Finite Group Databasebeta

← C2×#64103⋊C3 #13152437

Subgroups — C2×#64103⋊C3 #13152437

Order 1728.

1289 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#3752⋊C3C211yesmaximal, characteristic, complemented
5C2×#3752⋊C3C211yesmaximal, characteristic, complemented
6C2×#3752⋊C3C211yesmaximal, characteristic, complemented
7C2×#3750⋊C3C211yesmaximal, characteristic, complemented
8#64103⋊C3C218yesmaximal, direct product factor
9C2×#64103C311yesmaximal, characteristic, complemented
10C2×C3×#894—31nomaximal
11C23×(C3⋊D3)⋊C3C2211yescharacteristic
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×C4⋊((C3⋊D3)⋊C3)—24no—
19C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
20C2×C4×(C3⋊D3)⋊C3—24no—
21C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
22C4⋊C4×H(3,𝔽3)C2214yescomplemented
23#3752⋊C3C2214yescomplemented
24#3752⋊C3C2214yescomplemented
25#3752⋊C3C2214yescomplemented
26#3750⋊C3C2214yescomplemented
27C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
28C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
29C22×C3×D12—31no—
30C22×C3×C4×D3—31no—
31C2×C32×C4⋊C4—21no—
32C2×C32×C4⋊C4D311yescharacteristic, complemented
33C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
34C2×#3752C2×C311yescharacteristic, complemented
35C2×#3752C2×C311yescharacteristic, complemented
36C2×#3752C2×C311yescharacteristic, complemented
37C2×#3750C2×C311yescharacteristic, complemented
38#64103C2×C318yescomplemented
39C2×C3×(C2×D3)⋊C4—31no—
40C2×C3×(C2×D3)⋊C4—31no—
41C2×C3×(C2×D3)⋊C4—31no—
42C2×C3×Dic3⋊C4—31no—
43C3×#894—38no—
44C23×H(3,𝔽3)C2311yescharacteristic
45C22×(C3⋊D3)⋊C3—22no—
46C22×(C3⋊D3)⋊C3—22no—
47C22×(C3⋊D3)⋊C3—22no—
48C22×(C3⋊D3)⋊C3—22no—
49C22×(C3⋊D3)⋊C3—22no—
50C22×(C3⋊D3)⋊C3D414yes—
51C22×(C3⋊D3)⋊C3C2312yes—
52C22×(C3⋊D3)⋊C3C2312yes—
53C2×C4×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)—22no—
55C2×C4×H(3,𝔽3)—22no—
56C2×C4×H(3,𝔽3)C2312yes—
57C2×C4×H(3,𝔽3)C2312yes—
58C2×C4×H(3,𝔽3)C2312yes—
59C2×(C3⋊Dic3)⋊C3—22no—
60C2×(C3⋊Dic3)⋊C3—22no—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C4⋊((C3⋊D3)⋊C3)—28no—
64C4×(C3⋊D3)⋊C3—28no—
65C2×C3×#123—91no—
66C2×#894—31no—
67C2×#894—91no—
68C23×C3⋊D3C22×C311yescharacteristic
69C23×C3⋊D3C22×C311yescharacteristic
70C23×C3×D3—31no—
71C23×C3×D3—31no—
72C22×C32×C4—21no—
73C22×C32×C4—21no—
74C22×C32×C4—21no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C32×C4C2×D311yescharacteristic
78C22×C32×C4C22×C311yescharacteristic
79C22×C32×C4C22×C311yescharacteristic
80C22×C32×C4C22×C311yescharacteristic
81C22×C3⋊Dic3C22×C311yescharacteristic
82C22×C3⋊Dic3C22×C311yescharacteristic
83C2×C4⋊(C3⋊D3)—24no—
84C2×C4⋊(C3⋊D3)C22×C314yes—
85C2×C4×C3⋊D3—24no—
86C2×C4×C3⋊D3C22×C314yescomplemented
87C22×C3×Dic3—31no—
88C22×C3×Dic3—31no—
89C2×C3×D12—34no—
90C2×C3×D12—64no—
91C2×C3×C4×D3—34no—
92C2×C3×C4×D3—64no—
93C32×C4⋊C4—24no—
94C32×C4⋊C4C2×D314yescomplemented
95C32×C4⋊C4C22×C314yescomplemented
96#3752C22×C314yescomplemented
97#3752C22×C314yescomplemented
98#3752C22×C314yescomplemented
99#3750C22×C314yescomplemented
100C3×(C2×D3)⋊C4—34no—
101C3×(C2×D3)⋊C4—34no—
102C3×(C2×D3)⋊C4—34no—
103C3×Dic3⋊C4—34no—
104C22×H(3,𝔽3)C2×D412yes—
105C22×H(3,𝔽3)#4112yes—
106C22×H(3,𝔽3)#4112yes—
107C22×H(3,𝔽3)C2411yescharacteristic
108C2×(C3⋊D3)⋊C3—28no—
109C2×(C3⋊D3)⋊C3—22no—
110C2×(C3⋊D3)⋊C3—22no—
111C2×(C3⋊D3)⋊C3—22no—
112C2×(C3⋊D3)⋊C3—22no—
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3—44no—
115C2×(C3⋊D3)⋊C3C2×D414yes—
116C4×H(3,𝔽3)—22no—
117C4×H(3,𝔽3)—22no—
118C4×H(3,𝔽3)—22no—
119(C3⋊Dic3)⋊C3—22no—
120(C3⋊Dic3)⋊C3—22no—
121C22×C3×D4—91no—
122C23×C3×C4—91no—
123C22×D12—31no—
124C22×D12—91no—
125C22×C4×D3—31no—
126C22×C4×D3—91no—
127C3×#123—98no—
128C2×C3×C4⋊C4—31no—
129C2×C3×C4⋊C4—31no—
130C2×C3×C4⋊C4—61no—
131C2×C3×C4⋊C4—91no—
132C2×C3×C4⋊C4C3×D311yescharacteristic
133C2×C3×C22⋊C4—91no—
134C2×C3×C22⋊C4—91no—
135C2×C3×C22⋊C4—91no—
136C2×(C2×D3)⋊C4—31no—
137C2×(C2×D3)⋊C4—31no—
138C2×(C2×D3)⋊C4—31no—
139C2×(C2×D3)⋊C4—91no—
140C2×(C2×D3)⋊C4—91no—
141C2×(C2×D3)⋊C4—91no—
142C2×Dic3⋊C4—31no—
143C2×Dic3⋊C4—91no—
144#894—38no—
145#894—98no—
146C23×C32—21no—
147C23×C32C22×D311yescharacteristic
148C23×C32C23×C311yescharacteristic
149C22×C3⋊D3—22no—
150C22×C3⋊D3—22no—
151C22×C3⋊D3—22no—
152C22×C3⋊D3—22no—
153C22×C3⋊D3—22no—
154C22×C3⋊D3C3×D414yes—
155C22×C3⋊D3C23×C312yes—
156C22×C3⋊D3C23×C312yes—
157C22×C3×D3—34no—
158C22×C3×D3—32no—
159C22×C3×D3—32no—
160C22×C3×D3—62no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C22×C3×D3—62no—
164C22×C3×D3—62no—
165C2×C32×C4—24no—
166C2×C32×C4—22no—
167C2×C32×C4—22no—
168C2×C32×C4—22no—
169C2×C32×C4—22no—
170C2×C32×C4—22no—
171C2×C32×C4—22no—
172C2×C32×C4—22no—
173C2×C32×C4—22no—
174C2×C32×C4—22no—
175C2×C32×C4—42no—
176C2×C32×C4—42no—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D312yes—
179C2×C32×C4C22×D312yes—
180C2×C32×C4C23×C312yes—
181C2×C32×C4C23×C312yes—
182C2×C32×C4C23×C312yes—
183C2×C3⋊Dic3—22no—
184C2×C3⋊Dic3—22no—
185C2×C3⋊Dic3C23×C312yes—
186C2×C3⋊Dic3C23×C312yes—
187C4⋊(C3⋊D3)—28no—
188C4×C3⋊D3—28no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C3×D12—68no—
194C3×C4×D3—68no—
195C2×#123—91noSylow
196C2×H(3,𝔽3)#12314yescomplemented
197C2×H(3,𝔽3)C22×D411yescharacteristic
198C2×H(3,𝔽3)C2×#4111yescharacteristic
199C2×H(3,𝔽3)C2×#4111yescharacteristic
200(C3⋊D3)⋊C3—24no—
201(C3⋊D3)⋊C3—42no—
202C24×C3—91no—
203C24×C3—91no—
204C23×D3—31no—
205C23×D3—31no—
206C23×D3—91no—
207C23×D3—91no—
208C2×C3×D4—94no—
209C2×C3×D4—184no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—31no—
215C22×C3×C4—31no—
216C22×C3×C4—61no—
217C22×C3×C4—61no—
218C22×C3×C4—61no—
219C22×C3×C4—94no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C22×C3×C4—184no—
223C22×C3×C4C2×C3×D311yescharacteristic
224C22×C3×C4C2×C3×D311yescharacteristic
225C22×C3×C4C2×C3×D311yescharacteristic
226C22×Dic3—31no—
227C22×Dic3—31no—
228C22×Dic3—91no—
229C22×Dic3—91no—
230C2×D12—34no—
231C2×D12—64no—
232C2×D12—94no—
233C2×D12—184no—
234C2×C4×D3—34no—
235C2×C4×D3—64no—
236C2×C4×D3—94no—
237C2×C4×D3—184no—
238C3×C4⋊C4—34no—
239C3×C4⋊C4—34no—
240C3×C4⋊C4—64no—
241C3×C4⋊C4—94no—
242C3×C4⋊C4C2×C3×D314yes—
243C3×C22⋊C4—94no—
244C3×C22⋊C4—94no—
245C3×C22⋊C4—94no—
246(C2×D3)⋊C4—34no—
247(C2×D3)⋊C4—34no—
248(C2×D3)⋊C4—34no—
249(C2×D3)⋊C4—94no—
250(C2×D3)⋊C4—94no—
251(C2×D3)⋊C4—94no—
252Dic3⋊C4—34no—
253Dic3⋊C4—94no—
254C22×C32—21no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32#23512yes—
259C22×C32D3×D412yes—
260C22×C32#23912yes—
261C22×C32C2×C3×D412yes—
262C22×C32C3×#4112yes—
263C22×C32C3×#4112yes—
264C22×C32C23×D311yescharacteristic
265C22×C32C24×C311yescharacteristic
266C2×C3⋊D3—28no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—22no—
270C2×C3⋊D3—22no—
271C2×C3⋊D3—22no—
272C2×C3⋊D3—44no—
273C2×C3⋊D3C2×C3×D414yes—
274C2×C3×D3—34no—
275C2×C3×D3—68no—
276C2×C3×D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—124no—
282C32×C4—24no—
283C32×C4—22no—
284C32×C4—22no—
285C32×C4—22no—
286C32×C4—22no—
287C32×C4—22no—
288C32×C4—22no—
289C32×C4—42no—
290C32×C4—42no—
291C3⋊Dic3—22no—
292C3⋊Dic3—22no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C22×D4—91no—
296C23×C4—91no—
297#123—98no—
298C2×C4⋊C4—91no—
299C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
300C2×C22⋊C4—91no—
301C2×C22⋊C4—91no—
302C2×C22⋊C4—91no—
303H(3,𝔽3)C2×#12311yescharacteristic, complemented, Sylow
304C23×C3—31no—
305C23×C3—31no—
306C23×C3—61no—
307C23×C3—92no—
308C23×C3—94no—
309C23×C3—92no—
310C23×C3—182no—
311C23×C3—182no—
312C23×C3—182no—
313C23×C3—182no—
314C23×C3—182no—
315C23×C3C22×C3×D311yescharacteristic
316C22×D3—32no—
317C22×D3—34no—
318C22×D3—32no—
319C22×D3—62no—
320C22×D3—62no—
321C22×D3—62no—
322C22×D3—62no—
323C22×D3—62no—
324C22×D3—94no—
325C22×D3—92no—
326C22×D3—92no—
327C22×D3—182no—
328C22×D3—182no—
329C22×D3—182no—
330C22×D3—182no—
331C22×D3—182no—
332C3×D4—188no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—32no—
337C2×C3×C4—32no—
338C2×C3×C4—32no—
339C2×C3×C4—32no—
340C2×C3×C4—32no—
341C2×C3×C4—32no—
342C2×C3×C4—62no—
343C2×C3×C4—62no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—64no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—62no—
352C2×C3×C4—92no—
353C2×C3×C4—92no—
354C2×C3×C4—122no—
355C2×C3×C4—122no—
356C2×C3×C4—188no—
357C2×C3×C4—182no—
358C2×C3×C4—182no—
359C2×C3×C4C22×C3×D312yes—
360C2×C3×C4C22×C3×D312yes—
361C2×C3×C4C22×C3×D312yes—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—62no—
365C2×Dic3—62no—
366C2×Dic3—92no—
367C2×Dic3—92no—
368C2×Dic3—182no—
369C2×Dic3—182no—
370D12—68no—
371D12—188no—
372C4×D3—68no—
373C4×D3—188no—
374C2×C32—24no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32#89414yescomplemented
379C2×C32C3×#12314yescomplemented
380C2×C32C2×#23511yescharacteristic
381C2×C32C2×D3×D411yescharacteristic
382C2×C32C2×#23911yescharacteristic
383C2×C32C22×C3×D411yescharacteristic
384C2×C32C2×C3×#4111yescharacteristic
385C2×C32C2×C3×#4111yescharacteristic
386C3⋊D3—24no—
387C3⋊D3—42no—
388C3×D3—64no—
389C3×D3—122no—
390C24—91no—
391C24—91no—
392C2×D4—94no—
393C2×D4—184no—
394C22×C4—91no—
395C22×C4—91no—
396C22×C4—94no—
397C22×C4—184no—
398C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
399C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
400C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
401C4⋊C4—94no—
402C4⋊C4C2×(C3⋊D3)⋊C314yescomplemented
403C22⋊C4—94no—
404C22⋊C4—94no—
405C22⋊C4—94no—
406C22×C3—31no—
407C22×C3—31no—
408C22×C3—32no—
409C22×C3—32no—
410C22×C3—32no—
411C22×C3—32no—
412C22×C3—32no—
413C22×C3—32no—
414C22×C3—61no—
415C22×C3—62no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—94no—
419C22×C3—182no—
420C22×C3—182no—
421C22×C3—188no—
422C22×C3—182no—
423C22×C3—182no—
424C22×C3—182no—
425C22×C3—364no—
426C22×C3C3×#23512yes—
427C22×C3C3×D3×D412yes—
428C22×C3C3×#23912yes—
429C22×C3C23×C3×D311yescharacteristic
430C2×D3—34no—
431C2×D3—68no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—94no—
438C2×D3—124no—
439C2×D3—182no—
440C2×D3—182no—
441C2×D3—188no—
442C2×D3—182no—
443C2×D3—182no—
444C2×D3—182no—
445C2×D3—364no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—62no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—62no—
455C3×C4—64no—
456C3×C4—122no—
457C3×C4—122no—
458C3×C4—182no—
459C3×C4—182no—
460Dic3—62no—
461Dic3—62no—
462Dic3—182no—
463Dic3—182no—
464C32—21no—
465C32C2×#89411yescharacteristic, complemented
466C32C2×C3×#12311yescharacteristic, complemented
467C23—92no—
468C23—94no—
469C23—92no—
470C23—182no—
471C23—182no—
472C23—182no—
473C23—182no—
474C23—182no—
475C23C22×(C3⋊D3)⋊C311yescharacteristic
476D4—188no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—92no—
481C2×C4—92no—
482C2×C4—188no—
483C2×C4—182no—
484C2×C4—182no—
485C2×C4C22×(C3⋊D3)⋊C312yes—
486C2×C4C22×(C3⋊D3)⋊C312yes—
487C2×C4C22×(C3⋊D3)⋊C312yes—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—34no—
495C2×C3—34no—
496C2×C3—61no—
497C2×C3—61no—
498C2×C3—61no—
499C2×C3—64no—
500C2×C3—184no—
501C2×C3—362no—
502C2×C3C3×#89414yes—
503C2×C3C2×C3×#23511yescharacteristic
504C2×C3C2×C3×D3×D411yescharacteristic
505C2×C3C2×C3×#23911yescharacteristic
506D3—64no—
507D3—122no—
508D3—184no—
509D3—362no—
510C22—94no—
511C22—182no—
512C22—182no—
513C22—188no—
514C22—182no—
515C22—182no—
516C22—182no—
517C22—364no—
518C22#3828⋊C312yes—
519C22D4×(C3⋊D3)⋊C312yes—
520C22#8872212yes—
521C22C23×(C3⋊D3)⋊C311yescharacteristic
522C4—22no—
523C4—22no—
524C4—22no—
525C4—182no—
526C4—182no—
527C3—31no—
528C3—31no—
529C3—61no—
530C3C2×C3×#89411yescharacteristic
531C2—184no—
532C2—362no—
533C2#64103⋊C314yesdirect product factor
534C2C2×#3828⋊C311yescharacteristic
535C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
536C2C2×#8872211yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×(C3⋊D3)⋊C3C2semidirect
3. C2×C4⋊C4×H(3,𝔽3)C2semidirect
4. C2×#3752⋊C3C2semidirect
5. C2×#3752⋊C3C2semidirect
6. C2×#3752⋊C3C2semidirect
7. C2×#3750⋊C3C2semidirect
8. #64103⋊C3C2direct (N × H)
9. C2×#64103C3semidirect
21. C2×C4×(C3⋊D3)⋊C3C22semidirect
22. C4⋊C4×H(3,𝔽3)C22semidirect
23. #3752⋊C3C22semidirect
24. #3752⋊C3C22semidirect
25. #3752⋊C3C22semidirect
26. #3750⋊C3C22semidirect
28. C22×C4×C3⋊D3C2×C3semidirect
32. C2×C32×C4⋊C4D3semidirect
33. C2×C32×C4⋊C4C2×C3semidirect
34. C2×#3752C2×C3semidirect
35. C2×#3752C2×C3semidirect
36. C2×#3752C2×C3semidirect
37. C2×#3750C2×C3semidirect
38. #64103C2×C3semidirect
86. C2×C4×C3⋊D3C22×C3semidirect
94. C32×C4⋊C4C2×D3semidirect
95. C32×C4⋊C4C22×C3semidirect
96. #3752C22×C3semidirect
97. #3752C22×C3semidirect
98. #3752C22×C3semidirect
99. #3750C22×C3semidirect
196. C2×H(3,𝔽3)#123semidirect
299. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
303. H(3,𝔽3)C2×#123semidirect
378. C2×C32#894semidirect
379. C2×C32C3×#123semidirect
402. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
465. C32C2×#894semidirect
466. C32C2×C3×#123semidirect
533. C2#64103⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C22×C4×(C3⋊D3)⋊C32
C2×C4⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
C2×#3750⋊C32
#64103⋊C32
C2×#641033
C2×C3×#8943