Finite Group Databasebeta

← C3×#11046 #10629653

Subgroups — C3×#11046 #10629653

Order 768.

1178 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#99C212yesmaximal
2C3×#2315C211yesmaximal, characteristic, complemented
3C3×#2308C211yesmaximal, characteristic
4C2×C3×#342C212yesmaximal, complemented
5C2×C3×#342C211yesmaximal, characteristic, complemented
6#11046C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C22⋊C4C2211yescharacteristic
9C22×C3×C22⋊C4C2212yes—
10C22×C3×C22⋊C4C2212yes—
11C22×C3×C22⋊C4C2211yescharacteristic
12C2×C3×#99—24no—
13C2×C3×#99—24no—
14C2×C3×#99—24no—
15C3×#386—22no—
16C3×#386—22no—
17C3×#385—24no—
18C3×#381C412yescomplemented
19C3×#380—22no—
20C3×#379C412yescomplemented
21C3×#342—24no—
22C3×#342—22no—
23C22×#99C2×C312yes—
24#2315C2×C311yescharacteristic, complemented
25#2308C2×C311yescharacteristic
26C2×#342C2×C312yescomplemented
27C2×#342C2×C311yescharacteristic, complemented
28C25×C3C2×C411yescharacteristic
29C25×C3C2311yescharacteristic
30C22×C3×D4—21no—
31C22×C3×D4—21no—
32C22×C3×D4—22no—
33C22×C3×D4—22no—
34C22×C3×D4—21no—
35C22×C3×D4—21no—
36C22×C3×D4—22no—
37C22×C3×D4—22no—
38C22×C3×D4D412yes—
39C22×C3×D4Q812yes—
40C23×C3×C4—22no—
41C23×C3×C4—21no—
42C23×C3×C4—22no—
43C23×C3×C4C2×C411yescharacteristic
44C23×C3×C4D412yes—
45C2×C3×C4⋊C4—24no—
46C2×C3×C22⋊C4—22no—
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—22no—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—24no—
62C2×C3×C22⋊C4C2×C412yes—
63C2×C3×C22⋊C4C2×C412yes—
64C2×C3×C22⋊C4D414yes—
65C2×C3×C22⋊C4D414yes—
66C2×C3×C22⋊C4D412yes—
67C3×#99—216no—
68C3×#95—24no—
69C3×#95—24no—
70C23×D4C22×C311yescharacteristic
71C22×C22⋊C4C22×C311yescharacteristic
72C22×C22⋊C4C22×C312yes—
73C22×C22⋊C4C22×C312yes—
74C22×C22⋊C4C22×C311yescharacteristic
75C2×#99—24no—
76C2×#99—24no—
77C2×#99—24no—
78#386—22no—
79#386—22no—
80#385—24no—
81#381C3×C412yescomplemented
82#380—22no—
83#379C3×C412yescomplemented
84#342—24no—
85#342—22no—
86C24×C3—21no—
87C24×C3—21no—
88C24×C3—22no—
89C24×C3—22no—
90C24×C3—82no—
91C24×C3—82no—
92C24×C3C4⋊C412yescomplemented
93C24×C3C22×C411yescharacteristic
94C24×C3C2×D411yescharacteristic
95C24×C3C2×D411yescharacteristic
96C24×C3C2×D412yes—
97C24×C3C2×D411yescharacteristic
98C24×C3C2×D412yes—
99C24×C3C2×Q811yescharacteristic
100C24×C3#4112yes—
101C24×C3#4111yescharacteristic
102C24×C3#4111yescharacteristic
103C24×C3#4112yes—
104C2×C3×D4—24no—
105C2×C3×D4—22no—
106C2×C3×D4—22no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—24no—
110C2×C3×D4—24no—
111C2×C3×D4—24no—
112C2×C3×D4—41no—
113C2×C3×D4—42no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C2×C3×D4—41no—
117C2×C3×D4—42no—
118C2×C3×D4—42no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—41no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—24no—
130C22×C3×C4—24no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4—82no—
137C22×C3×C4—82no—
138C22×C3×C4—84no—
139C22×C3×C4—84no—
140C22×C3×C4C22⋊C414yes—
141C22×C3×C4C4⋊C412yescomplemented
142C3×C4⋊C4—84no—
143C3×C22⋊C4—24no—
144C3×C22⋊C4—24no—
145C3×C22⋊C4—24no—
146C3×C22⋊C4—24no—
147C3×C22⋊C4—24no—
148C3×C22⋊C4—24no—
149C3×C22⋊C4—24no—
150C3×C22⋊C4—42no—
151C3×C22⋊C4—42no—
152C3×C22⋊C4—41no—
153C3×C22⋊C4—42no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—42no—
156C3×C22⋊C4—84no—
157C25C2×C3×C411yescharacteristic
158C25C23×C311yescharacteristic
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C3×D412yes—
168C22×D4C3×Q812yes—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4C2×C3×C411yescharacteristic
173C23×C4C3×D412yes—
174C2×C4⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4C2×C3×C412yes—
192C2×C22⋊C4C2×C3×C412yes—
193C2×C22⋊C4C3×D414yes—
194C2×C22⋊C4C3×D414yes—
195C2×C22⋊C4C3×D412yes—
196#99—216no—
197#95—24no—
198#95—24no—
199C23×C3—22no—
200C23×C3—24no—
201C23×C3—42no—
202C23×C3—44no—
203C23×C3—42no—
204C23×C3—44no—
205C23×C3—42no—
206C23×C3—44no—
207C23×C3—41no—
208C23×C3—42no—
209C23×C3—41no—
210C23×C3—84no—
211C23×C3—81no—
212C23×C3—82no—
213C23×C3—81no—
214C23×C3—81no—
215C23×C3—82no—
216C23×C3—81no—
217C23×C3—82no—
218C23×C3—82no—
219C23×C3—84no—
220C23×C3C2×C22⋊C411yescharacteristic
221C23×C3C2×C4⋊C411yescharacteristic
222C23×C3#11711yescharacteristic
223C23×C3C4×D412yes—
224C23×C3C4×D412yes—
225C23×C3#12011yescharacteristic
226C23×C3#12012yes—
227C23×C3#12112yes—
228C23×C3#12111yescharacteristic
229C23×C3#12111yescharacteristic
230C23×C3#12112yes—
231C23×C3#12112yes—
232C23×C3C22⋊Q811yescharacteristic
233C23×C3C22⋊Q811yescharacteristic
234C23×C3C22⋊Q812yes—
235C23×C3C22⋊Q812yes—
236C23×C3#12311yescharacteristic
237C23×C3#12312yes—
238C23×C3#12312yes—
239C23×C3#12412yes—
240C23×C3#12411yescharacteristic
241C23×C3#12612yes—
242C23×C3#12711yescharacteristic
243C3×D4—42no—
244C3×D4—42no—
245C3×D4—44no—
246C3×D4—44no—
247C3×D4—42no—
248C3×D4—42no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—44no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—41no—
256C2×C3×C4—44no—
257C2×C3×C4—44no—
258C2×C3×C4—48no—
259C2×C3×C4—42no—
260C2×C3×C4—44no—
261C2×C3×C4—44no—
262C2×C3×C4—82no—
263C2×C3×C4—84no—
264C2×C3×C4—84no—
265C2×C3×C4—84no—
266C2×C3×C4—84no—
267C2×C3×C4—84no—
268C2×C3×C4—84no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—82no—
274C24—82no—
275C24C3×C4⋊C412yescomplemented
276C24C22×C3×C411yescharacteristic
277C24C2×C3×D411yescharacteristic
278C24C2×C3×D411yescharacteristic
279C24C2×C3×D412yes—
280C24C2×C3×D411yescharacteristic
281C24C2×C3×D412yes—
282C24C2×C3×Q811yescharacteristic
283C24C3×#4112yes—
284C24C3×#4111yescharacteristic
285C24C3×#4111yescharacteristic
286C24C3×#4112yes—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—41no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4C3×C22⋊C414yes—
324C22×C4C3×C4⋊C412yescomplemented
325C4⋊C4—84no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—42no—
339C22⋊C4—84no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—24no—
343C22×C3—22no—
344C22×C3—24no—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—42no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—42no—
358C22×C3—42no—
359C22×C3—41no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—41no—
363C22×C3—44no—
364C22×C3—42no—
365C22×C3—84no—
366C22×C3—82no—
367C22×C3—84no—
368C22×C3—82no—
369C22×C3#38011yescharacteristic
370C22×C3#38512yes—
371C22×C3#38611yescharacteristic
372C22×C3#38812yes—
373C22×C3#39011yescharacteristic
374C22×C3#39211yescharacteristic
375C22×C3#39311yescharacteristic
376C22×C3#39412yes—
377C22×C3#39612yes—
378C22×C3#39711yescharacteristic
379C22×C3#39911yescharacteristic
380C3×C4—44no—
381C3×C4—42no—
382C3×C4—84no—
383C3×C4—82no—
384C3×C4—84no—
385C23—22no—
386C23—24no—
387C23—42no—
388C23—44no—
389C23—42no—
390C23—44no—
391C23—42no—
392C23—44no—
393C23—41no—
394C23—42no—
395C23—41no—
396C23—84no—
397C23—81no—
398C23—82no—
399C23—81no—
400C23—81no—
401C23—82no—
402C23—81no—
403C23—82no—
404C23—82no—
405C23—84no—
406C23C2×C3×C22⋊C411yescharacteristic
407C23C2×C3×C4⋊C411yescharacteristic
408C23C3×#11711yescharacteristic
409C23C3×C4×D412yes—
410C23C3×C4×D412yes—
411C23C3×#12011yescharacteristic
412C23C3×#12012yes—
413C23C3×#12112yes—
414C23C3×#12111yescharacteristic
415C23C3×#12111yescharacteristic
416C23C3×#12112yes—
417C23C3×#12112yes—
418C23C3×C22⋊Q811yescharacteristic
419C23C3×C22⋊Q811yescharacteristic
420C23C3×C22⋊Q812yes—
421C23C3×C22⋊Q812yes—
422C23C3×#12311yescharacteristic
423C23C3×#12312yes—
424C23C3×#12312yes—
425C23C3×#12412yes—
426C23C3×#12411yescharacteristic
427C23C3×#12612yes—
428C23C3×#12711yescharacteristic
429D4—42no—
430D4—42no—
431D4—44no—
432D4—44no—
433D4—42no—
434D4—42no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—44no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—22no—
458C2×C3—21no—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—22no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—82no—
466C2×C3#143911yescharacteristic
467C2×C3#191312yes—
468C22—22no—
469C22—22no—
470C22—24no—
471C22—22no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22—42no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—42no—
486C22—42no—
487C22—41no—
488C22—42no—
489C22—42no—
490C22—41no—
491C22—44no—
492C22—42no—
493C22—84no—
494C22—82no—
495C22—84no—
496C22—82no—
497C22C3×#38011yescharacteristic
498C22C3×#38512yes—
499C22C3×#38611yescharacteristic
500C22C3×#38812yes—
501C22C3×#39011yescharacteristic
502C22C3×#39211yescharacteristic
503C22C3×#39311yescharacteristic
504C22C3×#39412yes—
505C22C3×#39612yes—
506C22C3×#39711yescharacteristic
507C22C3×#39911yescharacteristic
508C4—44no—
509C4—42no—
510C4—84no—
511C4—82no—
512C4—84no—
513C3#1104611yescharacteristic, direct product factor, Sylow
514C2—21no—
515C2—21no—
516C2—22no—
517C2—21no—
518C2—22no—
519C2—22no—
520C2—21no—
521C2—22no—
522C2—21no—
523C2—21no—
524C2—82no—
525C2C3×#143911yescharacteristic
526C2C3×#191312yes—
subgroupG/H
center, socleC22×C3#386
commutatorC24C22×C3×C4
Frattini, cosocleC25C23×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2315C2semidirect
4. C2×C3×#342C2semidirect
5. C2×C3×#342C2semidirect
6. #11046C3direct (N × H)
18. C3×#381C4semidirect
20. C3×#379C4semidirect
24. #2315C2×C3semidirect
26. C2×#342C2×C3semidirect
27. C2×#342C2×C3semidirect
81. #381C3×C4semidirect
83. #379C3×C4semidirect
92. C24×C3C4⋊C4semidirect
141. C22×C3×C4C4⋊C4semidirect
275. C24C3×C4⋊C4semidirect
324. C22×C4C3×C4⋊C4semidirect
513. C3#11046direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#992
C3×#23152
C3×#23082
C2×C3×#3422
#110463