Finite Group Databasebeta

← C2×#64159⋊C3 #13153713

Subgroups — C2×#64159⋊C3 #13153713

Order 1728.

1095 conjugacy classes of subgroups.

The subgroup lattice has 560 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#64159⋊C3C218yesmaximal, direct product factor
4C2×#3755⋊C3C211yesmaximal, characteristic, complemented
5C2×#3756⋊C3C211yesmaximal, characteristic, complemented
6C2×#3753⋊C3C211yesmaximal, characteristic, complemented
7C2×#3754⋊C3C211yesmaximal, characteristic, complemented
8C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
9C2×#64159C311yesmaximal, characteristic, complemented
10C2×C3×#950—31nomaximal
11#41×H(3,𝔽3)—22no—
12#41×H(3,𝔽3)C2214yescomplemented
13C2×Q8×H(3,𝔽3)C2211yescharacteristic
14C2×D4×H(3,𝔽3)—21no—
15C2×D4×H(3,𝔽3)C2211yescharacteristic
16C22×C4×H(3,𝔽3)—21no—
17C22×C4×H(3,𝔽3)C2211yescharacteristic
18C2×#638⋊C3—21no—
19#3828⋊C3—22no—
20#3828⋊C3C2214yescomplemented
21C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
22C2×C4×(C3⋊D3)⋊C3—21no—
23C2×#634⋊C3C2211yescharacteristic
24#3755⋊C3C2214yescomplemented
25#3756⋊C3C2214yescomplemented
26#3753⋊C3C2214yescomplemented
27#3754⋊C3C2214yescomplemented
28C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
29C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
30C2×(C3⋊(C3⋊C8))⋊C3C2214yescomplemented
31C2×C32×#41—21no—
32C2×C32×#41D311yescharacteristic, complemented
33C2×C32×#41C2×C311yescharacteristic, complemented
34C2×#3828C2×C311yescharacteristic, complemented
35C2×C3×#235—31no—
36#64159C2×C318yescomplemented
37C2×#3756C2×C311yescharacteristic, complemented
38C2×#3755C2×C311yescharacteristic, complemented
39C2×#3754C2×C311yescharacteristic, complemented
40C2×#3753C2×C311yescharacteristic, complemented
41C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
42C3×#950—38no—
43C2×C3×#216—31no—
44C2×C3×#215—31no—
45C2×C3×#214—31no—
46C2×C3×#213—31no—
47C22×C3×C3⋊C8—31no—
48C23×H(3,𝔽3)—21no—
49C23×H(3,𝔽3)D411yescharacteristic
50C22×(C3⋊D3)⋊C3—21no—
51Q8×H(3,𝔽3)—21no—
52Q8×H(3,𝔽3)C2312yes—
53D4×H(3,𝔽3)—24no—
54D4×H(3,𝔽3)—21no—
55D4×H(3,𝔽3)C2312yes—
56C2×C4×H(3,𝔽3)—21no—
57C2×C4×H(3,𝔽3)—24no—
58C2×C4×H(3,𝔽3)D411yescharacteristic
59C2×C4×H(3,𝔽3)D412yes—
60C2×C4×H(3,𝔽3)C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62#638⋊C3—24no—
63C2×(C3⋊Dic3)⋊C3—21no—
64C4⋊((C3⋊D3)⋊C3)—21no—
65C4⋊((C3⋊D3)⋊C3)C2312yes—
66C4×(C3⋊D3)⋊C3—24no—
67#634⋊C3—21no—
68#634⋊C3C2312yes—
69(C3⋊(C3⋊C8))⋊C3C2312yes—
70(C3⋊(C3⋊C8))⋊C3C2312yes—
71C2×C3×#135—91no—
72C2×#950—31no—
73C2×#950—91no—
74C32×#41—24no—
75C32×#41—24no—
76C32×#41—22no—
77C32×#41—22no—
78C32×#41C2×D314yescomplemented
79C32×#41C22×C314yescomplemented
80C2×C32×Q8—21no—
81C2×C32×Q8C2×D311yescharacteristic
82C2×C32×Q8C22×C311yescharacteristic
83C2×C32×D4—22no—
84C2×C32×D4—21no—
85C2×C32×D4—21no—
86C2×C32×D4—21no—
87C2×C32×D4C2×D311yescharacteristic
88C2×C32×D4C22×C311yescharacteristic
89C22×C32×C4—22no—
90C22×C32×C4—21no—
91C22×C32×C4—21no—
92C22×C32×C4—21no—
93C22×C32×C4C2×D311yescharacteristic
94C22×C32×C4C22×C311yescharacteristic
95C2×#638—21no—
96#3828—22no—
97#3828C22×C314yescomplemented
98C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
99C2×C4×C3⋊D3—21no—
100C2×#634C22×C311yescharacteristic
101C2×C3×#67—61no—
102C3×#235—34no—
103C3×#235—62no—
104C2×C3×D12—31no—
105C2×C3×C4×D3—61no—
106C2×C3×Dic6—31no—
107#3756C22×C314yescomplemented
108#3755C22×C314yescomplemented
109#3754C22×C314yescomplemented
110#3753C22×C314yescomplemented
111C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
112C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
113C2×C3⋊(C3⋊C8)C22×C314yescomplemented
114C3×#216—34no—
115C3×#215—34no—
116C3×#214—34no—
117C3×#213—34no—
118C2×C3×C3⋊C8—31no—
119C2×C3×C3⋊C8—31no—
120C2×C3×C3⋊C8—34no—
121C22×H(3,𝔽3)—22no—
122C22×H(3,𝔽3)—22no—
123C22×H(3,𝔽3)—42no—
124C22×H(3,𝔽3)C2×D412yes—
125C22×H(3,𝔽3)C2×D411yescharacteristic
126C2×(C3⋊D3)⋊C3—22no—
127C2×(C3⋊D3)⋊C3—42no—
128C4×H(3,𝔽3)—22no—
129C4×H(3,𝔽3)C2×D411yescharacteristic
130C4×H(3,𝔽3)C2×D412yes—
131C4×H(3,𝔽3)C2411yescharacteristic
132(C3⋊Dic3)⋊C3—22no—
133C2×C3×#41—31no—
134C2×C3×#41—31no—
135C2×C3×#41—61no—
136C2×C3×#41—91no—
137C2×C3×#41C3×D311yescharacteristic
138C2×#235—31no—
139C2×#235—91no—
140C3×#135—98no—
141C2×C3×Q16—91no—
142C2×C3×SD16—91no—
143C2×C3×SD16—91no—
144C2×C3×D8—91no—
145C22×C3×C8—91no—
146#950—38no—
147#950—98no—
148C2×#216—31no—
149C2×#216—91no—
150C2×#215—31no—
151C2×#215—91no—
152C2×#214—31no—
153C2×#214—91no—
154C2×#213—31no—
155C2×#213—91no—
156C22×C3⋊C8—31no—
157C22×C3⋊C8—91no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—21no—
162C23×C32#6711yescharacteristic
163C23×C32C3×D411yescharacteristic
164C22×C3⋊D3—21no—
165C22×C3×D3—61no—
166C32×Q8—22no—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×Q8—22no—
170C32×Q8C22×D312yes—
171C32×Q8C23×C312yes—
172C32×D4—28no—
173C32×D4—22no—
174C32×D4—24no—
175C32×D4—24no—
176C32×D4—21no—
177C32×D4—21no—
178C32×D4—22no—
179C32×D4C22×D312yes—
180C32×D4C23×C312yes—
181C2×C32×C4—22no—
182C2×C32×C4—28no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—24no—
186C2×C32×C4—24no—
187C2×C32×C4—21no—
188C2×C32×C4—22no—
189C2×C32×C4—21no—
190C2×C32×C4—22no—
191C2×C32×C4#6711yescharacteristic
192C2×C32×C4#6712yes—
193C2×C32×C4C3×D411yescharacteristic
194C2×C32×C4C3×D412yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D312yes—
197C2×C32×C4C23×C311yescharacteristic
198C2×C32×C4C23×C312yes—
199#638—24no—
200C2×C3⋊Dic3—21no—
201C4⋊(C3⋊D3)—21no—
202C4⋊(C3⋊D3)C23×C312yes—
203C4×C3⋊D3—24no—
204#634—21no—
205#634C23×C312yes—
206C3×#67—64no—
207C2×C3×Dic3—61no—
208C3×D12—32no—
209C3×D12—61no—
210C3×C4×D3—64no—
211C3×Dic6—32no—
212C3×Dic6—61no—
213C3⋊(C3⋊C8)C23×C312yes—
214C3⋊(C3⋊C8)C23×C312yes—
215C3×C3⋊C8—32no—
216C3×C3⋊C8—32no—
217C2×#135—91noSylow
218C2×H(3,𝔽3)—22no—
219C2×H(3,𝔽3)—42no—
220C2×H(3,𝔽3)#13512yescomplemented
221C2×H(3,𝔽3)C22×D411yescharacteristic
222(C3⋊D3)⋊C3—42no—
223C3×#41—22no—
224C3×#41—34no—
225C3×#41—34no—
226C3×#41—62no—
227C3×#41—62no—
228C3×#41—64no—
229C3×#41—64no—
230C3×#41—94no—
231C3×#41—182no—
232C3×#41C2×C3×D314yes—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—61no—
236C2×C3×Q8—91no—
237C2×C3×Q8C2×C3×D311yescharacteristic
238C2×C3×D4—21no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—61no—
242C2×C3×D4—61no—
243C2×C3×D4—62no—
244C2×C3×D4—61no—
245C2×C3×D4—91no—
246C2×C3×D4—181no—
247C2×C3×D4C2×C3×D311yescharacteristic
248C22×C3×C4—21no—
249C22×C3×C4—31no—
250C22×C3×C4—31no—
251C22×C3×C4—61no—
252C22×C3×C4—61no—
253C22×C3×C4—62no—
254C22×C3×C4—61no—
255C22×C3×C4—181no—
256C22×C3×C4C2×C3×D311yescharacteristic
257C2×#67—61no—
258C2×#67—181no—
259#235—34no—
260#235—62no—
261#235—94no—
262#235—182no—
263C2×D12—31no—
264C2×D12—91no—
265C2×C4×D3—61no—
266C2×C4×D3—181no—
267C2×Dic6—31no—
268C2×Dic6—91no—
269C3×Q16—94no—
270C3×SD16—94no—
271C3×SD16—94no—
272C3×D8—94no—
273C2×C3×C8—91no—
274C2×C3×C8—91no—
275C2×C3×C8—94no—
276#216—34no—
277#216—94no—
278#215—34no—
279#215—94no—
280#214—34no—
281#214—94no—
282#213—34no—
283#213—94no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—34no—
287C2×C3⋊C8—91no—
288C2×C3⋊C8—91no—
289C2×C3⋊C8—94no—
290C22×C32—22no—
291C22×C32—22no—
292C22×C32—24no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—21no—
296C22×C32—22no—
297C22×C32—42no—
298C22×C32—44no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32C2×#6712yes—
302C22×C32C2×#6711yescharacteristic
303C22×C32C2×C3×D412yes—
304C22×C32C2×C3×D411yescharacteristic
305C2×C3⋊D3—22no—
306C2×C3⋊D3—42no—
307C2×C3×D3—62no—
308C2×C3×D3—122no—
309C32×C4—24no—
310C32×C4—22no—
311C32×C4—22no—
312C32×C4—21no—
313C32×C4—22no—
314C32×C4—21no—
315C32×C4C2×#6711yescharacteristic
316C32×C4C2×#6712yes—
317C32×C4C2×C3×D411yescharacteristic
318C32×C4C2×C3×D412yes—
319C32×C4C23×D311yescharacteristic
320C32×C4C24×C311yescharacteristic
321C3⋊Dic3—22no—
322C3×Dic3—62no—
323C2×#41—91no—
324C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
325#135—98no—
326C2×Q16—91no—
327C2×SD16—91no—
328C2×SD16—91no—
329C2×D8—91no—
330C22×C8—91no—
331H(3,𝔽3)C2×#13511yescharacteristic, complemented, Sylow
332C23×C3—21no—
333C23×C3—31no—
334C23×C3—31no—
335C23×C3—61no—
336C23×C3—61no—
337C23×C3—62no—
338C23×C3—61no—
339C23×C3—181no—
340C23×C3C3×#6711yescharacteristic
341C22×D3—61no—
342C22×D3—181no—
343C3×Q8—21no—
344C3×Q8—32no—
345C3×Q8—32no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—62no—
349C3×Q8—62no—
350C3×Q8—92no—
351C3×Q8—181no—
352C3×Q8C22×C3×D312yes—
353C3×D4—24no—
354C3×D4—21no—
355C3×D4—32no—
356C3×D4—32no—
357C3×D4—61no—
358C3×D4—61no—
359C3×D4—64no—
360C3×D4—64no—
361C3×D4—68no—
362C3×D4—62no—
363C3×D4—62no—
364C3×D4—92no—
365C3×D4—181no—
366C3×D4—184no—
367C3×D4C22×C3×D312yes—
368C2×C3×C4—21no—
369C2×C3×C4—24no—
370C2×C3×C4—32no—
371C2×C3×C4—32no—
372C2×C3×C4—31no—
373C2×C3×C4—31no—
374C2×C3×C4—31no—
375C2×C3×C4—32no—
376C2×C3×C4—31no—
377C2×C3×C4—32no—
378C2×C3×C4—64no—
379C2×C3×C4—64no—
380C2×C3×C4—61no—
381C2×C3×C4—61no—
382C2×C3×C4—62no—
383C2×C3×C4—68no—
384C2×C3×C4—62no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—62no—
388C2×C3×C4—184no—
389C2×C3×C4—181no—
390C2×C3×C4C3×#6712yes—
391C2×C3×C4C3×#6711yescharacteristic
392C2×C3×C4C22×C3×D311yescharacteristic
393C2×C3×C4C22×C3×D312yes—
394#67—64no—
395#67—184no—
396C2×Dic3—61no—
397C2×Dic3—181no—
398D12—32no—
399D12—61no—
400D12—92no—
401D12—181no—
402C4×D3—64no—
403C4×D3—184no—
404Dic6—32no—
405Dic6—61no—
406Dic6—92no—
407Dic6—181no—
408C3×C8—92no—
409C3×C8—92no—
410C3⋊C8—32no—
411C3⋊C8—32no—
412C3⋊C8—92no—
413C3⋊C8—92no—
414C2×C32—22no—
415C2×C32—22no—
416C2×C32—22no—
417C2×C32—21no—
418C2×C32—42no—
419C2×C32—44no—
420C2×C32—42no—
421C2×C32—42no—
422C2×C32#95012yescomplemented
423C2×C32C3×#13512yescomplemented
424C2×C32C22×#6711yescharacteristic
425C2×C32C22×C3×D411yescharacteristic
426C3⋊D3—42no—
427C3×D3—122no—
428#41—22no—
429#41—94no—
430#41—182no—
431#41C2×(C3⋊D3)⋊C314yescomplemented
432C2×Q8—91no—
433C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
434C2×D4—21no—
435C2×D4—91no—
436C2×D4—181no—
437C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
438C22×C4—21no—
439C22×C4—181no—
440C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
441Q16—94no—
442SD16—94no—
443SD16—94no—
444D8—94no—
445C2×C8—91no—
446C2×C8—91no—
447C2×C8—94no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—31no—
451C22×C3—31no—
452C22×C3—32no—
453C22×C3—32no—
454C22×C3—42no—
455C22×C3—62no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3—62no—
459C22×C3—64no—
460C22×C3—61no—
461C22×C3—62no—
462C22×C3—122no—
463C22×C3—122no—
464C22×C3—122no—
465C22×C3—124no—
466C22×C3—182no—
467C22×C3—362no—
468C22×C3C2×C3×#6712yes—
469C22×C3C2×C3×#6711yescharacteristic
470C2×D3—62no—
471C2×D3—122no—
472C2×D3—182no—
473C2×D3—362no—
474C3×C4—22no—
475C3×C4—31no—
476C3×C4—31no—
477C3×C4—31no—
478C3×C4—32no—
479C3×C4—32no—
480C3×C4—31no—
481C3×C4—62no—
482C3×C4—62no—
483C3×C4—64no—
484C3×C4—61no—
485C3×C4—62no—
486C3×C4—61no—
487C3×C4—182no—
488C3×C4C2×C3×#6711yescharacteristic
489C3×C4C2×C3×#6712yes—
490C3×C4C23×C3×D311yescharacteristic
491Dic3—62no—
492Dic3—182no—
493C32—21no—
494C32C2×#95011yescharacteristic, complemented
495C32C2×C3×#13511yescharacteristic, complemented
496C23—21no—
497C23—181no—
498C23#638⋊C311yescharacteristic
499Q8—21no—
500Q8—92no—
501Q8—181no—
502Q8C22×(C3⋊D3)⋊C312yes—
503D4—24no—
504D4—21no—
505D4—92no—
506D4—184no—
507D4—181no—
508D4C22×(C3⋊D3)⋊C312yes—
509C2×C4—21no—
510C2×C4—24no—
511C2×C4—181no—
512C2×C4—184no—
513C2×C4#638⋊C312yes—
514C2×C4#638⋊C311yescharacteristic
515C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
516C2×C4C22×(C3⋊D3)⋊C312yes—
517C8—92no—
518C8—92no—
519C2×C3—22no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—32no—
523C2×C3—32no—
524C2×C3—42no—
525C2×C3—62no—
526C2×C3—62no—
527C2×C3—61no—
528C2×C3—62no—
529C2×C3—122no—
530C2×C3—122no—
531C2×C3—122no—
532C2×C3—124no—
533C2×C3—362no—
534C2×C3C3×#95012yes—
535C2×C3C22×C3×#6711yescharacteristic
536D3—122no—
537D3—362no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—182no—
542C22—362no—
543C22C2×#638⋊C312yes—
544C22C2×#638⋊C311yescharacteristic
545C4—22no—
546C4—182no—
547C4C2×#638⋊C312yes—
548C4C2×#638⋊C311yescharacteristic
549C4C23×(C3⋊D3)⋊C311yescharacteristic
550C3—31no—
551C3—31no—
552C3—61no—
553C3C2×C3×#95011yescharacteristic
554C2—22no—
555C2—42no—
556C2—362no—
557C2#64159⋊C312yesdirect product factor
558C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC2×C4#638⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingC2×#41×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C2×C3×#135
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×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. #64159⋊C3C2direct (N × H)
4. C2×#3755⋊C3C2semidirect
5. C2×#3756⋊C3C2semidirect
6. C2×#3753⋊C3C2semidirect
7. C2×#3754⋊C3C2semidirect
8. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
9. C2×#64159C3semidirect
12. #41×H(3,𝔽3)C22semidirect
20. #3828⋊C3C22semidirect
24. #3755⋊C3C22semidirect
25. #3756⋊C3C22semidirect
26. #3753⋊C3C22semidirect
27. #3754⋊C3C22semidirect
30. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
32. C2×C32×#41D3semidirect
33. C2×C32×#41C2×C3semidirect
34. C2×#3828C2×C3semidirect
36. #64159C2×C3semidirect
37. C2×#3756C2×C3semidirect
38. C2×#3755C2×C3semidirect
39. C2×#3754C2×C3semidirect
40. C2×#3753C2×C3semidirect
41. C22×C3⋊(C3⋊C8)C2×C3semidirect
78. C32×#41C2×D3semidirect
79. C32×#41C22×C3semidirect
97. #3828C22×C3semidirect
107. #3756C22×C3semidirect
108. #3755C22×C3semidirect
109. #3754C22×C3semidirect
110. #3753C22×C3semidirect
113. C2×C3⋊(C3⋊C8)C22×C3semidirect
220. C2×H(3,𝔽3)#135semidirect
324. C2×#41(C3⋊D3)⋊C3semidirect
331. H(3,𝔽3)C2×#135semidirect
422. C2×C32#950semidirect
423. C2×C32C3×#135semidirect
431. #41C2×(C3⋊D3)⋊C3semidirect
494. C32C2×#950semidirect
495. C32C2×C3×#135semidirect
557. C2#64159⋊C3direct (N × H)

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
#64159⋊C32
C2×#3755⋊C32
C2×#3756⋊C32
C2×#3753⋊C32
C2×#3754⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
C2×#641593
C2×C3×#9503