Finite Group Databasebeta

← C2×C4⋊#6258 #13164476

Subgroups — C2×C4⋊#6258 #13164476

Order 1728.

1128 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 91 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×#6258C212yesmaximal, complemented
2C2×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C4⋊#6258C214yesmaximal, direct product factor
4C2×C3×C4⋊S4—33nomaximal
5C2×C3×#930—31nomaximal
6C22×C22⋊H(3,𝔽3)C2211yescharacteristic
7C2×#6258—24no—
8C2×#6258C2214yescomplemented
9C2×C4⋊#272—41nomaximal
10C4×C22⋊H(3,𝔽3)C2212yescomplemented
11C22×C3×S4—36no—
12C23×C32×C4D311yescharacteristic, complemented
13C22×C3×#67—32no—
14C22×C3×D12—31no—
15C2×C3×C4×A4D313yescomplemented
16C3×C4⋊S4—312no—
17C3×#930—34no—
18C3×#930—34no—
19C2×C3×(C2×D3)⋊C4—32no—
20C2×C3×C4⋊Dic3—31no—
21C22×#272—42no—
22C2×C22⋊H(3,𝔽3)D412yescomplemented
23C2×C22⋊H(3,𝔽3)C2311yescharacteristic
24#6258—24no—
25C2×C4×H(3,𝔽3)—41no—
26C4⋊#272—44no—
27C2×C4⋊S4—33no—
28C2×C3×#121—91no—
29C2×#930—31no—
30C24×C32C2×D311yescharacteristic
31C23×C3×D3—32no—
32C22×C3×A4C2×D313yes—
33C2×C3×S4—312no—
34C2×C3×S4—612no—
35C22×C32×C4—31no—
36C22×C32×C4—31no—
37C22×C32×C4—32no—
38C22×C32×C4C2×D312yescomplemented
39C2×C3×#67—34no—
40C2×C3×#67—34no—
41C2×C3×#67—64no—
42C2×C3×#67—64no—
43C22×C3×Dic3—32no—
44C2×C3×D12—34no—
45C2×C3×D12—61no—
46C2×C3×D12—62no—
47C2×C3×D12—61no—
48C2×C3×D12—123no—
49C3×C4×A4C2×D316yescomplemented
50C3×(C2×D3)⋊C4—38no—
51C3×C4⋊Dic3—34no—
52C22×H(3,𝔽3)—41no—
53C2×#272—44no—
54C2×#272—84no—
55C22⋊H(3,𝔽3)C2×D411yescharacteristic, complemented
56C4×H(3,𝔽3)—42no—
57C22×S4—36no—
58C22×C3×D4—91no—
59C22×C3×D4—92no—
60C23×C3×C4—31no—
61C23×C3×C4C3⋊D311yescharacteristic
62C22×#67—32no—
63C22×D12—31no—
64C2×C4×A4—33no—
65C4⋊S4—312no—
66C3×#121—94no—
67C3×#121—94no—
68C2×C3×C4⋊C4—91no—
69C2×C3×C22⋊C4—92no—
70#930—34no—
71#930—34no—
72C2×(C2×D3)⋊C4—32no—
73C2×C4⋊Dic3—31no—
74C23×C32—31no—
75C23×C32—31no—
76C23×C32—32no—
77C23×C32D1212yescomplemented
78C23×C32C22×D311yescharacteristic
79C22×C3×D3—34no—
80C22×C3×D3—64no—
81C22×C3×D3—62no—
82C22×C3×D3—64no—
83C22×C3×D3—62no—
84C22×C3×D3—126no—
85C2×C3×A4D1216yescomplemented
86C2×C3×A4C22×D313yes—
87C3×S4—612no—
88C2×C32×C4—31no—
89C2×C32×C4—32no—
90C2×C32×C4—32no—
91C2×C32×C4—32no—
92C2×C32×C4—43no—
93C2×C32×C4—61no—
94C2×C32×C4S411yescharacteristic, complemented
95C3×#67—68no—
96C3×#67—64no—
97C3×#67—64no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—64no—
100C3×D12—64no—
101C3×D12—64no—
102C3×D12—1212no—
103C2×#121—91noSylow
104C2×H(3,𝔽3)—42no—
105C2×H(3,𝔽3)—41no—
106#272—84no—
107C24×C3—31no—
108C24×C3—92no—
109C24×C3C2×C3⋊D311yescharacteristic
110C23×D3—32no—
111C22×A4—33no—
112C2×S4—312no—
113C2×S4—612no—
114C2×C3×D4—94no—
115C2×C3×D4—94no—
116C2×C3×D4—94no—
117C2×C3×D4—181no—
118C2×C3×D4—182no—
119C2×C3×D4—184no—
120C2×C3×D4—184no—
121C2×C3×D4—181no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—32no—
126C22×C3×C4—31no—
127C22×C3×C4—32no—
128C22×C3×C4—32no—
129C22×C3×C4—61no—
130C22×C3×C4—61no—
131C22×C3×C4—62no—
132C22×C3×C4—92no—
133C22×C3×C4C2×C3⋊D312yes—
134C2×#67—34no—
135C2×#67—34no—
136C2×#67—64no—
137C2×#67—64no—
138C22×Dic3—32no—
139C2×D12—34no—
140C2×D12—61no—
141C2×D12—62no—
142C2×D12—61no—
143C2×D12—123no—
144C4×A4—36no—
145C3×C4⋊C4—94no—
146C3×C22⋊C4—98no—
147(C2×D3)⋊C4—38no—
148C4⋊Dic3—34no—
149C22×C32—32no—
150C22×C32—31no—
151C22×C32—32no—
152C22×C32—32no—
153C22×C32—31no—
154C22×C32—31no—
155C22×C32—43no—
156C22×C32—61no—
157C22×C32C2×D1211yescharacteristic, complemented
158C22×C32C2×S411yescharacteristic
159C2×C3×D3—64no—
160C2×C3×D3—64no—
161C2×C3×D3—64no—
162C2×C3×D3—1212no—
163C2×C3×D3—124no—
164C2×C3×D3—124no—
165C2×C3×D3—2412no—
166C3×A4C2×D1213yescomplemented
167C32×C4—32no—
168C32×C4—46no—
169C32×C4C2×S412yescomplemented
170C3×Dic3—64no—
171C22×D4—91no—
172C22×D4—92no—
173C23×C4#27211yescharacteristic, complemented
174#121—94no—
175#121—94no—
176C2×C4⋊C4—91no—
177C2×C22⋊C4—92no—
178H(3,𝔽3)—41noSylow
179C23×C3—31no—
180C23×C3—31no—
181C23×C3—31no—
182C23×C3—32no—
183C23×C3—31no—
184C23×C3—32no—
185C23×C3—32no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3—61no—
189C23×C3—62no—
190C23×C3—94no—
191C23×C3—184no—
192C23×C3—182no—
193C23×C3—184no—
194C23×C3—182no—
195C23×C3C4⋊(C3⋊D3)12yes—
196C23×C3C22×C3⋊D311yescharacteristic
197C22×D3—34no—
198C22×D3—62no—
199C22×D3—62no—
200C22×D3—64no—
201C22×D3—64no—
202C22×D3—126no—
203C2×A4—36no—
204C2×A4—33no—
205S4—612no—
206C3×D4—184no—
207C3×D4—188no—
208C3×D4—184no—
209C3×D4—184no—
210C3×D4—184no—
211C2×C3×C4—32no—
212C2×C3×C4—31no—
213C2×C3×C4—32no—
214C2×C3×C4—31no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—31no—
220C2×C3×C4—62no—
221C2×C3×C4—62no—
222C2×C3×C4—62no—
223C2×C3×C4—61no—
224C2×C3×C4—61no—
225C2×C3×C4—62no—
226C2×C3×C4—61no—
227C2×C3×C4—94no—
228C2×C3×C4—123no—
229C2×C3×C4—184no—
230C2×C3×C4C3⋊S411yescharacteristic
231#67—68no—
232#67—64no—
233#67—64no—
234C2×Dic3—34no—
235C2×Dic3—64no—
236D12—64no—
237D12—64no—
238D12—1212no—
239C2×C32—31no—
240C2×C32—31no—
241C2×C32—32no—
242C2×C32—46no—
243C2×C32—43no—
244C2×C32C4⋊S412yescomplemented
245C2×C32C22×S411yescharacteristic
246C3×D3—124no—
247C3×D3—2412no—
248C24—92no—
249C24C2×#27211yescharacteristic
250C2×D4—94no—
251C2×D4—94no—
252C2×D4—94no—
253C2×D4—181no—
254C2×D4—182no—
255C2×D4—184no—
256C2×D4—184no—
257C2×D4—181no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—32no—
261C22×C4—92no—
262C22×C4C2×#27212yescomplemented
263C4⋊C4—94no—
264C22⋊C4—98no—
265C22×C3—32no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—32no—
269C22×C3—31no—
270C22×C3—32no—
271C22×C3—32no—
272C22×C3—32no—
273C22×C3—31no—
274C22×C3—31no—
275C22×C3—32no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—62no—
282C22×C3—61no—
283C22×C3—62no—
284C22×C3—62no—
285C22×C3—61no—
286C22×C3—61no—
287C22×C3—62no—
288C22×C3—123no—
289C22×C3—184no—
290C22×C3—184no—
291C22×C3—184no—
292C22×C3—364no—
293C22×C3—364no—
294C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
295C22×C3C2×C3⋊S411yescharacteristic
296C2×D3—64no—
297C2×D3—64no—
298C2×D3—64no—
299C2×D3—1212no—
300C2×D3—124no—
301C2×D3—124no—
302C2×D3—2412no—
303A4—33no—
304C3×C4—32no—
305C3×C4—32no—
306C3×C4—32no—
307C3×C4—62no—
308C3×C4—126no—
309C3×C4—184no—
310C3×C4C2×C3⋊S412yes—
311Dic3—64no—
312C32—43no—
313C32C2×C4⋊S411yescharacteristic, complemented
314C23—31no—
315C23—31no—
316C23—32no—
317C23—94no—
318C23—182no—
319C23—184no—
320C23—184no—
321C23—182no—
322C23C4⋊#27212yescomplemented
323C23C22×#27211yescharacteristic
324D4—184no—
325D4—188no—
326D4—184no—
327D4—184no—
328D4—184no—
329C2×C4—31no—
330C2×C4—32no—
331C2×C4—32no—
332C2×C4—32no—
333C2×C4—61no—
334C2×C4—94no—
335C2×C4—184no—
336C2×C4#625811yescharacteristic, complemented
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—32no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—32no—
343C2×C3—31no—
344C2×C3—32no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—62no—
348C2×C3—123no—
349C2×C3—126no—
350C2×C3—364no—
351C2×C3C4⋊(C3⋊S4)12yes—
352C2×C3C22×C3⋊S411yescharacteristic
353D3—124no—
354D3—2412no—
355C22—32no—
356C22—31no—
357C22—31no—
358C22—31no—
359C22—32no—
360C22—32no—
361C22—61no—
362C22—184no—
363C22—184no—
364C22—184no—
365C22—364no—
366C22—364no—
367C22C2×C4⋊#27211yescharacteristic, complemented
368C22C2×#625811yescharacteristic
369C4—32no—
370C4—184no—
371C4C2×#625812yescomplemented
372C3—31no—
373C3—123no—
374C3C2×C4⋊(C3⋊S4)11yescharacteristic
375C2—31no—
376C2—31no—
377C2—32no—
378C2—364no—
379C2C4⋊#625812yesdirect product factor
380C2C22×#625811yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊S4
commutator, cosocleC2×C22⋊H(3,𝔽3)C23
FrattiniC2×C3C22×C3⋊S4
FittingC23×C32×C4D3
socleC24×C3C2×C3⋊D3
hypercenterC2×C3×C4C3⋊S4
hypocenterC22⋊H(3,𝔽3)C2×D4
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6258C2semidirect
2. C2×C4×C22⋊H(3,𝔽3)C2semidirect
3. C4⋊#6258C2direct (N × H)
8. C2×#6258C22semidirect
10. C4×C22⋊H(3,𝔽3)C22semidirect
12. C23×C32×C4D3semidirect
15. C2×C3×C4×A4D3semidirect
22. C2×C22⋊H(3,𝔽3)D4semidirect
38. C22×C32×C4C2×D3semidirect
49. C3×C4×A4C2×D3semidirect
55. C22⋊H(3,𝔽3)C2×D4semidirect
77. C23×C32D12semidirect
85. C2×C3×A4D12semidirect
94. C2×C32×C4S4semidirect
157. C22×C32C2×D12semidirect
166. C3×A4C2×D12semidirect
169. C32×C4C2×S4semidirect
173. C23×C4#272semidirect
244. C2×C32C4⋊S4semidirect
262. C22×C4C2×#272semidirect
313. C32C2×C4⋊S4semidirect
322. C23C4⋊#272semidirect
336. C2×C4#6258semidirect
367. C22C2×C4⋊#272semidirect
371. C4C2×#6258semidirect
379. 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
C4⋊#62582
C2×C3×C4⋊S43
C2×C3×#9303
C2×C4⋊#2724