Finite Group Databasebeta

← C2×C3×#5244 #11865249

Subgroups — C2×C3×#5244 #11865249

Order 1152.

2485 conjugacy classes of subgroups; only the 330 normal subgroups are shown.

normal subgroups only

This group has 2485 conjugacy classes of subgroups — too many to draw. Only its 330 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
1C2×C32×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×C3×#937C213yesmaximal, complemented
3C3×#5244C218yesmaximal, direct product factor
4C2×C3×#834C213yesmaximal, complemented
5C2×C3×B(4,𝔽2)—31nomaximal
6C2×#5244C311yesmaximal, characteristic, direct product factor
7C24×C3×D3C2211yescharacteristic, complemented
8C32×H(5,𝔽2)—26no—
9C32×H(5,𝔽2)C2214yescomplemented
10C2×C32×#41—23no—
11C22×C32×D4—23no—
12C22×C32×D4C2213yes—
13C22×C3×#67—23no—
14C2×C3×#217C2213yescomplemented
15C3×#937—26no—
16C3×#937C22112yescomplemented
17C2×C3×(C2×D3)⋊C4—23no—
18C3×#834C22112yescomplemented
19C2×C3×H(5,𝔽2)—21no—
20C2×C3×H(5,𝔽2)D311yescharacteristic, complemented
21C2×C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
22C2×C3×#120—33no—
23C2×#937C2×C313yescomplemented
24C3×B(4,𝔽2)—38no—
25C2×C3×#99—33no—
26#5244C2×C318yesdirect product factor
27C2×#834C2×C313yescomplemented
28C24×C32—21no—
29C24×C32D413yescomplemented
30C24×C32C2311yescharacteristic
31C23×C3×D3—23no—
32C23×C3×D3—23no—
33C23×C3×D3—44no—
34C23×C3×D3C2312yescomplemented
35C32×#41—212no—
36C32×#41—212no—
37C2×C32×Q8—21no—
38C2×C32×D4—23no—
39C2×C32×D4—212no—
40C2×C32×D4—26no—
41C2×C32×D4—26no—
42C2×C32×D4—212no—
43C2×C32×D4D416yescomplemented
44C2×C32×D4C2316yes—
45C22×C32×C4—23no—
46C22×C32×C4D413yescomplemented
47C2×C3×#67—212no—
48C2×C3×#67—412no—
49C22×C3×Dic3—23no—
50C3×#217—23no—
51C3×#217C2316yescomplemented
52C3×(C2×D3)⋊C4—212no—
53C2×B(4,𝔽2)—31noSylow
54C25×C3—31no—
55C24×D3C22×C311yescharacteristic, complemented
56C3×H(5,𝔽2)—212no—
57C3×H(5,𝔽2)—26no—
58C3×H(5,𝔽2)—26no—
59C3×H(5,𝔽2)—24no—
60C3×H(5,𝔽2)C2×D314yescomplemented
61C3×H(5,𝔽2)C22×C314yescomplemented
62C2×C3×#41—26no—
63C2×C3×#41—23no—
64C2×C3×#41—23no—
65C22×C3×D4—26no—
66C22×C3×D4—23no—
67C22×C3×D4—23no—
68C22×C3×D4—23no—
69C22×C3×D4—63no—
70C22×C3×D4C2×D313yes—
71C22×C3×D4C22×C313yes—
72C22×#67—23no—
73C3×#120—312no—
74C3×#120—66no—
75C2×C3×C22⋊C4—33no—
76C2×C3×C22⋊C4—63no—
77C2×#217C22×C313yescomplemented
78#937—26no—
79#937C22×C3112yescomplemented
80C2×(C2×D3)⋊C4—23no—
81C3×#99—312no—
82#834C22×C3112yescomplemented
83C23×C32—24no—
84C23×C32—23no—
85C23×C32—23no—
86C23×C32—42no—
87C23×C32—46no—
88C23×C32—82no—
89C23×C32C2×D413yes—
90C23×C32C2×D413yes—
91C23×C32C2×D416yescomplemented
92C23×C32C2411yescharacteristic
93C22×C3×D3—212no—
94C22×C3×D3—26no—
95C22×C3×D3—44no—
96C22×C3×D3—412no—
97C22×C3×D3—46no—
98C22×C3×D3—43no—
99C22×C3×D3—41no—
100C32×Q8—24no—
101C32×D4—212no—
102C32×D4—212no—
103C32×D4—212no—
104C2×C32×C4—212no—
105C2×C32×C4—23no—
106C2×C32×C4—23no—
107C2×C32×C4C2×D416yescomplemented
108C3×#67—424no—
109C2×C3×Dic3—26no—
110C2×C3×Dic3—46no—
111C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
112C2×#120—33no—
113B(4,𝔽2)—38no—
114C2×#99—33no—
115C24×C3—22no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—23no—
119C24×C3—21no—
120C24×C3—32no—
121C24×C3—63no—
122C24×C3—63no—
123C24×C3—124no—
124C24×C3#6713yescomplemented
125C24×C3C3×D413yescomplemented
126C24×C3C22×D311yescharacteristic
127C24×C3C23×C311yescharacteristic
128C23×D3—23no—
129C23×D3—23no—
130C23×D3—44no—
131C23×D3C23×C312yescomplemented
132C3×#41—224no—
133C3×#41—212no—
134C3×#41—212no—
135C3×#41—224no—
136C3×#41—212no—
137C3×#41—212no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×D4—26no—
142C2×C3×D4—23no—
143C2×C3×D4—23no—
144C2×C3×D4—212no—
145C2×C3×D4—224no—
146C2×C3×D4—212no—
147C2×C3×D4—212no—
148C2×C3×D4—212no—
149C2×C3×D4—224no—
150C2×C3×D4—26no—
151C2×C3×D4—26no—
152C2×C3×D4—212no—
153C2×C3×D4—26no—
154C2×C3×D4—26no—
155C2×C3×D4—212no—
156C2×C3×D4—26no—
157C2×C3×D4—26no—
158C2×C3×D4—612no—
159C2×C3×D4—1212no—
160C2×C3×D4#6716yescomplemented
161C2×C3×D4C3×D416yescomplemented
162C2×C3×D4C22×D316yes—
163C2×C3×D4C23×C316yes—
164C22×C3×C4—26no—
165C22×C3×C4—23no—
166C22×C3×C4—23no—
167C22×C3×C4—23no—
168C22×C3×C4—63no—
169C22×C3×C4#6713yescomplemented
170C22×C3×C4C3×D413yescomplemented
171C2×#67—212no—
172C2×#67—412no—
173C22×Dic3—23no—
174C3×C22⋊C4—36no—
175C3×C22⋊C4—63no—
176C3×C22⋊C4—612no—
177#217—23no—
178#217C23×C316yescomplemented
179(C2×D3)⋊C4—212no—
180C22×C32—26no—
181C22×C32—26no—
182C22×C32—46no—
183C22×C32—43no—
184C22×C32—41no—
185C22×C32—46no—
186C22×C32—83no—
187C22×C32—84no—
188C22×C32#12013yes—
189C22×C32#12011yescharacteristic
190C22×C32C22×D413yes—
191C2×C3×D3—212no—
192C2×C3×D3—46no—
193C2×C3×D3—412no—
194C2×C3×D3—42no—
195C2×C3×D3—44no—
196C32×C4—26no—
197C3×Dic3—46no—
198C25—31no—
199H(5,𝔽2)—26no—
200H(5,𝔽2)C2×C3×D314yescomplemented
201C2×#41—23no—
202C22×D4—23no—
203C22×D4—63no—
204C22×D4C2×C3×D313yes—
205#120—312no—
206#120—66no—
207C2×C22⋊C4—33no—
208C2×C22⋊C4—63no—
209#99—312no—
210C23×C3—26no—
211C23×C3—28no—
212C23×C3—24no—
213C23×C3—24no—
214C23×C3—26no—
215C23×C3—23no—
216C23×C3—23no—
217C23×C3—23no—
218C23×C3—23no—
219C23×C3—21no—
220C23×C3—23no—
221C23×C3—23no—
222C23×C3—26no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—46no—
226C23×C3—46no—
227C23×C3—612no—
228C23×C3—66no—
229C23×C3—82no—
230C23×C3—86no—
231C23×C3—84no—
232C23×C3—82no—
233C23×C3—82no—
234C23×C3—126no—
235C23×C3—124no—
236C23×C3—1212no—
237C23×C3—123no—
238C23×C3—121no—
239C23×C3C2×#6713yes—
240C23×C3C2×#6713yes—
241C23×C3C2×#6716yescomplemented
242C23×C3C2×C3×D413yes—
243C23×C3C2×C3×D413yes—
244C23×C3C2×C3×D416yescomplemented
245C23×C3C23×D311yescharacteristic
246C23×C3C24×C311yescharacteristic
247C22×D3—212no—
248C22×D3—26no—
249C22×D3—44no—
250C22×D3—412no—
251C22×D3—46no—
252C22×D3—43no—
253C22×D3—41no—
254C3×Q8—28no—
255C3×Q8—24no—
256C3×Q8—24no—
257C3×D4—224no—
258C3×D4—212no—
259C3×D4—212no—
260C3×D4—224no—
261C3×D4—224no—
262C3×D4—212no—
263C3×D4—212no—
264C3×D4—212no—
265C3×D4—212no—
266C3×D4—1224no—
267C2×C3×C4—26no—
268C2×C3×C4—224no—
269C2×C3×C4—212no—
270C2×C3×C4—212no—
271C2×C3×C4—26no—
272C2×C3×C4—23no—
273C2×C3×C4—23no—
274C2×C3×C4—23no—
275C2×C3×C4—23no—
276C2×C3×C4—26no—
277C2×C3×C4—66no—
278C2×C3×C4—126no—
279C2×C3×C4C2×#6716yescomplemented
280C2×C3×C4C2×C3×D416yescomplemented
281#67—424no—
282C2×Dic3—26no—
283C2×Dic3—46no—
284C2×C32—23no—
285C2×C32—23no—
286C2×C32—46no—
287C2×C32B(4,𝔽2)12yescomplemented
288C2×C32C2×#12011yescharacteristic
289C3×D3—44no—
290C24—21no—
291C24—32no—
292C24—63no—
293C24—63no—
294C24—124no—
295C24C3×#6713yescomplemented
296C24C22×C3×D311yescharacteristic
297#41—212no—
298#41—212no—
299C2×Q8—21no—
300C2×D4—23no—
301C2×D4—212no—
302C2×D4—26no—
303C2×D4—26no—
304C2×D4—212no—
305C2×D4—612no—
306C2×D4—1212no—
307C2×D4C3×#6716yescomplemented
308C2×D4C22×C3×D316yes—
309C22×C4—23no—
310C22×C4—63no—
311C22×C4C3×#6713yescomplemented
312C22⋊C4—36no—
313C22⋊C4—63no—
314C22⋊C4—612no—
315C22×C3—26no—
316C22×C3—26no—
317C22×C3—212no—
318C22×C3—26no—
319C22×C3—26no—
320C22×C3—21no—
321C22×C3—23no—
322C22×C3—23no—
323C22×C3—46no—
324C22×C3—43no—
325C22×C3—41no—
326C22×C3—43no—
327C22×C3—41no—
328C22×C3—46no—
329C22×C3—46no—
330C22×C3—412no—
331C22×C3—46no—
332C22×C3—46no—
333C22×C3—612no—
334C22×C3—81no—
335C22×C3—83no—
336C22×C3—86no—
337C22×C3—83no—
338C22×C3—83no—
339C22×C3—86no—
340C22×C3—88no—
341C22×C3—84no—
342C22×C3—84no—
343C22×C3—126no—
344C22×C3—1212no—
345C22×C3—124no—
346C22×C3—122no—
347C22×C3#95313yes—
348C22×C3#95311yescharacteristic
349C22×C3C3×#12013yes—
350C22×C3C3×#12011yescharacteristic
351C22×C3C22×#6713yes—
352C22×C3C22×C3×D413yes—
353C2×D3—212no—
354C2×D3—46no—
355C2×D3—412no—
356C2×D3—44no—
357C2×D3—42no—
358C3×C4—212no—
359C3×C4—26no—
360C3×C4—26no—
361C3×C4—126no—
362Dic3—46no—
363C32C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
364C23—24no—
365C23—23no—
366C23—23no—
367C23—42no—
368C23—46no—
369C23—612no—
370C23—66no—
371C23—82no—
372C23—126no—
373C23—124no—
374C23—1212no—
375C23—123no—
376C23—121no—
377C23C2×C3×#6713yes—
378C23C2×C3×#6713yes—
379C23C2×C3×#6716yescomplemented
380C23C23×C3×D311yescharacteristic
381Q8—24no—
382D4—212no—
383D4—212no—
384D4—212no—
385D4—1224no—
386C2×C4—212no—
387C2×C4—23no—
388C2×C4—23no—
389C2×C4—66no—
390C2×C4—126no—
391C2×C4C2×C3×#6716yescomplemented
392C2×C3—23no—
393C2×C3—23no—
394C2×C3—23no—
395C2×C3—23no—
396C2×C3—22no—
397C2×C3—21no—
398C2×C3—43no—
399C2×C3—43no—
400C2×C3—412no—
401C2×C3—46no—
402C2×C3—46no—
403C2×C3—124no—
404C2×C3#524412yesdirect product factor
405C2×C3C3×B(4,𝔽2)12yescomplemented
406C2×C3C2×#95311yescharacteristic
407C2×C3C2×C3×#12011yescharacteristic
408D3—44no—
409C22—26no—
410C22—26no—
411C22—41no—
412C22—43no—
413C22—46no—
414C22—46no—
415C22—612no—
416C22—83no—
417C22—84no—
418C22—126no—
419C22—1212no—
420C22—124no—
421C22—122no—
422C22C3×#95313yes—
423C22C3×#95311yescharacteristic
424C22C22×C3×#6713yes—
425C4—26no—
426C4—126no—
427C3—21no—
428C3C2×#524411yescharacteristic, direct product factor
429C3C2×C3×B(4,𝔽2)11yescharacteristic, complemented
430C2—23no—
431C2—23no—
432C2—46no—
433C2—124no—
434C2C3×#524412yesdirect product factor
435C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC22×C3#953
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC2×C32×H(5,𝔽2)C2
socleC22×C32#120
hypercenterC2×C3×H(5,𝔽2)D3
hypocenterC3C2×C3×B(4,𝔽2)
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C32×H(5,𝔽2)C2semidirect
2. C2×C3×#937C2semidirect
3. C3×#5244C2direct (N × H)
4. C2×C3×#834C2semidirect
6. C2×#5244C3direct (N × H)
7. C24×C3×D3C22semidirect
9. C32×H(5,𝔽2)C22semidirect
14. C2×C3×#217C22semidirect
16. C3×#937C22semidirect
18. C3×#834C22semidirect
20. C2×C3×H(5,𝔽2)D3semidirect
21. C2×C3×H(5,𝔽2)C2×C3semidirect
23. C2×#937C2×C3semidirect
26. #5244C2×C3direct (N × H)
27. C2×#834C2×C3semidirect
29. C24×C32D4semidirect
34. C23×C3×D3C23semidirect
43. C2×C32×D4D4semidirect
46. C22×C32×C4D4semidirect
51. C3×#217C23semidirect
55. C24×D3C22×C3semidirect
60. C3×H(5,𝔽2)C2×D3semidirect
61. C3×H(5,𝔽2)C22×C3semidirect
77. C2×#217C22×C3semidirect
79. #937C22×C3semidirect
82. #834C22×C3semidirect
91. C23×C32C2×D4semidirect
107. C2×C32×C4C2×D4semidirect
111. C2×H(5,𝔽2)C3×D3semidirect
124. C24×C3#67semidirect
125. C24×C3C3×D4semidirect
131. C23×D3C23×C3semidirect
160. C2×C3×D4#67semidirect
161. C2×C3×D4C3×D4semidirect
169. C22×C3×C4#67semidirect
170. C22×C3×C4C3×D4semidirect
178. #217C23×C3semidirect
200. H(5,𝔽2)C2×C3×D3semidirect
241. C23×C3C2×#67semidirect
244. C23×C3C2×C3×D4semidirect
279. C2×C3×C4C2×#67semidirect
280. C2×C3×C4C2×C3×D4semidirect
287. C2×C32B(4,𝔽2)semidirect
295. C24C3×#67semidirect
307. C2×D4C3×#67semidirect
311. C22×C4C3×#67semidirect
363. C32C2×B(4,𝔽2)semidirect
379. C23C2×C3×#67semidirect
391. C2×C4C2×C3×#67semidirect
404. C2×C3#5244direct (N × H)
405. C2×C3C3×B(4,𝔽2)semidirect
428. C3C2×#5244direct (N × H)
429. C3C2×C3×B(4,𝔽2)semidirect
434. C2C3×#5244direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×H(5,𝔽2)2
C2×C3×#9372
C3×#52442
C2×C3×#8342
C2×C3×B(4,𝔽2)3
C2×#52443