Finite Group Databasebeta

← C2×C3×#85559 #72276682

Subgroups — C2×C3×#85559 #72276682

Order 2304.

1088 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#85559C212yesmaximal, direct product factor
2C2×C3×#5464C211yesmaximal, characteristic, complemented
3C2×#85559C311yesmaximal, characteristic, direct product factor
4C2×C3×#2199—31nomaximal
5C2×C3×#1020—42nomaximal
6C3×#5464C2211yescharacteristic, complemented
7C2×C3×#857—41nomaximal
8C2×C3×H(3,𝔽4)D311yescharacteristic, complemented
9C2×#5464—21no—
10C2×#5464C2×C311yescharacteristic, complemented
11#85559C2×C312yesdirect product factor
12C2×C3×B(4,𝔽2)—32no—
13C2×C3×#455—31no—
14C3×#2199—34no—
15C3×#2199—32no—
16C3×#2199—32no—
17C2×C3×#354—31no—
18C2×C3×AΣL(1,𝔽16)—32no—
19C2×C3×C22⋊A4—42no—
20C3×#1020—44no—
21C2×C3×C42⋊C3—41no—
22C2×C3×C42⋊C3—81no—
23C3×#857—42no—
24C2×#2199—31noSylow
25C2×#1020—42no—
26C2×C3×H(5,𝔽2)—31no—
27C2×C3×#137—61no—
28C3×H(3,𝔽4)—32no—
29C3×H(3,𝔽4)—31no—
30C3×H(3,𝔽4)—61no—
31C3×H(3,𝔽4)C2×D311yescharacteristic, complemented
32C2×C3×#124—31no—
33C2×C3×#124—61no—
34C2×C3×#120—32no—
35C2×C3×#120—62no—
36#5464—21no—
37#5464C22×C311yescharacteristic, complemented
38C2×#857—41no—
39C3×B(4,𝔽2)—34no—
40C3×B(4,𝔽2)—34no—
41C3×B(4,𝔽2)—62no—
42C3×B(4,𝔽2)—62no—
43C3×#455—32no—
44C3×#455—32no—
45C3×#455—61no—
46C3×#455—61no—
47C2×C3×#104—61no—
48C2×C3×#100—31no—
49C2×C3×#99—32no—
50C2×C3×#99—62no—
51C3×#354—32no—
52C3×#354—32no—
53C3×AΣL(1,𝔽16)—34no—
54C3×AΣL(1,𝔽16)—34no—
55C3×C22⋊A4—42no—
56C2×C3×S4—161no—
57C2×C3×S4—164no—
58C3×C42⋊C3—41no—
59C3×C42⋊C3—81no—
60C2×H(3,𝔽4)C3×D311yescharacteristic, complemented
61C2×B(4,𝔽2)—32no—
62C2×#455—31no—
63#2199—34no—
64#2199—32no—
65#2199—32no—
66C2×#354—31no—
67C2×AΣL(1,𝔽16)—32no—
68C25×C3S412yescomplemented
69C2×C22⋊A4—42no—
70C2×C22⋊A4—82no—
71#1020—44no—
72C3×H(5,𝔽2)—32no—
73C3×H(5,𝔽2)—62no—
74C3×H(5,𝔽2)—61no—
75C3×H(5,𝔽2)—122no—
76C2×C3×#41—61no—
77C2×C3×#41—121no—
78C22×C3×Q8—31no—
79C22×C3×D4—31no—
80C22×C3×D4—62no—
81C22×C3×D4—61no—
82C22×C3×D4—121no—
83C22×C3×D4—122no—
84C3×#137—62no—
85C3×#137—62no—
86C3×#137—122no—
87C2×C3×Q16—121no—
88C2×C3×SD16—121no—
89C2×C3×M16—61no—
90C3×#124—31no—
91C3×#124—31no—
92C3×#124—31no—
93C3×#124—31no—
94C3×#124—61no—
95C3×#124—61no—
96C3×#124—62no—
97C3×#124—61no—
98C3×#124—62no—
99C3×#124—62no—
100C3×#124—61no—
101C3×#120—32no—
102C3×#120—32no—
103C3×#120—32no—
104C3×#120—32no—
105C3×#120—62no—
106C3×#120—62no—
107C3×#120—64no—
108C3×#120—64no—
109C3×#120—124no—
110C2×C3×C22⋊C4—32no—
111C2×C3×C22⋊C4—62no—
112C2×C3×C22⋊C4—62no—
113C2×C3×C22⋊C4—122no—
114C2×C3×C42—21no—
115C2×C3×C42S411yescharacteristic, complemented
116C2×C42⋊C3—41no—
117C2×C42⋊C3—81no—
118C2×C42⋊C3—82no—
119C2×C42⋊C3—81no—
120#857—42no—
121C3×#104—62no—
122C3×#104—62no—
123C3×#100—32no—
124C3×#100—61no—
125C3×#99—34no—
126C3×#99—64no—
127C3×#99—64no—
128C3×#99—62no—
129C2×C3×A4—164no—
130C2×C3×A4—161no—
131C2×C3×A4—322no—
132C3×S4—162no—
133C3×S4—168no—
134C2×H(5,𝔽2)—31no—
135C2×#137—61no—
136H(3,𝔽4)—32no—
137H(3,𝔽4)—31no—
138H(3,𝔽4)—61no—
139H(3,𝔽4)C2×C3×D311yescharacteristic, complemented
140C2×#124—31no—
141C2×#124—61no—
142C2×#120—32no—
143C2×#120—62no—
144B(4,𝔽2)—34no—
145B(4,𝔽2)—34no—
146B(4,𝔽2)—62no—
147B(4,𝔽2)—62no—
148#455—32no—
149#455—32no—
150#455—61no—
151#455—61no—
152C2×#104—61no—
153C2×#100—31no—
154C2×#99—32no—
155C2×#99—62no—
156#354—32no—
157#354—32no—
158AΣL(1,𝔽16)—34no—
159AΣL(1,𝔽16)—34no—
160C24×C3—32no—
161C24×C3—32no—
162C24×C3—62no—
163C24×C3—122no—
164C24×C3—122no—
165C24×C3C2×S412yescomplemented
166C22⋊A4—42no—
167C22⋊A4—82no—
168C2×S4—161no—
169C2×S4—164no—
170C3×#41—62no—
171C3×#41—62no—
172C3×#41—121no—
173C3×#41—122no—
174C3×#41—121no—
175C3×#41—124no—
176C3×#41—122no—
177C2×C3×Q8—31no—
178C2×C3×Q8—31no—
179C2×C3×Q8—61no—
180C2×C3×Q8—61no—
181C2×C3×Q8—121no—
182C2×C3×Q8—121no—
183C2×C3×D4—31no—
184C2×C3×D4—32no—
185C2×C3×D4—31no—
186C2×C3×D4—61no—
187C2×C3×D4—64no—
188C2×C3×D4—64no—
189C2×C3×D4—61no—
190C2×C3×D4—61no—
191C2×C3×D4—62no—
192C2×C3×D4—121no—
193C2×C3×D4—121no—
194C2×C3×D4—122no—
195C2×C3×D4—122no—
196C2×C3×D4—122no—
197C2×C3×D4—122no—
198C2×C3×D4—122no—
199C2×C3×D4—124no—
200C2×C3×D4—121no—
201C2×C3×D4—124no—
202C2×C3×D4—124no—
203C2×C3×D4—241no—
204C2×C3×D4—241no—
205C2×C3×D4—244no—
206C2×C3×D4—244no—
207C22×C3×C4—31no—
208C22×C3×C4—62no—
209C22×C3×C4—61no—
210C22×C3×C4—121no—
211C22×C3×C4—122no—
212C3×Q16—122no—
213C3×Q16—122no—
214C3×SD16—122no—
215C3×SD16—122no—
216C3×M16—62no—
217C3×M16—121no—
218C2×C3×C8—121no—
219C3×C22⋊C4—32no—
220C3×C22⋊C4—32no—
221C3×C22⋊C4—62no—
222C3×C22⋊C4—62no—
223C3×C22⋊C4—62no—
224C3×C22⋊C4—62no—
225C3×C22⋊C4—62no—
226C3×C22⋊C4—64no—
227C3×C22⋊C4—124no—
228C3×C22⋊C4—124no—
229C3×C22⋊C4—122no—
230C3×C42—21no—
231C3×C42—31no—
232C3×C42—61no—
233C3×C42C2×S411yescharacteristic, complemented
234C42⋊C3—41no—
235C42⋊C3—81no—
236C42⋊C3—82no—
237C42⋊C3—81no—
238C2×C3×D3—641no—
239C3×A4—164no—
240C3×A4—161no—
241C3×A4—322no—
242C25C3×S412yescomplemented
243H(5,𝔽2)—32no—
244H(5,𝔽2)—62no—
245H(5,𝔽2)—61no—
246H(5,𝔽2)—122no—
247C2×#41—61no—
248C2×#41—121no—
249C22×Q8—31no—
250C22×D4—31no—
251C22×D4—62no—
252C22×D4—61no—
253C22×D4—121no—
254C22×D4—122no—
255#137—62no—
256#137—62no—
257#137—122no—
258C2×Q16—121no—
259C2×SD16—121no—
260C2×M16—61no—
261#124—31no—
262#124—31no—
263#124—31no—
264#124—31no—
265#124—61no—
266#124—61no—
267#124—62no—
268#124—61no—
269#124—62no—
270#124—62no—
271#124—61no—
272#120—32no—
273#120—32no—
274#120—32no—
275#120—32no—
276#120—62no—
277#120—62no—
278#120—64no—
279#120—64no—
280#120—124no—
281C2×C22⋊C4—32no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—122no—
285C2×C42—21no—
286C2×C42C3×S411yescharacteristic, complemented
287#104—62no—
288#104—62no—
289#100—32no—
290#100—61no—
291#99—34no—
292#99—64no—
293#99—64no—
294#99—62no—
295C23×C3—32no—
296C23×C3—32no—
297C23×C3—44no—
298C23×C3—62no—
299C23×C3—62no—
300C23×C3—64no—
301C23×C3—82no—
302C23×C3—122no—
303C23×C3—122no—
304C23×C3—121no—
305C23×C3—122no—
306C23×C3—124no—
307C23×C3—122no—
308C23×C3—122no—
309C23×C3—242no—
310C23×C3—242no—
311C23×C3—242no—
312C23×C3—242no—
313C23×C3#102011yescharacteristic
314C2×A4—164no—
315C2×A4—161no—
316C2×A4—322no—
317C2×A4—324no—
318C2×A4—324no—
319C2×A4—321no—
320S4—162no—
321S4—168no—
322C3×Q8—62no—
323C3×Q8—121no—
324C3×Q8—121no—
325C3×Q8—121no—
326C3×Q8—241no—
327C3×D4—62no—
328C3×D4—121no—
329C3×D4—121no—
330C3×D4—122no—
331C3×D4—121no—
332C3×D4—122no—
333C3×D4—124no—
334C3×D4—124no—
335C3×D4—122no—
336C3×D4—241no—
337C3×D4—241no—
338C3×D4—242no—
339C3×D4—248no—
340C3×D4—248no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—64no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—122no—
349C2×C3×C4—122no—
350C2×C3×C4—121no—
351C2×C3×C4—121no—
352C2×C3×C4—121no—
353C2×C3×C4—122no—
354C2×C3×C4—124no—
355C2×C3×C4—242no—
356C2×C3×C4—242no—
357C3×C8—122no—
358C2×C32—641no—
359C3×D3—642no—
360C24—32no—
361C24—32no—
362C24—62no—
363C24—122no—
364C24—122no—
365C24C2×C3×S412yescomplemented
366#41—62no—
367#41—62no—
368#41—121no—
369#41—122no—
370#41—121no—
371#41—124no—
372#41—122no—
373C2×Q8—31no—
374C2×Q8—31no—
375C2×Q8—61no—
376C2×Q8—61no—
377C2×Q8—121no—
378C2×Q8—121no—
379C2×D4—31no—
380C2×D4—32no—
381C2×D4—31no—
382C2×D4—61no—
383C2×D4—64no—
384C2×D4—64no—
385C2×D4—61no—
386C2×D4—61no—
387C2×D4—62no—
388C2×D4—121no—
389C2×D4—121no—
390C2×D4—122no—
391C2×D4—122no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—122no—
395C2×D4—124no—
396C2×D4—121no—
397C2×D4—124no—
398C2×D4—124no—
399C2×D4—241no—
400C2×D4—241no—
401C2×D4—244no—
402C2×D4—244no—
403C22×C4—31no—
404C22×C4—62no—
405C22×C4—61no—
406C22×C4—121no—
407C22×C4—122no—
408Q16—122no—
409Q16—122no—
410SD16—122no—
411SD16—122no—
412M16—62no—
413M16—121no—
414C2×C8—121no—
415C22⋊C4—32no—
416C22⋊C4—32no—
417C22⋊C4—62no—
418C22⋊C4—62no—
419C22⋊C4—62no—
420C22⋊C4—62no—
421C22⋊C4—62no—
422C22⋊C4—64no—
423C22⋊C4—124no—
424C22⋊C4—124no—
425C22⋊C4—122no—
426C42—21no—
427C42—31no—
428C42—61no—
429C42C2×C3×S411yescharacteristic, complemented
430C22×C3—31no—
431C22×C3—31no—
432C22×C3—44no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—82no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—122no—
439C22×C3—122no—
440C22×C3—124no—
441C22×C3—122no—
442C22×C3—242no—
443C22×C3—241no—
444C22×C3—242no—
445C22×C3—244no—
446C22×C3—241no—
447C22×C3—482no—
448C22×C3C2×#102011yescharacteristic
449C2×D3—641no—
450A4—164no—
451A4—161no—
452A4—322no—
453A4—324no—
454A4—324no—
455A4—321no—
456C3×C4—61no—
457C3×C4—61no—
458C3×C4—122no—
459C3×C4—121no—
460C3×C4—121no—
461C3×C4—244no—
462C32—641noSylow
463C23—32no—
464C23—32no—
465C23—44no—
466C23—62no—
467C23—62no—
468C23—64no—
469C23—82no—
470C23—122no—
471C23—122no—
472C23—121no—
473C23—122no—
474C23—124no—
475C23—122no—
476C23—122no—
477C23—242no—
478C23—242no—
479C23—242no—
480C23—242no—
481C23C3×#102011yescharacteristic
482Q8—62no—
483Q8—121no—
484Q8—121no—
485Q8—121no—
486Q8—241no—
487D4—62no—
488D4—121no—
489D4—121no—
490D4—122no—
491D4—121no—
492D4—122no—
493D4—124no—
494D4—124no—
495D4—122no—
496D4—241no—
497D4—241no—
498D4—242no—
499D4—248no—
500D4—248no—
501C2×C4—31no—
502C2×C4—31no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—64no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—121no—
513C2×C4—122no—
514C2×C4—124no—
515C2×C4—242no—
516C2×C4—242no—
517C8—122no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—122no—
521C2×C3—122no—
522C2×C3—242no—
523C2×C3—641no—
524C2×C3—1281no—
525C2×C3#8555911yescharacteristic, direct product factor
526D3—642no—
527C22—31no—
528C22—31no—
529C22—44no—
530C22—62no—
531C22—62no—
532C22—82no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—122no—
537C22—124no—
538C22—122no—
539C22—242no—
540C22—241no—
541C22—242no—
542C22—244no—
543C22—241no—
544C22—482no—
545C22C2×C3×#102011yescharacteristic
546C4—61no—
547C4—61no—
548C4—122no—
549C4—121no—
550C4—121no—
551C4—244no—
552C3—641no—
553C3—1281no—
554C3C2×#8555911yescharacteristic, direct product factor
555C2—31no—
556C2—31no—
557C2—122no—
558C2—122no—
559C2—242no—
560C2C3×#8555911yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#85559
commutator, cosocle, hypocenter#5464C22×C3
FrattiniC22C2×C3×#1020
FittingC2×C3×H(3,𝔽4)D3
socleC23×C3#1020
LCSG ▷ #5464
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×#85559C2direct (N × H)
2. C2×C3×#5464C2semidirect
3. C2×#85559C3direct (N × H)
6. C3×#5464C22semidirect
8. C2×C3×H(3,𝔽4)D3semidirect
10. C2×#5464C2×C3semidirect
11. #85559C2×C3direct (N × H)
31. C3×H(3,𝔽4)C2×D3semidirect
37. #5464C22×C3semidirect
60. C2×H(3,𝔽4)C3×D3semidirect
68. C25×C3S4semidirect
115. C2×C3×C42S4semidirect
139. H(3,𝔽4)C2×C3×D3semidirect
165. C24×C3C2×S4semidirect
233. C3×C42C2×S4semidirect
242. C25C3×S4semidirect
286. C2×C42C3×S4semidirect
365. C24C2×C3×S4semidirect
429. C42C2×C3×S4semidirect
525. C2×C3#85559direct (N × H)
554. C3C2×#85559direct (N × H)
560. C2C3×#85559direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#855592
C2×C3×#54642
C2×#855593
C2×C3×#21993
C2×C3×#10204
C2×C3×#8574