Finite Group Databasebeta

← C2×C3×#87588 #72274479

Subgroups — C2×C3×#87588 #72274479

Order 2304.

13778 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 13778 conjugacy classes of subgroups — too many to draw. Only its 74 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×#87588C212yesmaximal, direct product factor
2C2×C3×C22⋊(C22⋊A4)C211yesmaximal, characteristic, complemented
3C2×#87588C311yesmaximal, characteristic, direct product factor
4C2×C3×#2846—31nomaximal
5C3×C22⋊(C22⋊A4)C2211yescharacteristic, complemented
6C2×C3×#1020—47nomaximal
7C27×C3D311yescharacteristic, complemented
8C2×C22⋊(C22⋊A4)—21no—
9C2×C22⋊(C22⋊A4)C2×C311yescharacteristic, complemented
10#87588C2×C312yesdirect product factor
11C22×C3×#120—37no—
12C3×#2846—314no—
13C3×#2846—32no—
14C2×C3×#379—37no—
15C2×C3×C22⋊A4—47no—
16C2×C3×C22⋊A4—87no—
17C3×#1020—414no—
18C2×#2846—31noSylow
19C26×C3—37no—
20C26×C3—37no—
21C26×C3—67no—
22C26×C3—67no—
23C26×C3C2×D311yescharacteristic, complemented
24C22⋊(C22⋊A4)—21no—
25C22⋊(C22⋊A4)C22×C311yescharacteristic, complemented
26C2×#1020—47no—
27C23×C3×D4—37no—
28C2×C3×#120—342no—
29C2×C3×#120—314no—
30C2×C3×#120—621no—
31C2×C3×#120—621no—
32C2×C3×#120—67no—
33C2×C3×#120—67no—
34C22×C3×C22⋊C4—321no—
35C3×#379—342no—
36C3×#379—314no—
37C3×C22⋊A4—47no—
38C3×C22⋊A4—87no—
39C2×C3×S4—167no—
40C27C3×D311yescharacteristic, complemented
41C22×#120—37no—
42#2846—314no—
43#2846—32no—
44C2×#379—37no—
45C25×C3—27no—
46C25×C3—37no—
47C25×C3—321no—
48C25×C3—321no—
49C25×C3—321no—
50C25×C3—37no—
51C25×C3—37no—
52C25×C3—31no—
53C25×C3—67no—
54C25×C3—67no—
55C25×C3—642no—
56C25×C3—621no—
57C25×C3—621no—
58C25×C3—642no—
59C25×C3—642no—
60C25×C3—621no—
61C25×C3—684no—
62C25×C3—628no—
63C25×C3—684no—
64C25×C3S417yescomplemented
65C2×C22⋊A4—47no—
66C2×C22⋊A4—87no—
67C2×C22⋊A4—814no—
68C2×C22⋊A4—87no—
69#1020—414no—
70C22×C3×D4—314no—
71C22×C3×D4—314no—
72C22×C3×D4—621no—
73C22×C3×D4—621no—
74C22×C3×D4—621no—
75C22×C3×D4—621no—
76C23×C3×C4—37no—
77C3×#120—642no—
78C3×#120—684no—
79C3×#120—642no—
80C3×#120—642no—
81C3×#120—614no—
82C2×C3×C22⋊C4—342no—
83C2×C3×C22⋊C4—342no—
84C2×C3×C22⋊C4—621no—
85C2×C3×C22⋊C4—621no—
86C2×C3×C22⋊C4—621no—
87C2×C3×C22⋊C4—621no—
88C2×C3×A4—167no—
89C2×C3×A4—327no—
90C3×S4—1614no—
91C26—37no—
92C26—37no—
93C26—67no—
94C26—67no—
95C26C2×C3×D311yescharacteristic, complemented
96C23×D4—37no—
97C2×#120—342no—
98C2×#120—314no—
99C2×#120—621no—
100C2×#120—621no—
101C2×#120—67no—
102C2×#120—67no—
103C22×C22⋊C4—321no—
104#379—342no—
105#379—314no—
106C24×C3—27no—
107C24×C3—31no—
108C24×C3—37no—
109C24×C3—37no—
110C24×C3—321no—
111C24×C3—321no—
112C24×C3—321no—
113C24×C3—321no—
114C24×C3—321no—
115C24×C3—342no—
116C24×C3—321no—
117C24×C3—321no—
118C24×C3—32no—
119C24×C3—67no—
120C24×C3—628no—
121C24×C3—67no—
122C24×C3—621no—
123C24×C3—642no—
124C24×C3—621no—
125C24×C3—628no—
126C24×C3—628no—
127C24×C3—642no—
128C24×C3—684no—
129C24×C3—684no—
130C24×C3—642no—
131C24×C3—642no—
132C24×C3—684no—
133C24×C3—684no—
134C24×C3—684no—
135C24×C3—684no—
136C24×C3—621no—
137C24×C3—642no—
138C24×C3—621no—
139C24×C3—621no—
140C24×C3—67no—
141C24×C3—67no—
142C24×C3—628no—
143C24×C3—67no—
144C24×C3—628no—
145C24×C3—684no—
146C24×C3—684no—
147C24×C3—628no—
148C24×C3—684no—
149C24×C3—684no—
150C24×C3—628no—
151C24×C3—6168no—
152C24×C3—6168no—
153C24×C3—684no—
154C24×C3—656no—
155C24×C3—68no—
156C24×C3—67no—
157C24×C3C2×S417yescomplemented
158C22⋊A4—47no—
159C22⋊A4—87no—
160C22⋊A4—814no—
161C22⋊A4—87no—
162C2×S4—167no—
163C2×C3×D4—642no—
164C2×C3×D4—642no—
165C2×C3×D4—642no—
166C2×C3×D4—642no—
167C2×C3×D4—1221no—
168C2×C3×D4—1242no—
169C2×C3×D4—1221no—
170C2×C3×D4—1221no—
171C2×C3×D4—127no—
172C22×C3×C4—314no—
173C22×C3×C4—621no—
174C22×C3×C4—621no—
175C3×C22⋊C4—642no—
176C3×C22⋊C4—642no—
177C3×C22⋊C4—642no—
178C3×C22⋊C4—642no—
179C2×C3×D3—641no—
180C3×A4—167no—
181C3×A4—327no—
182C25—27no—
183C25—37no—
184C25—321no—
185C25—321no—
186C25—321no—
187C25—37no—
188C25—37no—
189C25—31no—
190C25—67no—
191C25—67no—
192C25—642no—
193C25—621no—
194C25—621no—
195C25—642no—
196C25—642no—
197C25—621no—
198C25—684no—
199C25—628no—
200C25—684no—
201C25C3×S417yescomplemented
202C22×D4—314no—
203C22×D4—314no—
204C22×D4—621no—
205C22×D4—621no—
206C22×D4—621no—
207C22×D4—621no—
208C23×C4—37no—
209#120—642no—
210#120—684no—
211#120—642no—
212#120—642no—
213#120—614no—
214C2×C22⋊C4—342no—
215C2×C22⋊C4—342no—
216C2×C22⋊C4—621no—
217C2×C22⋊C4—621no—
218C2×C22⋊C4—621no—
219C2×C22⋊C4—621no—
220C23×C3—27no—
221C23×C3—31no—
222C23×C3—37no—
223C23×C3—37no—
224C23×C3—321no—
225C23×C3—321no—
226C23×C3—321no—
227C23×C3—321no—
228C23×C3—342no—
229C23×C3—321no—
230C23×C3—321no—
231C23×C3—321no—
232C23×C3—67no—
233C23×C3—628no—
234C23×C3—67no—
235C23×C3—628no—
236C23×C3—642no—
237C23×C3—621no—
238C23×C3—621no—
239C23×C3—642no—
240C23×C3—628no—
241C23×C3—628no—
242C23×C3—684no—
243C23×C3—684no—
244C23×C3—642no—
245C23×C3—642no—
246C23×C3—621no—
247C23×C3—684no—
248C23×C3—684no—
249C23×C3—684no—
250C23×C3—684no—
251C23×C3—621no—
252C23×C3—621no—
253C23×C3—642no—
254C23×C3—614no—
255C23×C3—628no—
256C23×C3—684no—
257C23×C3—684no—
258C23×C3—628no—
259C23×C3—6168no—
260C23×C3—6168no—
261C23×C3—684no—
262C23×C3—684no—
263C23×C3—684no—
264C23×C3—68no—
265C23×C3—656no—
266C23×C3—628no—
267C23×C3—67no—
268C23×C3—67no—
269C23×C3—1221no—
270C23×C3—127no—
271C23×C3#102017yescomplemented
272C2×A4—167no—
273C2×A4—327no—
274C2×A4—3214no—
275C2×A4—327no—
276S4—1614no—
277C3×D4—1214no—
278C3×D4—1242no—
279C3×D4—1242no—
280C3×D4—1214no—
281C2×C3×C4—642no—
282C2×C3×C4—1221no—
283C2×C3×C4—127no—
284C2×C32—641no—
285C3×D3—642no—
286C24—27no—
287C24—31no—
288C24—37no—
289C24—37no—
290C24—321no—
291C24—321no—
292C24—321no—
293C24—321no—
294C24—321no—
295C24—342no—
296C24—321no—
297C24—321no—
298C24—32no—
299C24—67no—
300C24—628no—
301C24—67no—
302C24—621no—
303C24—642no—
304C24—621no—
305C24—628no—
306C24—628no—
307C24—642no—
308C24—684no—
309C24—684no—
310C24—642no—
311C24—642no—
312C24—684no—
313C24—684no—
314C24—684no—
315C24—684no—
316C24—621no—
317C24—642no—
318C24—621no—
319C24—621no—
320C24—67no—
321C24—67no—
322C24—628no—
323C24—67no—
324C24—628no—
325C24—684no—
326C24—684no—
327C24—628no—
328C24—684no—
329C24—684no—
330C24—628no—
331C24—6168no—
332C24—6168no—
333C24—684no—
334C24—656no—
335C24—68no—
336C24—67no—
337C24C2×C3×S417yescomplemented
338C2×D4—642no—
339C2×D4—642no—
340C2×D4—642no—
341C2×D4—642no—
342C2×D4—1221no—
343C2×D4—1242no—
344C2×D4—1221no—
345C2×D4—1221no—
346C2×D4—127no—
347C22×C4—314no—
348C22×C4—621no—
349C22×C4—621no—
350C22⋊C4—642no—
351C22⋊C4—642no—
352C22⋊C4—642no—
353C22⋊C4—642no—
354C22×C3—27no—
355C22×C3—37no—
356C22×C3—37no—
357C22×C3—321no—
358C22×C3—37no—
359C22×C3—321no—
360C22×C3—321no—
361C22×C3—621no—
362C22×C3—642no—
363C22×C3—621no—
364C22×C3—642no—
365C22×C3—642no—
366C22×C3—621no—
367C22×C3—684no—
368C22×C3—67no—
369C22×C3—628no—
370C22×C3—684no—
371C22×C3—67no—
372C22×C3—1214no—
373C22×C3—247no—
374C22×C3—241no—
375C22×C3C2×#102017yescomplemented
376C2×D3—641no—
377A4—167no—
378A4—327no—
379A4—3214no—
380A4—327no—
381C3×C4—1214no—
382C32—641noSylow
383C23—27no—
384C23—31no—
385C23—37no—
386C23—37no—
387C23—321no—
388C23—321no—
389C23—321no—
390C23—321no—
391C23—342no—
392C23—321no—
393C23—321no—
394C23—321no—
395C23—67no—
396C23—628no—
397C23—67no—
398C23—628no—
399C23—642no—
400C23—621no—
401C23—621no—
402C23—642no—
403C23—628no—
404C23—628no—
405C23—684no—
406C23—684no—
407C23—642no—
408C23—642no—
409C23—621no—
410C23—684no—
411C23—684no—
412C23—684no—
413C23—684no—
414C23—621no—
415C23—621no—
416C23—642no—
417C23—614no—
418C23—628no—
419C23—684no—
420C23—684no—
421C23—628no—
422C23—6168no—
423C23—6168no—
424C23—684no—
425C23—684no—
426C23—684no—
427C23—68no—
428C23—656no—
429C23—628no—
430C23—67no—
431C23—67no—
432C23—1221no—
433C23—127no—
434C23C3×#102017yescomplemented
435D4—1214no—
436D4—1242no—
437D4—1242no—
438D4—1214no—
439C2×C4—642no—
440C2×C4—1221no—
441C2×C4—127no—
442C2×C3—37no—
443C2×C3—37no—
444C2×C3—67no—
445C2×C3—67no—
446C2×C3—242no—
447C2×C3—641no—
448C2×C3—1281no—
449C2×C3#8758811yescharacteristic, direct product factor
450D3—642no—
451C22—27no—
452C22—37no—
453C22—37no—
454C22—321no—
455C22—37no—
456C22—321no—
457C22—321no—
458C22—621no—
459C22—642no—
460C22—621no—
461C22—642no—
462C22—642no—
463C22—621no—
464C22—684no—
465C22—67no—
466C22—628no—
467C22—684no—
468C22—67no—
469C22—1214no—
470C22—247no—
471C22—241no—
472C22C2×C3×#102017yescomplemented
473C4—1214no—
474C3—641no—
475C3—1281no—
476C3C2×#8758811yescharacteristic, direct product factor
477C2—37no—
478C2—37no—
479C2—67no—
480C2—67no—
481C2—242no—
482C2C3×#8758811yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#87588
commutator, cosocle, hypocenterC22⋊(C22⋊A4)C22×C3
FrattiniC1G
Fitting, socleC27×C3D3
LCSG ▷ C22⋊(C22⋊A4)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×#87588C2direct (N × H)
2. C2×C3×C22⋊(C22⋊A4)C2semidirect
3. C2×#87588C3direct (N × H)
5. C3×C22⋊(C22⋊A4)C22semidirect
7. C27×C3D3semidirect
9. C2×C22⋊(C22⋊A4)C2×C3semidirect
10. #87588C2×C3direct (N × H)
23. C26×C3C2×D3semidirect
25. C22⋊(C22⋊A4)C22×C3semidirect
40. C27C3×D3semidirect
64. C25×C3S4semidirect
95. C26C2×C3×D3semidirect
157. C24×C3C2×S4semidirect
201. C25C3×S4semidirect
271. C23×C3#1020semidirect
337. C24C2×C3×S4semidirect
375. C22×C3C2×#1020semidirect
434. C23C3×#1020semidirect
449. C2×C3#87588direct (N × H)
472. C22C2×C3×#1020semidirect
476. C3C2×#87588direct (N × H)
482. C2C3×#87588direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#875882
C2×C3×C22⋊(C22⋊A4)2
C2×#875883
C2×C3×#28463
C2×C3×#10204