Finite Group Databasebeta

← C3×#27682 #10646289

Subgroups — C3×#27682 #10646289

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C22⋊Q8C211yesmaximal, characteristic
2C3×#2285C211yesmaximal, characteristic
3C3×#2101C212yesmaximal, complemented
4C3×#2097C212yesmaximal, complemented
5C3×#1948C211yesmaximal, characteristic, complemented
6C3×#1947C211yesmaximal, characteristic, complemented
7C3×#1873C212yesmaximal
8C3×#1854C212yesmaximal
9C3×#1803C211yesmaximal, characteristic, complemented
10C3×#1803C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic
12#27682C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×#117C2212yes—
16C22×C3×C42C2211yescharacteristic
17C2×C3×#107C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#103C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×C22⋊C8C2212yes—
24C3×#391C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#387—21no—
27C3×#385—21no—
28C3×#384C2211yescharacteristic
29C3×#384C2211yescharacteristic
30C3×#382C2212yes—
31C3×#380—21no—
32C3×#380—21no—
33C3×#380C2211yescharacteristic
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C22⋊C4—21no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C4×C22⋊Q8C2×C311yescharacteristic
45#2285C2×C311yescharacteristic
46#2101C2×C312yescomplemented
47#2097C2×C312yescomplemented
48#1948C2×C311yescharacteristic, complemented
49#1947C2×C311yescharacteristic, complemented
50#1873C2×C312yes—
51#1854C2×C312yes—
52#1803C2×C311yescharacteristic, complemented
53#1803C2×C311yescharacteristic, complemented
54#1797C2×C311yescharacteristic
55C22×C3×Q8C2311yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×C3×C8C2312yes—
60C22×C3×C8C2312yes—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8Q814yes—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×#117—42no—
66C3×#117—42no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—22no—
83C2×C3×C42—22no—
84C2×C3×C42—21no—
85C2×C3×C42—21no—
86C2×C3×C42—22no—
87C2×C3×C42D411yescharacteristic
88C2×C3×C42D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C3×#107—42no—
92C3×#106—42no—
93C3×C4⋊C8—22no—
94C3×C4⋊C8—22no—
95C3×#103—42no—
96C3×#103—42no—
97C3×C22⋊C8—24no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C2×C22⋊Q8C22×C311yescharacteristic
106C2×C4×Q8C22×C311yescharacteristic
107C2×#117C22×C312yes—
108C22×C42C22×C311yescharacteristic
109C2×#107C22×C312yes—
110C2×#106C22×C312yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×C4⋊C8C22×C311yescharacteristic
113C2×#103C22×C312yes—
114C2×#103C22×C312yes—
115C2×C22⋊C8C22×C312yes—
116#391C22×C311yescharacteristic
117#389C22×C311yescharacteristic
118#387—21no—
119#385—21no—
120#384C22×C311yescharacteristic
121#384C22×C311yescharacteristic
122#382C22×C312yes—
123#380—21no—
124#380—21no—
125#380C22×C311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C22×C311yescharacteristic
130C4×C22⋊C4—21no—
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#340C22×C312yes—
135#336C22×C312yes—
136C24×C3C2×D411yescharacteristic
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8C2×Q812yes—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—41no—
158C22×C3×C4—41no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×Q812yes—
165C22×C3×C4#4112yes—
166C22×C3×C4C2411yescharacteristic
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×Q812yes—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—44no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—21no—
196C3×C42—21no—
197C3×C42—44no—
198C3×C42—42no—
199C3×C42—42no—
200C3×C42#4112yes—
201C3×C42#4112yes—
202C22×Q8C23×C311yescharacteristic
203C23×C4C3×D411yescharacteristic
204C23×C4C3×D411yescharacteristic
205C23×C4C23×C311yescharacteristic
206C22×C8C23×C312yes—
207C22×C8C23×C312yes—
208C22⋊Q8—22no—
209C22⋊Q8C3×Q814yes—
210C4×Q8—22no—
211C4×Q8—22no—
212#117—42no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C312yes—
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C4⋊C4C23×C311yescharacteristic
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C3×D411yescharacteristic
235C2×C42C3×D411yescharacteristic
236C2×C42C23×C311yescharacteristic
237C2×C42C23×C311yescharacteristic
238#107—42no—
239#106—42no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242#103—42no—
243#103—42no—
244C22⋊C8—24no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3C22⋊Q812yes—
259C23×C3C22×D411yescharacteristic
260C3×Q8—22no—
261C3×Q8—22no—
262C3×Q8—41no—
263C3×Q8—41no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—44no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—44no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4C22⋊Q812yes—
299C2×C3×C4#12312yes—
300C2×C3×C4#12312yes—
301C2×C3×C4#13512yes—
302C2×C3×C4#13712yes—
303C2×C3×C4C22×Q811yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4H(5,𝔽2)11yescharacteristic
308C2×C3×C4H(5,𝔽2)11yescharacteristic
309C3×C8—42no—
310C3×C8—42no—
311C24C2×C3×D411yescharacteristic
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8C2×C3×Q812yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D411yescharacteristic
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×Q812yes—
340C22×C4C3×#4112yes—
341C22×C4C24×C311yescharacteristic
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4C2×C3×Q812yes—
365C22⋊C4—24no—
366C22⋊C4—44no—
367C22⋊C4—42no—
368C42—22no—
369C42—22no—
370C42—21no—
371C42—21no—
372C42—44no—
373C42—42no—
374C42—42no—
375C42C3×#4112yes—
376C42C3×#4112yes—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3C2×C22⋊Q811yescharacteristic
385C22×C3C2×#12311yescharacteristic
386C22×C3#53511yescharacteristic
387C22×C3C2×#13511yescharacteristic
388C22×C3C2×#13711yescharacteristic
389C22×C3#57511yescharacteristic
390C22×C3#57511yescharacteristic
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—42no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—42no—
397C3×C4—81no—
398C3×C4—82no—
399C3×C4—81no—
400C3×C4—82no—
401C3×C4#48514yes—
402C3×C4#55311yescharacteristic
403C3×C4#55311yescharacteristic
404C3×C4#55411yescharacteristic
405C3×C4#55411yescharacteristic
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23C3×C22⋊Q812yes—
413C23C22×C3×D411yescharacteristic
414Q8—22no—
415Q8—22no—
416Q8—41no—
417Q8—41no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4C3×C22⋊Q812yes—
453C2×C4C3×#12312yes—
454C2×C4C3×#12312yes—
455C2×C4C3×#13512yes—
456C2×C4C3×#13712yes—
457C2×C4C22×C3×Q811yescharacteristic
458C2×C4C2×C3×#4111yescharacteristic
459C2×C4C2×C3×#4111yescharacteristic
460C2×C4C2×C3×#4111yescharacteristic
461C2×C4C3×H(5,𝔽2)11yescharacteristic
462C2×C4C3×H(5,𝔽2)11yescharacteristic
463C8—42no—
464C8—42no—
465C2×C3—42no—
466C2×C3#248711yescharacteristic
467C2×C3#307611yescharacteristic
468C2×C3#307811yescharacteristic
469C2×C3C2×#48511yescharacteristic
470C2×C3#309311yescharacteristic
471C2×C3#321211yescharacteristic
472C2×C3#321611yescharacteristic
473C22—22no—
474C22—22no—
475C22—22no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22C2×C3×C22⋊Q811yescharacteristic
481C22C2×C3×#12311yescharacteristic
482C22C3×#53511yescharacteristic
483C22C2×C3×#13511yescharacteristic
484C22C2×C3×#13711yescharacteristic
485C22C3×#57511yescharacteristic
486C22C3×#57511yescharacteristic
487C4—22no—
488C4—22no—
489C4—42no—
490C4—42no—
491C4—41no—
492C4—41no—
493C4—81no—
494C4—82no—
495C4—82no—
496C4—81no—
497C4C3×#48514yes—
498C4C3×#55311yescharacteristic
499C4C3×#55311yescharacteristic
500C4C3×#55411yescharacteristic
501C4C3×#55411yescharacteristic
502C3#2768211yescharacteristic, direct product factor, Sylow
503C2—42no—
504C2C3×#248711yescharacteristic
505C2C3×#307611yescharacteristic
506C2C3×#307811yescharacteristic
507C2C2×C3×#48511yescharacteristic
508C2C3×#309311yescharacteristic
509C2C3×#321211yescharacteristic
510C2C3×#321611yescharacteristic
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
3. C3×#2101C2semidirect
4. C3×#2097C2semidirect
5. C3×#1948C2semidirect
6. C3×#1947C2semidirect
9. C3×#1803C2semidirect
10. C3×#1803C2semidirect
12. #27682C3direct (N × H)
46. #2101C2×C3semidirect
47. #2097C2×C3semidirect
48. #1948C2×C3semidirect
49. #1947C2×C3semidirect
52. #1803C2×C3semidirect
53. #1803C2×C3semidirect
502. C3#27682direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×C22⋊Q82
C3×#22852
C3×#21012
C3×#20972
C3×#19482
C3×#19472
C3×#18732
C3×#18542
C3×#18032
C3×#17972
#276823