Finite Group Databasebeta

← C3×#34976 #10653583

Subgroups — C3×#34976 #10653583

Order 768.

2810 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#41C212yesmaximal, complemented
2C3×#2609C214yesmaximal, complemented
3C3×#2582C211yesmaximal, characteristic, complemented
4C3×#2538C212yesmaximal, complemented
5C3×#2537C211yesmaximal, characteristic, complemented
6C3×#2485C214yesmaximal, complemented
7C3×#2407C214yesmaximal, complemented
8C3×#2392C214yesmaximal, complemented
9C3×C4×C22⋊Q8C214yesmaximal, complemented
10C3×C4×#121C214yesmaximal, complemented
11C3×#2285C211yesmaximal, characteristic, complemented
12#34976C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C2×C3×C22⋊Q8C2214yes—
15C2×C3×#121C2214yes—
16C3×C4×#41—216no—
17C2×C3×C4×Q8C2212yes—
18C2×C3×C4×Q8C2212yes—
19C2×C3×C4×D4C2214yes—
20C2×C3×C4×D4C2214yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2212yes—
23C2×C3×C4×D4C2214yes—
24C2×C3×#117C2212yes—
25C2×C3×#117C2214yes—
26C2×C3×#117C2212yes—
27C22×C3×C42C2214yes—
28C22×C3×C42C2211yescharacteristic
29C3×#398C2214yes—
30C3×#397C2214yes—
31C3×#396C2218yes—
32C3×#394C2218yes—
33C3×#394C2214yes—
34C3×#392C2214yes—
35C3×#391C2214yes—
36C3×#390C2214yes—
37C3×#389C2214yes—
38C3×#389C2214yes—
39C3×#388C2218yes—
40C3×#387C2218yes—
41C3×#386C2218yes—
42C3×#385C2218yes—
43C3×#384C2211yescharacteristic
44C3×#384C2211yescharacteristic
45C3×#382C2212yes—
46C3×#380C2214yes—
47C3×#380C2214yes—
48C3×#380C2214yes—
49C3×C4×C4⋊C4C2214yes—
50C3×C4×C4⋊C4C2214yes—
51C3×C4×C22⋊C4C2218yes—
52C2×C4×#41C2×C312yescomplemented
53#2609C2×C314yescomplemented
54#2582C2×C311yescharacteristic, complemented
55#2538C2×C312yescomplemented
56#2537C2×C311yescharacteristic, complemented
57#2485C2×C314yescomplemented
58#2407C2×C314yescomplemented
59#2392C2×C314yescomplemented
60C4×C22⋊Q8C2×C314yescomplemented
61C4×#121C2×C314yescomplemented
62#2285C2×C311yescharacteristic, complemented
63C2×C3×#41—28no—
64C2×C3×#41—28no—
65C2×C3×#41—28no—
66C22×C3×Q8C2312yes—
67C22×C3×D4C2314yes—
68C22×C3×D4C2314yes—
69C22×C3×D4C2312yes—
70C23×C3×C4C2314yes—
71C23×C3×C4C2314yes—
72C23×C3×C4C2314yes—
73C23×C3×C4C2312yes—
74C23×C3×C4C2311yescharacteristic
75C3×C22⋊Q8—216no—
76C3×#121—216no—
77C3×C4×Q8—28no—
78C3×C4×Q8—28no—
79C3×C4×D4—216no—
80C3×C4×D4—216no—
81C3×C4×D4—28no—
82C3×C4×D4—28no—
83C3×C4×D4—48no—
84C3×#117—28no—
85C3×#117—216no—
86C3×#117—44no—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2318yes—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C4⋊C4C2314yes—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2314yes—
97C2×C3×C22⋊C4C2318yes—
98C2×C3×C22⋊C4C2318yes—
99C2×C3×C22⋊C4C2314yes—
100C2×C3×C22⋊C4C2314yes—
101C2×C3×C22⋊C4C2314yes—
102C2×C3×C22⋊C4C2314yes—
103C2×C3×C22⋊C4C2318yes—
104C2×C3×C42—216no—
105C2×C3×C42D418yescomplemented
106C2×C3×C42C2314yes—
107C2×C3×C42C2314yes—
108C2×C3×C42C2314yes—
109C2×C3×C42C2314yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2312yes—
113C3×#95C2318yes—
114C3×#95C2318yes—
115C3×#95C2314yes—
116C3×#95C2318yes—
117C3×#95C2314yes—
118C22×#41C22×C312yes—
119C2×C22⋊Q8C22×C314yes—
120C2×#121C22×C314yes—
121C4×#41—216no—
122C2×C4×Q8C22×C312yes—
123C2×C4×Q8C22×C312yes—
124C2×C4×D4C22×C314yes—
125C2×C4×D4C22×C314yes—
126C2×C4×D4C22×C312yes—
127C2×C4×D4C22×C312yes—
128C2×C4×D4C22×C314yes—
129C2×#117C22×C314yes—
130C2×#117C22×C312yes—
131C2×#117C22×C312yes—
132C22×C42C22×C314yes—
133C22×C42C22×C311yescharacteristic
134#398C22×C314yes—
135#397C22×C314yes—
136#396C22×C318yes—
137#394C22×C314yes—
138#394C22×C318yes—
139#392C22×C314yes—
140#391C22×C314yes—
141#390C22×C314yes—
142#389C22×C314yes—
143#389C22×C314yes—
144#388C22×C318yes—
145#387C22×C318yes—
146#386C22×C318yes—
147#385C22×C318yes—
148#384C22×C311yescharacteristic
149#384C22×C311yescharacteristic
150#382C22×C312yes—
151#380C22×C314yes—
152#380C22×C314yes—
153#380C22×C314yes—
154C4×C4⋊C4C22×C314yes—
155C4×C4⋊C4C22×C314yes—
156C4×C22⋊C4C22×C318yes—
157C24×C3C2411yescharacteristic
158C24×C3C2414yes—
159C3×#41—264no—
160C2×C3×Q8—24no—
161C2×C3×Q8—24no—
162C2×C3×Q8#4118yescomplemented
163C2×C3×D4—28no—
164C2×C3×D4—28no—
165C2×C3×D4—216no—
166C2×C3×D4—24no—
167C2×C3×D4—28no—
168C2×C3×D4—24no—
169C2×C3×D4—44no—
170C2×C3×D4#4118yescomplemented
171C22×C3×C4—28no—
172C22×C3×C4—28no—
173C22×C3×C4—28no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—28no—
179C22×C3×C4—28no—
180C22×C3×C4—28no—
181C22×C3×C4—28no—
182C22×C3×C4—28no—
183C22×C3×C4—28no—
184C22×C3×C4C2×D418yes—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2414yes—
187C22×C3×C4C2414yes—
188C22×C3×C4C2414yes—
189C22×C3×C4C2414yes—
190C22×C3×C4C2412yes—
191C22×C3×C4C2411yescharacteristic
192C22×C3×C4C2414yes—
193C22×C3×C4C2412yes—
194C22×C3×C4C2411yescharacteristic
195C3×C4⋊C4—28no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—28no—
200C3×C4⋊C4—28no—
201C3×C4⋊C4—216no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—44no—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—216no—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—216no—
210C3×C22⋊C4—44no—
211C3×C22⋊C4—48no—
212C3×C42—28no—
213C3×C42—28no—
214C3×C42—28no—
215C3×C42—28no—
216C3×C42C2×D418yes—
217C3×C42C2×D414yes—
218C3×C42C2×D414yes—
219C2×#41—28no—
220C2×#41—28no—
221C2×#41—28no—
222C22×Q8C23×C312yes—
223C22×D4C23×C314yes—
224C22×D4C23×C314yes—
225C22×D4C23×C312yes—
226C23×C4C23×C314yes—
227C23×C4C23×C314yes—
228C23×C4C23×C314yes—
229C23×C4C23×C311yescharacteristic
230C23×C4C23×C312yes—
231C22⋊Q8—216no—
232#121—216no—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×D4—216no—
236C4×D4—216no—
237C4×D4—28no—
238C4×D4—28no—
239C4×D4—48no—
240#117—28no—
241#117—216no—
242#117—44no—
243C2×C4⋊C4C23×C312yes—
244C2×C4⋊C4C23×C312yes—
245C2×C4⋊C4C23×C314yes—
246C2×C4⋊C4C23×C314yes—
247C2×C4⋊C4C23×C314yes—
248C2×C4⋊C4C23×C314yes—
249C2×C4⋊C4C23×C318yes—
250C2×C4⋊C4C23×C312yes—
251C2×C4⋊C4C23×C312yes—
252C2×C4⋊C4C23×C314yes—
253C2×C22⋊C4C23×C314yes—
254C2×C22⋊C4C23×C314yes—
255C2×C22⋊C4C23×C314yes—
256C2×C22⋊C4C23×C318yes—
257C2×C22⋊C4C23×C318yes—
258C2×C22⋊C4C23×C314yes—
259C2×C22⋊C4C23×C318yes—
260C2×C42—216no—
261C2×C42C3×D418yescomplemented
262C2×C42C23×C314yes—
263C2×C42C23×C314yes—
264C2×C42C23×C314yes—
265C2×C42C23×C314yes—
266C2×C42C23×C311yescharacteristic
267C2×C42C23×C311yescharacteristic
268C2×C42C23×C312yes—
269#95C23×C318yes—
270#95C23×C318yes—
271#95C23×C314yes—
272#95C23×C318yes—
273#95C23×C314yes—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—21no—
281C23×C3—24no—
282C23×C3—21no—
283C23×C3—24no—
284C23×C3C22×D412yes—
285C23×C3C2×#4118yes—
286C23×C3C2511yescharacteristic
287C3×Q8—216no—
288C3×D4—232no—
289C3×D4—216no—
290C3×D4—416no—
291C2×C3×C4—232no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—216no—
299C2×C3×C4—28no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—28no—
304C2×C3×C4—28no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—416no—
311C2×C3×C4—416no—
312C2×C3×C4C22×D414yes—
313C2×C3×C4C22×D412yes—
314C2×C3×C4C22×D414yes—
315C2×C3×C4C22×D412yes—
316C2×C3×C4C2×#4118yes—
317C2×C3×C4C2×#4114yes—
318C2×C3×C4C2×#4114yes—
319C2×C3×C4H(5,𝔽2)12yes—
320C2×C3×C42−1+412yes—
321C24C24×C311yescharacteristic
322C24C24×C314yes—
323#41—264no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8C3×#4118yescomplemented
327C2×D4—28no—
328C2×D4—28no—
329C2×D4—216no—
330C2×D4—24no—
331C2×D4—28no—
332C2×D4—24no—
333C2×D4—44no—
334C2×D4C3×#4118yescomplemented
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4C2×C3×D418yes—
349C22×C4C2×C3×D414yes—
350C22×C4C24×C314yes—
351C22×C4C24×C314yes—
352C22×C4C24×C314yes—
353C22×C4C24×C314yes—
354C22×C4C24×C312yes—
355C22×C4C24×C311yescharacteristic
356C22×C4C24×C314yes—
357C22×C4C24×C312yes—
358C22×C4C24×C311yescharacteristic
359C4⋊C4—28no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—28no—
364C4⋊C4—28no—
365C4⋊C4—216no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C22⋊C4—28no—
370C22⋊C4—216no—
371C22⋊C4—28no—
372C22⋊C4—28no—
373C22⋊C4—216no—
374C22⋊C4—44no—
375C22⋊C4—48no—
376C42—28no—
377C42—28no—
378C42—28no—
379C42—28no—
380C42C2×C3×D418yes—
381C42C2×C3×D414yes—
382C42C2×C3×D414yes—
383C22×C3—24no—
384C22×C3—28no—
385C22×C3—22no—
386C22×C3—28no—
387C22×C3—28no—
388C22×C3—44no—
389C22×C3—44no—
390C22×C3—41no—
391C22×C3—42no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—41no—
395C22×C3C23×D411yescharacteristic
396C22×C3C22×#4112yes—
397C22×C3C22×#4112yes—
398C22×C3C2×H(5,𝔽2)11yescharacteristic
399C22×C3C2×2−1+411yescharacteristic
400C3×C4—24no—
401C3×C4—28no—
402C3×C4—22no—
403C3×C4—44no—
404C3×C4—44no—
405C3×C4—44no—
406C3×C4—44no—
407C3×C4#52518yes—
408C3×C4#53714yes—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—21no—
416C23—24no—
417C23—21no—
418C23—24no—
419C23C22×C3×D412yes—
420C23C2×C3×#4118yes—
421C23C25×C311yescharacteristic
422Q8—216no—
423D4—232no—
424D4—216no—
425D4—416no—
426C2×C4—232no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—216no—
434C2×C4—28no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—416no—
446C2×C4—416no—
447C2×C4C22×C3×D414yes—
448C2×C4C22×C3×D412yes—
449C2×C4C22×C3×D414yes—
450C2×C4C22×C3×D412yes—
451C2×C4C2×C3×#4118yes—
452C2×C4C2×C3×#4114yes—
453C2×C4C2×C3×#4114yes—
454C2×C4C3×H(5,𝔽2)12yes—
455C2×C4C3×2−1+412yes—
456C2×C3—42no—
457C2×C3—48no—
458C2×C3C2×#52512yes—
459C2×C3C2×#53711yescharacteristic
460C2×C3#348111yescharacteristic
461C2×C3#348212yes—
462C2×C3#348311yescharacteristic
463C22—24no—
464C22—28no—
465C22—22no—
466C22—28no—
467C22—28no—
468C22—44no—
469C22—44no—
470C22—41no—
471C22—42no—
472C22—44no—
473C22—44no—
474C22—41no—
475C22C23×C3×D411yescharacteristic
476C22C22×C3×#4112yes—
477C22C22×C3×#4112yes—
478C22C2×C3×H(5,𝔽2)11yescharacteristic
479C22C2×C3×2−1+411yescharacteristic
480C4—24no—
481C4—28no—
482C4—22no—
483C4—44no—
484C4—44no—
485C4—44no—
486C4—44no—
487C4C3×#52518yes—
488C4C3×#53714yes—
489C3#3497611yescharacteristic, direct product factor, Sylow
490C2—42no—
491C2—48no—
492C2C2×C3×#52512yes—
493C2C2×C3×#53711yescharacteristic
494C2C3×#348111yescharacteristic
495C2C3×#348212yes—
496C2C3×#348311yescharacteristic
subgroupG/H
centerC22×C3×C4C24
commutator, Frattini, cosocleC23C25×C3
socleC23×C3C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×#41C2semidirect
2. C3×#2609C2semidirect
3. C3×#2582C2semidirect
4. C3×#2538C2semidirect
5. C3×#2537C2semidirect
6. C3×#2485C2semidirect
7. C3×#2407C2semidirect
8. C3×#2392C2semidirect
9. C3×C4×C22⋊Q8C2semidirect
10. C3×C4×#121C2semidirect
11. C3×#2285C2semidirect
12. #34976C3direct (N × H)
52. C2×C4×#41C2×C3semidirect
53. #2609C2×C3semidirect
54. #2582C2×C3semidirect
55. #2538C2×C3semidirect
56. #2537C2×C3semidirect
57. #2485C2×C3semidirect
58. #2407C2×C3semidirect
59. #2392C2×C3semidirect
60. C4×C22⋊Q8C2×C3semidirect
61. C4×#121C2×C3semidirect
62. #2285C2×C3semidirect
105. C2×C3×C42D4semidirect
162. C2×C3×Q8#41semidirect
170. C2×C3×D4#41semidirect
261. C2×C42C3×D4semidirect
326. C2×Q8C3×#41semidirect
334. C2×D4C3×#41semidirect
489. C3#34976direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C3×#26092
C3×#25822
C3×#25382
C3×#25372
C3×#24852
C3×#24072
C3×#23922
C3×C4×C22⋊Q82
C3×C4×#1212
C3×#22852
#349763