Finite Group Databasebeta

← C3×#34512 #10653119

Subgroups — C3×#34512 #10653119

Order 768.

2576 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×H(3,𝔽4)C211yesmaximal, characteristic
2C3×#2375C213yesmaximal, complemented
3C3×#2372C213yesmaximal, complemented
4C3×#2367C216yesmaximal, complemented
5C3×#2358C213yesmaximal, complemented
6C3×#2337C216yesmaximal, complemented
7C3×#2326C213yesmaximal, complemented
8C3×#2321C213yesmaximal, complemented
9C3×C4×#120C213yesmaximal, complemented
10#34512C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic, complemented
12C3×H(3,𝔽4)C4116yescomplemented
13C2×C3×#124C2216yes—
14C2×C3×#124C2213yes—
15C2×C3×#120C2213yes—
16C2×C3×#120C2213yes—
17C2×C3×C4×Q8C2213yescomplemented
18C2×C3×C4×D4C2216yes—
19C2×C3×C4×D4C2213yes—
20C22×C3×C22⋊C4C2213yescomplemented
21C3×#391C2216yescomplemented
22C3×#391C2213yescomplemented
23C3×#390C2216yes—
24C3×#390C2213yes—
25C3×#388C2216yescomplemented
26C3×#388C2216yescomplemented
27C3×#388C2216yescomplemented
28C3×#388C2216yescomplemented
29C3×#388C2216yescomplemented
30C3×#388C2216yescomplemented
31C3×#386C2213yes—
32C3×#386C2213yes—
33C3×#386C2216yes—
34C3×#386C2216yes—
35C3×#385C2213yescomplemented
36C3×#385C2216yescomplemented
37C3×#385C2216yescomplemented
38C3×#385C2213yescomplemented
39C3×#384C2211yescharacteristic, complemented
40C3×#383C2212yescomplemented
41C3×#379C2213yescomplemented
42C3×#379C2211yescharacteristic, complemented
43C3×C4×C4⋊C4C2216yescomplemented
44C3×C4×C22⋊C4C2213yescomplemented
45C3×C4×C22⋊C4C2216yescomplemented
46C3×C4×C22⋊C4C2216yescomplemented
47C3×C4×C22⋊C4C2213yescomplemented
48C3×#376C2213yescomplemented
49C2×H(3,𝔽4)C2×C311yescharacteristic
50#2375C2×C313yescomplemented
51#2372C2×C313yescomplemented
52#2367C2×C316yescomplemented
53#2358C2×C313yescomplemented
54#2337C2×C316yescomplemented
55#2326C2×C313yescomplemented
56#2321C2×C313yescomplemented
57C4×#120C2×C313yescomplemented
58C25×C3C2311yescharacteristic
59C25×C3C2311yescharacteristic
60C22×C3×Q8C2313yes—
61C22×C3×D4C2313yes—
62C22×C3×D4C2316yes—
63C23×C3×C4—46no—
64C23×C3×C4C2313yes—
65C23×C3×C4C2313yes—
66C23×C3×C4C2313yes—
67C23×C3×C4C2313yes—
68C3×#124C2×C4148yescomplemented
69C3×#124C2×C4124yescomplemented
70C3×#120C2×C4124yescomplemented
71C3×#120C2×C4124yescomplemented
72C3×C4×Q8—46no—
73C3×C4×D4—46no—
74C3×C4×D4—412no—
75C2×C3×C4⋊C4C2316yes—
76C2×C3×C4⋊C4C2316yes—
77C2×C3×C4⋊C4C2313yes—
78C2×C3×C4⋊C4C2313yes—
79C2×C3×C22⋊C4—46no—
80C2×C3×C22⋊C4C2316yes—
81C2×C3×C22⋊C4C2313yes—
82C2×C3×C22⋊C4C2316yes—
83C2×C3×C22⋊C4C2313yes—
84C2×C3×C22⋊C4C2313yes—
85C2×C3×C22⋊C4C2316yes—
86C2×C3×C22⋊C4C2313yes—
87C2×C3×C22⋊C4C2316yes—
88C2×C3×C22⋊C4C2316yes—
89C2×C3×C22⋊C4C2313yes—
90C2×C3×C22⋊C4C2313yes—
91C2×C3×C22⋊C4C2316yes—
92C2×C3×C42C2313yes—
93C2×C3×C42C2313yes—
94C2×C3×C42C2316yes—
95C2×C3×C42C2316yes—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2312yes—
98C3×#95C2316yes—
99C3×#95C2316yes—
100C3×#95C2316yes—
101C3×#95C2316yes—
102C3×#95C2316yes—
103C3×#95C2316yes—
104C24×C4C22×C311yescharacteristic, complemented
105H(3,𝔽4)C3×C4116yescomplemented
106C2×#124C22×C313yes—
107C2×#124C22×C316yes—
108C2×#120C22×C313yes—
109C2×#120C22×C313yes—
110C2×C4×Q8C22×C313yescomplemented
111C2×C4×D4C22×C313yes—
112C2×C4×D4C22×C316yes—
113C22×C22⋊C4C22×C313yescomplemented
114#391C22×C313yescomplemented
115#391C22×C316yescomplemented
116#390C22×C316yes—
117#390C22×C313yes—
118#388C22×C316yescomplemented
119#388C22×C316yescomplemented
120#388C22×C316yescomplemented
121#388C22×C316yescomplemented
122#388C22×C316yescomplemented
123#388C22×C316yescomplemented
124#386C22×C316yes—
125#386C22×C313yes—
126#386C22×C316yes—
127#386C22×C313yes—
128#385C22×C313yescomplemented
129#385C22×C316yescomplemented
130#385C22×C316yescomplemented
131#385C22×C313yescomplemented
132#384C22×C311yescharacteristic, complemented
133#383C22×C312yescomplemented
134#379C22×C311yescharacteristic, complemented
135#379C22×C313yescomplemented
136C4×C4⋊C4C22×C316yescomplemented
137C4×C22⋊C4C22×C313yescomplemented
138C4×C22⋊C4C22×C313yescomplemented
139C4×C22⋊C4C22×C316yescomplemented
140C4×C22⋊C4C22×C316yescomplemented
141#376C22×C313yescomplemented
142C24×C3—43no—
143C24×C3—43no—
144C24×C3—43no—
145C24×C3—43no—
146C24×C3C22×C414yes—
147C24×C3C22×C414yescomplemented
148C24×C3C2413yes—
149C24×C3C2413yes—
150C2×C3×Q8—43no—
151C2×C3×Q8—43no—
152C2×C3×Q8C22×C4112yescomplemented
153C2×C3×D4—43no—
154C2×C3×D4—46no—
155C2×C3×D4—43no—
156C2×C3×D4—46no—
157C2×C3×D4C22×C4124yes—
158C2×C3×D4C22×C4112yes—
159C22×C3×C4—212no—
160C22×C3×C4—26no—
161C22×C3×C4—26no—
162C22×C3×C4—43no—
163C22×C3×C4—46no—
164C22×C3×C4—43no—
165C22×C3×C4—46no—
166C22×C3×C4—46no—
167C22×C3×C4—416no—
168C22×C3×C4C2416yes—
169C22×C3×C4C2411yescharacteristic
170C22×C3×C4C2413yes—
171C22×C3×C4C2413yes—
172C22×C3×C4C2413yes—
173C22×C3×C4C2416yes—
174C22×C3×C4C2413yes—
175C3×C4⋊C4—212no—
176C3×C4⋊C4—46no—
177C3×C4⋊C4—43no—
178C3×C4⋊C4—43no—
179C3×C22⋊C4—212no—
180C3×C22⋊C4—46no—
181C3×C22⋊C4—46no—
182C3×C22⋊C4—43no—
183C3×C22⋊C4—43no—
184C3×C22⋊C4—43no—
185C3×C22⋊C4—43no—
186C3×C22⋊C4—46no—
187C3×C22⋊C4C22×C4124yes—
188C3×C22⋊C4C22×C4112yes—
189C3×C22⋊C4C22×C4112yescomplemented
190C3×C22⋊C4C22×C4124yescomplemented
191C3×C42—26no—
192C3×C42—43no—
193C3×C42—46no—
194C3×C42—46no—
195C3×C42C22×C414yescomplemented
196C3×C42C22×C418yescomplemented
197C25C23×C311yescharacteristic
198C25C23×C311yescharacteristic
199C22×Q8C23×C313yes—
200C22×D4C23×C313yes—
201C22×D4C23×C316yes—
202C23×C4—46no—
203C23×C4C23×C313yes—
204C23×C4C23×C313yes—
205C23×C4C23×C313yes—
206C23×C4C23×C313yes—
207#124C2×C3×C4124yescomplemented
208#124C2×C3×C4148yescomplemented
209#120C2×C3×C4124yescomplemented
210#120C2×C3×C4124yescomplemented
211C4×Q8—46no—
212C4×D4—46no—
213C4×D4—412no—
214C2×C4⋊C4C23×C316yes—
215C2×C4⋊C4C23×C313yes—
216C2×C4⋊C4C23×C313yes—
217C2×C4⋊C4C23×C316yes—
218C2×C22⋊C4—46no—
219C2×C22⋊C4C23×C316yes—
220C2×C22⋊C4C23×C316yes—
221C2×C22⋊C4C23×C313yes—
222C2×C22⋊C4C23×C316yes—
223C2×C22⋊C4C23×C313yes—
224C2×C22⋊C4C23×C313yes—
225C2×C22⋊C4C23×C316yes—
226C2×C22⋊C4C23×C313yes—
227C2×C22⋊C4C23×C316yes—
228C2×C22⋊C4C23×C313yes—
229C2×C22⋊C4C23×C316yes—
230C2×C22⋊C4C23×C313yes—
231C2×C42C23×C313yes—
232C2×C42C23×C316yes—
233C2×C42C23×C316yes—
234C2×C42C23×C313yes—
235C2×C42C23×C311yescharacteristic
236C2×C42C23×C312yes—
237#95C23×C316yes—
238#95C23×C316yes—
239#95C23×C316yes—
240#95C23×C316yes—
241#95C23×C316yes—
242#95C23×C316yes—
243C23×C3—26no—
244C23×C3—23no—
245C23×C3—26no—
246C23×C3—23no—
247C23×C3—26no—
248C23×C3—23no—
249C23×C3—23no—
250C23×C3—26no—
251C23×C3—412no—
252C23×C3—412no—
253C23×C3—44no—
254C23×C3—412no—
255C23×C3—82no—
256C23×C3—86no—
257C23×C3C23×C416yes—
258C23×C3C23×C416yes—
259C23×C3C2511yescharacteristic
260C3×Q8—412no—
261C3×D4—412no—
262C3×D4—424no—
263C2×C3×C4—23no—
264C2×C3×C4—26no—
265C2×C3×C4—26no—
266C2×C3×C4—23no—
267C2×C3×C4—26no—
268C2×C3×C4—23no—
269C2×C3×C4—26no—
270C2×C3×C4—26no—
271C2×C3×C4—23no—
272C2×C3×C4—23no—
273C2×C3×C4—26no—
274C2×C3×C4—26no—
275C2×C3×C4—26no—
276C2×C3×C4—412no—
277C2×C3×C4—412no—
278C2×C3×C4—86no—
279C2×C3×C4—86no—
280C2×C3×C4C23×C416yes—
281C2×C3×C4C23×C4112yes—
282C2×C3×C4H(5,𝔽2)16yes—
283C24—43no—
284C24—43no—
285C24—43no—
286C24—43no—
287C24C22×C3×C414yes—
288C24C22×C3×C414yescomplemented
289C24C24×C313yes—
290C24C24×C313yes—
291C2×Q8—43no—
292C2×Q8—43no—
293C2×Q8C22×C3×C4112yescomplemented
294C2×D4—43no—
295C2×D4—46no—
296C2×D4—43no—
297C2×D4—46no—
298C2×D4C22×C3×C4112yes—
299C2×D4C22×C3×C4124yes—
300C22×C4—212no—
301C22×C4—26no—
302C22×C4—26no—
303C22×C4—43no—
304C22×C4—46no—
305C22×C4—43no—
306C22×C4—46no—
307C22×C4—46no—
308C22×C4—416no—
309C22×C4C24×C316yes—
310C22×C4C24×C311yescharacteristic
311C22×C4C24×C313yes—
312C22×C4C24×C313yes—
313C22×C4C24×C313yes—
314C22×C4C24×C316yes—
315C22×C4C24×C313yes—
316C4⋊C4—212no—
317C4⋊C4—43no—
318C4⋊C4—43no—
319C4⋊C4—46no—
320C22⋊C4—212no—
321C22⋊C4—46no—
322C22⋊C4—43no—
323C22⋊C4—43no—
324C22⋊C4—46no—
325C22⋊C4—46no—
326C22⋊C4—43no—
327C22⋊C4—43no—
328C22⋊C4C22×C3×C4124yes—
329C22⋊C4C22×C3×C4112yes—
330C22⋊C4C22×C3×C4124yescomplemented
331C22⋊C4C22×C3×C4112yescomplemented
332C42—26no—
333C42—46no—
334C42—46no—
335C42—43no—
336C42C22×C3×C414yescomplemented
337C42C22×C3×C418yescomplemented
338C22×C3—26no—
339C22×C3—212no—
340C22×C3—212no—
341C22×C3—26no—
342C22×C3—416no—
343C22×C3—43no—
344C22×C3—43no—
345C22×C3—43no—
346C22×C3—43no—
347C22×C3—46no—
348C22×C3—46no—
349C22×C3—88no—
350C22×C3C24×C411yescharacteristic
351C22×C3C2×H(5,𝔽2)13yes—
352C22×C3#58513yes—
353C3×C4—26no—
354C3×C4—212no—
355C3×C4—26no—
356C3×C4—41no—
357C3×C4—43no—
358C3×C4—43no—
359C3×C4—46no—
360C23—26no—
361C23—23no—
362C23—26no—
363C23—23no—
364C23—26no—
365C23—23no—
366C23—23no—
367C23—26no—
368C23—412no—
369C23—412no—
370C23—44no—
371C23—412no—
372C23—82no—
373C23—86no—
374C23C23×C3×C416yes—
375C23C23×C3×C416yes—
376C23C25×C311yescharacteristic
377Q8—412no—
378D4—412no—
379D4—424no—
380C2×C4—23no—
381C2×C4—26no—
382C2×C4—26no—
383C2×C4—23no—
384C2×C4—26no—
385C2×C4—23no—
386C2×C4—26no—
387C2×C4—26no—
388C2×C4—23no—
389C2×C4—23no—
390C2×C4—26no—
391C2×C4—26no—
392C2×C4—26no—
393C2×C4—412no—
394C2×C4—412no—
395C2×C4—86no—
396C2×C4—86no—
397C2×C4C23×C3×C416yes—
398C2×C4C23×C3×C4112yes—
399C2×C4C3×H(5,𝔽2)16yes—
400C2×C3—46no—
401C2×C3—46no—
402C2×C3C4×H(5,𝔽2)13yes—
403C2×C3#353211yescharacteristic
404C2×C3#354013yes—
405C22—26no—
406C22—212no—
407C22—212no—
408C22—26no—
409C22—416no—
410C22—43no—
411C22—43no—
412C22—43no—
413C22—43no—
414C22—46no—
415C22—46no—
416C22—88no—
417C22C24×C3×C411yescharacteristic
418C22C2×C3×H(5,𝔽2)13yes—
419C22C3×#58513yes—
420C4—26no—
421C4—212no—
422C4—26no—
423C4—41no—
424C4—43no—
425C4—46no—
426C4—43no—
427C3#3451211yescharacteristic, direct product factor, Sylow
428C2—46no—
429C2—46no—
430C2C3×C4×H(5,𝔽2)13yes—
431C2C3×#353211yescharacteristic
432C2C3×#354013yes—
subgroupG/H
center, socleC23×C3C25
commutatorC22C24×C3×C4
Frattini, cosocleC23C25×C3
prime radicalC2×C3×H(3,𝔽4)C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2375C2semidirect
3. C3×#2372C2semidirect
4. C3×#2367C2semidirect
5. C3×#2358C2semidirect
6. C3×#2337C2semidirect
7. C3×#2326C2semidirect
8. C3×#2321C2semidirect
9. C3×C4×#120C2semidirect
10. #34512C3direct (N × H)
11. C24×C3×C4C22semidirect
12. C3×H(3,𝔽4)C4semidirect
17. C2×C3×C4×Q8C22semidirect
20. C22×C3×C22⋊C4C22semidirect
21. C3×#391C22semidirect
22. C3×#391C22semidirect
25. C3×#388C22semidirect
26. C3×#388C22semidirect
27. C3×#388C22semidirect
28. C3×#388C22semidirect
29. C3×#388C22semidirect
30. C3×#388C22semidirect
35. C3×#385C22semidirect
36. C3×#385C22semidirect
37. C3×#385C22semidirect
38. C3×#385C22semidirect
39. C3×#384C22semidirect
40. C3×#383C22semidirect
41. C3×#379C22semidirect
42. C3×#379C22semidirect
43. C3×C4×C4⋊C4C22semidirect
44. C3×C4×C22⋊C4C22semidirect
45. C3×C4×C22⋊C4C22semidirect
46. C3×C4×C22⋊C4C22semidirect
47. C3×C4×C22⋊C4C22semidirect
48. C3×#376C22semidirect
50. #2375C2×C3semidirect
51. #2372C2×C3semidirect
52. #2367C2×C3semidirect
53. #2358C2×C3semidirect
54. #2337C2×C3semidirect
55. #2326C2×C3semidirect
56. #2321C2×C3semidirect
57. C4×#120C2×C3semidirect
68. C3×#124C2×C4semidirect
69. C3×#124C2×C4semidirect
70. C3×#120C2×C4semidirect
71. C3×#120C2×C4semidirect
104. C24×C4C22×C3semidirect
105. H(3,𝔽4)C3×C4semidirect
110. C2×C4×Q8C22×C3semidirect
113. C22×C22⋊C4C22×C3semidirect
114. #391C22×C3semidirect
115. #391C22×C3semidirect
118. #388C22×C3semidirect
119. #388C22×C3semidirect
120. #388C22×C3semidirect
121. #388C22×C3semidirect
122. #388C22×C3semidirect
123. #388C22×C3semidirect
128. #385C22×C3semidirect
129. #385C22×C3semidirect
130. #385C22×C3semidirect
131. #385C22×C3semidirect
132. #384C22×C3semidirect
133. #383C22×C3semidirect
134. #379C22×C3semidirect
135. #379C22×C3semidirect
136. C4×C4⋊C4C22×C3semidirect
137. C4×C22⋊C4C22×C3semidirect
138. C4×C22⋊C4C22×C3semidirect
139. C4×C22⋊C4C22×C3semidirect
140. C4×C22⋊C4C22×C3semidirect
141. #376C22×C3semidirect
147. C24×C3C22×C4semidirect
152. C2×C3×Q8C22×C4semidirect
189. C3×C22⋊C4C22×C4semidirect
190. C3×C22⋊C4C22×C4semidirect
195. C3×C42C22×C4semidirect
196. C3×C42C22×C4semidirect
207. #124C2×C3×C4semidirect
208. #124C2×C3×C4semidirect
209. #120C2×C3×C4semidirect
210. #120C2×C3×C4semidirect
288. C24C22×C3×C4semidirect
293. C2×Q8C22×C3×C4semidirect
330. C22⋊C4C22×C3×C4semidirect
331. C22⋊C4C22×C3×C4semidirect
336. C42C22×C3×C4semidirect
337. C42C22×C3×C4semidirect
427. C3#34512direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×H(3,𝔽4)2
C3×#23752
C3×#23722
C3×#23672
C3×#23582
C3×#23372
C3×#23262
C3×#23212
C3×C4×#1202
#345123