Finite Group Databasebeta

← C2×C3×#5419 #11922785

Subgroups — C2×C3×#5419 #11922785

Order 1152.

781 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic
2C22×C3×#226C211yesmaximal, characteristic, complemented
3C3×#5419C214yesmaximal, direct product factor
4C2×C3×#859C211yesmaximal, characteristic, complemented
5C2×#5419C311yesmaximal, characteristic, direct product factor
6C2×C3×#451—31nomaximal
7C22×C3×SL(2,𝔽3)—22no—
8C22×C3×SL(2,𝔽3)C2212yes—
9C22×C3×SL(2,𝔽3)C2211yescharacteristic
10C2×C3×#226—22no—
11C2×C3×#226C2212yescomplemented
12C2×C3×#217—41nomaximal
13C3×#859C2212yescomplemented
14C23×C3×Q8D311yescharacteristic
15C23×SL(2,𝔽3)—21no—
16C23×SL(2,𝔽3)C2×C311yescharacteristic
17C22×#226C2×C311yescharacteristic, complemented
18C22×C3×Q16—31no—
19C22×C3×Q16—31no—
20C2×C3×C22⋊Q8—31no—
21#5419C2×C314yesdirect product factor
22C2×#859C2×C311yescharacteristic, complemented
23C3×#451—34no—
24C3×#451—34no—
25C2×C3×#103—31no—
26C2×C3×#103—31no—
27C2×C3×C22⋊C8—31no—
28C24×C32—41no—
29C22×C3×Dic3—41no—
30C22×C3×Dic3—41no—
31C2×C3×SL(2,𝔽3)—22no—
32C2×C3×SL(2,𝔽3)D412yes—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3×#226—22no—
35C3×#217—44no—
36C2×#451—31noSylow
37C22×C3×Q8—22no—
38C22×C3×Q8—31no—
39C22×C3×Q8—32no—
40C22×C3×Q8—31no—
41C22×C3×Q8—62no—
42C22×C3×Q8C2×D312yes—
43C22×C3×Q8C2×D311yescharacteristic
44C23×C3×C4—31no—
45C22×SL(2,𝔽3)—24no—
46C22×SL(2,𝔽3)—22no—
47C22×SL(2,𝔽3)—22no—
48C22×SL(2,𝔽3)—21no—
49C22×SL(2,𝔽3)C22×C312yes—
50C22×SL(2,𝔽3)C22×C311yescharacteristic
51C2×#226—22no—
52C2×#226C22×C312yescomplemented
53C2×C3×Q16—34no—
54C2×C3×Q16—32no—
55C2×C3×Q16—32no—
56C2×C3×Q16—62no—
57C2×C3×Q16—64no—
58C2×C3×Q16—62no—
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×#217—41no—
67#859C22×C312yescomplemented
68C3×#103—34no—
69C3×#103—32no—
70C3×#103—32no—
71C3×C22⋊C8—34no—
72C23×C32—44no—
73C23×C32—42no—
74C23×C32—41no—
75C23×C32—82no—
76C23×C32—82no—
77C2×C3×Dic3—42no—
78C2×C3×Dic3—42no—
79C2×C3×Dic3—82no—
80C2×C3×Dic3—82no—
81C3×SL(2,𝔽3)C2×D411yescharacteristic
82C23×Q8C3×D311yescharacteristic
83C22×Q16—31no—
84C22×Q16—31no—
85C2×C22⋊Q8—31no—
86#451—34no—
87#451—34no—
88C2×#103—31no—
89C2×#103—31no—
90C2×C22⋊C8—31no—
91C24×C3—41no—
92C24×C3—81no—
93C24×C3S411yescharacteristic
94C2×C3×Q8—22no—
95C2×C3×Q8—31no—
96C2×C3×Q8—32no—
97C2×C3×Q8—32no—
98C2×C3×Q8—34no—
99C2×C3×Q8—32no—
100C2×C3×Q8—61no—
101C2×C3×Q8—61no—
102C2×C3×Q8—62no—
103C2×C3×Q8—62no—
104C2×C3×Q8—62no—
105C2×C3×Q8—64no—
106C2×C3×Q8—62no—
107C2×C3×Q8—64no—
108C2×C3×Q8#6712yes—
109C2×C3×Q8C22×D311yescharacteristic
110C22×C3×C4—31no—
111C22×C3×C4—32no—
112C22×C3×C4—32no—
113C22×C3×C4—31no—
114C22×C3×C4—62no—
115C22×C3×C4—62no—
116C22×C3×C4—61no—
117C22×C3×C4—61no—
118C22×Dic3—41no—
119C22×Dic3—41no—
120C2×SL(2,𝔽3)—24no—
121C2×SL(2,𝔽3)—22no—
122C2×SL(2,𝔽3)—22no—
123C2×SL(2,𝔽3)—21no—
124C2×SL(2,𝔽3)C3×D412yes—
125C2×SL(2,𝔽3)C23×C311yescharacteristic
126#226—22no—
127C3×Q16—64no—
128C3×Q16—62no—
129C3×Q16—68no—
130C3×Q16—62no—
131C2×C3×C8—32no—
132C2×C3×C8—32no—
133C2×C3×C8—62no—
134C2×C3×C8—62no—
135C3×C4⋊C4—32no—
136C3×C4⋊C4—61no—
137C3×C4⋊C4—64no—
138C3×C22⋊C4—64no—
139#217—44no—
140C22×C32—42no—
141C22×C32—44no—
142C22×C32—42no—
143C22×C32—42no—
144C22×C32—41no—
145C22×C32—82no—
146C22×C32—88no—
147C22×C32—82no—
148C3×Dic3—82no—
149C3×Dic3—82no—
150C22×Q8—22no—
151C22×Q8—31no—
152C22×Q8—32no—
153C22×Q8—31no—
154C22×Q8—62no—
155C22×Q8C2×C3×D312yes—
156C22×Q8C2×C3×D311yescharacteristic
157C23×C4—31no—
158C2×Q16—34no—
159C2×Q16—32no—
160C2×Q16—32no—
161C2×Q16—62no—
162C2×Q16—64no—
163C2×Q16—62no—
164C22×C8—31no—
165C22×C8—31no—
166C22⋊Q8—34no—
167C22⋊Q8—62no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—61no—
170C2×C22⋊C4—61no—
171#103—34no—
172#103—32no—
173#103—32no—
174C22⋊C8—34no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—41no—
178C23×C3—42no—
179C23×C3—44no—
180C23×C3—82no—
181C23×C3—82no—
182C23×C3—84no—
183C23×C3—84no—
184C23×C3—82no—
185C23×C3—81no—
186C23×C3—84no—
187C23×C3#22614yescomplemented
188C23×C3C2×S412yes—
189C23×C3C2×S411yescharacteristic
190C3×Q8—32no—
191C3×Q8—31no—
192C3×Q8—34no—
193C3×Q8—61no—
194C3×Q8—62no—
195C3×Q8—64no—
196C3×Q8—62no—
197C3×Q8—62no—
198C3×Q8—122no—
199C3×Q8C2×#6711yescharacteristic
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—32no—
205C2×C3×C4—62no—
206C2×C3×C4—62no—
207C2×C3×C4—64no—
208C2×C3×C4—62no—
209C2×C3×C4—62no—
210C2×C3×C4—62no—
211C2×C3×C4—122no—
212C2×C3×C4—122no—
213C2×Dic3—42no—
214C2×Dic3—42no—
215C2×Dic3—82no—
216C2×Dic3—82no—
217SL(2,𝔽3)—21no—
218SL(2,𝔽3)C2×C3×D411yescharacteristic
219C3×C8—62no—
220C3×C8—62no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—44no—
225C2×C32—84no—
226C24C3×S411yescharacteristic
227C2×Q8—22no—
228C2×Q8—31no—
229C2×Q8—32no—
230C2×Q8—32no—
231C2×Q8—34no—
232C2×Q8—32no—
233C2×Q8—61no—
234C2×Q8—61no—
235C2×Q8—62no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8—64no—
239C2×Q8—62no—
240C2×Q8—64no—
241C2×Q8C3×#6712yes—
242C2×Q8C22×C3×D311yescharacteristic
243C22×C4—31no—
244C22×C4—32no—
245C22×C4—31no—
246C22×C4—32no—
247C22×C4—62no—
248C22×C4—62no—
249C22×C4—61no—
250C22×C4—61no—
251Q16—64no—
252Q16—62no—
253Q16—68no—
254Q16—62no—
255C2×C8—32no—
256C2×C8—32no—
257C2×C8—62no—
258C2×C8—62no—
259C4⋊C4—32no—
260C4⋊C4—61no—
261C4⋊C4—64no—
262C22⋊C4—64no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—28no—
266C22×C3—42no—
267C22×C3—41no—
268C22×C3—42no—
269C22×C3—44no—
270C22×C3—42no—
271C22×C3—88no—
272C22×C3—82no—
273C22×C3—82no—
274C22×C3—816no—
275C22×C3—84no—
276C22×C3—84no—
277C22×C3—82no—
278C22×C3—81no—
279C22×C3—82no—
280C22×C3—84no—
281C22×C3—82no—
282C22×C3C2×#22612yes—
283C22×C3C2×#22614yescomplemented
284C22×C3#98312yes—
285C22×C3GL(2,ℤ/4)12yes—
286C22×C3C22×S411yescharacteristic
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—32no—
290C3×C4—62no—
291C3×C4—122no—
292C3×C4—122no—
293Dic3—82no—
294Dic3—82no—
295C32—41noSylow
296C23—22no—
297C23—22no—
298C23C3×#22614yescomplemented
299C23C2×C3×S412yes—
300C23C2×C3×S411yescharacteristic
301Q8—32no—
302Q8—31no—
303Q8—34no—
304Q8—61no—
305Q8—62no—
306Q8—64no—
307Q8—62no—
308Q8—62no—
309Q8—122no—
310Q8C2×C3×#6711yescharacteristic
311C2×C4—32no—
312C2×C4—32no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—32no—
316C2×C4—62no—
317C2×C4—62no—
318C2×C4—64no—
319C2×C4—62no—
320C2×C4—62no—
321C2×C4—62no—
322C2×C4—122no—
323C2×C4—122no—
324C8—62no—
325C8—62no—
326C2×C3—24no—
327C2×C3—44no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—41no—
331C2×C3—84no—
332C2×C3—88no—
333C2×C3—84no—
334C2×C3—81no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3#541914yesdirect product factor
338C2×C3C22×#22611yescharacteristic
339C2×C3C2×#98311yescharacteristic
340C2×C3C2×GL(2,ℤ/4)11yescharacteristic
341C22—28no—
342C22—22no—
343C22—22no—
344C22C2×C3×#22612yes—
345C22C2×C3×#22614yescomplemented
346C22C3×#98312yes—
347C22C3×GL(2,ℤ/4)12yes—
348C22C22×C3×S411yescharacteristic
349C4—31no—
350C4—31no—
351C4—32no—
352C4—62no—
353C4—122no—
354C4—122no—
355C3—41no—
356C3—81no—
357C3C2×#541911yescharacteristic, direct product factor
358C2—24no—
359C2C3×#541914yesdirect product factor
360C2C22×C3×#22611yescharacteristic
361C2C2×C3×#98311yescharacteristic
362C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC23×C3C2×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×Q8D3
prime radicalC23×C3×SL(2,𝔽3)C2
hypercenterC24×C3S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×#226C2semidirect
3. C3×#5419C2direct (N × H)
4. C2×C3×#859C2semidirect
5. C2×#5419C3direct (N × H)
11. C2×C3×#226C22semidirect
13. C3×#859C22semidirect
17. C22×#226C2×C3semidirect
21. #5419C2×C3direct (N × H)
22. C2×#859C2×C3semidirect
52. C2×#226C22×C3semidirect
67. #859C22×C3semidirect
187. C23×C3#226semidirect
283. C22×C3C2×#226semidirect
298. C23C3×#226semidirect
337. C2×C3#5419direct (N × H)
345. C22C2×C3×#226semidirect
357. C3C2×#5419direct (N × H)
359. C2C3×#5419direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C22×C3×#2262
C3×#54192
C2×C3×#8592
C2×#54193
C2×C3×#4513
C2×C3×#2174