Finite Group Databasebeta

← C2×C3×#5425 #11922800

Subgroups — C2×C3×#5425 #11922800

Order 1152.

681 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 92 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×#41⋊C3C211yesmaximal, characteristic
2C22×C3×#226C211yesmaximal, characteristic, complemented
3C3×#5425C214yesmaximal, direct product factor
4C2×C3×#859C211yesmaximal, characteristic, complemented
5C2×#5425C311yesmaximal, characteristic, direct product factor
6C2×C3×#452—31nomaximal
7C2×C3×#41⋊C3—22no—
8C2×C3×#41⋊C3C2212yes—
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C2×C3×#226—22no—
11C2×C3×#226C2212yescomplemented
12C2×C3×Dic3⋊C4—41nomaximal
13C3×#859C2212yescomplemented
14C22×C3×#41D311yescharacteristic
15C22×#41⋊C3—21no—
16C22×#41⋊C3C2×C311yescharacteristic
17C22×#226C2×C311yescharacteristic, complemented
18C22×C3×Q16—31no—
19C22×C3×SD16—31no—
20C2×C3×C22⋊Q8—31no—
21#5425C2×C314yesdirect product factor
22C2×#859C2×C311yescharacteristic, complemented
23C3×#452—34no—
24C3×#452—34no—
25C2×C3×#103—31no—
26C2×C3×#102—31no—
27C2×C3×C22⋊C8—31no—
28C22×C32×C4—41no—
29C22×C3×Dic3—41no—
30C22×C3×Dic3—41no—
31C3×#41⋊C3—22no—
32C2×C3×SL(2,𝔽3)D412yes—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3×#226—22no—
35C3×Dic3⋊C4—44no—
36C2×#452—31noSylow
37C2×C3×#41—22no—
38C2×C3×#41—32no—
39C2×C3×#41—62no—
40C2×C3×#41C2×D312yes—
41C22×C3×Q8—31no—
42C22×C3×Q8C2×D311yescharacteristic
43C22×C3×D4—31no—
44C23×C3×C4—31no—
45C2×#41⋊C3—24no—
46C2×#41⋊C3—22no—
47C2×#41⋊C3—22no—
48C2×#41⋊C3C22×C312yes—
49C22×SL(2,𝔽3)—21no—
50C22×SL(2,𝔽3)C22×C311yescharacteristic
51C2×#226—22no—
52C2×#226C22×C312yescomplemented
53C2×C3×Q16—32no—
54C2×C3×Q16—32no—
55C2×C3×Q16—62no—
56C2×C3×Q16—62no—
57C2×C3×SD16—34no—
58C2×C3×SD16—64no—
59C22×C3×C8—31no—
60C22×C3×C8—31no—
61C3×C22⋊Q8—34no—
62C3×C22⋊Q8—62no—
63C2×C3×C4⋊C4—31no—
64C2×C3×C4⋊C4—61no—
65C2×C3×C22⋊C4—61no—
66C2×Dic3⋊C4—41no—
67#859C22×C312yescomplemented
68C3×#103—32no—
69C3×#103—32no—
70C3×#102—34no—
71C3×C22⋊C8—34no—
72C23×C32—41no—
73C2×C32×C4—42no—
74C2×C32×C4—82no—
75C2×C3×Dic3—42no—
76C2×C3×Dic3—42no—
77C2×C3×Dic3—44no—
78C2×C3×Dic3—82no—
79C3×SL(2,𝔽3)C2×D411yescharacteristic
80C22×#41C3×D311yescharacteristic
81C22×Q16—31no—
82C22×SD16—31no—
83C2×C22⋊Q8—31no—
84#452—34no—
85#452—34no—
86C2×#103—31no—
87C2×#102—31no—
88C2×C22⋊C8—31no—
89C24×C3—31no—
90C3×#41—22no—
91C3×#41—64no—
92C3×#41—62no—
93C3×#41—64no—
94C2×C3×Q8—31no—
95C2×C3×Q8—32no—
96C2×C3×Q8—32no—
97C2×C3×Q8—61no—
98C2×C3×Q8—62no—
99C2×C3×Q8—62no—
100C2×C3×Q8#6712yes—
101C2×C3×Q8C22×D311yescharacteristic
102C2×C3×D4—32no—
103C2×C3×D4—34no—
104C2×C3×D4—61no—
105C2×C3×D4—62no—
106C22×C3×C4—31no—
107C22×C3×C4—32no—
108C22×C3×C4—32no—
109C22×C3×C4—41no—
110C22×C3×C4—62no—
111C22×C3×C4—62no—
112C22×C3×C4—61no—
113C22×C3×C4—61no—
114C22×C3×C4—81no—
115C22×C3×C4S411yescharacteristic
116C22×Dic3—41no—
117C22×Dic3—41no—
118#41⋊C3—24no—
119#41⋊C3—22no—
120C2×SL(2,𝔽3)—22no—
121C2×SL(2,𝔽3)—21no—
122C2×SL(2,𝔽3)C3×D412yes—
123C2×SL(2,𝔽3)C23×C311yescharacteristic
124#226—22no—
125C3×Q16—64no—
126C3×Q16—62no—
127C3×Q16—62no—
128C3×SD16—68no—
129C2×C3×C8—32no—
130C2×C3×C8—32no—
131C2×C3×C8—62no—
132C2×C3×C8—62no—
133C3×C4⋊C4—32no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—64no—
136C3×C22⋊C4—64no—
137Dic3⋊C4—44no—
138C22×C32—42no—
139C22×C32—42no—
140C22×C32—42no—
141C22×C32—41no—
142C32×C4—82no—
143C3×Dic3—44no—
144C3×Dic3—82no—
145C2×#41—22no—
146C2×#41—32no—
147C2×#41—62no—
148C2×#41C2×C3×D312yes—
149C22×Q8—31no—
150C22×Q8C2×C3×D311yescharacteristic
151C22×D4—31no—
152C23×C4—31no—
153C2×Q16—32no—
154C2×Q16—32no—
155C2×Q16—62no—
156C2×Q16—62no—
157C2×SD16—34no—
158C2×SD16—64no—
159C22×C8—31no—
160C22×C8—31no—
161C22⋊Q8—34no—
162C22⋊Q8—62no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—61no—
165C2×C22⋊C4—61no—
166#103—32no—
167#103—32no—
168#102—34no—
169C22⋊C8—34no—
170C23×C3—32no—
171C23×C3—41no—
172C23×C3—62no—
173C23×C3—62no—
174C23×C3—62no—
175C23×C3—81no—
176C23×C3C2×S411yescharacteristic
177C3×Q8—32no—
178C3×Q8—31no—
179C3×Q8—61no—
180C3×Q8—62no—
181C3×Q8—62no—
182C3×Q8—122no—
183C3×Q8C2×#6711yescharacteristic
184C3×D4—34no—
185C3×D4—62no—
186C3×D4—64no—
187C2×C3×C4—22no—
188C2×C3×C4—32no—
189C2×C3×C4—32no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—42no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—64no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—82no—
199C2×C3×C4—84no—
200C2×C3×C4—82no—
201C2×C3×C4—122no—
202C2×C3×C4—122no—
203C2×C3×C4C2×S412yes—
204C2×Dic3—42no—
205C2×Dic3—42no—
206C2×Dic3—44no—
207C2×Dic3—82no—
208SL(2,𝔽3)—21no—
209SL(2,𝔽3)C2×C3×D411yescharacteristic
210C3×C8—62no—
211C3×C8—62no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—44no—
216C24—31no—
217#41—22no—
218#41—64no—
219#41—62no—
220#41—64no—
221C2×Q8—31no—
222C2×Q8—32no—
223C2×Q8—32no—
224C2×Q8—61no—
225C2×Q8—62no—
226C2×Q8—62no—
227C2×Q8C3×#6712yes—
228C2×Q8C22×C3×D311yescharacteristic
229C2×D4—32no—
230C2×D4—34no—
231C2×D4—61no—
232C2×D4—62no—
233C22×C4—31no—
234C22×C4—32no—
235C22×C4—32no—
236C22×C4—62no—
237C22×C4—62no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4C3×S411yescharacteristic
241Q16—64no—
242Q16—62no—
243Q16—62no—
244SD16—68no—
245C2×C8—32no—
246C2×C8—32no—
247C2×C8—62no—
248C2×C8—62no—
249C4⋊C4—32no—
250C4⋊C4—61no—
251C4⋊C4—64no—
252C22⋊C4—64no—
253C22×C3—42no—
254C22×C3—41no—
255C22×C3—42no—
256C22×C3—42no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—62no—
260C22×C3—82no—
261C22×C3—81no—
262C22×C3—82no—
263C22×C3—82no—
264C22×C3—124no—
265C22×C3#98412yes—
266C22×C3#98512yes—
267C22×C3GL(2,ℤ/4)12yes—
268C22×C3C22×S411yescharacteristic
269C3×C4—22no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—32no—
273C3×C4—82no—
274C3×C4—84no—
275C3×C4—122no—
276C3×C4—122no—
277Dic3—44no—
278Dic3—82no—
279C32—41noSylow
280C23—32no—
281C23—62no—
282C23—62no—
283C23—62no—
284C23C2×C3×S411yescharacteristic
285Q8—32no—
286Q8—31no—
287Q8—61no—
288Q8—62no—
289Q8—62no—
290Q8—122no—
291Q8C2×C3×#6711yescharacteristic
292D4—34no—
293D4—62no—
294D4—64no—
295C2×C4—22no—
296C2×C4—32no—
297C2×C4—32no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—64no—
303C2×C4—62no—
304C2×C4—62no—
305C2×C4—122no—
306C2×C4—122no—
307C2×C4C2×C3×S412yes—
308C8—62no—
309C8—62no—
310C2×C3—44no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—84no—
315C2×C3—81no—
316C2×C3—81no—
317C2×C3—81no—
318C2×C3—122no—
319C2×C3#542514yesdirect product factor
320C2×C3C2×#98411yescharacteristic
321C2×C3C2×#98511yescharacteristic
322C2×C3C2×GL(2,ℤ/4)11yescharacteristic
323C22—62no—
324C22—62no—
325C22—62no—
326C22—124no—
327C22C3×#98412yes—
328C22C3×#98512yes—
329C22C3×GL(2,ℤ/4)12yes—
330C22C22×C3×S411yescharacteristic
331C4—22no—
332C4—31no—
333C4—31no—
334C4—32no—
335C4—122no—
336C4—122no—
337C3—41no—
338C3—81no—
339C3C2×#542511yescharacteristic, direct product factor
340C2—122no—
341C2C3×#542514yesdirect product factor
342C2C2×C3×#98411yescharacteristic
343C2C2×C3×#98511yescharacteristic
344C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC22×C3×#41D3
prime radicalC22×C3×#41⋊C3C2
hypercenterC22×C3×C4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#226C2semidirect
3. C3×#5425C2direct (N × H)
4. C2×C3×#859C2semidirect
5. C2×#5425C3direct (N × H)
11. C2×C3×#226C22semidirect
13. C3×#859C22semidirect
17. C22×#226C2×C3semidirect
21. #5425C2×C3direct (N × H)
22. C2×#859C2×C3semidirect
52. C2×#226C22×C3semidirect
67. #859C22×C3semidirect
319. C2×C3#5425direct (N × H)
339. C3C2×#5425direct (N × H)
341. C2C3×#5425direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C22×C3×#2262
C3×#54252
C2×C3×#8592
C2×#54253
C2×C3×#4523
C2×C3×Dic3⋊C44