Finite Group Databasebeta

← C3×#25667 #10644274

Subgroups — C3×#25667 #10644274

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2C3×#3141C211yesmaximal, characteristic, complemented
3C3×#3079C211yesmaximal, characteristic, complemented
4C3×#2862C211yesmaximal, characteristic, complemented
5C3×#2040C214yesmaximal, complemented
6C3×#1997C211yesmaximal, characteristic, complemented
7C3×#1962C211yesmaximal, characteristic, complemented
8C3×#1860C214yesmaximal, complemented
9C3×C4×#104C211yesmaximal, characteristic, complemented
10#25667C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#556—21no—
12C3×#555—21no—
13C3×#554—21no—
14C3×#553—22no—
15C3×#548—22no—
16C3×#547—21no—
17C3×#542—22no—
18C3×#537C2212yes—
19C3×#533—21no—
20C3×#532—21no—
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×C4⋊Q8C2211yescharacteristic
24C3×#520C2212yes—
25C3×C4×#41C2211yescharacteristic
26C3×#489—22no—
27C3×#487—21no—
28C3×#486—21no—
29C3×#477—21no—
30C3×#476—21no—
31C3×#475—21no—
32C3×#474—21no—
33C3×#428C2212yes—
34C3×#423C2211yescharacteristic
35C2×C3×#104C2212yes—
36C2×C3×#104C2214yes—
37C3×#417C2214yes—
38C3×#417C2212yes—
39C3×C4×M16C2211yescharacteristic
40C3×#391—22no—
41C3×#391—22no—
42C3×#382C2214yes—
43C3×C43C2211yescharacteristic
44C3×#343C2214yes—
45C3×#339C2212yes—
46#3511C2×C311yescharacteristic
47#3141C2×C311yescharacteristic, complemented
48#3079C2×C311yescharacteristic, complemented
49#2862C2×C311yescharacteristic, complemented
50#2040C2×C314yescomplemented
51#1997C2×C311yescharacteristic, complemented
52#1962C2×C311yescharacteristic, complemented
53#1860C2×C314yescomplemented
54C4×#104C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C2×C3×#41C2311yescharacteristic
57C2×C3×#41C2312yes—
58C22×C3×Q8—21no—
59C22×C3×Q8D412yes—
60C2×C3×M16C2312yes—
61C2×C3×M16C2312yes—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8D411yescharacteristic
71C3×#127D411yescharacteristic
72C3×#126—24no—
73C3×#125—22no—
74C3×#124—22no—
75C3×#124—24no—
76C3×#124D412yes—
77C3×#123—24no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×#121—22no—
83C3×#121—22no—
84C3×#121—22no—
85C3×#121—21no—
86C3×#121—21no—
87C3×#121—24no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8D411yescharacteristic
91C3×C4×Q8D411yescharacteristic
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4—24no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4D411yescharacteristic
100C3×C4×D4D411yescharacteristic
101C3×#117—22no—
102C3×#117—22no—
103C3×#117—21no—
104C3×#117C2311yescharacteristic
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4D412yes—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C42—21no—
111C2×C3×C42—21no—
112C2×C3×C42—21no—
113C2×C3×C42C2312yes—
114C2×C3×C42C2311yescharacteristic
115C3×#107—22no—
116C3×#106—22no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8—21no—
119C3×#104—24no—
120C3×#104—28no—
121C3×#103—22no—
122C3×#103—24no—
123C3×#102—22no—
124C3×#102—24no—
125C3×#97—21no—
126C3×C4×C8—21no—
127C3×#95—28no—
128#556—21no—
129#555—21no—
130#554—21no—
131#553—22no—
132#548—22no—
133#547—21no—
134#542—22no—
135#537C22×C312yes—
136#533—21no—
137#532—21no—
138#532C22×C311yescharacteristic
139C2×C4⋊Q8C22×C311yescharacteristic
140C2×C4⋊Q8C22×C311yescharacteristic
141#520C22×C312yes—
142C4×#41C22×C311yescharacteristic
143#489—22no—
144#487—21no—
145#486—21no—
146#477—21no—
147#476—21no—
148#475—21no—
149#474—21no—
150#428C22×C312yes—
151#423C22×C311yescharacteristic
152C2×#104C22×C314yes—
153C2×#104C22×C312yes—
154#417C22×C312yes—
155#417C22×C314yes—
156C4×M16C22×C311yescharacteristic
157#391—22no—
158#391—22no—
159#382C22×C314yes—
160C43C22×C311yescharacteristic
161#343C22×C314yes—
162#339C22×C312yes—
163C3×#41—24no—
164C3×#41—22no—
165C3×#41—24no—
166C3×#41—22no—
167C3×#41—84no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—44no—
171C2×C3×Q8—44no—
172C2×C3×Q8—41no—
173C2×C3×Q8—41no—
174C2×C3×Q8C2×D412yes—
175C2×C3×Q8C2×D411yescharacteristic
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4C2×D412yes—
182C2×C3×D4C2×D411yescharacteristic
183C22×C3×C4—22no—
184C22×C3×C4—24no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D414yes—
193C22×C3×C4C2411yescharacteristic
194C3×M16—24no—
195C3×M16—22no—
196C3×M16—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—41no—
212C3×C4⋊C4—41no—
213C3×C4⋊C4C2×D411yescharacteristic
214C3×C4⋊C4C2×D411yescharacteristic
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—22no—
223C3×C42—24no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—22no—
228C3×C42—42no—
229C3×C42—42no—
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C3×C42C2×D411yescharacteristic
234C3×C42#4114yescomplemented
235C2×#41—22no—
236C2×#41C23×C311yescharacteristic
237C2×#41C23×C312yes—
238C22×Q8—21no—
239C22×Q8C3×D412yes—
240C2×M16C23×C312yes—
241C2×M16C23×C312yes—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8C3×D411yescharacteristic
251#127C3×D411yescharacteristic
252#126—24no—
253#125—22no—
254#124—22no—
255#124—24no—
256#124C3×D412yes—
257#123—24no—
258C22⋊Q8—22no—
259C22⋊Q8—24no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262#121—22no—
263#121—22no—
264#121—22no—
265#121—21no—
266#121—21no—
267#121—24no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8C3×D411yescharacteristic
271C4×Q8C3×D411yescharacteristic
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—24no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4C3×D411yescharacteristic
280C4×D4C3×D411yescharacteristic
281#117—22no—
282#117—22no—
283#117—21no—
284#117C23×C311yescharacteristic
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C3×D412yes—
288C2×C4⋊C4C23×C312yes—
289C2×C4⋊C4C23×C312yes—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C23×C312yes—
294C2×C42C23×C311yescharacteristic
295#107—22no—
296#106—22no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299#104—28no—
300#104—24no—
301#103—22no—
302#103—24no—
303#102—22no—
304#102—24no—
305#97—21no—
306C4×C8—21no—
307#95—28no—
308C23×C3—21no—
309C23×C3—22no—
310C23×C3C22×D411yescharacteristic
311C3×Q8—21no—
312C3×Q8—21no—
313C3×Q8—24no—
314C3×Q8—44no—
315C3×Q8—82no—
316C3×Q8—82no—
317C3×Q8—82no—
318C3×D4—21no—
319C3×D4—21no—
320C3×D4—24no—
321C3×D4—44no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—84no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—44no—
342C2×C3×C4—44no—
343C2×C3×C4—42no—
344C2×C3×C4—84no—
345C2×C3×C4#12011yescharacteristic
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C2×#4112yes—
352C2×C3×C42−1+412yes—
353C3×C8—42no—
354C3×C8—42no—
355#41—24no—
356#41—22no—
357#41—24no—
358#41—22no—
359#41—84no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8C2×C3×D412yes—
367C2×Q8C2×C3×D411yescharacteristic
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4C2×C3×D412yes—
374C2×D4C2×C3×D411yescharacteristic
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D414yes—
385C22×C4C24×C311yescharacteristic
386M16—24no—
387M16—22no—
388M16—22no—
389C2×C8—22no—
390C2×C8—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4C2×C3×D411yescharacteristic
406C4⋊C4C2×C3×D411yescharacteristic
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C42—24no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—22no—
420C42—42no—
421C42—42no—
422C42C2×C3×D411yescharacteristic
423C42C2×C3×D411yescharacteristic
424C42C2×C3×D411yescharacteristic
425C42C2×C3×D411yescharacteristic
426C42C3×#4114yescomplemented
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—42no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—42no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3C2×#12011yescharacteristic
436C22×C3#54812yes—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—44no—
440C3×C4—42no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—44no—
444C3×C4—41no—
445C3×C4—82no—
446C3×C4#45412yes—
447C3×C4#45512yes—
448C3×C4#53211yescharacteristic
449C3×C4#53611yescharacteristic
450C3×C4#54712yes—
451C23—21no—
452C23—22no—
453C23C22×C3×D411yescharacteristic
454Q8—21no—
455Q8—21no—
456Q8—24no—
457Q8—44no—
458Q8—82no—
459Q8—82no—
460Q8—82no—
461D4—21no—
462D4—21no—
463D4—24no—
464D4—44no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—84no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—84no—
488C2×C4C3×#12011yescharacteristic
489C2×C4C3×#12011yescharacteristic
490C2×C4C3×#12011yescharacteristic
491C2×C4C3×#12011yescharacteristic
492C2×C4C22×C3×D411yescharacteristic
493C2×C4C22×C3×D411yescharacteristic
494C2×C4C2×C3×#4112yes—
495C2×C4C3×2−1+412yes—
496C8—42no—
497C8—42no—
498C2×C3—22no—
499C2×C3—81no—
500C2×C3—82no—
501C2×C3#243211yescharacteristic
502C2×C3C2×#45511yescharacteristic
503C2×C3C2×#45411yescharacteristic
504C22—21no—
505C22—21no—
506C22—42no—
507C22—41no—
508C22—41no—
509C22—42no—
510C22—41no—
511C22—42no—
512C22C2×C3×#12011yescharacteristic
513C22C3×#54812yes—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—42no—
518C4—41no—
519C4—41no—
520C4—44no—
521C4—41no—
522C4—82no—
523C4C3×#45412yes—
524C4C3×#45512yes—
525C4C3×#53211yescharacteristic
526C4C3×#53611yescharacteristic
527C4C3×#54712yes—
528C3#2566711yescharacteristic, direct product factor, Sylow
529C2—22no—
530C2—81no—
531C2—82no—
532C2C3×#243211yescharacteristic
533C2C2×C3×#45511yescharacteristic
534C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3511C2
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×#3141C2semidirect
3. C3×#3079C2semidirect
4. C3×#2862C2semidirect
5. C3×#2040C2semidirect
6. C3×#1997C2semidirect
7. C3×#1962C2semidirect
8. C3×#1860C2semidirect
9. C3×C4×#104C2semidirect
10. #25667C3direct (N × H)
47. #3141C2×C3semidirect
48. #3079C2×C3semidirect
49. #2862C2×C3semidirect
50. #2040C2×C3semidirect
51. #1997C2×C3semidirect
52. #1962C2×C3semidirect
53. #1860C2×C3semidirect
54. C4×#104C2×C3semidirect
234. C3×C42#41semidirect
426. C42C3×#41semidirect
528. C3#25667direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35112
C3×#31412
C3×#30792
C3×#28622
C3×#20402
C3×#19972
C3×#19622
C3×#18602
C3×C4×#1042
#256673