Finite Group Databasebeta

← C3×#10912 #10629519

Subgroups — C3×#10912 #10629519

Order 768.

1266 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#385C211yesmaximal, characteristic
2C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
3C2×C3×#342C212yesmaximal
4C2×C3×#342C212yesmaximal
5C3×C22⋊#95C211yesmaximal, characteristic
6#10912C311yesmaximal, characteristic, direct product factor, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×C4⋊C4C412yescomplemented
9C22×C3×C22⋊C4C2211yescharacteristic
10C22×C3×C22⋊C4C2212yes—
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4C2212yes—
13C22×C3×C42—21no—
14C3×#385—21no—
15C3×#385—22no—
16C3×#385—21no—
17C3×C4×C22⋊C4—24no—
18C2×C3×#95—21no—
19C2×C3×#95—22no—
20C2×C3×#95C412yescomplemented
21C3×#342—24no—
22C3×#342C418yescomplemented
23C2×#385C2×C311yescharacteristic
24C2×C4×C22⋊C4C2×C311yescharacteristic
25C2×#342C2×C312yes—
26C2×#342C2×C312yes—
27C22⋊#95C2×C311yescharacteristic
28C25×C3C2311yescharacteristic
29C23×C3×C4—22no—
30C23×C3×C4—22no—
31C23×C3×C4—22no—
32C23×C3×C4—22no—
33C23×C3×C4—44no—
34C23×C3×C4C2×C412yes—
35C23×C3×C4C2×C411yescharacteristic
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C412yes—
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D412yes—
40C23×C3×C4D411yescharacteristic
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4—22no—
43C2×C3×C4⋊C4—44no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—22no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—22no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4C2×C414yes—
60C2×C3×C22⋊C4C2×C414yes—
61C2×C3×C22⋊C4C2×C414yes—
62C2×C3×C22⋊C4D414yes—
63C2×C3×C22⋊C4Q814yes—
64C2×C3×C42—24no—
65C2×C3×C42—44no—
66C3×#95—24no—
67C3×#95—22no—
68C3×#95—22no—
69C3×#95—44no—
70C24×C4C22×C311yescharacteristic
71C22×C4⋊C4C3×C412yescomplemented
72C22×C22⋊C4C22×C311yescharacteristic
73C22×C22⋊C4C22×C312yes—
74C22×C22⋊C4C22×C311yescharacteristic
75C22×C22⋊C4C22×C312yes—
76C22×C42—21no—
77#385—21no—
78#385—22no—
79#385—21no—
80C4×C22⋊C4—24no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95C3×C412yescomplemented
84#342—24no—
85#342C3×C418yescomplemented
86C24×C3—42no—
87C24×C3—42no—
88C24×C3C22×C411yescharacteristic
89C24×C3C22×C412yes—
90C24×C3C2×D411yescharacteristic
91C24×C3C2×D411yescharacteristic
92C24×C3C2×D412yes—
93C24×C3C2×Q812yes—
94C24×C3#4112yes—
95C24×C3#4112yes—
96C24×C3#4112yes—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—24no—
100C22×C3×C4—22no—
101C22×C3×C4—22no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—21no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4—24no—
108C22×C3×C4—21no—
109C22×C3×C4—24no—
110C22×C3×C4—24no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—24no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—42no—
117C22×C3×C4—42no—
118C22×C3×C4—42no—
119C22×C3×C4—48no—
120C22×C3×C4—48no—
121C22×C3×C4—44no—
122C22×C3×C4—44no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4—44no—
126C22×C3×C4—44no—
127C22×C3×C4—42no—
128C22×C3×C4—42no—
129C22×C3×C4—42no—
130C22×C3×C4—42no—
131C22×C3×C4C4214yescomplemented
132C22×C3×C4C22⋊C412yes—
133C22×C3×C4C22⋊C412yes—
134C3×C4⋊C4—44no—
135C3×C4⋊C4—44no—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—24no—
139C3×C22⋊C4—24no—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—24no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—42no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—42no—
147C3×C22⋊C4—42no—
148C3×C22⋊C4C4⋊C418yescomplemented
149C3×C42—48no—
150C25C23×C311yescharacteristic
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—44no—
156C23×C4C2×C3×C412yes—
157C23×C4C2×C3×C411yescharacteristic
158C23×C4C2×C3×C411yescharacteristic
159C23×C4C2×C3×C412yes—
160C23×C4C3×D411yescharacteristic
161C23×C4C3×D412yes—
162C23×C4C3×D411yescharacteristic
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—44no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×C3×C414yes—
182C2×C22⋊C4C2×C3×C414yes—
183C2×C22⋊C4C2×C3×C414yes—
184C2×C22⋊C4C3×D414yes—
185C2×C22⋊C4C3×Q814yes—
186C2×C42—24no—
187C2×C42—44no—
188#95—24no—
189#95—22no—
190#95—22no—
191#95—44no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—22no—
196C23×C3—22no—
197C23×C3—22no—
198C23×C3—42no—
199C23×C3—42no—
200C23×C3—44no—
201C23×C3—42no—
202C23×C3—44no—
203C23×C3—42no—
204C23×C3—44no—
205C23×C3—44no—
206C23×C3C2×C4211yescharacteristic
207C23×C3C2×C22⋊C411yescharacteristic
208C23×C3C2×C4⋊C412yes—
209C23×C3#11711yescharacteristic
210C23×C3#11712yes—
211C23×C3C4×D412yes—
212C23×C3C4×D412yes—
213C23×C3C4×D412yes—
214C23×C3C4×D412yes—
215C23×C3C4×Q812yes—
216C23×C3C4×Q812yes—
217C23×C3#12011yescharacteristic
218C23×C3#12112yes—
219C23×C3C22⋊Q812yes—
220C23×C3C22⋊Q812yes—
221C23×C3#12311yescharacteristic
222C23×C3#12312yes—
223C23×C3#12411yescharacteristic
224C23×C3#12512yes—
225C23×C3#12612yes—
226C23×C3C4⋊Q811yescharacteristic
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—24no—
232C2×C3×C4—28no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—48no—
237C2×C3×C4—48no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—48no—
241C2×C3×C4—48no—
242C2×C3×C4—44no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—44no—
248C2×C3×C4—48no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—44no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—82no—
255C2×C3×C4—82no—
256C2×C3×C4—84no—
257C2×C3×C4—82no—
258C2×C3×C4—82no—
259C2×C3×C4#9914yes—
260C24—42no—
261C24—42no—
262C24C22×C3×C411yescharacteristic
263C24C22×C3×C412yes—
264C24C2×C3×D411yescharacteristic
265C24C2×C3×D411yescharacteristic
266C24C2×C3×D412yes—
267C24C2×C3×Q812yes—
268C24C3×#4112yes—
269C24C3×#4112yes—
270C24C3×#4112yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—48no—
294C22×C4—48no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4C3×C4214yescomplemented
306C22×C4C3×C22⋊C412yes—
307C22×C4C3×C22⋊C412yes—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4C3×C4⋊C418yescomplemented
323C42—48no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—21no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—21no—
332C22×C3—24no—
333C22×C3—22no—
334C22×C3—24no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—24no—
339C22×C3—24no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3C4×C22⋊C411yescharacteristic
349C22×C3C4×C4⋊C412yes—
350C22×C3#38512yes—
351C22×C3#38611yescharacteristic
352C22×C3#38712yes—
353C22×C3#38712yes—
354C22×C3#38812yes—
355C22×C3#38912yes—
356C22×C3#39111yescharacteristic
357C22×C3C2×#9912yes—
358C22×C3#41012yes—
359C3×C4—24no—
360C3×C4—48no—
361C3×C4—42no—
362C3×C4—44no—
363C3×C4—84no—
364C3×C4—81no—
365C3×C4—81no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—42no—
380C23C2×C3×C4211yescharacteristic
381C23C2×C3×C22⋊C411yescharacteristic
382C23C2×C3×C4⋊C412yes—
383C23C3×#11711yescharacteristic
384C23C3×#11712yes—
385C23C3×C4×D412yes—
386C23C3×C4×D412yes—
387C23C3×C4×D412yes—
388C23C3×C4×D412yes—
389C23C3×C4×Q812yes—
390C23C3×C4×Q812yes—
391C23C3×#12011yescharacteristic
392C23C3×#12112yes—
393C23C3×C22⋊Q812yes—
394C23C3×C22⋊Q812yes—
395C23C3×#12311yescharacteristic
396C23C3×#12312yes—
397C23C3×#12411yescharacteristic
398C23C3×#12512yes—
399C23C3×#12612yes—
400C23C3×C4⋊Q811yescharacteristic
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4C3×#9914yes—
434C2×C3—22no—
435C2×C3—22no—
436C2×C3—22no—
437C2×C3—22no—
438C2×C3—22no—
439C2×C3—22no—
440C2×C3#143611yescharacteristic
441C2×C3C4×#9912yes—
442C2×C3#189811yescharacteristic
443C2×C3#189911yescharacteristic
444C2×C3#199612yes—
445C22—24no—
446C22—22no—
447C22—24no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—24no—
452C22—24no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22C3×C4×C22⋊C411yescharacteristic
470C22C3×C4×C4⋊C412yes—
471C22C3×#38512yes—
472C22C3×#38611yescharacteristic
473C22C3×#38712yes—
474C22C3×#38712yes—
475C22C3×#38812yes—
476C22C3×#38912yes—
477C22C3×#39111yescharacteristic
478C22C2×C3×#9912yes—
479C22C3×#41012yes—
480C4—24no—
481C4—48no—
482C4—44no—
483C4—42no—
484C4—84no—
485C4—81no—
486C4—81no—
487C3#1091211yescharacteristic, direct product factor, Sylow
488C2—22no—
489C2—22no—
490C2—22no—
491C2—22no—
492C2—22no—
493C2—22no—
494C2C3×#143611yescharacteristic
495C2C3×C4×#9912yes—
496C2C3×#189811yescharacteristic
497C2C3×#189911yescharacteristic
498C2C3×#199612yes—
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC23C2×C3×C42
Frattini, cosocleC25C23×C3
prime radicalC25×C3C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
6. #10912C3direct (N × H)
8. C22×C3×C4⋊C4C4semidirect
20. C2×C3×#95C4semidirect
22. C3×#342C4semidirect
71. C22×C4⋊C4C3×C4semidirect
83. C2×#95C3×C4semidirect
85. #342C3×C4semidirect
131. C22×C3×C4C42semidirect
148. C3×C22⋊C4C4⋊C4semidirect
305. C22×C4C3×C42semidirect
322. C22⋊C4C3×C4⋊C4semidirect
487. C3#10912direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3852
C2×C3×C4×C22⋊C42
C2×C3×#3422
C3×C22⋊#952
#109123