Finite Group Databasebeta

← C2×#64259⋊C3 #13164394

Subgroups — C2×#64259⋊C3 #13164394

Order 1728.

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

normal subgroups only

This group has 688 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
1C2×C4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×(C3⋊SL(2,ℤ/4))⋊C3C212yesmaximal
3#64259⋊C3C214yesmaximal, direct product factor
4C2×#64259C311yesmaximal, characteristic, complemented
5C2×C3×#924—31nomaximal
6C22×C22⋊H(3,𝔽3)C2211yescharacteristic
7C2×#634⋊C3—41nomaximal
8C4×C22⋊H(3,𝔽3)C2212yes—
9(C3⋊SL(2,ℤ/4))⋊C3C2214yes—
10C23×C32×C4D311yescharacteristic
11C22×C3×Dic6—31no—
12C2×C3×C4×A4—21no—
13C2×C3×C4×A4C2×C311yescharacteristic
14C2×C3⋊SL(2,ℤ/4)C2×C312yes—
15#64259C2×C314yescomplemented
16C2×C3×#217—32no—
17C2×C3×C4⋊Dic3—31no—
18C3×#924—34no—
19C3×#924—34no—
20C2×C3×Dic3⋊C4—32no—
21C2×C22⋊H(3,𝔽3)Q812yescomplemented
22C2×C22⋊H(3,𝔽3)C2311yescharacteristic
23C2×C4×H(3,𝔽3)—41no—
24C2×(C3⋊Dic3)⋊C3—42no—
25#634⋊C3—44no—
26C2×A4⋊Q8—91no—
27C2×C3×C22⋊Q8—91no—
28C2×#924—31no—
29C24×C32C2×D311yescharacteristic
30C22×C3×A4—21no—
31C22×C3×A4C22×C311yescharacteristic
32C22×C32×C4—31no—
33C22×C32×C4—31no—
34C22×C32×C4—32no—
35C22×C32×C4C2×D312yes—
36C2×#634—41no—
37C22×C3×Dic3—32no—
38C22×C3×Dic3—32no—
39C2×C3×Dic6—34no—
40C2×C3×Dic6—61no—
41C2×C3×Dic6—62no—
42C2×C3×Dic6—61no—
43C3×C4×A4—22no—
44C3×C4×A4C22×C312yes—
45C3⋊SL(2,ℤ/4)C22×C314yes—
46C3×#217—34no—
47C3×#217—34no—
48C3×C4⋊Dic3—34no—
49C3×Dic3⋊C4—38no—
50C22×H(3,𝔽3)—41no—
51C22⋊H(3,𝔽3)C2×Q811yescharacteristic, complemented
52C4×H(3,𝔽3)—42no—
53(C3⋊Dic3)⋊C3—44no—
54C22×C3×Q8—91no—
55C23×C3×C4—31no—
56C23×C3×C4C3×D311yescharacteristic
57C22×Dic6—31no—
58C2×C4×A4—31no—
59C2×C4×A4—61no—
60C2×SL(2,ℤ/4)—92no—
61A4⋊Q8—94no—
62C3×C22⋊Q8—94no—
63C3×C22⋊Q8—94no—
64C2×C3×C4⋊C4—92no—
65C2×C3×C4⋊C4—91no—
66C2×C3×C22⋊C4—92no—
67C2×#217—32no—
68C2×C4⋊Dic3—31no—
69#924—34no—
70#924—34no—
71C2×Dic3⋊C4—32no—
72C23×C32—31no—
73C23×C32—32no—
74C23×C32—31no—
75C23×C32Dic612yescomplemented
76C23×C32C22×D311yescharacteristic
77C2×C3×A4—22no—
78C2×C3×A4—21no—
79C2×C3×A4C3×Q812yescomplemented
80C2×C3×A4C23×C311yescharacteristic
81C2×C32×C4—31no—
82C2×C32×C4—32no—
83C2×C32×C4—32no—
84C2×C32×C4—32no—
85C2×C32×C4—41no—
86C2×C32×C4—61no—
87C2×C32×C4—81no—
88C2×C32×C4S411yescharacteristic
89C2×C3⋊Dic3—42no—
90#634—44no—
91C2×C3×Dic3—34no—
92C2×C3×Dic3—34no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—64no—
95C2×C3×Dic3—62no—
96C3×Dic6—64no—
97C3×Dic6—64no—
98C2×C22⋊Q8—91noSylow
99C2×H(3,𝔽3)—42no—
100C2×H(3,𝔽3)—41no—
101C24×C3—31no—
102C24×C3C2×C3×D311yescharacteristic
103C22×A4—31no—
104C22×A4—61no—
105C2×C3×Q8—94no—
106C2×C3×Q8—181no—
107C2×C3×Q8—182no—
108C2×C3×Q8—181no—
109C22×C3×C4—31no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—32no—
113C22×C3×C4—31no—
114C22×C3×C4—32no—
115C22×C3×C4—32no—
116C22×C3×C4—61no—
117C22×C3×C4—61no—
118C22×C3×C4—62no—
119C22×C3×C4—92no—
120C22×C3×C4—92no—
121C22×C3×C4C2×C3×D312yes—
122C22×Dic3—32no—
123C22×Dic3—32no—
124C2×Dic6—34no—
125C2×Dic6—61no—
126C2×Dic6—62no—
127C2×Dic6—61no—
128C2×Dic6—361no—
129C4×A4—32no—
130C4×A4—62no—
131SL(2,ℤ/4)—94no—
132C3×C4⋊C4—98no—
133C3×C4⋊C4—94no—
134C3×C22⋊C4—94no—
135C3×C22⋊C4—94no—
136#217—34no—
137#217—34no—
138C4⋊Dic3—34no—
139Dic3⋊C4—38no—
140C22×C32—32no—
141C22×C32—31no—
142C22×C32—32no—
143C22×C32—32no—
144C22×C32—31no—
145C22×C32—31no—
146C22×C32—41no—
147C22×C32—61no—
148C22×C32—81no—
149C22×C32C2×Dic611yescharacteristic, complemented
150C22×C32C2×S411yescharacteristic
151C3×A4—21no—
152C3×A4C2×C3×Q811yescharacteristic, complemented
153C32×C4—32no—
154C32×C4—42no—
155C32×C4—82no—
156C32×C4C2×S412yes—
157C3⋊Dic3—44no—
158C3×Dic3—64no—
159C3×Dic3—64no—
160C22×Q8—91no—
161C23×C4(C3⋊D3)⋊C311yescharacteristic
162C22⋊Q8—94no—
163C22⋊Q8—94no—
164C2×C4⋊C4—92no—
165C2×C4⋊C4—91no—
166C2×C22⋊C4—92no—
167H(3,𝔽3)—41noSylow
168C23×C3—31no—
169C23×C3—31no—
170C23×C3—31no—
171C23×C3—32no—
172C23×C3—31no—
173C23×C3—32no—
174C23×C3—32no—
175C23×C3—31no—
176C23×C3—61no—
177C23×C3—61no—
178C23×C3—62no—
179C23×C3C3×Dic612yes—
180C23×C3C22×C3×D311yescharacteristic
181C2×A4—32no—
182C2×A4—31no—
183C2×A4—62no—
184C2×A4—61no—
185C3×Q8—184no—
186C3×Q8—184no—
187C2×C3×C4—32no—
188C2×C3×C4—31no—
189C2×C3×C4—32no—
190C2×C3×C4—31no—
191C2×C3×C4—32no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—32no—
195C2×C3×C4—31no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—62no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—62no—
202C2×C3×C4—61no—
203C2×C3×C4—94no—
204C2×C3×C4—94no—
205C2×C3×C4—121no—
206C2×C3×C4—184no—
207C2×C3×C4—182no—
208C2×C3×C4—182no—
209C2×C3×C4—241no—
210C2×C3×C4C3×S411yescharacteristic
211C2×Dic3—34no—
212C2×Dic3—34no—
213C2×Dic3—62no—
214C2×Dic3—64no—
215C2×Dic3—62no—
216C2×Dic3—362no—
217Dic6—64no—
218Dic6—64no—
219Dic6—364no—
220C2×C32—31no—
221C2×C32—31no—
222C2×C32—32no—
223C2×C32—41no—
224C2×C32—42no—
225C2×C32—81no—
226C2×C32—82no—
227C2×C32A4⋊Q812yescomplemented
228C2×C32C22×S411yescharacteristic
229C24C2×(C3⋊D3)⋊C311yescharacteristic
230C2×Q8—94no—
231C2×Q8—181no—
232C2×Q8—182no—
233C2×Q8—181no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—92no—
238C22×C4—92no—
239C22×C4C2×(C3⋊D3)⋊C312yes—
240C4⋊C4—98no—
241C4⋊C4—94no—
242C22⋊C4—94no—
243C22⋊C4—94no—
244C22×C3—32no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—31no—
248C22×C3—32no—
249C22×C3—32no—
250C22×C3—31no—
251C22×C3—31no—
252C22×C3—32no—
253C22×C3—32no—
254C22×C3—32no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—62no—
261C22×C3—61no—
262C22×C3—62no—
263C22×C3—62no—
264C22×C3—61no—
265C22×C3—61no—
266C22×C3—62no—
267C22×C3—121no—
268C22×C3—241no—
269C22×C3C2×C3×Dic611yescharacteristic
270C22×C3C2×C3×S411yescharacteristic
271A4—31no—
272A4—61no—
273C3×C4—32no—
274C3×C4—32no—
275C3×C4—32no—
276C3×C4—62no—
277C3×C4—122no—
278C3×C4—184no—
279C3×C4—184no—
280C3×C4—242no—
281C3×C4C2×C3×S412yes—
282Dic3—64no—
283Dic3—64no—
284Dic3—364no—
285C32—41no—
286C32—81no—
287C32C2×A4⋊Q811yescharacteristic, complemented
288C23—31no—
289C23—31no—
290C23—32no—
291C23#634⋊C312yescomplemented
292C23C22×(C3⋊D3)⋊C311yescharacteristic
293Q8—184no—
294Q8—184no—
295C2×C4—31no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—61no—
300C2×C4—94no—
301C2×C4—94no—
302C2×C4—182no—
303C2×C4—184no—
304C2×C4—182no—
305C2×C4(C3⋊S4)⋊C311yescharacteristic
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—32no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—32no—
312C2×C3—31no—
313C2×C3—32no—
314C2×C3—61no—
315C2×C3—61no—
316C2×C3—62no—
317C2×C3—121no—
318C2×C3—122no—
319C2×C3—241no—
320C2×C3—242no—
321C2×C3C3×A4⋊Q812yes—
322C2×C3C22×C3×S411yescharacteristic
323C22—32no—
324C22—31no—
325C22—31no—
326C22—31no—
327C22—32no—
328C22—32no—
329C22—61no—
330C22C2×#634⋊C311yescharacteristic, complemented
331C22C2×(C3⋊S4)⋊C311yescharacteristic
332C4—32no—
333C4—184no—
334C4—184no—
335C4C2×(C3⋊S4)⋊C312yes—
336C3—31no—
337C3—121no—
338C3—241no—
339C3C2×C3×A4⋊Q811yescharacteristic
340C2—31no—
341C2—31no—
342C2—32no—
343C2#64259⋊C312yesdirect product factor
344C2C22×(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
prime radicalC22×C22⋊H(3,𝔽3)C22
hypercenterC2×C4(C3⋊S4)⋊C3
hypocenterC3×A4C2×C3×Q8
LCSG ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #64259⋊C3C2direct (N × H)
4. C2×#64259C3semidirect
15. #64259C2×C3semidirect
21. C2×C22⋊H(3,𝔽3)Q8semidirect
51. C22⋊H(3,𝔽3)C2×Q8semidirect
75. C23×C32Dic6semidirect
79. C2×C3×A4C3×Q8semidirect
149. C22×C32C2×Dic6semidirect
152. C3×A4C2×C3×Q8semidirect
227. C2×C32A4⋊Q8semidirect
287. C32C2×A4⋊Q8semidirect
291. C23#634⋊C3semidirect
330. C22C2×#634⋊C3semidirect
343. C2#64259⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊H(3,𝔽3)2
C2×(C3⋊SL(2,ℤ/4))⋊C32
#64259⋊C32
C2×#642593
C2×C3×#9243
C2×#634⋊C34