
416C22×C4C2×(C3⋊D3)⋊C312yes—
417C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
418C4⋊C4—94no—
419C22⋊C4—94no—
420C22⋊C4—94no—
421C42—91no—
422C42—91no—
423C42—92no—
424C22×C3—22no—
425C22×C3—24no—
426C22×C3—22no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—32no—
430C22×C3—32no—
431C22×C3—62no—
432C22×C3—62no—
433C22×C3—64no—
434C22×C3—62no—
435C22×C3—62no—
436C22×C3—64no—
437C22×C3—64no—
438C22×C3—61no—
439C22×C3—62no—
440C22×C3—122no—
441C22×C3—124no—
442C22×C3—182no—
443C22×C3—184no—
444C22×C3C2×C3×#6712yes—
445C22×C3C23×C3×D311yescharacteristic
446C2×D3—64no—
447C2×D3—62no—
448C2×D3—182no—
449C2×D3—184no—
450C3×C4—22no—
451C3×C4—32no—
452C3×C4—32no—
453C3×C4—32no—
454C3×C4—32no—
455C3×C4—62no—
456C3×C4—62no—
457C3×C4—64no—
458C3×C4—62no—
459C3×C4—62no—
460C3×C4—94no—
461C3×C4—182no—
462C3×C4—182no—
463C3×C4C2×C3×#6712yes—
464C3×C4C2×C3×#6712yes—
465Dic3—34no—
466Dic3—62no—
467Dic3—62no—
468Dic3—94no—
469Dic3—182no—
470Dic3—182no—
471C32—21no—
472C32#583211yescharacteristic, complemented
473C32C3×#53211yescharacteristic, complemented
474C23—92no—
475C23C22×(C3⋊D3)⋊C312yes—
476C23C22×(C3⋊D3)⋊C311yescharacteristic
477Q8—22no—
478Q8—182no—
479D4—24no—
480D4—22no—
481D4—188no—
482D4—184no—
483D4—182no—
484C2×C4—24no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4—92no—
488C2×C4—184no—
489C2×C4—184no—
490C2×C4#638⋊C314yescomplemented
491C2×C4C22×(C3⋊D3)⋊C312yes—
492C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
493C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
494C2×C3—22no—
495C2×C3—32no—
496C2×C3—32no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—42no—
500C2×C3—62no—
501C2×C3—62no—
502C2×C3—62no—
503C2×C3—61no—
504C2×C3—122no—
505C2×C3—122no—
506C2×C3—122no—
507C2×C3—124no—
508C2×C3—362no—
509C2×C3C3×D3×#4112yes—
510C2×C3C22×C3×#6711yescharacteristic
511D3—122no—
512D3—362no—
513C22—22no—
514C22—24no—
515C22—22no—
516C22—184no—
517C22—182no—
518C22C2×#638⋊C312yes—
519C22C23×(C3⋊D3)⋊C311yescharacteristic
520C4—22no—
521C4—94no—
522C4—182no—
523C4—182no—
524C4C2×#638⋊C312yes—
525C4C2×#638⋊C312yes—
526C3—31no—
527C3—31no—
528C3—61no—
529C3C3×#583211yescharacteristic
530C2—22no—
531C2—42no—
532C2—362no—
533C2#41×(C3⋊D3)⋊C312yes—
534C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×#41×H(3,𝔽3)C2
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C3×#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×H(3,𝔽3)C2semidirect
2. C2×#3828⋊C3C2semidirect
3. #64156⋊C3C2semidirect
4. #64153⋊C3C2semidirect
5. #64149⋊C3C2semidirect
6. #64148⋊C3C2semidirect
7. #64145⋊C3C2semidirect
8. C4×#638⋊C3C2semidirect
9. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
10. #10596303C3semidirect
32. C2×C32×#41D3semidirect
33. C2×C32×#41C2×C3semidirect
34. C2×#3828C2×C3semidirect
36. #64156C2×C3semidirect
37. #64153C2×C3semidirect
38. #64149C2×C3semidirect
39. #64148C2×C3semidirect
40. #64145C2×C3semidirect
41. C4×#638C2×C3semidirect
42. C2×C4×C3⋊Dic3C2×C3semidirect
57. C2×C4×H(3,𝔽3)D4semidirect
129. (C3⋊Dic3)⋊C3#41semidirect
182. C2×C32×C4#67semidirect
183. C2×C32×C4C3×D4semidirect
307. C3⋊Dic3C3×#41semidirect
312. C2×#41(C3⋊D3)⋊C3semidirect
320. H(3,𝔽3)#532semidirect
472. C32#5832semidirect
473. C32C3×#532semidirect
490. C2×C4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#41×H(3,𝔽3)2
C2×#3828⋊C32
#64156⋊C32
#64153⋊C32
#64149⋊C32
#64148⋊C32
#64145⋊C32
C4×#638⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#105963033
C3×#58323