Finite Group Databasebeta

← C2×C3×#5430 #11922815

Subgroups — C2×C3×#5430 #11922815

Order 1152.

637 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×2−1+4⋊C3C211yesmaximal, characteristic
2C2×C3×#984C211yesmaximal, characteristic, complemented
3C3×#5430C214yesmaximal, direct product factor
4C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
5C2×#5430C311yesmaximal, characteristic, direct product factor
6C2×C3×#456—31nomaximal
7C3×2−1+4⋊C3—21no—
8C3×2−1+4⋊C3C2212yes—
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)—21no—
11C3×#984—21no—
12C3×#984C2212yescomplemented
13C2×C3×#226—21no—
14C2×C3×#214—41nomaximal
15C3×GU(2,𝔽3)C2212yescomplemented
16C2×C3×2−1+4D311yescharacteristic
17C2×2−1+4⋊C3—21no—
18C2×2−1+4⋊C3C2×C311yescharacteristic
19C2×#984C2×C311yescharacteristic, complemented
20C2×C3×#137—31no—
21C2×C3×#137—31no—
22C2×C3×C4⋊Q8—31no—
23#5430C2×C314yesdirect product factor
24C2×GU(2,𝔽3)C2×C311yescharacteristic, complemented
25C3×#456—34no—
26C3×#456—34no—
27C2×C3×#104—31no—
28C2×C3×#104—31no—
29C2×C3×#101—31no—
30C2×C32×D4—41no—
31C2×C3×Dic6—41no—
32C3×#41⋊C3D411yescharacteristic
33C3×#41⋊C3C2311yescharacteristic
34C2×C3×SL(2,𝔽3)—22no—
35C2×C3×SL(2,𝔽3)D411yescharacteristic
36C3×#226—22no—
37C3×#214—44no—
38C2×C3×C3⋊C8—41no—
39C2×#456—31noSylow
40C3×2−1+4—21no—
41C3×2−1+4—32no—
42C3×2−1+4—61no—
43C3×2−1+4C2×D312yes—
44C2×C3×#41—31no—
45C2×C3×#41—61no—
46C2×C3×#41C2×D311yescharacteristic
47C22×C3×Q8—21no—
48C22×C3×Q8—31no—
49C22×C3×Q8—31no—
502−1+4⋊C3—22no—
512−1+4⋊C3—21no—
522−1+4⋊C3—22no—
532−1+4⋊C3C22×C312yes—
54C2×#41⋊C3—21no—
55C2×#41⋊C3C22×C311yescharacteristic
56C22×SL(2,𝔽3)—22no—
57C22×SL(2,𝔽3)—21no—
58#984—21no—
59#984C22×C312yescomplemented
60C2×#226—21no—
61C3×#137—32no—
62C3×#137—34no—
63C3×#137—32no—
64C3×#137—61no—
65C3×#137—62no—
66C3×#137—61no—
67C2×C3×Q16—61no—
68C2×C3×Q16—61no—
69C2×C3×SD16—61no—
70C2×C3×SD16—61no—
71C2×C3×M16—31no—
72C2×C3×M16—31no—
73C3×C4⋊Q8—34no—
74C3×C4⋊Q8—62no—
75C2×C3×C4⋊C4—62no—
76C2×C3×C42—31no—
77C2×#214—41no—
78GU(2,𝔽3)C22×C312yescomplemented
79C3×#104—32no—
80C3×#104—34no—
81C3×#104—32no—
82C3×#101—34no—
83C23×C32—81no—
84C32×D4—42no—
85C32×D4—81no—
86C2×C32×C4—41no—
87C2×C3×Dic3—81no—
88C3×Dic6—42no—
89C3×Dic6—81no—
90C3×SL(2,𝔽3)C2×D411yescharacteristic
91C3×C3⋊C8—42no—
92C2×2−1+4C3×D311yescharacteristic
93C2×#137—31no—
94C2×#137—31no—
95C2×C4⋊Q8—31no—
96#456—34no—
97#456—34no—
98C2×#104—31no—
99C2×#104—31no—
100C2×#101—31no—
101C3×#41—31no—
102C3×#41—31no—
103C3×#41—32no—
104C3×#41—32no—
105C3×#41—62no—
106C3×#41—61no—
107C3×#41—61no—
108C3×#41—64no—
109C3×#41—62no—
110C3×#41#6711yescharacteristic
111C3×#41C22×D311yescharacteristic
112C2×C3×Q8—22no—
113C2×C3×Q8—32no—
114C2×C3×Q8—32no—
115C2×C3×Q8—32no—
116C2×C3×Q8—31no—
117C2×C3×Q8—61no—
118C2×C3×Q8—61no—
119C2×C3×Q8—61no—
120C2×C3×Q8—61no—
121C2×C3×Q8—61no—
122C2×C3×Q8—61no—
123C2×C3×Q8—61no—
124C2×C3×Q8—62no—
125C2×C3×Q8—61no—
126C2×C3×Q8#6711yescharacteristic
127C2×C3×D4—31no—
128C2×C3×D4—41no—
129C2×C3×D4—61no—
130C2×C3×D4—81no—
131C2×C3×D4S411yescharacteristic
132C22×C3×C4—31no—
133C22×C3×C4—32no—
134C22×C3×C4—61no—
135C22×C3×C4—61no—
136C22×C3×C4—61no—
137C2×Dic6—41no—
138#41⋊C3—21no—
139#41⋊C3—21no—
140#41⋊C3C3×D411yescharacteristic
141#41⋊C3C23×C311yescharacteristic
142C2×SL(2,𝔽3)—24no—
143C2×SL(2,𝔽3)—22no—
144C2×SL(2,𝔽3)—21no—
145C2×SL(2,𝔽3)C3×D411yescharacteristic
146#226—22no—
147C3×Q16—62no—
148C3×Q16—64no—
149C3×Q16—62no—
150C3×SD16—62no—
151C3×SD16—62no—
152C3×SD16—64no—
153C3×M16—32no—
154C3×M16—32no—
155C3×M16—61no—
156C3×M16—61no—
157C2×C3×C8—61no—
158C2×C3×C8—61no—
159C3×C4⋊C4—68no—
160C3×C42—32no—
161C3×C42—61no—
162#214—44no—
163C2×C3⋊C8—41no—
164C22×C32—41no—
165C22×C32—82no—
166C22×C32—162no—
167C32×C4—41no—
168C32×C4—41no—
169C3×Dic3—82no—
1702−1+4—21no—
1712−1+4—32no—
1722−1+4—61no—
1732−1+4C2×C3×D312yes—
174C2×#41—31no—
175C2×#41—61no—
176C2×#41C2×C3×D311yescharacteristic
177C22×Q8—21no—
178C22×Q8—31no—
179C22×Q8—31no—
180#137—32no—
181#137—34no—
182#137—32no—
183#137—61no—
184#137—62no—
185#137—61no—
186C2×Q16—61no—
187C2×Q16—61no—
188C2×SD16—61no—
189C2×SD16—61no—
190C2×M16—31no—
191C2×M16—31no—
192C4⋊Q8—34no—
193C4⋊Q8—62no—
194C2×C4⋊C4—62no—
195C2×C42—31no—
196#104—32no—
197#104—34no—
198#104—32no—
199#101—34no—
200C23×C3—21no—
201C23×C3—31no—
202C23×C3—81no—
203C23×C3—82no—
204C3×Q8—31no—
205C3×Q8—32no—
206C3×Q8—62no—
207C3×Q8—62no—
208C3×Q8—62no—
209C3×Q8—61no—
210C3×Q8—62no—
211C3×Q8—121no—
212C3×Q8—121no—
213C3×Q8C2×#6711yescharacteristic
214C3×D4—21no—
215C3×D4—31no—
216C3×D4—31no—
217C3×D4—42no—
218C3×D4—61no—
219C3×D4—62no—
220C3×D4—62no—
221C3×D4—81no—
222C3×D4—82no—
223C3×D4—82no—
224C3×D4C2×S412yes—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—34no—
231C2×C3×C4—41no—
232C2×C3×C4—62no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—62no—
236C2×C3×C4—64no—
237C2×C3×C4—61no—
238C2×C3×C4—62no—
239C2×C3×C4—81no—
240C2×C3×C4—121no—
241C2×C3×C4—121no—
242C2×C3×C4C2×S411yescharacteristic
243C2×Dic3—81no—
244Dic6—42no—
245Dic6—81no—
246SL(2,𝔽3)—21no—
247SL(2,𝔽3)C2×C3×D411yescharacteristic
248C3×C8—62no—
249C3×C8—62no—
250C3⋊C8—42no—
251C2×C32—41no—
252C2×C32—42no—
253C2×C32—162no—
254#41—31no—
255#41—31no—
256#41—32no—
257#41—32no—
258#41—62no—
259#41—61no—
260#41—61no—
261#41—64no—
262#41—62no—
263#41C3×#6711yescharacteristic
264#41C22×C3×D311yescharacteristic
265C2×Q8—22no—
266C2×Q8—32no—
267C2×Q8—32no—
268C2×Q8—32no—
269C2×Q8—31no—
270C2×Q8—61no—
271C2×Q8—61no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—61no—
276C2×Q8—61no—
277C2×Q8—62no—
278C2×Q8—61no—
279C2×Q8C3×#6711yescharacteristic
280C2×D4—31no—
281C2×D4—61no—
282C2×D4C3×S411yescharacteristic
283C22×C4—31no—
284C22×C4—32no—
285C22×C4—61no—
286C22×C4—61no—
287C22×C4—61no—
288Q16—62no—
289Q16—64no—
290Q16—62no—
291SD16—62no—
292SD16—62no—
293SD16—64no—
294M16—32no—
295M16—32no—
296M16—61no—
297M16—61no—
298C2×C8—61no—
299C2×C8—61no—
300C4⋊C4—68no—
301C42—32no—
302C42—61no—
303C22×C3—22no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—62no—
309C22×C3—82no—
310C22×C3—84no—
311C22×C3—81no—
312C22×C3—162no—
313C22×C3—164no—
314C22×C3GL(2,ℤ/4)11yescharacteristic
315C3×C4—31no—
316C3×C4—31no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—64no—
320C3×C4—62no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4—122no—
324C3×C4GL(2,ℤ/4)11yescharacteristic
325C3×C4C22×S411yescharacteristic
326Dic3—82no—
327C32—41noSylow
328C23—21no—
329C23—31no—
330Q8—31no—
331Q8—32no—
332Q8—62no—
333Q8—62no—
334Q8—62no—
335Q8—61no—
336Q8—62no—
337Q8—121no—
338Q8—121no—
339Q8C2×C3×#6711yescharacteristic
340D4—21no—
341D4—31no—
342D4—31no—
343D4—61no—
344D4—62no—
345D4—62no—
346D4C2×C3×S412yes—
347C2×C4—31no—
348C2×C4—31no—
349C2×C4—31no—
350C2×C4—31no—
351C2×C4—31no—
352C2×C4—34no—
353C2×C4—62no—
354C2×C4—62no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—64no—
358C2×C4—61no—
359C2×C4—62no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4C2×C3×S411yescharacteristic
363C8—62no—
364C8—62no—
365C2×C3—42no—
366C2×C3—42no—
367C2×C3—41no—
368C2×C3—61no—
369C2×C3—61no—
370C2×C3—82no—
371C2×C3—81no—
372C2×C3—162no—
373C2×C3—164no—
374C2×C3#543012yesdirect product factor
375C2×C3C2×GL(2,ℤ/4)11yescharacteristic
376C22—22no—
377C22—31no—
378C22—31no—
379C22—42no—
380C22—62no—
381C22C3×GL(2,ℤ/4)11yescharacteristic
382C4—31no—
383C4—31no—
384C4—64no—
385C4—62no—
386C4—122no—
387C4C3×GL(2,ℤ/4)11yescharacteristic
388C4C22×C3×S411yescharacteristic
389C3—41no—
390C3—81no—
391C3C2×#543011yescharacteristic, direct product factor
392C2—42no—
393C2—61no—
394C2—61no—
395C2C3×#543012yesdirect product factor
396C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3GL(2,ℤ/4)
commutator, cosocle#41⋊C3C23×C3
FrattiniC4C22×C3×S4
FittingC2×C3×2−1+4D3
prime radicalC2×C3×2−1+4⋊C3C2
hypercenterC2×C3×D4S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#984C2semidirect
3. C3×#5430C2direct (N × H)
4. C2×C3×GU(2,𝔽3)C2semidirect
5. C2×#5430C3direct (N × H)
12. C3×#984C22semidirect
15. C3×GU(2,𝔽3)C22semidirect
19. C2×#984C2×C3semidirect
23. #5430C2×C3direct (N × H)
24. C2×GU(2,𝔽3)C2×C3semidirect
59. #984C22×C3semidirect
78. GU(2,𝔽3)C22×C3semidirect
374. C2×C3#5430direct (N × H)
391. C3C2×#5430direct (N × H)
395. C2C3×#5430direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×2−1+4⋊C32
C2×C3×#9842
C3×#54302
C2×C3×GU(2,𝔽3)2
C2×#54303
C2×C3×#4563
C2×C3×#2144