Finite Group Databasebeta

← C2×#64087⋊C3 #13152356

Subgroups — C2×#64087⋊C3 #13152356

Order 1728.

1259 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
4C2×#3757⋊C3C211yesmaximal, characteristic
5C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
6C2×#3750⋊C3C211yesmaximal, characteristic, complemented
7C2×#3750⋊C3C211yesmaximal, characteristic, complemented
8#64087⋊C3C218yesmaximal, direct product factor
9C2×#64087C311yesmaximal, characteristic, complemented
10C2×C3×#878—31nomaximal
11C24×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3—24no—
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C22×(C3⋊Dic3)⋊C3C2214yes—
19C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×#634⋊C3—24no—
21C2×#634⋊C3C2214yescomplemented
22C22⋊C4×H(3,𝔽3)C2214yes—
23#3757⋊C3C2214yes—
24C4⋊((C3⋊Dic3)⋊C3)C2214yescomplemented
25#3750⋊C3C2214yescomplemented
26#3750⋊C3C2214yescomplemented
27C23×C3⋊Dic3C2×C311yescharacteristic
28C22×#634C2×C311yescharacteristic, complemented
29C23×C3×Dic3—31no—
30C22×C3×Dic6—31no—
31C2×C32×C22⋊C4—21no—
32C2×C32×C22⋊C4D311yescharacteristic
33C2×C32×C22⋊C4C2×C311yescharacteristic
34C2×#3757C2×C311yescharacteristic
35C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
36C2×#3750C2×C311yescharacteristic, complemented
37C2×#3750C2×C311yescharacteristic, complemented
38#64087C2×C318yescomplemented
39C2×C3×#217—31no—
40C2×C3×C4⋊Dic3—31no—
41C2×C3×Dic3⋊C4—31no—
42C2×C3×Dic3⋊C4—31no—
43C3×#878—38no—
44C23×H(3,𝔽3)—22no—
45C23×H(3,𝔽3)—22no—
46C23×H(3,𝔽3)Q814yescomplemented
47C23×H(3,𝔽3)C2312yes—
48C23×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)—22no—
50C2×C4×H(3,𝔽3)—22no—
51C2×C4×H(3,𝔽3)C2312yes—
52C2×C4×H(3,𝔽3)C2312yes—
53C2×(C3⋊Dic3)⋊C3—22no—
54C2×(C3⋊Dic3)⋊C3—22no—
55C2×(C3⋊Dic3)⋊C3—22no—
56C2×(C3⋊Dic3)⋊C3—28no—
57C2×(C3⋊Dic3)⋊C3D414yes—
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62#634⋊C3—28no—
63C2×C3×C22⋊Q8—91no—
64C2×#878—31no—
65C2×#878—91no—
66C24×C32—21no—
67C24×C32C2×D311yescharacteristic
68C24×C32C22×C311yescharacteristic
69C22×C32×C4—21no—
70C22×C32×C4—21no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C22×C311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C3⋊Dic3—24no—
76C22×C3⋊Dic3C22×C314yes—
77C22×C3⋊Dic3C22×C311yescharacteristic
78C22×C3⋊Dic3C22×C311yescharacteristic
79C22×C3⋊Dic3C22×C311yescharacteristic
80C22×C3⋊Dic3C22×C311yescharacteristic
81C2×#634—24no—
82C2×#634C22×C314yescomplemented
83C22×C3×Dic3—31no—
84C22×C3×Dic3—34no—
85C22×C3×Dic3—31no—
86C22×C3×Dic3—31no—
87C22×C3×Dic3—31no—
88C22×C3×Dic3—64no—
89C2×C3×Dic6—34no—
90C2×C3×Dic6—64no—
91C32×C22⋊C4—24no—
92C32×C22⋊C4C2×D314yes—
93C32×C22⋊C4C22×C314yes—
94#3757C22×C314yes—
95C4⋊(C3⋊Dic3)C22×C314yescomplemented
96#3750C22×C314yescomplemented
97#3750C22×C314yescomplemented
98C3×#217—34no—
99C3×C4⋊Dic3—34no—
100C3×Dic3⋊C4—34no—
101C3×Dic3⋊C4—34no—
102C22×H(3,𝔽3)—28no—
103C22×H(3,𝔽3)—22no—
104C22×H(3,𝔽3)—22no—
105C22×H(3,𝔽3)C2×D412yes—
106C22×H(3,𝔽3)C2×Q812yes—
107C22×H(3,𝔽3)C2×Q814yescomplemented
108C22×H(3,𝔽3)#4112yes—
109C22×H(3,𝔽3)C2411yescharacteristic
110C4×H(3,𝔽3)—22no—
111C4×H(3,𝔽3)—22no—
112(C3⋊Dic3)⋊C3—22no—
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3C2×D414yes—
116C22×C3×Q8—91no—
117C23×C3×C4—91no—
118C23×Dic3—31no—
119C23×Dic3—91no—
120C22×Dic6—31no—
121C22×Dic6—91no—
122C3×C22⋊Q8—98no—
123C2×C3×C4⋊C4—91no—
124C2×C3×C4⋊C4—91no—
125C2×C3×C4⋊C4—91no—
126C2×C3×C22⋊C4—31no—
127C2×C3×C22⋊C4—31no—
128C2×C3×C22⋊C4—61no—
129C2×C3×C22⋊C4—91no—
130C2×C3×C22⋊C4C3×D311yescharacteristic
131C2×#217—31no—
132C2×#217—91no—
133C2×C4⋊Dic3—31no—
134C2×C4⋊Dic3—91no—
135C2×Dic3⋊C4—31no—
136C2×Dic3⋊C4—31no—
137C2×Dic3⋊C4—91no—
138C2×Dic3⋊C4—91no—
139#878—38no—
140#878—98no—
141C23×C32—22no—
142C23×C32—22no—
143C23×C32—22no—
144C23×C32—22no—
145C23×C32—24no—
146C23×C32—21no—
147C23×C32—22no—
148C23×C32—42no—
149C23×C32—42no—
150C23×C32Dic614yescomplemented
151C23×C32C3×Q814yescomplemented
152C23×C32C22×D312yes—
153C23×C32C22×D311yescharacteristic
154C23×C32C23×C312yes—
155C23×C32C23×C311yescharacteristic
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—22no—
161C2×C32×C4—22no—
162C2×C32×C4—42no—
163C2×C32×C4—42no—
164C2×C32×C4C22×D312yes—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C23×C312yes—
167C2×C32×C4C23×C312yes—
168C2×C3⋊Dic3—22no—
169C2×C3⋊Dic3—22no—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3—28no—
172C2×C3⋊Dic3C3×D414yes—
173C2×C3⋊Dic3C23×C312yes—
174C2×C3⋊Dic3C23×C312yes—
175C2×C3⋊Dic3C23×C312yes—
176C2×C3⋊Dic3C23×C312yes—
177#634—28no—
178C2×C3×Dic3—34no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—68no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—62no—
187C3×Dic6—68no—
188C2×C22⋊Q8—91noSylow
189C2×H(3,𝔽3)—24no—
190C2×H(3,𝔽3)C22⋊Q814yescomplemented
191C2×H(3,𝔽3)C22×D411yescharacteristic
192C2×H(3,𝔽3)C22×Q811yescharacteristic
193C2×H(3,𝔽3)C2×#4111yescharacteristic
194C24×C3—31no—
195C24×C3—31no—
196C24×C3—61no—
197C24×C3C2×C3×D311yescharacteristic
198C2×C3×Q8—94no—
199C2×C3×Q8—184no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—61no—
205C22×C3×C4—61no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C22×C3×C4—94no—
211C22×C3×C4—184no—
212C22×C3×C4C2×C3×D311yescharacteristic
213C22×C3×C4C2×C3×D311yescharacteristic
214C22×Dic3—31no—
215C22×Dic3—34no—
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—31no—
219C22×Dic3—64no—
220C22×Dic3—91no—
221C22×Dic3—94no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C22×Dic3—91no—
225C22×Dic3—184no—
226C2×Dic6—34no—
227C2×Dic6—64no—
228C2×Dic6—94no—
229C2×Dic6—184no—
230C3×C4⋊C4—94no—
231C3×C4⋊C4—94no—
232C3×C4⋊C4—94no—
233C3×C22⋊C4—34no—
234C3×C22⋊C4—34no—
235C3×C22⋊C4—64no—
236C3×C22⋊C4—94no—
237C3×C22⋊C4C2×C3×D314yes—
238#217—34no—
239#217—94no—
240C4⋊Dic3—34no—
241C4⋊Dic3—94no—
242Dic3⋊C4—34no—
243Dic3⋊C4—34no—
244Dic3⋊C4—94no—
245Dic3⋊C4—94no—
246C22×C32—28no—
247C22×C32—28no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—22no—
254C22×C32—24no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—48no—
258C22×C32—42no—
259C22×C32—42no—
260C22×C32C2×Dic612yes—
261C22×C32C2×Dic614yescomplemented
262C22×C32D3×D412yes—
263C22×C32#23712yes—
264C22×C32C2×C3×D412yes—
265C22×C32C2×C3×Q812yes—
266C22×C32C2×C3×Q814yescomplemented
267C22×C32C3×#4112yes—
268C22×C32C23×D311yescharacteristic
269C22×C32C24×C311yescharacteristic
270C32×C4—22no—
271C32×C4—22no—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—42no—
275C32×C4—42no—
276C3⋊Dic3—22no—
277C3⋊Dic3—22no—
278C3⋊Dic3—22no—
279C3⋊Dic3C2×C3×D414yes—
280C3×Dic3—34no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C22×Q8—91no—
285C23×C4—91no—
286C22⋊Q8—98no—
287C2×C4⋊C4—91no—
288C2×C4⋊C4—91no—
289C2×C4⋊C4—91no—
290C2×C22⋊C4—91no—
291C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic
292H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—34no—
296C23×C3—34no—
297C23×C3—32no—
298C23×C3—32no—
299C23×C3—31no—
300C23×C3—31no—
301C23×C3—62no—
302C23×C3—62no—
303C23×C3—62no—
304C23×C3—62no—
305C23×C3—64no—
306C23×C3—62no—
307C23×C3—61no—
308C23×C3—122no—
309C23×C3—122no—
310C23×C3C3×Dic614yes—
311C23×C3C22×C3×D312yes—
312C23×C3C22×C3×D311yescharacteristic
313C3×Q8—188no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—32no—
317C2×C3×C4—32no—
318C2×C3×C4—32no—
319C2×C3×C4—32no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—92no—
327C2×C3×C4—94no—
328C2×C3×C4—92no—
329C2×C3×C4—92no—
330C2×C3×C4—92no—
331C2×C3×C4—122no—
332C2×C3×C4—122no—
333C2×C3×C4—188no—
334C2×C3×C4—182no—
335C2×C3×C4—182no—
336C2×C3×C4—182no—
337C2×C3×C4C22×C3×D312yes—
338C2×C3×C4C22×C3×D312yes—
339C2×Dic3—32no—
340C2×Dic3—34no—
341C2×Dic3—32no—
342C2×Dic3—32no—
343C2×Dic3—32no—
344C2×Dic3—68no—
345C2×Dic3—62no—
346C2×Dic3—62no—
347C2×Dic3—62no—
348C2×Dic3—94no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—92no—
353C2×Dic3—188no—
354C2×Dic3—182no—
355C2×Dic3—182no—
356C2×Dic3—182no—
357Dic6—68no—
358Dic6—188no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—44no—
366C2×C32#87814yescomplemented
367C2×C32C3×C22⋊Q814yescomplemented
368C2×C32C22×Dic611yescharacteristic
369C2×C32C2×D3×D411yescharacteristic
370C2×C32C2×#23711yescharacteristic
371C2×C32C22×C3×D411yescharacteristic
372C2×C32C22×C3×Q811yescharacteristic
373C2×C32C2×C3×#4111yescharacteristic
374C24C2×(C3⋊D3)⋊C311yescharacteristic
375C2×Q8—94no—
376C2×Q8—184no—
377C22×C4—91no—
378C22×C4—94no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—91no—
382C22×C4—184no—
383C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
384C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
385C4⋊C4—94no—
386C4⋊C4—94no—
387C4⋊C4—94no—
388C22⋊C4—94no—
389C22⋊C4C2×(C3⋊D3)⋊C314yes—
390C22×C3—28no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—32no—
396C22×C3—32no—
397C22×C3—34no—
398C22×C3—34no—
399C22×C3—32no—
400C22×C3—32no—
401C22×C3—32no—
402C22×C3—32no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—68no—
408C22×C3—68no—
409C22×C3—61no—
410C22×C3—62no—
411C22×C3—64no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—122no—
415C22×C3—122no—
416C22×C3—128no—
417C22×C3C2×C3×Dic612yes—
418C22×C3C2×C3×Dic614yes—
419C22×C3C3×D3×D412yes—
420C22×C3C3×#23712yes—
421C22×C3C23×C3×D311yescharacteristic
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—62no—
425C3×C4—62no—
426C3×C4—62no—
427C3×C4—62no—
428C3×C4—94no—
429C3×C4—122no—
430C3×C4—122no—
431C3×C4—182no—
432C3×C4—182no—
433C3×C4—182no—
434Dic3—34no—
435Dic3—62no—
436Dic3—62no—
437Dic3—62no—
438Dic3—94no—
439Dic3—182no—
440Dic3—182no—
441Dic3—182no—
442C32—21no—
443C32C2×#87811yescharacteristic, complemented
444C32C2×C3×C22⋊Q811yescharacteristic, complemented
445C23—22no—
446C23—22no—
447C23#634⋊C314yescomplemented
448C23C22×(C3⋊D3)⋊C312yes—
449C23C22×(C3⋊D3)⋊C311yescharacteristic
450Q8—188no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—92no—
454C2×C4—94no—
455C2×C4—92no—
456C2×C4—92no—
457C2×C4—92no—
458C2×C4—188no—
459C2×C4—182no—
460C2×C4—182no—
461C2×C4—182no—
462C2×C4C22×(C3⋊D3)⋊C312yes—
463C2×C4C22×(C3⋊D3)⋊C312yes—
464C2×C3—24no—
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—31no—
471C2×C3—34no—
472C2×C3—34no—
473C2×C3—64no—
474C2×C3—64no—
475C2×C3—61no—
476C2×C3—61no—
477C2×C3—61no—
478C2×C3—64no—
479C2×C3—124no—
480C2×C3C3×#87814yes—
481C2×C3C22×C3×Dic611yescharacteristic
482C2×C3C2×C3×D3×D411yescharacteristic
483C2×C3C2×C3×#23711yescharacteristic
484C22—22no—
485C22—22no—
486C22—28no—
487C22C2×#634⋊C312yes—
488C22C2×#634⋊C314yescomplemented
489C22D4×(C3⋊D3)⋊C312yes—
490C22#3830⋊C312yes—
491C22C23×(C3⋊D3)⋊C311yescharacteristic
492C4—22no—
493C4—22no—
494C4—94no—
495C4—182no—
496C4—182no—
497C4—182no—
498C3—31no—
499C3—31no—
500C3—61no—
501C3C2×C3×#87811yescharacteristic
502C2—24no—
503C2#64087⋊C314yesdirect product factor
504C2C22×#634⋊C311yescharacteristic
505C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
506C2C2×#3830⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC24×H(3,𝔽3)C22
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
5. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
6. C2×#3750⋊C3C2semidirect
7. C2×#3750⋊C3C2semidirect
8. #64087⋊C3C2direct (N × H)
9. C2×#64087C3semidirect
21. C2×#634⋊C3C22semidirect
24. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
25. #3750⋊C3C22semidirect
26. #3750⋊C3C22semidirect
28. C22×#634C2×C3semidirect
35. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
36. C2×#3750C2×C3semidirect
37. C2×#3750C2×C3semidirect
38. #64087C2×C3semidirect
46. C23×H(3,𝔽3)Q8semidirect
82. C2×#634C22×C3semidirect
95. C4⋊(C3⋊Dic3)C22×C3semidirect
96. #3750C22×C3semidirect
97. #3750C22×C3semidirect
107. C22×H(3,𝔽3)C2×Q8semidirect
150. C23×C32Dic6semidirect
151. C23×C32C3×Q8semidirect
190. C2×H(3,𝔽3)C22⋊Q8semidirect
261. C22×C32C2×Dic6semidirect
266. C22×C32C2×C3×Q8semidirect
292. H(3,𝔽3)C2×C22⋊Q8semidirect
366. C2×C32#878semidirect
367. C2×C32C3×C22⋊Q8semidirect
443. C32C2×#878semidirect
444. C32C2×C3×C22⋊Q8semidirect
447. C23#634⋊C3semidirect
488. C22C2×#634⋊C3semidirect
503. C2#64087⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×(C3⋊Dic3)⋊C32
C22×#634⋊C32
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
#64087⋊C32
C2×#640873
C2×C3×#8783