Finite Group Databasebeta

← C2×C4×#6258 #13164475

Subgroups — C2×C4×#6258 #13164475

Order 1728.

1128 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C22×#6258C211yesmaximal, characteristic
2C2×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#88613C211yesmaximal, characteristic, complemented
4C4×#6258C214yesmaximal, direct product factor
5C2×C3×C4×S4—33nomaximal
6C2×C3×C4×#67—31nomaximal
7C22×C22⋊H(3,𝔽3)C2211yescharacteristic
8C2×#6258C414yesdirect product factor
9C2×#6258C2212yes—
10C2×C4×#272—41nomaximal
11C4×C22⋊H(3,𝔽3)C2212yescomplemented
12#88613C2212yescomplemented
13C22×C3×S4—33no—
14C23×C32×C4D311yescharacteristic, complemented
15C22×C3×#67—31no—
16C22×C3×C4×D3—31no—
17C2×C3×C4×A4D313yescomplemented
18C2×C3×SL(2,ℤ/4)—33no—
19C3×C4×S4—312no—
20C2×C3×#217—31no—
21C3×C4×#67—34no—
22C3×C4×#67—34no—
23C2×C3×(C2×D3)⋊C4—31no—
24C2×C3×Dic3⋊C4—31no—
25C2×C3×C4×Dic3—31no—
26C22×#272—41no—
27C2×C22⋊H(3,𝔽3)C2×C412yescomplemented
28C2×C22⋊H(3,𝔽3)C2311yescharacteristic
29#6258C2×C414yesdirect product factor
30C2×C4×H(3,𝔽3)—41no—
31C2×#1084—41no—
32C4×#272—44no—
33C2×C4×S4—33no—
34C2×C3×C4×D4—91no—
35C2×C4×#67—31no—
36C24×C32C2×D311yescharacteristic
37C23×C3×D3—31no—
38C22×C3×A4C2×D313yes—
39C2×C3×S4—312no—
40C2×C3×S4—36no—
41C22×C32×C4—31no—
42C22×C32×C4—31no—
43C22×C32×C4—32no—
44C22×C32×C4C2×D312yescomplemented
45C2×C3×#67—32no—
46C2×C3×#67—34no—
47C2×C3×#67—34no—
48C2×C3×#67—32no—
49C22×C3×Dic3—31no—
50C22×C3×Dic3—31no—
51C22×C3×Dic3—31no—
52C2×C3×C4×D3—34no—
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—62no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—123no—
57C3×C4×A4C2×D316yescomplemented
58C3×SL(2,ℤ/4)—36no—
59C3×#217—32no—
60C3×#217—32no—
61C3×(C2×D3)⋊C4—34no—
62C3×Dic3⋊C4—34no—
63C3×C4×Dic3—34no—
64C22×H(3,𝔽3)—41no—
65C2×#272—44no—
66C2×#272—42no—
67C22⋊H(3,𝔽3)C22×C411yescharacteristic, complemented
68C4×H(3,𝔽3)—42no—
69#1084—42no—
70C22×S4—33no—
71C22×C3×D4—91no—
72C23×C3×C4—31no—
73C23×C3×C4—91no—
74C23×C3×C4C3⋊D311yescharacteristic
75C22×#67—31no—
76C22×C4×D3—31no—
77C2×C4×A4—33no—
78C2×SL(2,ℤ/4)—33no—
79C4×S4—312no—
80C3×C4×D4—94no—
81C3×C4×D4—94no—
82C2×C3×C4⋊C4—91no—
83C2×C3×C22⋊C4—91no—
84C2×C3×C22⋊C4—91no—
85C2×C3×C42—91no—
86C2×#217—31no—
87C4×#67—34no—
88C4×#67—34no—
89C2×(C2×D3)⋊C4—31no—
90C2×Dic3⋊C4—31no—
91C2×C4×Dic3—31no—
92C23×C32—31no—
93C23×C32—31no—
94C23×C32—32no—
95C23×C32C4×D312yescomplemented
96C23×C32C22×D311yescharacteristic
97C22×C3×D3—32no—
98C22×C3×D3—34no—
99C22×C3×D3—62no—
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C22×C3×D3—123no—
103C2×C3×A4C4×D316yescomplemented
104C2×C3×A4C22×D313yes—
105C3×S4—312no—
106C2×C32×C4—31no—
107C2×C32×C4—32no—
108C2×C32×C4—32no—
109C2×C32×C4—32no—
110C2×C32×C4—43no—
111C2×C32×C4—61no—
112C2×C32×C4S411yescharacteristic, complemented
113C3×#67—38no—
114C3×#67—34no—
115C3×#67—34no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—34no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—123no—
124C3×C4×D3—64no—
125C3×C4×D3—64no—
126C3×C4×D3—1212no—
127C2×C4×D4—91noSylow
128C2×H(3,𝔽3)—42no—
129C2×H(3,𝔽3)—41no—
130#272—44no—
131C24×C3—31no—
132C24×C3—91no—
133C24×C3C2×C3⋊D311yescharacteristic
134C23×D3—31no—
135C22×A4—33no—
136C2×S4—36no—
137C2×S4—312no—
138C2×C3×D4—92no—
139C2×C3×D4—94no—
140C2×C3×D4—94no—
141C2×C3×D4—92no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—32no—
146C22×C3×C4—31no—
147C22×C3×C4—32no—
148C22×C3×C4—32no—
149C22×C3×C4—61no—
150C22×C3×C4—61no—
151C22×C3×C4—62no—
152C22×C3×C4—94no—
153C22×C3×C4—91no—
154C22×C3×C4—91no—
155C22×C3×C4—91no—
156C22×C3×C4—181no—
157C22×C3×C4—182no—
158C22×C3×C4—181no—
159C22×C3×C4C2×C3⋊D312yes—
160C2×#67—32no—
161C2×#67—32no—
162C2×#67—34no—
163C2×#67—34no—
164C22×Dic3—31no—
165C22×Dic3—31no—
166C22×Dic3—31no—
167C2×C4×D3—34no—
168C2×C4×D3—61no—
169C2×C4×D3—62no—
170C2×C4×D3—61no—
171C2×C4×D3—123no—
172C4×A4—36no—
173SL(2,ℤ/4)—36no—
174C3×C4⋊C4—94no—
175C3×C22⋊C4—94no—
176C3×C22⋊C4—92no—
177C3×C22⋊C4—92no—
178C3×C42—94no—
179#217—32no—
180#217—32no—
181(C2×D3)⋊C4—34no—
182Dic3⋊C4—34no—
183C4×Dic3—34no—
184C22×C32—32no—
185C22×C32—31no—
186C22×C32—32no—
187C22×C32—32no—
188C22×C32—31no—
189C22×C32—31no—
190C22×C32—43no—
191C22×C32—61no—
192C22×C32C2×C4×D311yescharacteristic, complemented
193C22×C32C2×S411yescharacteristic
194C2×C3×D3—34no—
195C2×C3×D3—62no—
196C2×C3×D3—64no—
197C2×C3×D3—64no—
198C2×C3×D3—62no—
199C2×C3×D3—126no—
200C2×C3×D3—1212no—
201C3×A4C2×C4×D313yescomplemented
202C32×C4—32no—
203C32×C4—46no—
204C32×C4C2×S412yescomplemented
205C3×Dic3—34no—
206C3×Dic3—62no—
207C3×Dic3—62no—
208C3×Dic3—126no—
209C22×D4—91no—
210C23×C4—91no—
211C23×C4#27211yescharacteristic, complemented
212C4×D4—94no—
213C4×D4—94no—
214C2×C4⋊C4—91no—
215C2×C22⋊C4—91no—
216C2×C22⋊C4—91no—
217C2×C42—91no—
218H(3,𝔽3)—41noSylow
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—31no—
222C23×C3—32no—
223C23×C3—31no—
224C23×C3—32no—
225C23×C3—32no—
226C23×C3—31no—
227C23×C3—61no—
228C23×C3—61no—
229C23×C3—62no—
230C23×C3—92no—
231C23×C3—94no—
232C23×C3—181no—
233C23×C3—182no—
234C23×C3—181no—
235C23×C3C4×C3⋊D312yes—
236C23×C3C22×C3⋊D311yescharacteristic
237C22×D3—32no—
238C22×D3—34no—
239C22×D3—61no—
240C22×D3—61no—
241C22×D3—62no—
242C22×D3—123no—
243C2×A4—36no—
244C2×A4—33no—
245S4—312no—
246C3×D4—98no—
247C3×D4—94no—
248C3×D4—94no—
249C2×C3×C4—32no—
250C2×C3×C4—31no—
251C2×C3×C4—32no—
252C2×C3×C4—31no—
253C2×C3×C4—32no—
254C2×C3×C4—32no—
255C2×C3×C4—32no—
256C2×C3×C4—32no—
257C2×C3×C4—31no—
258C2×C3×C4—62no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—62no—
264C2×C3×C4—61no—
265C2×C3×C4—92no—
266C2×C3×C4—94no—
267C2×C3×C4—92no—
268C2×C3×C4—92no—
269C2×C3×C4—123no—
270C2×C3×C4—184no—
271C2×C3×C4—182no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4—184no—
275C2×C3×C4C3⋊S411yescharacteristic
276#67—38no—
277#67—34no—
278#67—34no—
279C2×Dic3—32no—
280C2×Dic3—34no—
281C2×Dic3—32no—
282C2×Dic3—32no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—62no—
286C2×Dic3—123no—
287C4×D3—64no—
288C4×D3—64no—
289C4×D3—1212no—
290C2×C32—31no—
291C2×C32—31no—
292C2×C32—32no—
293C2×C32—46no—
294C2×C32—43no—
295C2×C32C4×S412yescomplemented
296C2×C32C22×S411yescharacteristic
297C3×D3—64no—
298C3×D3—1212no—
299C24—91no—
300C24C2×#27211yescharacteristic
301C2×D4—92no—
302C2×D4—94no—
303C2×D4—94no—
304C2×D4—92no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—32no—
308C22×C4—94no—
309C22×C4—91no—
310C22×C4—91no—
311C22×C4—91no—
312C22×C4—181no—
313C22×C4—182no—
314C22×C4—181no—
315C22×C4C2×#27212yescomplemented
316C4⋊C4—94no—
317C22⋊C4—94no—
318C22⋊C4—92no—
319C22⋊C4—92no—
320C42—94no—
321C22×C3—32no—
322C22×C3—31no—
323C22×C3—31no—
324C22×C3—32no—
325C22×C3—31no—
326C22×C3—32no—
327C22×C3—32no—
328C22×C3—32no—
329C22×C3—31no—
330C22×C3—31no—
331C22×C3—32no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—62no—
338C22×C3—61no—
339C22×C3—62no—
340C22×C3—62no—
341C22×C3—61no—
342C22×C3—61no—
343C22×C3—62no—
344C22×C3—94no—
345C22×C3—123no—
346C22×C3—182no—
347C22×C3—184no—
348C22×C3—184no—
349C22×C3—182no—
350C22×C3C2×C4×C3⋊D311yescharacteristic
351C22×C3C2×C3⋊S411yescharacteristic
352C2×D3—34no—
353C2×D3—62no—
354C2×D3—62no—
355C2×D3—64no—
356C2×D3—64no—
357C2×D3—126no—
358C2×D3—1212no—
359A4—33no—
360C3×C4—32no—
361C3×C4—32no—
362C3×C4—32no—
363C3×C4—62no—
364C3×C4—94no—
365C3×C4—126no—
366C3×C4—182no—
367C3×C4—182no—
368C3×C4C2×C3⋊S412yes—
369Dic3—34no—
370Dic3—62no—
371Dic3—62no—
372Dic3—126no—
373C32—43no—
374C32C2×C4×S411yescharacteristic, complemented
375C23—31no—
376C23—31no—
377C23—32no—
378C23—92no—
379C23—94no—
380C23—181no—
381C23—182no—
382C23—181no—
383C23C4×#27212yescomplemented
384C23C22×#27211yescharacteristic
385D4—98no—
386D4—94no—
387D4—94no—
388C2×C4—31no—
389C2×C4—32no—
390C2×C4—32no—
391C2×C4—32no—
392C2×C4—61no—
393C2×C4—92no—
394C2×C4—94no—
395C2×C4—92no—
396C2×C4—92no—
397C2×C4—184no—
398C2×C4—181no—
399C2×C4—182no—
400C2×C4—181no—
401C2×C4—184no—
402C2×C4#625811yescharacteristic, direct product factor
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—32no—
406C2×C3—31no—
407C2×C3—31no—
408C2×C3—32no—
409C2×C3—31no—
410C2×C3—32no—
411C2×C3—61no—
412C2×C3—61no—
413C2×C3—62no—
414C2×C3—123no—
415C2×C3—126no—
416C2×C3—184no—
417C2×C3C4×C3⋊S412yes—
418C2×C3C22×C3⋊S411yescharacteristic
419D3—64no—
420D3—1212no—
421C22—32no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—32no—
426C22—32no—
427C22—61no—
428C22—94no—
429C22—182no—
430C22—184no—
431C22—182no—
432C22—184no—
433C22C2×C4×#27211yescharacteristic, complemented
434C22C2×#625811yescharacteristic
435C4—32no—
436C4—94no—
437C4—182no—
438C4—182no—
439C4C2×#625812yesdirect product factor
440C3—31no—
441C3—123no—
442C3C2×C4×C3⋊S411yescharacteristic
443C2—31no—
444C2—31no—
445C2—32no—
446C2—184no—
447C2C4×#625812yesdirect product factor
448C2C22×#625811yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4C3⋊S4
commutator, hypocenterC22⋊H(3,𝔽3)C22×C4
FrattiniC2×C3C22×C3⋊S4
FittingC23×C32×C4D3
socleC24×C3C2×C3⋊D3
prime radicalC22×#6258C2
cosocleC2×C22⋊H(3,𝔽3)C23
LCSG ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊H(3,𝔽3)C2semidirect
3. C2×#88613C2semidirect
4. C4×#6258C2direct (N × H)
8. C2×#6258C4direct (N × H)
11. C4×C22⋊H(3,𝔽3)C22semidirect
12. #88613C22semidirect
14. C23×C32×C4D3semidirect
17. C2×C3×C4×A4D3semidirect
27. C2×C22⋊H(3,𝔽3)C2×C4semidirect
29. #6258C2×C4direct (N × H)
44. C22×C32×C4C2×D3semidirect
57. C3×C4×A4C2×D3semidirect
67. C22⋊H(3,𝔽3)C22×C4semidirect
95. C23×C32C4×D3semidirect
103. C2×C3×A4C4×D3semidirect
112. C2×C32×C4S4semidirect
192. C22×C32C2×C4×D3semidirect
201. C3×A4C2×C4×D3semidirect
204. C32×C4C2×S4semidirect
211. C23×C4#272semidirect
295. C2×C32C4×S4semidirect
315. C22×C4C2×#272semidirect
374. C32C2×C4×S4semidirect
383. C23C4×#272semidirect
402. C2×C4#6258direct (N × H)
433. C22C2×C4×#272semidirect
439. C4C2×#6258direct (N × H)
447. C2C4×#6258direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62582
C2×C4×C22⋊H(3,𝔽3)2
C2×#886132
C4×#62582
C2×C3×C4×S43
C2×C3×C4×#673
C2×C4×#2724