Finite Group Databasebeta

← C3×C4×2−1+4⋊C3 #11919688

Subgroups — C3×C4×2−1+4⋊C3 #11919688

Order 1152.

892 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×2−1+4⋊C3C211yesmaximal, characteristic
2C3×#520⋊C3C211yesmaximal, characteristic, complemented
3C3×#519⋊C3C212yesmaximal, complemented
4C3×C4×#41⋊C3C211yesmaximal, characteristic, complemented
5C2×C3×C4×SL(2,𝔽3)C212yesmaximal, complemented
6C3×C4×2−1+4C311yesmaximal, characteristic, complemented
7C4×2−1+4⋊C3C313yesmaximal, direct product factor
8C3×2−1+4⋊C3C414yesdirect product factor
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2212yes—
11C32×C4×D4—41nomaximal
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2212yes—
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C2×C3×2−1+4C2×C311yescharacteristic
16C2×2−1+4⋊C3C2×C313yes—
17C3×#520—31no—
18C3×#520—32no—
19C3×#520C2×C311yescharacteristic, complemented
20C3×#519—31no—
21C3×#519C2×C312yescomplemented
22C3×C4×#41—31no—
23C3×C4×#41—32no—
24C3×C4×#41C2×C311yescharacteristic, complemented
25C2×C3×C4×Q8—31no—
26C2×C3×C4×Q8C2×C312yescomplemented
27#520⋊C3C2×C313yescomplemented
28#519⋊C3C2×C316yescomplemented
29C4×#41⋊C3C2×C313yescomplemented
30C2×C4×SL(2,𝔽3)C2×C316yescomplemented
31C2×C32×D4—41no—
32C22×C32×C4—42no—
33C3×#41⋊C3C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2×C414yescomplemented
35C2×C3×SL(2,𝔽3)C2311yescharacteristic
36C32×C4⋊C4—41no—
37C32×C22⋊C4—42no—
38C32×C42—41no—
39C4×2−1+4C3211yescharacteristic, complemented, Sylow
40C3×2−1+4—34no—
41C3×2−1+4C3×C414yescomplemented
42C2×C3×#41—31no—
43C2×C3×#41—32no—
44C2×C3×#41C22×C311yescharacteristic
45C22×C3×Q8—31no—
46C22×C3×Q8C22×C312yes—
472−1+4⋊C3C3×C4112yesdirect product factor
48C2×#41⋊C3C22×C313yes—
49C22×SL(2,𝔽3)C22×C316yes—
50C3×C4×Q8—32no—
51C3×C4×Q8—31no—
52C3×C4×Q8—32no—
53C3×C4×Q8—32no—
54C3×C4×Q8—31no—
55C3×C4×Q8—32no—
56C3×C4×Q8—32no—
57C3×C4×Q8C22×C311yescharacteristic
58C3×C4×Q8C22×C312yes—
59C3×C4×Q8C22×C311yescharacteristic
60C3×C4×D4—34no—
61C3×C4×D4—31no—
62C3×C4×D4—32no—
63C3×C4×D4—31no—
64C3×C4×D4—31no—
65C3×C4×D4—32no—
66C3×C4×D4—32no—
67C3×C4×D4—43no—
68C3×C4×D4A411yescharacteristic
69C3×#117—34no—
70C3×#117—31no—
71C3×#117—32no—
72C3×#117—32no—
73C3×#117—31no—
74C2×C3×C4⋊C4—32no—
75C2×C3×C4⋊C4—32no—
76C2×C3×C4⋊C4—31no—
77C2×C3×C42—32no—
78C2×C3×C42—32no—
79C2×C3×C42—31no—
80C4×SL(2,𝔽3)C22×C313yes—
81C4×SL(2,𝔽3)C22×C316yes—
82C4×SL(2,𝔽3)C22×C313yes—
83C23×C32—42no—
84C32×D4—44no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4—42no—
88C2×C32×C4—41no—
89C2×C32×C4—84no—
90C3×SL(2,𝔽3)C22×C411yescharacteristic
91C2×2−1+4C2×C3211yescharacteristic
92#520—31no—
93#520—32no—
94#520C2×C3211yescharacteristic, complemented
95#519—31no—
96#519C2×C3212yescomplemented
97C4×#41—31no—
98C4×#41—32no—
99C4×#41C2×C3211yescharacteristic, complemented
100C2×C4×Q8—31no—
101C2×C4×Q8C2×C3212yescomplemented
102C3×#41—38no—
103C3×#41—34no—
104C3×#41—34no—
105C3×#41—38no—
106C3×#41—32no—
107C3×#41C2×C3×C412yescomplemented
108C2×C3×Q8—34no—
109C2×C3×Q8—34no—
110C2×C3×Q8—31no—
111C2×C3×Q8—32no—
112C2×C3×Q8—34no—
113C2×C3×Q8C2×C3×C414yescomplemented
114C2×C3×Q8C23×C311yescharacteristic
115C2×C3×D4—31no—
116C2×C3×D4—32no—
117C2×C3×D4—43no—
118C2×C3×D4C2×A411yescharacteristic
119C22×C3×C4—32no—
120C22×C3×C4—32no—
121C22×C3×C4—31no—
122C22×C3×C4—32no—
123C22×C3×C4—32no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—46no—
127C22×C3×C4C2×A412yes—
128#41⋊C3C2×C3×C416yescomplemented
129C2×SL(2,𝔽3)C2×C3×C4112yescomplemented
130C2×SL(2,𝔽3)C23×C313yes—
131C3×C4⋊C4—31no—
132C3×C4⋊C4—32no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—34no—
135C3×C4⋊C4—32no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—32no—
138C3×C4⋊C4—43no—
139C3×C4⋊C4C2×A411yescharacteristic
140C3×C22⋊C4—32no—
141C3×C22⋊C4—32no—
142C3×C22⋊C4—32no—
143C3×C22⋊C4—46no—
144C3×C22⋊C4C2×A412yes—
145C3×C42—34no—
146C3×C42—32no—
147C3×C42—31no—
148C3×C42—32no—
149C3×C42—31no—
150C3×C42—32no—
151C3×C42—31no—
152C3×C42—43no—
153C3×C42C2×A411yescharacteristic
154C22×C32—44no—
155C22×C32—41no—
156C22×C32—82no—
157C22×C32—82no—
158C32×C4—42no—
159C32×C4—42no—
160C32×C4—81no—
161C32×C4—82no—
1622−1+4—34no—
1632−1+4C32×C414yescomplemented
164C2×#41—31no—
165C2×#41—32no—
166C2×#41C22×C3211yescharacteristic
167C22×Q8—31no—
168C22×Q8C22×C3212yes—
169C4×Q8—32no—
170C4×Q8—31no—
171C4×Q8—32no—
172C4×Q8—32no—
173C4×Q8—31no—
174C4×Q8—32no—
175C4×Q8—32no—
176C4×Q8C22×C3211yescharacteristic
177C4×Q8C22×C3212yes—
178C4×Q8C22×C3211yescharacteristic
179C4×D4—34no—
180C4×D4—31no—
181C4×D4—32no—
182C4×D4—31no—
183C4×D4—31no—
184C4×D4—32no—
185C4×D4—32no—
186C4×D4C3×A411yescharacteristic
187#117—34no—
188#117—31no—
189#117—32no—
190#117—32no—
191#117—31no—
192C2×C4⋊C4—32no—
193C2×C4⋊C4—32no—
194C2×C4⋊C4—31no—
195C2×C42—32no—
196C2×C42—32no—
197C2×C42—31no—
198C23×C3—31no—
199C23×C3—46no—
200C23×C3C22×A412yes—
201C3×Q8—34no—
202C3×Q8—31no—
203C3×Q8—34no—
204C3×Q8—34no—
205C3×Q8C22×C3×C411yescharacteristic
206C3×D4—32no—
207C3×D4—32no—
208C3×D4—38no—
209C3×D4—412no—
210C3×D4C4×A414yes—
211C2×C3×C4—24no—
212C2×C3×C4—38no—
213C2×C3×C4—34no—
214C2×C3×C4—34no—
215C2×C3×C4—32no—
216C2×C3×C4—31no—
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—31no—
220C2×C3×C4—32no—
221C2×C3×C4—31no—
222C2×C3×C4—43no—
223C2×C3×C4—43no—
224C2×C3×C4—43no—
225C2×C3×C4—46no—
226C2×C3×C4—62no—
227C2×C3×C4—812no—
228C2×C3×C4—121no—
229C2×C3×C4—122no—
230C2×C3×C4—122no—
231C2×C3×C4C22×A412yes—
232C2×C3×C4C22×A411yescharacteristic
233C2×C3×C4C22×A411yescharacteristic
234C2×C3×C4C22×A411yescharacteristic
235SL(2,𝔽3)C22×C3×C413yes—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—84no—
240#41—38no—
241#41—34no—
242#41—34no—
243#41—38no—
244#41—32no—
245#41C2×C32×C412yescomplemented
246C2×Q8—34no—
247C2×Q8—34no—
248C2×Q8—31no—
249C2×Q8—32no—
250C2×Q8—34no—
251C2×Q8C2×C32×C414yescomplemented
252C2×Q8C23×C3211yescharacteristic
253C2×D4—31no—
254C2×D4—32no—
255C2×D4C2×C3×A411yescharacteristic
256C22×C4—32no—
257C22×C4—32no—
258C22×C4—31no—
259C22×C4—32no—
260C22×C4—32no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4C2×C3×A412yes—
264C4⋊C4—31no—
265C4⋊C4—32no—
266C4⋊C4—31no—
267C4⋊C4—34no—
268C4⋊C4—32no—
269C4⋊C4—31no—
270C4⋊C4—32no—
271C4⋊C4C2×C3×A411yescharacteristic
272C22⋊C4—32no—
273C22⋊C4—32no—
274C22⋊C4—32no—
275C22⋊C4C2×C3×A412yes—
276C42—34no—
277C42—31no—
278C42—32no—
279C42—31no—
280C42—32no—
281C42—32no—
282C42—31no—
283C42C2×C3×A411yescharacteristic
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—32no—
287C22×C3—412no—
288C22×C3—43no—
289C22×C3—61no—
290C22×C3—61no—
291C22×C3—86no—
292C22×C3—86no—
293C22×C3C2×C4×A414yes—
294C22×C3C23×A411yescharacteristic
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—34no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—46no—
301C3×C4—46no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—62no—
305C3×C4—86no—
306C3×C4—83no—
307C3×C4C2×C4×A412yes—
308C3×C42−1+4⋊C312yesdirect product factor
309C32—41noSylow
310C23—31no—
311C23C22×C3×A412yes—
312Q8—34no—
313Q8—31no—
314Q8—34no—
315Q8—34no—
316Q8C22×C32×C411yescharacteristic
317D4—32no—
318D4—32no—
319D4—38no—
320D4C3×C4×A414yes—
321C2×C4—24no—
322C2×C4—38no—
323C2×C4—34no—
324C2×C4—34no—
325C2×C4—31no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—31no—
330C2×C4—32no—
331C2×C4—31no—
332C2×C4—62no—
333C2×C4—121no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4C22×C3×A412yes—
337C2×C4C22×C3×A411yescharacteristic
338C2×C4C22×C3×A411yescharacteristic
339C2×C4C22×C3×A411yescharacteristic
340C2×C3—24no—
341C2×C3—43no—
342C2×C3—43no—
343C2×C3—43no—
344C2×C3—62no—
345C2×C3—812no—
346C2×C3C22×C4×A411yescharacteristic
347C2×C3C2×2−1+4⋊C311yescharacteristic
348C2×C3#594511yescharacteristic
349C22—22no—
350C22—22no—
351C22—32no—
352C22—61no—
353C22—61no—
354C22C2×C3×C4×A414yes—
355C22C23×C3×A411yescharacteristic
356C4—22no—
357C4—21no—
358C4—31no—
359C4—34no—
360C4—31no—
361C4—61no—
362C4—61no—
363C4—62no—
364C4C2×C3×C4×A412yes—
365C4C3×2−1+4⋊C312yesdirect product factor
366C3—43no—
367C3C4×2−1+4⋊C311yescharacteristic, direct product factor
368C2—24no—
369C2—62no—
370C2C22×C3×C4×A411yescharacteristic
371C2C2×C3×2−1+4⋊C311yescharacteristic
372C2C3×#594511yescharacteristic
subgroupG/H
centerC2×C3×C4C22×A4
commutator, hypocenterQ8C22×C32×C4
FrattiniC22C23×C3×A4
FittingC3×C4×2−1+4C3
socleC22×C3C23×A4
prime radicalC2×C3×2−1+4⋊C3C2
cosocleC2×Q8C23×C32
hypercenterC3×C4×D4A4
LCSG ▷ Q8
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. C3×#519⋊C3C2semidirect
4. C3×C4×#41⋊C3C2semidirect
5. C2×C3×C4×SL(2,𝔽3)C2semidirect
6. C3×C4×2−1+4C3semidirect
7. C4×2−1+4⋊C3C3direct (N × H)
8. C3×2−1+4⋊C3C4direct (N × H)
19. C3×#520C2×C3semidirect
21. C3×#519C2×C3semidirect
24. C3×C4×#41C2×C3semidirect
26. C2×C3×C4×Q8C2×C3semidirect
27. #520⋊C3C2×C3semidirect
28. #519⋊C3C2×C3semidirect
29. C4×#41⋊C3C2×C3semidirect
30. C2×C4×SL(2,𝔽3)C2×C3semidirect
33. C3×#41⋊C3C2×C4semidirect
34. C2×C3×SL(2,𝔽3)C2×C4semidirect
39. C4×2−1+4C32semidirect
41. C3×2−1+4C3×C4semidirect
47. 2−1+4⋊C3C3×C4direct (N × H)
94. #520C2×C32semidirect
96. #519C2×C32semidirect
99. C4×#41C2×C32semidirect
101. C2×C4×Q8C2×C32semidirect
107. C3×#41C2×C3×C4semidirect
113. C2×C3×Q8C2×C3×C4semidirect
128. #41⋊C3C2×C3×C4semidirect
129. C2×SL(2,𝔽3)C2×C3×C4semidirect
163. 2−1+4C32×C4semidirect
245. #41C2×C32×C4semidirect
251. C2×Q8C2×C32×C4semidirect
308. C3×C42−1+4⋊C3direct (N × H)
365. C4C3×2−1+4⋊C3direct (N × H)
367. C3C4×2−1+4⋊C3direct (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
C3×#520⋊C32
C3×#519⋊C32
C3×C4×#41⋊C32
C2×C3×C4×SL(2,𝔽3)2
C3×C4×2−1+43
C4×2−1+4⋊C33
C32×C4×D44