Finite Group Databasebeta

← C3×#32923 #10651530

Subgroups — C3×#32923 #10651530

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×2−1+4C211yesmaximal, characteristic
2C3×#3039C212yesmaximal, complemented
3C2×C3×#456C212yesmaximal
4C3×#2546C211yesmaximal, characteristic, complemented
5C3×#1886C212yesmaximal, complemented
6C3×#1882C214yesmaximal, complemented
7C3×#1806C212yesmaximal, complemented
8C3×#1801C211yesmaximal, characteristic, complemented
9#32923C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×2−1+4C2211yescharacteristic
11C2×C3×#137C2214yes—
12C2×C3×C4⋊Q8C2212yes—
13C2×C3×C4⋊Q8C2211yescharacteristic
14C3×#520—22no—
15C3×#520—22no—
16C3×#520C2212yes—
17C3×#519—22no—
18C3×#519C2211yescharacteristic
19C3×C4×#41—22no—
20C3×C4×#41—22no—
21C3×C4×#41C2212yes—
22C2×C3×C4×Q8—22no—
23C2×C3×C4×Q8C2211yescharacteristic
24C3×#462—22no—
25C3×#462—22no—
26C3×#461—22no—
27C3×#461—22no—
28C3×#456—28no—
29C3×#423C2212yes—
30C2×C3×#104C2214yes—
31C3×#419C2214yes—
32C2×C3×#101C2212yes—
33C3×#398—22no—
34C3×#391C2212yes—
35C3×#389—22no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×#343C2212yes—
38C3×#339C2214yes—
39C4×2−1+4C2×C311yescharacteristic
40#3039C2×C312yescomplemented
41C2×#456C2×C312yes—
42#2546C2×C311yescharacteristic, complemented
43#1886C2×C312yescomplemented
44#1882C2×C314yescomplemented
45#1806C2×C312yescomplemented
46#1801C2×C311yescharacteristic, complemented
47C3×2−1+4—24no—
48C3×2−1+4—22no—
49C3×2−1+4D414yes—
50C2×C3×#41—22no—
51C2×C3×#41—22no—
52C2×C3×#41C2312yes—
53C22×C3×Q8—22no—
54C22×C3×Q8C2312yes—
55C22×C3×Q8C2311yescharacteristic
56C3×#137—28no—
57C3×#137—28no—
58C2×C3×Q16—24no—
59C2×C3×SD16—24no—
60C2×C3×M16C2314yes—
61C3×C4⋊Q8—24no—
62C3×C4⋊Q8—24no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×C4×Q8—22no—
66C3×C4×Q8—24no—
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8—22no—
73C3×C4×Q8D412yes—
74C3×C4×Q8D412yes—
75C3×C4×D4—22no—
76C3×C4×D4—24no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4—24no—
81C3×C4×D4—22no—
82C3×C4×D4D412yes—
83C3×C4×D4D412yes—
84C3×#117—24no—
85C3×#117—24no—
86C3×#117—22no—
87C3×#117—22no—
88C3×#117—22no—
89C3×#117C2312yes—
90C2×C3×C4⋊C4—24no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—24no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4D412yes—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C42—24no—
99C2×C3×C42—22no—
100C2×C3×C42D412yes—
101C2×C3×C42C2312yes—
102C2×C3×C42C2311yescharacteristic
103C3×C4⋊C8—22no—
104C3×C4⋊C8—22no—
105C3×#104—28no—
106C3×#103—24no—
107C3×#103—24no—
108C3×#101—24no—
109C3×#95—21no—
110C3×#95—24no—
111C2×2−1+4C22×C311yescharacteristic
112C2×#137C22×C314yes—
113C2×C4⋊Q8C22×C312yes—
114C2×C4⋊Q8C22×C311yescharacteristic
115#520—22no—
116#520—22no—
117#520C22×C312yes—
118#519—22no—
119#519C22×C311yescharacteristic
120C4×#41—22no—
121C4×#41—22no—
122C4×#41C22×C312yes—
123C2×C4×Q8—22no—
124C2×C4×Q8C22×C311yescharacteristic
125#462—22no—
126#462—22no—
127#461—22no—
128#461—22no—
129#456—28no—
130#423C22×C312yes—
131C2×#104C22×C314yes—
132#419C22×C314yes—
133C2×#101C22×C312yes—
134#398—22no—
135#391C22×C312yes—
136#389—22no—
137C4×C4⋊C4C22×C311yescharacteristic
138#343C22×C312yes—
139#339C22×C314yes—
140C3×#41—28no—
141C3×#41—28no—
142C3×#41—28no—
143C3×#41—28no—
144C3×#41—22no—
145C3×#41—22no—
146C3×#41C2×D414yes—
147C3×#41#4114yes—
148C2×C3×Q8—22no—
149C2×C3×Q8—28no—
150C2×C3×Q8—28no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—24no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—24no—
157C2×C3×Q8C2×D412yes—
158C2×C3×Q8C2×D412yes—
159C2×C3×Q8#4112yes—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4C2×D412yes—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×Q16—48no—
176C3×SD16—48no—
177C3×M16—24no—
178C3×M16—24no—
179C2×C3×C8—24no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—41no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—41no—
191C3×C4⋊C4—48no—
192C3×C4⋊C4C2×D412yes—
193C3×C4⋊C4C2×D412yes—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—22no—
198C3×C42—24no—
199C3×C42—24no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—22no—
206C3×C42—22no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
2092−1+4—24no—
2102−1+4—22no—
2112−1+4C3×D414yes—
212C2×#41—22no—
213C2×#41—22no—
214C2×#41C23×C312yes—
215C22×Q8—22no—
216C22×Q8C23×C312yes—
217C22×Q8C23×C311yescharacteristic
218#137—28no—
219#137—28no—
220C2×Q16—24no—
221C2×SD16—24no—
222C2×M16C23×C314yes—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4×Q8—22no—
228C4×Q8—24no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8C3×D412yes—
236C4×Q8C3×D412yes—
237C4×D4—22no—
238C4×D4—24no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—24no—
243C4×D4—22no—
244C4×D4C3×D412yes—
245C4×D4C3×D412yes—
246#117—24no—
247#117—24no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117C23×C312yes—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C3×D412yes—
259C2×C4⋊C4C23×C311yescharacteristic
260C2×C42—24no—
261C2×C42—22no—
262C2×C42C3×D412yes—
263C2×C42C23×C312yes—
264C2×C42C23×C311yescharacteristic
265C4⋊C8—22no—
266C4⋊C8—22no—
267#104—28no—
268#103—24no—
269#103—24no—
270#101—24no—
271#95—21no—
272#95—24no—
273C23×C3—22no—
274C23×C3C22×D411yescharacteristic
275C3×Q8—28no—
276C3×Q8—24no—
277C3×Q8—22no—
278C3×Q8—24no—
279C3×Q8—44no—
280C3×Q8—44no—
281C3×Q8#12114yes—
282C3×D4—28no—
283C3×D4—24no—
284C3×D4—24no—
285C3×D4—22no—
286C3×D4#12114yes—
287C2×C3×C4—216no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—44no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—82no—
307C2×C3×C4—84no—
308C2×C3×C4#12011yescharacteristic
309C2×C3×C4#12012yes—
310C2×C3×C4#12011yescharacteristic
311C2×C3×C4#12114yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D412yes—
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4111yescharacteristic
316C3×C8—44no—
317#41—28no—
318#41—28no—
319#41—28no—
320#41—28no—
321#41—22no—
322#41—22no—
323#41C2×C3×D414yes—
324#41C3×#4114yes—
325C2×Q8—22no—
326C2×Q8—28no—
327C2×Q8—28no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8C2×C3×D412yes—
335C2×Q8C2×C3×D412yes—
336C2×Q8C3×#4112yes—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4C2×C3×D412yes—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D412yes—
349C22×C4C2×C3×D412yes—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C24×C311yescharacteristic
352Q16—48no—
353SD16—48no—
354M16—24no—
355M16—24no—
356C2×C8—24no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—48no—
369C4⋊C4C2×C3×D412yes—
370C4⋊C4C2×C3×D412yes—
371C22⋊C4—24no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C42—24no—
376C42—24no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42C2×C3×D412yes—
385C42C2×C3×D412yes—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—24no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3C2×#12011yescharacteristic
392C22×C3C2×#12111yescharacteristic
393C22×C3#52511yescharacteristic
394C3×C4—24no—
395C3×C4—24no—
396C3×C4—21no—
397C3×C4—22no—
398C3×C4—44no—
399C3×C4—42no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—82no—
403C3×C4#45612yescomplemented
404C3×C4C2×#12112yes—
405C3×C4#52512yes—
406C23—22no—
407C23C22×C3×D411yescharacteristic
408Q8—28no—
409Q8—24no—
410Q8—22no—
411Q8—24no—
412Q8—44no—
413Q8—44no—
414Q8C3×#12114yes—
415D4—28no—
416D4—24no—
417D4—24no—
418D4—22no—
419D4C3×#12114yes—
420C2×C4—216no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4C3×#12011yescharacteristic
442C2×C4C3×#12012yes—
443C2×C4C3×#12011yescharacteristic
444C2×C4C3×#12114yes—
445C2×C4C22×C3×D411yescharacteristic
446C2×C4C22×C3×D412yes—
447C2×C4C2×C3×#4112yes—
448C2×C4C2×C3×#4111yescharacteristic
449C8—44no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—44no—
453C2×C3#284711yescharacteristic
454C2×C3C2×#45611yescharacteristic
455C2×C3#302211yescharacteristic
456C22—21no—
457C22—21no—
458C22—24no—
459C22—42no—
460C22—42no—
461C22C2×C3×#12011yescharacteristic
462C22C2×C3×#12111yescharacteristic
463C22C3×#52511yescharacteristic
464C4—21no—
465C4—22no—
466C4—24no—
467C4—24no—
468C4—44no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—82no—
473C4C3×#45612yescomplemented
474C4C2×C3×#12112yes—
475C4C3×#52512yes—
476C3#3292311yescharacteristic, direct product factor, Sylow
477C2—21no—
478C2—21no—
479C2—44no—
480C2C3×#284711yescharacteristic
481C2C2×C3×#45611yescharacteristic
482C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3039C2semidirect
4. C3×#2546C2semidirect
5. C3×#1886C2semidirect
6. C3×#1882C2semidirect
7. C3×#1806C2semidirect
8. C3×#1801C2semidirect
9. #32923C3direct (N × H)
40. #3039C2×C3semidirect
42. #2546C2×C3semidirect
43. #1886C2×C3semidirect
44. #1882C2×C3semidirect
45. #1806C2×C3semidirect
46. #1801C2×C3semidirect
403. C3×C4#456semidirect
473. C4C3×#456semidirect
476. C3#32923direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×2−1+42
C3×#30392
C2×C3×#4562
C3×#25462
C3×#18862
C3×#18822
C3×#18062
C3×#18012
#329233