Finite Group Databasebeta

← C2×C3⋊(B(4,𝔽2)⋊C3) #11921376

Subgroups — C2×C3⋊(B(4,𝔽2)⋊C3) #11921376

Order 1152.

512 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 31 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×C3×#997C211yesmaximal, characteristic, complemented
2C3⋊(B(4,𝔽2)⋊C3)C212yesmaximal, direct product factor
3C2×#5244C311yesmaximal, characteristic, complemented
4C2×B(4,𝔽2)⋊C3—31nomaximal
5C22×D3×A4—41nomaximal
6C3×#997C2211yescharacteristic, complemented
7C2×C3×H(5,𝔽2)C2×C311yescharacteristic, complemented
8C2×#997—21no—
9C2×#997D311yescharacteristic, complemented
10C2×#937—31no—
11#5244—32no—
12#5244C2×C312yescomplemented
13C2×#834—31no—
14B(4,𝔽2)⋊C3—32no—
15C22×C3×A4—41no—
16C22×C3×A4—81no—
17C2×D3×A4—44no—
18C2×D3×A4—42no—
19C2×C3×SL(2,𝔽3)—81no—
20C2×B(4,𝔽2)—31noSylow
21C24×D3A411yescharacteristic, complemented
22C23×A4—121no—
23C3×H(5,𝔽2)—31no—
24C3×H(5,𝔽2)—61no—
25C3×H(5,𝔽2)—61no—
26C3×H(5,𝔽2)C22×C311yescharacteristic, complemented
27C2×C3×#41—61no—
28C22×C3×D4—31no—
29C22×C3×D4—61no—
30C22×#67—61no—
31#997—21no—
32#997C2×D311yescharacteristic, complemented
33C2×#217—31no—
34#937—32no—
35#937—32no—
36#937—61no—
37#937—61no—
38C2×(C2×D3)⋊C4—61no—
39#834—32no—
40#834—32no—
41C22×C3×D3—161no—
42C2×C3×A4—42no—
43C2×C3×A4—41no—
44C2×C3×A4—81no—
45C2×C3×A4—82no—
46D3×A4—44no—
47C3×SL(2,𝔽3)—81no—
48C2×H(5,𝔽2)C3×D311yescharacteristic, complemented
49C2×#120—91no—
50B(4,𝔽2)—32no—
51B(4,𝔽2)—92no—
52C2×#99—91no—
53C24×C3—21no—
54C24×C3—31no—
55C24×C3C2×A411yescharacteristic
56C23×D3—44no—
57C23×D3—61no—
58C23×D3—61no—
59C23×D3C2×A412yescomplemented
60C22×A4—41no—
61C22×A4—81no—
62C22×A4—82no—
63C22×A4—81no—
64C22×A4—122no—
65C22×A4—124no—
66C3×#41—61no—
67C3×#41—62no—
68C3×#41—61no—
69C3×#41—61no—
70C3×#41—62no—
71C3×#41—61no—
72C2×C3×Q8—21no—
73C2×C3×D4—31no—
74C2×C3×D4—31no—
75C2×C3×D4—31no—
76C2×C3×D4—31no—
77C2×C3×D4—61no—
78C2×C3×D4—62no—
79C2×C3×D4—62no—
80C2×C3×D4—62no—
81C2×C3×D4—61no—
82C2×C3×D4—61no—
83C2×C3×D4—62no—
84C2×C3×D4—61no—
85C2×C3×D4—61no—
86C22×C3×C4—31no—
87C22×C3×C4—61no—
88C2×#67—62no—
89C2×#67—62no—
90C2×#67—124no—
91C22×Dic3—61no—
92C2×SL(2,𝔽3)—81no—
93C2×SL(2,𝔽3)—82no—
94#217—32no—
95#217—61no—
96(C2×D3)⋊C4—62no—
97(C2×D3)⋊C4—62no—
98C22×C32—161no—
99C2×C3×D3—164no—
100C2×C3×D3—162no—
101C3×A4—41no—
102C3×A4—81no—
103C25—31no—
104H(5,𝔽2)—31no—
105H(5,𝔽2)—61no—
106H(5,𝔽2)—61no—
107H(5,𝔽2)C2×C3×D311yescharacteristic, complemented
108C2×#41—61no—
109C22×D4—31no—
110C22×D4—61no—
111C22×D4—181no—
112#120—92no—
113#120—92no—
114#120—181no—
115#120—181no—
116C2×C22⋊C4—91no—
117C2×C22⋊C4—181no—
118#99—92no—
119#99—92no—
120C23×C3—21no—
121C23×C3—31no—
122C23×C3—31no—
123C23×C3—31no—
124C23×C3—31no—
125C23×C3—42no—
126C23×C3—61no—
127C23×C3—61no—
128C23×C3—61no—
129C23×C3—82no—
130C23×C3—122no—
131C23×C3—481no—
132C23×C3C22×A411yescharacteristic
133C22×D3—41no—
134C22×D3—44no—
135C22×D3—62no—
136C22×D3—64no—
137C22×D3—121no—
138C22×D3—122no—
139C22×D3—124no—
140C2×A4—42no—
141C2×A4—41no—
142C2×A4—81no—
143C2×A4—82no—
144C2×A4—82no—
145C2×A4—82no—
146C2×A4—84no—
147C2×A4—81no—
148C2×A4—124no—
149C3×Q8—21no—
150C3×Q8—61no—
151C3×D4—62no—
152C3×D4—62no—
153C3×D4—62no—
154C3×D4—62no—
155C3×D4—61no—
156C3×D4—61no—
157C3×D4—62no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—61no—
161C2×C3×C4—62no—
162C2×C3×C4—61no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165#67—124no—
166#67—124no—
167C2×Dic3—62no—
168C2×Dic3—122no—
169SL(2,𝔽3)—81no—
170SL(2,𝔽3)—82no—
171C2×C32—162no—
172C2×C32—161no—
173C3×D3—164no—
174C24—21no—
175C24—31no—
176C24—32no—
177C24—124no—
178C24—181no—
179C24—181no—
180C24D3×A411yescharacteristic
181#41—61no—
182#41—62no—
183#41—61no—
184#41—61no—
185#41—62no—
186#41—61no—
187C2×Q8—21no—
188C2×D4—31no—
189C2×D4—31no—
190C2×D4—31no—
191C2×D4—31no—
192C2×D4—61no—
193C2×D4—62no—
194C2×D4—62no—
195C2×D4—62no—
196C2×D4—61no—
197C2×D4—61no—
198C2×D4—62no—
199C2×D4—61no—
200C2×D4—61no—
201C2×D4—182no—
202C2×D4—182no—
203C2×D4—364no—
204C22×C4—31no—
205C22×C4—61no—
206C22×C4—181no—
207C22⋊C4—92no—
208C22⋊C4—182no—
209C22⋊C4—182no—
210C22⋊C4—181no—
211C22×C3—31no—
212C22×C3—31no—
213C22×C3—41no—
214C22×C3—62no—
215C22×C3—61no—
216C22×C3—61no—
217C22×C3—81no—
218C22×C3—121no—
219C22×C3—121no—
220C22×C3—121no—
221C22×C3—122no—
222C22×C3—161no—
223C22×C3—241no—
224C22×C3—241no—
225C22×C3—321no—
226C22×C3—484no—
227C22×C3—482no—
228C22×C3#120⋊C311yescharacteristic
229C2×D3—44no—
230C2×D3—42no—
231C2×D3—64no—
232C2×D3—122no—
233C2×D3—124no—
234A4—41no—
235A4—81no—
236A4—81no—
237A4—82no—
238C3×C4—61no—
239C3×C4—61no—
240Dic3—122no—
241C32—161noSylow
242C23—21no—
243C23—31no—
244C23—31no—
245C23—31no—
246C23—31no—
247C23—42no—
248C23—61no—
249C23—61no—
250C23—61no—
251C23—82no—
252C23—124no—
253C23—122no—
254C23—121no—
255C23—182no—
256C23—184no—
257C23—361no—
258C23—362no—
259C23—364no—
260C23C2×D3×A411yescharacteristic
261Q8—21no—
262Q8—61no—
263D4—62no—
264D4—62no—
265D4—62no—
266D4—62no—
267D4—61no—
268D4—61no—
269D4—62no—
270D4—364no—
271D4—364no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—182no—
280C2×C4—362no—
281C2×C3—61no—
282C2×C3—61no—
283C2×C3—121no—
284C2×C3—121no—
285C2×C3—162no—
286C2×C3—161no—
287C2×C3—322no—
288C2×C3—321no—
289C2×C3—484no—
290C2×C3B(4,𝔽2)⋊C312yescomplemented
291C2×C3C2×#120⋊C311yescharacteristic
292D3—44no—
293C22—31no—
294C22—31no—
295C22—41no—
296C22—62no—
297C22—61no—
298C22—61no—
299C22—81no—
300C22—121no—
301C22—121no—
302C22—121no—
303C22—122no—
304C22—124no—
305C22—122no—
306C22—184no—
307C22—241no—
308C22—241no—
309C22—362no—
310C22—364no—
311C22C3⋊(#120⋊C3)11yescharacteristic
312C4—61no—
313C4—61no—
314C4—362no—
315C3—161no—
316C3—321no—
317C3C2×B(4,𝔽2)⋊C311yescharacteristic, complemented
318C2—61no—
319C2—61no—
320C2—121no—
321C2—121no—
322C2—124no—
323C2C3⋊(B(4,𝔽2)⋊C3)12yesdirect product factor
324C2C2×C3⋊(#120⋊C3)11yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊(#120⋊C3)
commutator, cosocle, hypocenterC3×H(5,𝔽2)C22×C3
FrattiniC23C2×D3×A4
FittingC2×C3×H(5,𝔽2)C2×C3
socleC22×C3#120⋊C3
LCSG ▷ C3×H(5,𝔽2)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×#997C2semidirect
2. C3⋊(B(4,𝔽2)⋊C3)C2direct (N × H)
3. C2×#5244C3semidirect
6. C3×#997C22semidirect
7. C2×C3×H(5,𝔽2)C2×C3semidirect
9. C2×#997D3semidirect
12. #5244C2×C3semidirect
21. C24×D3A4semidirect
26. C3×H(5,𝔽2)C22×C3semidirect
32. #997C2×D3semidirect
48. C2×H(5,𝔽2)C3×D3semidirect
59. C23×D3C2×A4semidirect
107. H(5,𝔽2)C2×C3×D3semidirect
290. C2×C3B(4,𝔽2)⋊C3semidirect
317. C3C2×B(4,𝔽2)⋊C3semidirect
323. C2C3⋊(B(4,𝔽2)⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9972
C3⋊(B(4,𝔽2)⋊C3)2
C2×#52443
C2×B(4,𝔽2)⋊C33
C22×D3×A44