Finite Group Databasebeta

← C3×#83998 #11868152

Subgroups — C3×#83998 #11868152

Order 1152.

1021 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#137C211yesmaximal, characteristic, complemented
2C3×#5822C211yesmaximal, characteristic
3C2×C3×#942C211yesmaximal, characteristic, complemented
4C3×#5745C211yesmaximal, characteristic, complemented
5C3×#5237C211yesmaximal, characteristic, complemented
6C2×C3×#836C211yesmaximal, characteristic, complemented
7C3×#5192C214yesmaximal, complemented
8C3×#5177C214yesmaximal, complemented
9C3×#5125C211yesmaximal, characteristic, complemented
10C3×#3069—31nomaximal
11#83998C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C22×C32×Q8C2211yescharacteristic
14C2×C3×#235C2211yescharacteristic
15C32×#137—22no—
16C32×#137—22no—
17C2×C32×Q16C2212yescomplemented
18C2×C32×SD16C2212yescomplemented
19C2×C32×M16C2211yescharacteristic
20C3×#951—21no—
21C3×#950—22no—
22C3×#949—21no—
23C2×C3×#216—21no—
24C3×#942—22no—
25C3×#942C2214yes—
26C3×#942C2212yes—
27C2×C3×#215—21no—
28C3×#932C2212yes—
29C2×C3×#208C2211yescharacteristic
30C2×C3×#208C2211yescharacteristic
31C2×C3×#208C2211yescharacteristic
32C3×#907—22no—
33C3×D3×M16—22no—
34C3×#901C2214yes—
35C3×#836C2214yescomplemented
36C3×#836C2212yescomplemented
37C3×#833C2212yescomplemented
38C3×#819C2214yescomplemented
39C3×#575—31no—
40C2×C3×#137—21no—
41C2×C3×#137—31no—
42C2×C3×#137D311yescharacteristic, complemented
43C2×C3×#137C2×C311yescharacteristic, complemented
44C3×#568—31no—
45#5822C2×C311yescharacteristic
46C2×#942C2×C311yescharacteristic, complemented
47#5745C2×C311yescharacteristic, complemented
48C3×#473—34no—
49C3×#471—34no—
50C3×#430—31no—
51C3×#413—31no—
52C2×C3×#101—31no—
53#5237C2×C311yescharacteristic, complemented
54C2×#836C2×C311yescharacteristic, complemented
55#5192C2×C314yescomplemented
56#5177C2×C314yescomplemented
57#5125C2×C311yescharacteristic, complemented
58C32×#41—21no—
59C32×#41—21no—
60C32×#41—21no—
61C32×#41—21no—
62C2×C32×Q8—21no—
63C2×C32×Q8—21no—
64C2×C32×Q8—21no—
65C2×C32×Q8—21no—
66C2×C32×Q8—21no—
67C2×C32×Q8C2312yes—
68C2×C32×Q8C2311yescharacteristic
69C2×C32×D4—21no—
70C2×C32×D4C2311yescharacteristic
71C22×C32×C4—21no—
72C22×C32×C4—21no—
73C22×C32×C4C2311yescharacteristic
74C2×C3×#67—21no—
75C3×#235—21no—
76C3×#235—21no—
77C3×#235D412yes—
78C3×#235C2312yes—
79C2×C3×D12D411yescharacteristic
80C2×C3×C4×D3—21no—
81C2×C3×Dic6D411yescharacteristic
82C32×Q16—24no—
83C32×SD16—24no—
84C32×M16D412yes—
85C32×M16D412yes—
86C2×C32×C8C2312yes—
87C3×#216—24no—
88C3×#216—22no—
89C3×#215—24no—
90C3×#215—22no—
91C3×#214—22no—
92C3×#213—22no—
93C3×#208—21no—
94C3×#208—21no—
95C3×#208—21no—
96C3×#208C2312yes—
97C3×#208C2312yes—
98C3×#208C2312yes—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—21no—
102C3×ΓL(1,𝔽25)—24no—
103C3×D3×C8—24no—
104#3069—31noSylow
105C2×C3×#41—21no—
106C2×C3×#41—31no—
107C2×C3×#41C2×D311yescharacteristic
108C2×C3×#41C22×C311yescharacteristic
109C22×C3×Q8—21no—
110C22×C3×Q8C2×D311yescharacteristic
111C22×C3×Q8C22×C311yescharacteristic
112C2×#235C22×C311yescharacteristic
113C3×#137—24no—
114C3×#137—24no—
115C3×#137—22no—
116C3×#137—22no—
117C3×#137—22no—
118C3×#137—22no—
119C3×#137—34no—
120C3×#137—32no—
121C3×#137—62no—
122C3×#137—61no—
123C3×AGL(1,ℤ/8)—32no—
124C3×AGL(1,ℤ/8)—61no—
125C3×#135—62no—
126C2×C3×Q16—22no—
127C2×C3×Q16—61no—
128C2×C3×Q16C2×D312yescomplemented
129C2×C3×Q16C22×C312yescomplemented
130C2×C3×SD16—22no—
131C2×C3×SD16—61no—
132C2×C3×SD16C2×D312yescomplemented
133C2×C3×SD16C22×C312yescomplemented
134C3×#131—34no—
135C3×#131—62no—
136C2×C3×M16—21no—
137C2×C3×M16—31no—
138C2×C3×M16—31no—
139C2×C3×M16—31no—
140C2×C3×M16—62no—
141C2×C3×M16C2×D311yescharacteristic
142C2×C3×M16C22×C311yescharacteristic
143#951—21no—
144#950—22no—
145#949—21no—
146C2×#216—21no—
147#942—22no—
148#942C22×C314yes—
149#942C22×C312yes—
150C2×#215—21no—
151#932C22×C312yes—
152C2×#208C22×C311yescharacteristic
153C2×#208C22×C311yescharacteristic
154C2×#208C22×C311yescharacteristic
155#907—22no—
156D3×M16—22no—
157#901C22×C314yes—
158C3×#108—34no—
159C3×#101—34no—
160C3×#101—32no—
161C3×#100—32no—
162#836C22×C314yescomplemented
163#836C22×C312yescomplemented
164#833C22×C312yescomplemented
165#819C22×C314yescomplemented
166C23×C32—21no—
167C23×C32#4111yescharacteristic
168C22×C3×D3—21no—
169C32×Q8—22no—
170C32×Q8—21no—
171C32×Q8—22no—
172C32×Q8—21no—
173C32×Q8—41no—
174C32×Q8—41no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—41no—
178C32×D4—41no—
179C2×C32×C4—21no—
180C2×C32×C4—22no—
181C2×C32×C4—41no—
182C2×C32×C4—41no—
183C2×C32×C4—41no—
184C2×C32×C4—41no—
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2×D411yescharacteristic
187C2×C32×C4C2×D411yescharacteristic
188C2×C32×C4C2×D411yescharacteristic
189C2×C32×C4#4111yescharacteristic
190C2×C32×C4C2411yescharacteristic
191C3×#67—22no—
192C3×#67—22no—
193C2×C3×Dic3—21no—
194C3×D12—21no—
195C3×D12C2×D412yes—
196C3×C4×D3—22no—
197C3×C4×D3—22no—
198C3×Dic6—21no—
199C3×Dic6C2×D412yes—
200C32×C8C2×D414yes—
201C3×C3⋊C8—22no—
202C3×C3⋊C8—22no—
203C3×C3⋊C8—22no—
204#575—31no—
205C2×#137—31no—
206C2×#137C3×D311yescharacteristic, complemented
207#568—31no—
208#473—34no—
209#471—34no—
210#430—31no—
211#413—31no—
212C2×#101—31no—
213C3×#41—22no—
214C3×#41—22no—
215C3×#41—21no—
216C3×#41—21no—
217C3×#41—21no—
218C3×#41—21no—
219C3×#41—21no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—32no—
224C3×#41—32no—
225C3×#41—41no—
226C3×#41—41no—
227C3×#41—61no—
228C3×#41—61no—
229C2×C3×Q8—21no—
230C2×C3×Q8—21no—
231C2×C3×Q8—22no—
232C2×C3×Q8—22no—
233C2×C3×Q8—21no—
234C2×C3×Q8—21no—
235C2×C3×Q8—21no—
236C2×C3×Q8—21no—
237C2×C3×Q8—21no—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—21no—
241C2×C3×Q8—22no—
242C2×C3×Q8—21no—
243C2×C3×Q8—31no—
244C2×C3×Q8—41no—
245C2×C3×Q8—41no—
246C2×C3×Q8—41no—
247C2×C3×Q8C22×D312yes—
248C2×C3×Q8C22×D311yescharacteristic
249C2×C3×Q8C23×C312yes—
250C2×C3×Q8C23×C311yescharacteristic
251C2×C3×D4—21no—
252C2×C3×D4—21no—
253C2×C3×D4—21no—
254C2×C3×D4—31no—
255C2×C3×D4—41no—
256C2×C3×D4—61no—
257C2×C3×D4C22×D311yescharacteristic
258C2×C3×D4C23×C311yescharacteristic
259C22×C3×C4—21no—
260C22×C3×C4—21no—
261C22×C3×C4—21no—
262C22×C3×C4—21no—
263C22×C3×C4—21no—
264C22×C3×C4—41no—
265C22×C3×C4—41no—
266C22×C3×C4—61no—
267C22×C3×C4C22×D311yescharacteristic
268C22×C3×C4C23×C311yescharacteristic
269C2×#67—21no—
270#235—21no—
271#235—21no—
272#235C3×D412yes—
273#235C23×C312yes—
274C2×D12C3×D411yescharacteristic
275C2×C4×D3—21no—
276C2×Dic6C3×D411yescharacteristic
277C3×Q16—28no—
278C3×Q16—24no—
279C3×Q16—24no—
280C3×Q16—64no—
281C3×Q16—62no—
282C3×SD16—28no—
283C3×SD16—24no—
284C3×SD16—24no—
285C3×SD16—62no—
286C3×SD16—64no—
287C3×SD16—62no—
288C3×D8—62no—
289C3×M16—22no—
290C3×M16—22no—
291C3×M16—32no—
292C3×M16—32no—
293C3×M16—32no—
294C3×M16—61no—
295C3×M16—61no—
296C3×M16—61no—
297C3×M16—64no—
298C3×M16#6712yes—
299C3×M16#6712yes—
300C3×M16C3×D412yes—
301C3×M16C3×D412yes—
302C2×C3×C8—22no—
303C2×C3×C8—61no—
304C2×C3×C8—61no—
305C2×C3×C8—61no—
306C2×C3×C8—64no—
307C2×C3×C8C22×D312yes—
308C2×C3×C8C23×C312yes—
309#216—24no—
310#216—22no—
311#215—24no—
312#215—22no—
313#214—22no—
314#213—22no—
315#208—21no—
316#208—21no—
317#208—21no—
318#208C23×C312yes—
319#208C23×C312yes—
320#208C23×C312yes—
321C2×C3⋊C8—21no—
322C2×C3⋊C8—21no—
323C2×C3⋊C8—21no—
324ΓL(1,𝔽25)—24no—
325D3×C8—24no—
326C22×C32—41no—
327C22×C32—41no—
328C22×C32—81no—
329C22×C32#12111yescharacteristic
330C22×C32#12111yescharacteristic
331C22×C32C2×#4111yescharacteristic
332C2×C3×D3—22no—
333C2×C3×D3—81no—
334C32×C4—41no—
335C32×C4—42no—
336C32×C4#12111yescharacteristic
337C32×C4#12111yescharacteristic
338C32×C4C22×D411yescharacteristic
339C32×C4C22×D411yescharacteristic
340C3×Dic3—22no—
341C2×#41—31no—
342C2×#41C2×C3×D311yescharacteristic
343C22×Q8C2×C3×D311yescharacteristic
344#137—22no—
345#137—22no—
346#137—34no—
347#137—32no—
348#137—62no—
349#137—61no—
350AGL(1,ℤ/8)—32no—
351AGL(1,ℤ/8)—61no—
352#135—62no—
353C2×Q16—61no—
354C2×Q16C2×C3×D312yescomplemented
355C2×SD16—61no—
356C2×SD16C2×C3×D312yescomplemented
357#131—34no—
358#131—62no—
359C2×M16—31no—
360C2×M16—31no—
361C2×M16—31no—
362C2×M16—62no—
363C2×M16C2×C3×D311yescharacteristic
364#108—34no—
365#101—34no—
366#101—32no—
367#100—32no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—41no—
372C23×C3—61no—
373C23×C3#23711yescharacteristic
374C23×C3C3×#4111yescharacteristic
375C22×D3—21no—
376C3×Q8—24no—
377C3×Q8—22no—
378C3×Q8—22no—
379C3×Q8—21no—
380C3×Q8—22no—
381C3×Q8—21no—
382C3×Q8—22no—
383C3×Q8—21no—
384C3×Q8—22no—
385C3×Q8—21no—
386C3×Q8—32no—
387C3×Q8—42no—
388C3×Q8—41no—
389C3×Q8—42no—
390C3×Q8—42no—
391C3×Q8—41no—
392C3×Q8—41no—
393C3×Q8—41no—
394C3×Q8—41no—
395C3×Q8—61no—
396C3×D4—22no—
397C3×D4—21no—
398C3×D4—21no—
399C3×D4—21no—
400C3×D4—21no—
401C3×D4—32no—
402C3×D4—41no—
403C3×D4—42no—
404C3×D4—42no—
405C3×D4—41no—
406C3×D4—41no—
407C3×D4—41no—
408C3×D4—41no—
409C3×D4—62no—
410C3×D4—62no—
411C3×D4—61no—
412C2×C3×C4—21no—
413C2×C3×C4—22no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—21no—
417C2×C3×C4—21no—
418C2×C3×C4—21no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—41no—
423C2×C3×C4—42no—
424C2×C3×C4—42no—
425C2×C3×C4—42no—
426C2×C3×C4—42no—
427C2×C3×C4—42no—
428C2×C3×C4—41no—
429C2×C3×C4—41no—
430C2×C3×C4—41no—
431C2×C3×C4—41no—
432C2×C3×C4—41no—
433C2×C3×C4—41no—
434C2×C3×C4—41no—
435C2×C3×C4—41no—
436C2×C3×C4—61no—
437C2×C3×C4—62no—
438C2×C3×C4—62no—
439C2×C3×C4D3×D411yescharacteristic
440C2×C3×C4D3×D411yescharacteristic
441C2×C3×C4#23711yescharacteristic
442C2×C3×C4C2×#6711yescharacteristic
443C2×C3×C4C2×#6711yescharacteristic
444C2×C3×C4C2×C3×D411yescharacteristic
445C2×C3×C4C2×C3×D411yescharacteristic
446C2×C3×C4C2×C3×D411yescharacteristic
447C2×C3×C4C2×C3×D411yescharacteristic
448C2×C3×C4C3×#4111yescharacteristic
449C2×C3×C4C23×D311yescharacteristic
450C2×C3×C4C24×C311yescharacteristic
451#67—22no—
452#67—22no—
453C2×Dic3—21no—
454D12—21no—
455D12C2×C3×D412yes—
456C4×D3—22no—
457C4×D3—22no—
458Dic6—21no—
459Dic6C2×C3×D412yes—
460C3×C8—24no—
461C3×C8—62no—
462C3×C8—62no—
463C3×C8—62no—
464C3×C8C2×#6714yes—
465C3×C8C2×C3×D414yes—
466C3⋊C8—22no—
467C3⋊C8—22no—
468C3⋊C8—22no—
469C2×C32—21no—
470C2×C32—21no—
471C2×C32—21no—
472C2×C32—81no—
473C2×C32C2×#12111yescharacteristic
474C3×D3—42no—
475#41—21no—
476#41—21no—
477#41—21no—
478#41—21no—
479#41—32no—
480#41—32no—
481#41—61no—
482#41—61no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—31no—
489C2×Q8C22×C3×D312yes—
490C2×Q8C22×C3×D311yescharacteristic
491C2×D4—21no—
492C2×D4—31no—
493C2×D4—61no—
494C2×D4C22×C3×D311yescharacteristic
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—61no—
498C22×C4C22×C3×D311yescharacteristic
499Q16—24no—
500Q16—64no—
501Q16—62no—
502SD16—24no—
503SD16—62no—
504SD16—64no—
505SD16—62no—
506D8—62no—
507M16—32no—
508M16—32no—
509M16—32no—
510M16—61no—
511M16—61no—
512M16—61no—
513M16—64no—
514M16C3×#6712yes—
515M16C3×#6712yes—
516C2×C8—61no—
517C2×C8—61no—
518C2×C8—61no—
519C2×C8—64no—
520C2×C8C22×C3×D312yes—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—41no—
529C22×C3—41no—
530C22×C3—62no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—82no—
534C22×C3—241no—
535C22×C3#93811yescharacteristic
536C22×C3#93811yescharacteristic
537C22×C3C3×#12111yescharacteristic
538C22×C3C3×#12111yescharacteristic
539C22×C3C2×#23711yescharacteristic
540C22×C3C2×C3×#4111yescharacteristic
541C2×D3—22no—
542C2×D3—81no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—42no—
548C3×C4—44no—
549C3×C4—41no—
550C3×C4—42no—
551C3×C4—41no—
552C3×C4—42no—
553C3×C4—62no—
554C3×C4#93811yescharacteristic
555C3×C4#93811yescharacteristic
556C3×C4C3×#12111yescharacteristic
557C3×C4C3×#12111yescharacteristic
558C3×C4C2×D3×D411yescharacteristic
559C3×C4C22×#6711yescharacteristic
560C3×C4C22×C3×D411yescharacteristic
561C3×C4C22×C3×D411yescharacteristic
562Dic3—22no—
563C32#306911yescharacteristic, complemented, Sylow
564C23—21no—
565C23—61no—
566C23C3×#23711yescharacteristic
567Q8—22no—
568Q8—21no—
569Q8—22no—
570Q8—21no—
571Q8—32no—
572Q8—41no—
573Q8—41no—
574Q8—61no—
575D4—21no—
576D4—21no—
577D4—32no—
578D4—41no—
579D4—41no—
580D4—62no—
581D4—62no—
582D4—61no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—61no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4C3×D3×D411yescharacteristic
593C2×C4C3×D3×D411yescharacteristic
594C2×C4C3×#23711yescharacteristic
595C2×C4C2×C3×#6711yescharacteristic
596C2×C4C2×C3×#6711yescharacteristic
597C2×C4C23×C3×D311yescharacteristic
598C8—62no—
599C8—62no—
600C8—62no—
601C8C2×C3×#6714yes—
602C2×C3—22no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—21no—
610C2×C3—41no—
611C2×C3—41no—
612C2×C3—82no—
613C2×C3—81no—
614C2×C3—81no—
615C2×C3—122no—
616C2×C3C2×#93811yescharacteristic
617C2×C3C2×C3×#12111yescharacteristic
618D3—42no—
619C22—41no—
620C22—41no—
621C22—62no—
622C22—81no—
623C22—241no—
624C22C3×#93811yescharacteristic
625C22C3×#93811yescharacteristic
626C22C2×C3×#23711yescharacteristic
627C4—41no—
628C4—42no—
629C4—62no—
630C4C3×#93811yescharacteristic
631C4C3×#93811yescharacteristic
632C4C2×C3×D3×D411yescharacteristic
633C4C22×C3×#6711yescharacteristic
634C3—21no—
635C3#8399811yescharacteristic, direct product factor
636C3C3×#306911yescharacteristic, complemented
637C2—21no—
638C2—21no—
639C2—21no—
640C2—81no—
641C2—122no—
642C2C2×C3×#93811yescharacteristic
subgroupG/H
centerC2×C3C2×#938
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#137C2
socleC2×C32C2×#121
prime radicalC3×#5822C2
hypercenterC2×C3×#137D3
hypocenterC3C3×#3069
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#137

Complements

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

NH ≅ G/Ntype
1. C2×C32×#137C2semidirect
3. C2×C3×#942C2semidirect
4. C3×#5745C2semidirect
5. C3×#5237C2semidirect
6. C2×C3×#836C2semidirect
7. C3×#5192C2semidirect
8. C3×#5177C2semidirect
9. C3×#5125C2semidirect
11. #83998C3direct (N × H)
17. C2×C32×Q16C22semidirect
18. C2×C32×SD16C22semidirect
35. C3×#836C22semidirect
36. C3×#836C22semidirect
37. C3×#833C22semidirect
38. C3×#819C22semidirect
42. C2×C3×#137D3semidirect
43. C2×C3×#137C2×C3semidirect
46. C2×#942C2×C3semidirect
47. #5745C2×C3semidirect
53. #5237C2×C3semidirect
54. C2×#836C2×C3semidirect
55. #5192C2×C3semidirect
56. #5177C2×C3semidirect
57. #5125C2×C3semidirect
128. C2×C3×Q16C2×D3semidirect
129. C2×C3×Q16C22×C3semidirect
132. C2×C3×SD16C2×D3semidirect
133. C2×C3×SD16C22×C3semidirect
162. #836C22×C3semidirect
163. #836C22×C3semidirect
164. #833C22×C3semidirect
165. #819C22×C3semidirect
206. C2×#137C3×D3semidirect
354. C2×Q16C2×C3×D3semidirect
356. C2×SD16C2×C3×D3semidirect
563. C32#3069semidirect
635. C3#83998direct (N × H)
636. C3C3×#3069semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1372
C3×#58222
C2×C3×#9422
C3×#57452
C3×#52372
C2×C3×#8362
C3×#51922
C3×#51772
C3×#51252
C3×#30693
#839983