Finite Group Databasebeta

← C2×C4⋊((C3⋊S4)⋊C3) #13164396

Subgroups — C2×C4⋊((C3⋊S4)⋊C3) #13164396

Order 1728.

944 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 86 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×(C3⋊S4)⋊C3C212yesmaximal, complemented
2C2×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C4⋊((C3⋊S4)⋊C3)C214yesmaximal, direct product factor
4C2×C4⋊(C3⋊S4)C311yesmaximal, characteristic, complemented
5C2×C3×#930—31nomaximal
6C22×C22⋊H(3,𝔽3)C2211yescharacteristic
7C2×(C3⋊S4)⋊C3—24no—
8C2×(C3⋊S4)⋊C3C2214yescomplemented
9C2×C4⋊((C3⋊D3)⋊C3)—41nomaximal
10C4×C22⋊H(3,𝔽3)C2212yescomplemented
11C22×C3⋊S4C2×C312yescomplemented
12C23×C32×C4D311yescharacteristic, complemented
13C22×C3×#67—32no—
14C22×C3×D12—31no—
15C2×C3×C4×A4—21no—
16C2×C3×C4×A4C2×C311yescharacteristic, complemented
17C4⋊(C3⋊S4)C2×C314yescomplemented
18C3×#930—34no—
19C3×#930—34no—
20C2×C3×(C2×D3)⋊C4—32no—
21C2×C3×C4⋊Dic3—31no—
22C22×(C3⋊D3)⋊C3—42no—
23C2×C22⋊H(3,𝔽3)D412yescomplemented
24C2×C22⋊H(3,𝔽3)C2311yescharacteristic
25(C3⋊S4)⋊C3—24no—
26C2×C4×H(3,𝔽3)—41no—
27C4⋊((C3⋊D3)⋊C3)—44no—
28C2×C4⋊S4—91no—
29C2×C3×#121—91no—
30C2×#930—31no—
31C24×C32C2×D311yescharacteristic
32C23×C3×D3—32no—
33C22×C3×A4—21no—
34C22×C3×A4C22×C311yescharacteristic
35C2×C3⋊S4—24no—
36C2×C3⋊S4C22×C314yescomplemented
37C22×C32×C4—31no—
38C22×C32×C4—31no—
39C22×C32×C4—32no—
40C22×C32×C4C2×D312yescomplemented
41C2×C4⋊(C3⋊D3)—41no—
42C2×C3×#67—34no—
43C2×C3×#67—34no—
44C2×C3×#67—64no—
45C2×C3×#67—64no—
46C22×C3×Dic3—32no—
47C2×C3×D12—34no—
48C2×C3×D12—61no—
49C2×C3×D12—62no—
50C2×C3×D12—61no—
51C3×C4×A4—22no—
52C3×C4×A4C22×C312yescomplemented
53C3×(C2×D3)⋊C4—38no—
54C3×C4⋊Dic3—34no—
55C22×H(3,𝔽3)—41no—
56C2×(C3⋊D3)⋊C3—44no—
57C2×(C3⋊D3)⋊C3—84no—
58C22⋊H(3,𝔽3)C2×D411yescharacteristic, complemented
59C4×H(3,𝔽3)—42no—
60C22×S4—92no—
61C22×C3×D4—91no—
62C22×C3×D4—92no—
63C23×C3×C4—31no—
64C23×C3×C4C3×D311yescharacteristic
65C22×#67—32no—
66C22×D12—31no—
67C2×C4×A4—31no—
68C2×C4×A4—61no—
69C4⋊S4—94no—
70C3×#121—94no—
71C3×#121—94no—
72C2×C3×C4⋊C4—91no—
73C2×C3×C22⋊C4—92no—
74#930—34no—
75#930—34no—
76C2×(C2×D3)⋊C4—32no—
77C2×C4⋊Dic3—31no—
78C23×C32—31no—
79C23×C32—31no—
80C23×C32—32no—
81C23×C32D1212yescomplemented
82C23×C32C22×D311yescharacteristic
83C22×C3⋊D3—42no—
84C22×C3×D3—34no—
85C22×C3×D3—64no—
86C22×C3×D3—62no—
87C22×C3×D3—64no—
88C22×C3×D3—62no—
89C2×C3×A4—22no—
90C2×C3×A4—21no—
91C2×C3×A4C3×D412yescomplemented
92C2×C3×A4C23×C311yescharacteristic
93C3⋊S4—24no—
94C2×C32×C4—31no—
95C2×C32×C4—32no—
96C2×C32×C4—32no—
97C2×C32×C4—32no—
98C2×C32×C4—41no—
99C2×C32×C4—61no—
100C2×C32×C4—81no—
101C2×C32×C4S411yescharacteristic, complemented
102C4⋊(C3⋊D3)—44no—
103C3×#67—68no—
104C3×#67—64no—
105C3×#67—64no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—64no—
108C3×D12—64no—
109C3×D12—64no—
110C2×#121—91noSylow
111C2×H(3,𝔽3)—42no—
112C2×H(3,𝔽3)—41no—
113(C3⋊D3)⋊C3—84no—
114C24×C3—31no—
115C24×C3—92no—
116C24×C3C2×C3×D311yescharacteristic
117C23×D3—32no—
118C22×A4—31no—
119C22×A4—61no—
120C2×S4—94no—
121C2×S4—184no—
122C2×C3×D4—94no—
123C2×C3×D4—94no—
124C2×C3×D4—94no—
125C2×C3×D4—181no—
126C2×C3×D4—182no—
127C2×C3×D4—181no—
128C2×C3×D4—184no—
129C2×C3×D4—184no—
130C22×C3×C4—31no—
131C22×C3×C4—31no—
132C22×C3×C4—31no—
133C22×C3×C4—32no—
134C22×C3×C4—31no—
135C22×C3×C4—32no—
136C22×C3×C4—32no—
137C22×C3×C4—61no—
138C22×C3×C4—61no—
139C22×C3×C4—62no—
140C22×C3×C4—92no—
141C22×C3×C4C2×C3×D312yes—
142C2×#67—34no—
143C2×#67—34no—
144C2×#67—64no—
145C2×#67—64no—
146C22×Dic3—32no—
147C2×D12—34no—
148C2×D12—61no—
149C2×D12—62no—
150C2×D12—61no—
151C2×D12—361no—
152C4×A4—32no—
153C4×A4—62no—
154C3×C4⋊C4—94no—
155C3×C22⋊C4—98no—
156(C2×D3)⋊C4—38no—
157C4⋊Dic3—34no—
158C22×C32—32no—
159C22×C32—31no—
160C22×C32—32no—
161C22×C32—32no—
162C22×C32—31no—
163C22×C32—31no—
164C22×C32—41no—
165C22×C32—61no—
166C22×C32—81no—
167C22×C32C2×D1211yescharacteristic, complemented
168C22×C32C2×S411yescharacteristic
169C2×C3⋊D3—44no—
170C2×C3⋊D3—84no—
171C2×C3×D3—64no—
172C2×C3×D3—64no—
173C2×C3×D3—64no—
174C2×C3×D3—124no—
175C2×C3×D3—124no—
176C3×A4—21no—
177C3×A4C2×C3×D411yescharacteristic, complemented
178C32×C4—32no—
179C32×C4—42no—
180C32×C4—82no—
181C32×C4C2×S412yescomplemented
182C3×Dic3—64no—
183C22×D4—91no—
184C22×D4—92no—
185C23×C4(C3⋊D3)⋊C311yescharacteristic, complemented
186#121—94no—
187#121—94no—
188C2×C4⋊C4—91no—
189C2×C22⋊C4—92no—
190H(3,𝔽3)—41noSylow
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—31no—
194C23×C3—32no—
195C23×C3—31no—
196C23×C3—32no—
197C23×C3—32no—
198C23×C3—31no—
199C23×C3—61no—
200C23×C3—61no—
201C23×C3—62no—
202C23×C3—94no—
203C23×C3—182no—
204C23×C3—184no—
205C23×C3—182no—
206C23×C3—184no—
207C23×C3C3×D1212yes—
208C23×C3C22×C3×D311yescharacteristic
209C22×D3—34no—
210C22×D3—62no—
211C22×D3—64no—
212C22×D3—64no—
213C22×D3—62no—
214C22×D3—362no—
215C2×A4—32no—
216C2×A4—31no—
217C2×A4—62no—
218C2×A4—61no—
219S4—184no—
220C3×D4—184no—
221C3×D4—184no—
222C3×D4—188no—
223C3×D4—184no—
224C3×D4—184no—
225C2×C3×C4—32no—
226C2×C3×C4—31no—
227C2×C3×C4—32no—
228C2×C3×C4—31no—
229C2×C3×C4—32no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—31no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—62no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—62no—
240C2×C3×C4—61no—
241C2×C3×C4—94no—
242C2×C3×C4—121no—
243C2×C3×C4—184no—
244C2×C3×C4—241no—
245C2×C3×C4C3×S411yescharacteristic
246#67—68no—
247#67—64no—
248#67—64no—
249C2×Dic3—34no—
250C2×Dic3—64no—
251D12—64no—
252D12—64no—
253D12—364no—
254C2×C32—31no—
255C2×C32—31no—
256C2×C32—32no—
257C2×C32—41no—
258C2×C32—42no—
259C2×C32—81no—
260C2×C32—82no—
261C2×C32C4⋊S412yescomplemented
262C2×C32C22×S411yescharacteristic
263C3⋊D3—84no—
264C3×D3—124no—
265C24—92no—
266C24C2×(C3⋊D3)⋊C311yescharacteristic
267C2×D4—94no—
268C2×D4—94no—
269C2×D4—94no—
270C2×D4—181no—
271C2×D4—182no—
272C2×D4—184no—
273C2×D4—184no—
274C2×D4—181no—
275C22×C4—31no—
276C22×C4—31no—
277C22×C4—32no—
278C22×C4—92no—
279C22×C4C2×(C3⋊D3)⋊C312yescomplemented
280C4⋊C4—94no—
281C22⋊C4—98no—
282C22×C3—32no—
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—31no—
286C22×C3—32no—
287C22×C3—32no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—32no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—31no—
296C22×C3—61no—
297C22×C3—61no—
298C22×C3—62no—
299C22×C3—61no—
300C22×C3—62no—
301C22×C3—62no—
302C22×C3—61no—
303C22×C3—61no—
304C22×C3—62no—
305C22×C3—121no—
306C22×C3—184no—
307C22×C3—184no—
308C22×C3—184no—
309C22×C3—241no—
310C22×C3—364no—
311C22×C3—364no—
312C22×C3C2×C3×D1211yescharacteristic
313C22×C3C2×C3×S411yescharacteristic
314C2×D3—64no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—124no—
318C2×D3—124no—
319C2×D3—364no—
320C2×D3—724no—
321A4—31no—
322A4—61no—
323C3×C4—32no—
324C3×C4—32no—
325C3×C4—32no—
326C3×C4—62no—
327C3×C4—122no—
328C3×C4—184no—
329C3×C4—242no—
330C3×C4C2×C3×S412yes—
331Dic3—64no—
332C32—41no—
333C32—81no—
334C32C2×C4⋊S411yescharacteristic, complemented
335C23—31no—
336C23—31no—
337C23—32no—
338C23—94no—
339C23—182no—
340C23—184no—
341C23—184no—
342C23—182no—
343C23C4⋊((C3⋊D3)⋊C3)12yescomplemented
344C23C22×(C3⋊D3)⋊C311yescharacteristic
345D4—184no—
346D4—188no—
347D4—184no—
348D4—184no—
349D4—184no—
350C2×C4—31no—
351C2×C4—32no—
352C2×C4—32no—
353C2×C4—32no—
354C2×C4—61no—
355C2×C4—94no—
356C2×C4—184no—
357C2×C4(C3⋊S4)⋊C311yescharacteristic, complemented
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—32no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—32no—
364C2×C3—31no—
365C2×C3—32no—
366C2×C3—61no—
367C2×C3—61no—
368C2×C3—62no—
369C2×C3—121no—
370C2×C3—122no—
371C2×C3—241no—
372C2×C3—242no—
373C2×C3—364no—
374C2×C3C3×C4⋊S412yes—
375C2×C3C22×C3×S411yescharacteristic
376D3—124no—
377D3—724no—
378C22—32no—
379C22—31no—
380C22—31no—
381C22—31no—
382C22—32no—
383C22—32no—
384C22—61no—
385C22—184no—
386C22—184no—
387C22—184no—
388C22—364no—
389C22—364no—
390C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic, complemented
391C22C2×(C3⋊S4)⋊C311yescharacteristic
392C4—32no—
393C4—184no—
394C4C2×(C3⋊S4)⋊C312yescomplemented
395C3—31no—
396C3—121no—
397C3—241no—
398C3C2×C3×C4⋊S411yescharacteristic
399C2—31no—
400C2—31no—
401C2—32no—
402C2—364no—
403C2C4⋊((C3⋊S4)⋊C3)12yesdirect product factor
404C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
centerC22C2×(C3⋊S4)⋊C3
commutator, cosocleC2×C3×A4C23×C3
FrattiniC2×C3C22×C3×S4
FittingC23×C32×C4D3
socleC24×C3C2×C3×D3
hypercenterC2×C4(C3⋊S4)⋊C3
hypocenterC3×A4C2×C3×D4
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×(C3⋊S4)⋊C3C2semidirect
2. C2×C4×C22⋊H(3,𝔽3)C2semidirect
3. C4⋊((C3⋊S4)⋊C3)C2direct (N × H)
4. C2×C4⋊(C3⋊S4)C3semidirect
8. C2×(C3⋊S4)⋊C3C22semidirect
10. C4×C22⋊H(3,𝔽3)C22semidirect
11. C22×C3⋊S4C2×C3semidirect
12. C23×C32×C4D3semidirect
16. C2×C3×C4×A4C2×C3semidirect
17. C4⋊(C3⋊S4)C2×C3semidirect
23. C2×C22⋊H(3,𝔽3)D4semidirect
36. C2×C3⋊S4C22×C3semidirect
40. C22×C32×C4C2×D3semidirect
52. C3×C4×A4C22×C3semidirect
58. C22⋊H(3,𝔽3)C2×D4semidirect
81. C23×C32D12semidirect
91. C2×C3×A4C3×D4semidirect
101. C2×C32×C4S4semidirect
167. C22×C32C2×D12semidirect
177. C3×A4C2×C3×D4semidirect
181. C32×C4C2×S4semidirect
185. C23×C4(C3⋊D3)⋊C3semidirect
261. C2×C32C4⋊S4semidirect
279. C22×C4C2×(C3⋊D3)⋊C3semidirect
334. C32C2×C4⋊S4semidirect
343. C23C4⋊((C3⋊D3)⋊C3)semidirect
357. C2×C4(C3⋊S4)⋊C3semidirect
390. C22C2×C4⋊((C3⋊D3)⋊C3)semidirect
394. C4C2×(C3⋊S4)⋊C3semidirect
403. C2C4⋊((C3⋊S4)⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×(C3⋊S4)⋊C32
C2×C4×C22⋊H(3,𝔽3)2
C4⋊((C3⋊S4)⋊C3)2
C2×C4⋊(C3⋊S4)3
C2×C3×#9303
C2×C4⋊((C3⋊D3)⋊C3)4