Finite Group Databasebeta

← C2×C3×#5930 #11923747

Subgroups — C2×C3×#5930 #11923747

Order 1152.

556 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 42 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×C22⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C3×#5930C212yesmaximal, direct product factor
3C2×#5930C311yesmaximal, characteristic, direct product factor
4C2×C3×#451—31nomaximal
5C3×C22⋊SL(2,𝔽3)C2211yescharacteristic
6C2×C3×SL(2,ℤ/4)—41nomaximal
7C2×C3×#226—42nomaximal
8C23×C3×Q8D311yescharacteristic
9C2×C22⋊SL(2,𝔽3)—21no—
10C2×C22⋊SL(2,𝔽3)C2×C311yescharacteristic
11#5930C2×C312yesdirect product factor
12C22×C3×Q16—32no—
13C2×C3×C22⋊Q8—31no—
14C3×#451—32no—
15C3×#451—34no—
16C3×#451—32no—
17C2×C3×#103—32no—
18C2×C3×C22⋊C8—31no—
19C22×C3×A4—41no—
20C2×C3×SL(2,𝔽3)—42no—
21C2×C3×SL(2,𝔽3)—81no—
22C3×SL(2,ℤ/4)—42no—
23C3×#226—44no—
24C2×#451—31noSylow
25C22×C3×Q8—31no—
26C22×C3×Q8—32no—
27C22×C3×Q8—32no—
28C22×C3×Q8—31no—
29C22×C3×Q8—61no—
30C22×C3×Q8—61no—
31C22×C3×Q8C2×D311yescharacteristic
32C23×C3×C4—31no—
33C22⋊SL(2,𝔽3)—21no—
34C22⋊SL(2,𝔽3)C22×C311yescharacteristic
35C2×SL(2,ℤ/4)—41no—
36C2×#226—42no—
37C2×C3×Q16—34no—
38C2×C3×Q16—34no—
39C2×C3×Q16—62no—
40C2×C3×Q16—62no—
41C2×C3×Q16—62no—
42C2×C3×Q16—62no—
43C22×C3×C8—32no—
44C3×C22⋊Q8—32no—
45C3×C22⋊Q8—32no—
46C3×C22⋊Q8—61no—
47C3×C22⋊Q8—61no—
48C2×C3×C4⋊C4—31no—
49C2×C3×C4⋊C4—61no—
50C2×C3×C22⋊C4—61no—
51C3×#103—34no—
52C3×#103—34no—
53C3×C22⋊C8—32no—
54C3×C22⋊C8—32no—
55C2×C3×A4—42no—
56C2×C3×A4—41no—
57C2×C3×Dic3—161no—
58C3×SL(2,𝔽3)—42no—
59C3×SL(2,𝔽3)—81no—
60C23×Q8C3×D311yescharacteristic
61C22×Q16—32no—
62C2×C22⋊Q8—31no—
63#451—32no—
64#451—34no—
65#451—32no—
66C2×#103—32no—
67C2×C22⋊C8—31no—
68C24×C3S411yescharacteristic
69C22×A4—41no—
70C22×A4—81no—
71C2×C3×Q8—21no—
72C2×C3×Q8—32no—
73C2×C3×Q8—32no—
74C2×C3×Q8—32no—
75C2×C3×Q8—32no—
76C2×C3×Q8—32no—
77C2×C3×Q8—32no—
78C2×C3×Q8—61no—
79C2×C3×Q8—61no—
80C2×C3×Q8—62no—
81C2×C3×Q8—62no—
82C2×C3×Q8—62no—
83C2×C3×Q8—61no—
84C2×C3×Q8—61no—
85C2×C3×Q8—62no—
86C2×C3×Q8—62no—
87C2×C3×Q8—61no—
88C2×C3×Q8—61no—
89C2×C3×Q8—62no—
90C2×C3×Q8S412yes—
91C22×C3×C4—31no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C22×C3×C4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—62no—
98C22×C3×C4—62no—
99C22×C3×C4—61no—
100C22×C3×C4—61no—
101C2×SL(2,𝔽3)—42no—
102C2×SL(2,𝔽3)—81no—
103C2×SL(2,𝔽3)—82no—
104C2×SL(2,𝔽3)—82no—
105SL(2,ℤ/4)—42no—
106#226—44no—
107C3×Q16—64no—
108C3×Q16—64no—
109C3×Q16—64no—
110C3×Q16—64no—
111C2×C3×C8—34no—
112C2×C3×C8—62no—
113C2×C3×C8—62no—
114C3×C4⋊C4—32no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—62no—
117C3×C4⋊C4—62no—
118C3×C22⋊C4—62no—
119C3×C22⋊C4—62no—
120C22×C32—161no—
121C3×A4—41no—
122C3×Dic3—162no—
123C22×Q8—31no—
124C22×Q8—32no—
125C22×Q8—32no—
126C22×Q8—31no—
127C22×Q8—61no—
128C22×Q8—61no—
129C22×Q8C2×C3×D311yescharacteristic
130C23×C4—31no—
131C2×Q16—34no—
132C2×Q16—34no—
133C2×Q16—62no—
134C2×Q16—62no—
135C2×Q16—62no—
136C2×Q16—62no—
137C22×C8—32no—
138C22⋊Q8—32no—
139C22⋊Q8—32no—
140C22⋊Q8—61no—
141C22⋊Q8—61no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—61no—
144C2×C22⋊C4—61no—
145#103—34no—
146#103—34no—
147C22⋊C8—32no—
148C22⋊C8—32no—
149C23×C3—31no—
150C23×C3—31no—
151C23×C3—32no—
152C23×C3#22612yescomplemented
153C23×C3C2×S411yescharacteristic
154C2×A4—41no—
155C2×A4—42no—
156C2×A4—81no—
157C2×A4—82no—
158C3×Q8—21no—
159C3×Q8—32no—
160C3×Q8—32no—
161C3×Q8—32no—
162C3×Q8—61no—
163C3×Q8—62no—
164C3×Q8—61no—
165C3×Q8—62no—
166C3×Q8—64no—
167C3×Q8—61no—
168C3×Q8—122no—
169C3×Q8C2×S412yes—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—32no—
173C2×C3×C4—32no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—122no—
185C2×C3×C4—121no—
186C2×C3×C4—121no—
187C2×Dic3—161no—
188SL(2,𝔽3)—42no—
189SL(2,𝔽3)—81no—
190SL(2,𝔽3)—82no—
191SL(2,𝔽3)—82no—
192C3×C8—64no—
193C2×C32—162no—
194C2×C32—161no—
195C24C3×S411yescharacteristic
196C2×Q8—21no—
197C2×Q8—32no—
198C2×Q8—32no—
199C2×Q8—32no—
200C2×Q8—32no—
201C2×Q8—32no—
202C2×Q8—32no—
203C2×Q8—61no—
204C2×Q8—61no—
205C2×Q8—62no—
206C2×Q8—62no—
207C2×Q8—62no—
208C2×Q8—61no—
209C2×Q8—61no—
210C2×Q8—62no—
211C2×Q8—62no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8C3×S412yes—
216C22×C4—31no—
217C22×C4—31no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—62no—
223C22×C4—62no—
224C22×C4—61no—
225C22×C4—61no—
226Q16—64no—
227Q16—64no—
228Q16—64no—
229Q16—64no—
230C2×C8—34no—
231C2×C8—62no—
232C2×C8—62no—
233C4⋊C4—32no—
234C4⋊C4—61no—
235C4⋊C4—62no—
236C4⋊C4—62no—
237C22⋊C4—62no—
238C22⋊C4—62no—
239C22×C3—32no—
240C22×C3—31no—
241C22×C3—32no—
242C22×C3—32no—
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—61no—
246C22×C3—161no—
247C22×C3—321no—
248C22×C3C2×#22611yescharacteristic, complemented
249C22×C3#102011yescharacteristic
250A4—41no—
251A4—81no—
252C3×C4—32no—
253C3×C4—32no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—122no—
257C3×C4—122no—
258Dic3—162no—
259C32—161noSylow
260C23—31no—
261C23—31no—
262C23—32no—
263C23C3×#22612yescomplemented
264C23C2×C3×S411yescharacteristic
265Q8—21no—
266Q8—32no—
267Q8—32no—
268Q8—32no—
269Q8—61no—
270Q8—62no—
271Q8—61no—
272Q8—62no—
273Q8—64no—
274Q8—61no—
275Q8—122no—
276Q8C2×C3×S412yes—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—62no—
290C2×C4—62no—
291C2×C4—121no—
292C2×C4—122no—
293C2×C4—121no—
294C8—64no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—32no—
298C2×C3—161no—
299C2×C3—162no—
300C2×C3—321no—
301C2×C3—322no—
302C2×C3#593012yesdirect product factor
303C2×C3C2×#102011yescharacteristic
304C22—31no—
305C22—32no—
306C22—31no—
307C22—31no—
308C22—32no—
309C22—32no—
310C22—61no—
311C22C2×C3×#22611yescharacteristic, complemented
312C22C3×#102011yescharacteristic
313C4—32no—
314C4—32no—
315C4—61no—
316C4—61no—
317C4—122no—
318C4—122no—
319C3—161no—
320C3—321no—
321C3C2×#593011yescharacteristic, direct product factor
322C2—31no—
323C2—31no—
324C2—32no—
325C2C3×#593012yesdirect product factor
326C2C2×C3×#102011yescharacteristic
subgroupG/H
center, hypercenterC22×C3#1020
commutator, cosocle, hypocenterC22⋊SL(2,𝔽3)C22×C3
FrattiniC2C2×C3×#1020
FittingC23×C3×Q8D3
socleC24×C3S4
prime radicalC2×C3×C22⋊SL(2,𝔽3)C2
LCSG ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×#5930C2direct (N × H)
3. C2×#5930C3direct (N × H)
11. #5930C2×C3direct (N × H)
152. C23×C3#226semidirect
248. C22×C3C2×#226semidirect
263. C23C3×#226semidirect
302. C2×C3#5930direct (N × H)
311. C22C2×C3×#226semidirect
321. C3C2×#5930direct (N × H)
325. C2C3×#5930direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊SL(2,𝔽3)2
C3×#59302
C2×#59303
C2×C3×#4513
C2×C3×SL(2,ℤ/4)4
C2×C3×#2264