Finite Group Databasebeta

← C3×#27653 #10646260

Subgroups — C3×#27653 #10646260

Order 768.

838 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#484C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic
3C3×C4×C22⋊Q8C211yesmaximal, characteristic
4C2×C3×#382C211yesmaximal, characteristic
5C3×#2085C212yesmaximal, complemented
6C3×#2074C212yesmaximal, complemented
7C3×#1949C212yesmaximal, complemented
8C3×#1824C211yesmaximal, characteristic
9C3×#1823C211yesmaximal, characteristic
10C3×#1805C212yesmaximal, complemented
11C3×#1797C211yesmaximal, characteristic
12#27653C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C3×#484—24no—
19C3×#483—24no—
20C2×C3×#107C2212yes—
21C2×C3×#106C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#389C2211yescharacteristic
29C3×#387—21no—
30C3×#385—21no—
31C3×#382—21no—
32C3×#382—21no—
33C3×#382C2212yes—
34C3×#382C2212yes—
35C3×#380C2211yescharacteristic
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C22⋊C4—21no—
39C2×C3×#95—21no—
40C2×C3×#95—21no—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#336C2212yes—
46C2×#484C2×C311yescharacteristic
47C2×#483C2×C311yescharacteristic
48C4×C22⋊Q8C2×C311yescharacteristic
49C2×#382C2×C311yescharacteristic
50#2085C2×C312yescomplemented
51#2074C2×C312yescomplemented
52#1949C2×C312yescomplemented
53#1824C2×C311yescharacteristic
54#1823C2×C311yescharacteristic
55#1805C2×C312yescomplemented
56#1797C2×C311yescharacteristic
57C22×C3×Q8C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C23×C3×C4D411yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×C22⋊Q8—22no—
66C3×C22⋊Q8D414yes—
67C3×C4×Q8—24no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—24no—
85C2×C3×C42—21no—
86C2×C3×C42—21no—
87C2×C3×C42D411yescharacteristic
88C2×C3×C42D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C3×#107—24no—
92C3×#106—24no—
93C3×C4⋊C8—24no—
94C3×#103—24no—
95C3×#103—24no—
96C3×C22⋊C8—22no—
97C3×C22⋊C8—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C2×C22⋊Q8C22×C311yescharacteristic
108C2×C4×Q8C22×C311yescharacteristic
109C22×C4⋊C4C22×C311yescharacteristic
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C42C22×C311yescharacteristic
112#484—24no—
113#483—24no—
114C2×#107C22×C312yes—
115C2×#106C22×C312yes—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C312yes—
118C2×#103C22×C312yes—
119C2×C22⋊C8C22×C311yescharacteristic
120C2×C22⋊C8C22×C311yescharacteristic
121#391C22×C311yescharacteristic
122#389C22×C311yescharacteristic
123#387—21no—
124#385—21no—
125#382—21no—
126#382—21no—
127#382C22×C312yes—
128#382C22×C312yes—
129#380C22×C311yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4C22×C311yescharacteristic
132C4×C22⋊C4—21no—
133C2×#95—21no—
134C2×#95—21no—
135#340C22×C312yes—
136#340C22×C312yes—
137#340C22×C312yes—
138#340C22×C312yes—
139#336C22×C312yes—
140C24×C3C2×D411yescharacteristic
141C2×C3×Q8—21no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8C2×D412yes—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4—42no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4#4112yes—
172C22×C3×C4C2411yescharacteristic
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4C2×D412yes—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—42no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—42no—
198C3×C42—42no—
199C3×C42#4114yes—
200C22×Q8C23×C311yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4C3×D411yescharacteristic
204C23×C4C3×D411yescharacteristic
205C23×C4C23×C311yescharacteristic
206C22×C8C23×C312yes—
207C22×C8C23×C312yes—
208C22⋊Q8—22no—
209C22⋊Q8C3×D414yes—
210C4×Q8—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C4⋊C4C23×C311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42—24no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42C3×D411yescharacteristic
231C2×C42C3×D411yescharacteristic
232C2×C42C23×C311yescharacteristic
233C2×C42C23×C311yescharacteristic
234#107—24no—
235#106—24no—
236C4⋊C8—24no—
237#103—24no—
238#103—24no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3#12112yes—
255C23×C3#13512yes—
256C23×C3#13712yes—
257C23×C3C22×D411yescharacteristic
258C3×Q8—24no—
259C3×Q8—42no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—84no—
295C2×C3×C4—84no—
296C2×C3×C4#12112yes—
297C2×C3×C4#12312yes—
298C2×C3×C4#12312yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4112yes—
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C42−1+412yes—
303C3×C8—42no—
304C3×C8—42no—
305C24C2×C3×D411yescharacteristic
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8C2×C3×D412yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C2×C3×D411yescharacteristic
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D412yes—
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C3×#4112yes—
337C22×C4C24×C311yescharacteristic
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C4⋊C4—24no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4C2×C3×D412yes—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—42no—
363C42—42no—
364C42C3×#4114yes—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3#46414yes—
372C22×C3C2×#12111yescharacteristic
373C22×C3C2×#12311yescharacteristic
374C22×C3#53611yescharacteristic
375C22×C3C2×#13511yescharacteristic
376C22×C3C2×#13711yescharacteristic
377C22×C3#57511yescharacteristic
378C22×C3#57511yescharacteristic
379C3×C4—22no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—81no—
387C3×C4—81no—
388C3×C4—82no—
389C3×C4—82no—
390C3×C4#54712yes—
391C3×C4#55212yes—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23C3×#12112yes—
397C23C3×#13512yes—
398C23C3×#13712yes—
399C23C22×C3×D411yescharacteristic
400Q8—24no—
401Q8—42no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4C3×#12112yes—
439C2×C4C3×#12312yes—
440C2×C4C3×#12312yes—
441C2×C4C22×C3×D411yescharacteristic
442C2×C4C2×C3×#4112yes—
443C2×C4C2×C3×#4111yescharacteristic
444C2×C4C3×2−1+412yes—
445C8—42no—
446C8—42no—
447C2×C3—24no—
448C2×C3#248611yescharacteristic
449C2×C3C2×#46411yescharacteristic
450C2×C3#304011yescharacteristic
451C2×C3#309111yescharacteristic
452C2×C3#309511yescharacteristic
453C2×C3#323211yescharacteristic
454C2×C3#323611yescharacteristic
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22C3×#46414yes—
462C22C2×C3×#12111yescharacteristic
463C22C2×C3×#12311yescharacteristic
464C22C3×#53611yescharacteristic
465C22C2×C3×#13511yescharacteristic
466C22C2×C3×#13711yescharacteristic
467C22C3×#57511yescharacteristic
468C22C3×#57511yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—42no—
473C4—42no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—82no—
478C4—82no—
479C4—81no—
480C4C3×#54712yes—
481C4C3×#55212yes—
482C3#2765311yescharacteristic, direct product factor, Sylow
483C2—24no—
484C2C3×#248611yescharacteristic
485C2C2×C3×#46411yescharacteristic
486C2C3×#304011yescharacteristic
487C2C3×#309111yescharacteristic
488C2C3×#309511yescharacteristic
489C2C3×#323211yescharacteristic
490C2C3×#323611yescharacteristic
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
5. C3×#2085C2semidirect
6. C3×#2074C2semidirect
7. C3×#1949C2semidirect
10. C3×#1805C2semidirect
12. #27653C3direct (N × H)
50. #2085C2×C3semidirect
51. #2074C2×C3semidirect
52. #1949C2×C3semidirect
55. #1805C2×C3semidirect
482. C3#27653direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4842
C2×C3×#4832
C3×C4×C22⋊Q82
C2×C3×#3822
C3×#20852
C3×#20742
C3×#19492
C3×#18242
C3×#18232
C3×#18052
C3×#17972
#276533