Finite Group Databasebeta

← C3×#32954 #10651561

Subgroups — C3×#32954 #10651561

Order 768.

1332 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×C3×#475C212yesmaximal, complemented
3C3×#2857C211yesmaximal, characteristic
4C3×#1865C214yesmaximal, complemented
5C3×#1853C212yesmaximal
6C3×#1803C212yesmaximal, complemented
7C3×#1797C211yesmaximal, characteristic
8C3×#1787C212yesmaximal
9#32954C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C22⋊Q8—21no—
13C2×C3×C4×Q8—28no—
14C2×C3×C4×Q8—21no—
15C2×C3×C4×Q8—21no—
16C2×C3×C4×Q8C2212yes—
17C2×C3×C4×Q8C2212yes—
18C22×C3×C4⋊C4—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C42—21no—
21C22×C3×C42C2211yescharacteristic
22C3×#475—28no—
23C2×C3×#106C2214yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2214yes—
26C2×C3×#103C2214yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#391—21no—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#384C2211yescharacteristic
32C3×#380—21no—
33C3×#380C2212yes—
34C3×#340C2214yes—
35C3×#336C2212yes—
36C22×C4×Q8C2×C311yescharacteristic
37C2×#475C2×C312yescomplemented
38#2857C2×C311yescharacteristic
39#1865C2×C314yescomplemented
40#1853C2×C312yes—
41#1803C2×C312yescomplemented
42#1797C2×C311yescharacteristic
43#1787C2×C312yes—
44C22×C3×Q8—24no—
45C22×C3×Q8—24no—
46C22×C3×Q8—21no—
47C22×C3×Q8—21no—
48C22×C3×Q8—22no—
49C22×C3×Q8Q814yescomplemented
50C22×C3×Q8C2312yes—
51C23×C3×C4—21no—
52C23×C3×C4—21no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×C3×C8C2314yes—
57C3×C4⋊Q8—24no—
58C3×C22⋊Q8—44no—
59C3×C4×Q8—28no—
60C3×C4×Q8—28no—
61C3×C4×Q8—24no—
62C3×C4×Q8—24no—
63C3×C4×Q8—24no—
64C3×C4×Q8D418yes—
65C2×C3×C4⋊C4—28no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C42—28no—
79C2×C3×C42—22no—
80C2×C3×C42—22no—
81C2×C3×C42—24no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×#106—28no—
87C3×C4⋊C8—24no—
88C3×#103—28no—
89C3×#103—28no—
90C3×C22⋊C8—24no—
91C3×#95—22no—
92C3×#95—22no—
93C23×Q8C22×C311yescharacteristic
94C2×C4⋊Q8C22×C311yescharacteristic
95C2×C22⋊Q8—21no—
96C2×C4×Q8—28no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8C22×C312yes—
100C2×C4×Q8C22×C312yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C42—21no—
104C22×C42C22×C311yescharacteristic
105#475—28no—
106C2×#106C22×C314yes—
107C2×C4⋊C8C22×C312yes—
108C2×#103C22×C314yes—
109C2×#103C22×C314yes—
110C2×C22⋊C8C22×C312yes—
111#391—21no—
112#389—21no—
113#389C22×C312yes—
114#384C22×C311yescharacteristic
115#380—21no—
116#380C22×C312yes—
117#340C22×C314yes—
118#336C22×C312yes—
119C24×C3C2×D411yescharacteristic
120C2×C3×Q8—24no—
121C2×C3×Q8—216no—
122C2×C3×Q8—24no—
123C2×C3×Q8—216no—
124C2×C3×Q8—24no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×Q8—42no—
131C2×C3×Q8C2×D414yes—
132C2×C3×Q8C2×Q814yes—
133C2×C3×Q8#4114yes—
134C22×C3×C4—24no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—24no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—41no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4SD1614yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×Q812yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C2×C3×C8—24no—
161C2×C3×C8—24no—
162C2×C3×C8—24no—
163C3×C4⋊C4—28no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—28no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—44no—
173C3×C4⋊C4C2×D414yes—
174C3×C22⋊C4—44no—
175C3×C42—28no—
176C3×C42—28no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42C2×D414yes—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—21no—
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8C3×Q814yescomplemented
187C22×Q8C23×C312yes—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4C3×D411yescharacteristic
191C23×C4C3×D411yescharacteristic
192C23×C4C23×C311yescharacteristic
193C22×C8C23×C314yes—
194C4⋊Q8—24no—
195C22⋊Q8—44no—
196C4×Q8—28no—
197C4×Q8—28no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8C3×D418yes—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C312yes—
214C2×C22⋊C4—22no—
215C2×C42—28no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42C3×D411yescharacteristic
220C2×C42C3×D411yescharacteristic
221C2×C42C23×C311yescharacteristic
222C2×C42C23×C311yescharacteristic
223#106—28no—
224C4⋊C8—24no—
225#103—28no—
226#103—28no—
227C22⋊C8—24no—
228#95—22no—
229#95—22no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3C22⋊Q812yes—
235C23×C3C2×SD1612yes—
236C23×C3#13712yes—
237C23×C3C22×D411yescharacteristic
238C3×Q8—216no—
239C3×Q8—24no—
240C3×Q8—24no—
241C3×Q8—24no—
242C3×Q8—84no—
243C3×Q8C22⋊Q818yescomplemented
244C2×C3×C4—216no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—28no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—44no—
262C2×C3×C4—48no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4#12012yes—
272C2×C3×C4#12012yes—
273C2×C3×C4C22⋊Q812yes—
274C2×C3×C4C2×SD1612yes—
275C2×C3×C4C2×SD1612yes—
276C2×C3×C4#13712yes—
277C2×C3×C4C22×D412yes—
278C2×C3×C4C22×Q811yescharacteristic
279C2×C3×C4C2×#4112yes—
280C2×C3×C4C2×#4111yescharacteristic
281C3×C8—44no—
282C24C2×C3×D411yescharacteristic
283C2×Q8—24no—
284C2×Q8—216no—
285C2×Q8—24no—
286C2×Q8—216no—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8C2×C3×D414yes—
295C2×Q8C2×C3×Q814yes—
296C2×Q8C3×#4114yes—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4C3×SD1614yes—
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C2×C3×D411yescharacteristic
319C22×C4C2×C3×D411yescharacteristic
320C22×C4C2×C3×Q812yes—
321C22×C4C3×#4112yes—
322C22×C4C24×C311yescharacteristic
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—28no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4C2×C3×D414yes—
337C22⋊C4—44no—
338C42—28no—
339C42—28no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42C2×C3×D414yes—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3#47514yescomplemented
351C22×C3C2×#12011yescharacteristic
352C22×C3C2×C22⋊Q811yescharacteristic
353C22×C3#52511yescharacteristic
354C22×C3C22×SD1611yescharacteristic
355C22×C3C2×#13711yescharacteristic
356C22×C3C2×#13711yescharacteristic
357C22×C3#57511yescharacteristic
358C3×C4—28no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—22no—
362C3×C4—41no—
363C3×C4—44no—
364C3×C4—82no—
365C3×C4—82no—
366C3×C4#44814yes—
367C3×C4C2×C22⋊Q812yes—
368C3×C4#52512yes—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23C3×C22⋊Q812yes—
374C23C2×C3×SD1612yes—
375C23C3×#13712yes—
376C23C22×C3×D411yescharacteristic
377Q8—216no—
378Q8—24no—
379Q8—24no—
380Q8—24no—
381Q8—84no—
382Q8C3×C22⋊Q818yescomplemented
383C2×C4—216no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—21no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—24no—
400C2×C4—44no—
401C2×C4—48no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4C3×#12012yes—
411C2×C4C3×#12012yes—
412C2×C4C3×C22⋊Q812yes—
413C2×C4C2×C3×SD1612yes—
414C2×C4C2×C3×SD1612yes—
415C2×C4C3×#13712yes—
416C2×C4C22×C3×D412yes—
417C2×C4C22×C3×Q811yescharacteristic
418C2×C4C2×C3×#4112yes—
419C2×C4C2×C3×#4111yescharacteristic
420C8—44no—
421C2×C3—24no—
422C2×C3#284811yescharacteristic
423C2×C3C2×#44811yescharacteristic
424C2×C3#300711yescharacteristic
425C2×C3C2×#47511yescharacteristic
426C2×C3#307811yescharacteristic
427C2×C3#310311yescharacteristic
428C2×C3#311311yescharacteristic
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22C3×#47514yescomplemented
436C22C2×C3×#12011yescharacteristic
437C22C2×C3×C22⋊Q811yescharacteristic
438C22C3×#52511yescharacteristic
439C22C22×C3×SD1611yescharacteristic
440C22C2×C3×#13711yescharacteristic
441C22C2×C3×#13711yescharacteristic
442C22C3×#57511yescharacteristic
443C4—22no—
444C4—22no—
445C4—28no—
446C4—22no—
447C4—41no—
448C4—44no—
449C4—82no—
450C4—82no—
451C4C3×#44814yes—
452C4C2×C3×C22⋊Q812yes—
453C4C3×#52512yes—
454C3#3295411yescharacteristic, direct product factor, Sylow
455C2—24no—
456C2C3×#284811yescharacteristic
457C2C2×C3×#44811yescharacteristic
458C2C3×#300711yescharacteristic
459C2C2×C3×#47511yescharacteristic
460C2C3×#307811yescharacteristic
461C2C3×#310311yescharacteristic
462C2C3×#311311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#475C2semidirect
4. C3×#1865C2semidirect
6. C3×#1803C2semidirect
9. #32954C3direct (N × H)
37. C2×#475C2×C3semidirect
39. #1865C2×C3semidirect
41. #1803C2×C3semidirect
49. C22×C3×Q8Q8semidirect
186. C22×Q8C3×Q8semidirect
243. C3×Q8C22⋊Q8semidirect
350. C22×C3#475semidirect
382. Q8C3×C22⋊Q8semidirect
435. C22C3×#475semidirect
454. C3#32954direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×C3×#4752
C3×#28572
C3×#18652
C3×#18532
C3×#18032
C3×#17972
C3×#17872
#329543