Finite Group Databasebeta

← C3×#25688 #10644295

Subgroups — C3×#25688 #10644295

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3509C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3C3×#3079C211yesmaximal, characteristic
4C3×#2862C211yesmaximal, characteristic, complemented
5C3×#2038C214yesmaximal
6C3×#1994C211yesmaximal, characteristic, complemented
7C3×#1966C211yesmaximal, characteristic, complemented
8C3×#1857C214yesmaximal, complemented
9C3×C4×#104C211yesmaximal, characteristic
10#25688C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#557—21no—
12C3×#557—21no—
13C3×#556—21no—
14C3×#556—21no—
15C3×#552—22no—
16C3×#552—22no—
17C3×#541—22no—
18C3×#536C2212yes—
19C3×#533—21no—
20C3×#533—21no—
21C3×#533C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×C4⋊Q8C2211yescharacteristic
24C3×#519C2212yes—
25C3×C4×#41C2211yescharacteristic
26C3×#501—22no—
27C3×#500—21no—
28C3×#498—21no—
29C3×#477—21no—
30C3×#476—21no—
31C3×#475—21no—
32C3×#474—21no—
33C3×#428C2214yes—
34C3×#428C2212yes—
35C3×#423C2211yescharacteristic
36C2×C3×#104C2212yes—
37C3×#417C2212yes—
38C2×C3×#101C2214yes—
39C3×C4×M16C2211yescharacteristic
40C3×#391—22no—
41C3×#391C2214yes—
42C3×#382—22no—
43C3×C43C2211yescharacteristic
44C3×#339C2214yes—
45C3×#339C2212yes—
46#3509C2×C311yescharacteristic
47#3165C2×C311yescharacteristic, complemented
48#3079C2×C311yescharacteristic
49#2862C2×C311yescharacteristic, complemented
50#2038C2×C314yes—
51#1994C2×C311yescharacteristic, complemented
52#1966C2×C311yescharacteristic, complemented
53#1857C2×C314yescomplemented
54C4×#104C2×C311yescharacteristic
55C2×C3×#41C2311yescharacteristic
56C22×C3×Q8—21no—
57C22×C3×Q8C2312yes—
58C2×C3×M16C2312yes—
59C2×C3×M16C2312yes—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—21no—
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⋊Q8D411yescharacteristic
70C3×C4⋊Q8D411yescharacteristic
71C3×#126—24no—
72C3×#126—24no—
73C3×#125—22no—
74C3×#125—22no—
75C3×#125—24no—
76C3×#125—24no—
77C3×#125D412yes—
78C3×#124—22no—
79C3×#123—24no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8—21no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—22no—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×Q8D411yescharacteristic
95C3×C4×Q8D411yescharacteristic
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4D411yescharacteristic
99C3×C4×D4D411yescharacteristic
100C3×#117—22no—
101C3×#117—22no—
102C3×#117—24no—
103C3×#117—21no—
104C3×#117C2311yescharacteristic
105C3×#117C2312yes—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4D412yes—
108C2×C3×C4⋊C4D412yes—
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×#107—24no—
117C3×#106—22no—
118C3×#106—24no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8—21no—
121C3×#104—24no—
122C3×#103—22no—
123C3×#102—22no—
124C3×#101—28no—
125C3×#97—21no—
126C3×C4×C8—21no—
127C3×#95—28no—
128#557—21no—
129#557—21no—
130#556—21no—
131#556—21no—
132#552—22no—
133#552—22no—
134#541—22no—
135#536C22×C312yes—
136#533—21no—
137#533—21no—
138#533C22×C311yescharacteristic
139C2×C4⋊Q8C22×C311yescharacteristic
140C2×C4⋊Q8C22×C311yescharacteristic
141#519C22×C312yes—
142C4×#41C22×C311yescharacteristic
143#501—22no—
144#500—21no—
145#498—21no—
146#477—21no—
147#476—21no—
148#475—21no—
149#474—21no—
150#428C22×C312yes—
151#428C22×C314yes—
152#423C22×C311yescharacteristic
153C2×#104C22×C312yes—
154#417C22×C312yes—
155C2×#101C22×C314yes—
156C4×M16C22×C311yescharacteristic
157#391—22no—
158#391C22×C314yes—
159#382—22no—
160C43C22×C311yescharacteristic
161#339C22×C312yes—
162#339C22×C314yes—
163C3×#41—22no—
164C3×#41—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—24no—
170C2×C3×Q8—24no—
171C2×C3×Q8—44no—
172C2×C3×Q8—41no—
173C2×C3×Q8—41no—
174C2×C3×Q8C2×D411yescharacteristic
175C2×C3×D4—21no—
176C2×C3×D4C2×D411yescharacteristic
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C3×M16—22no—
187C3×M16—24no—
188C3×M16#4114yes—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4—41no—
209C3×C4⋊C4C2×D411yescharacteristic
210C3×C4⋊C4C2×D412yes—
211C3×C4⋊C4C2×D411yescharacteristic
212C3×C4⋊C4C2×D412yes—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—21no—
219C3×C42—24no—
220C3×C42—24no—
221C3×C42—24no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—22no—
226C3×C42—22no—
227C3×C42—22no—
228C3×C42—42no—
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C42C2×D411yescharacteristic
232C3×C42C2×D411yescharacteristic
233C2×#41C23×C311yescharacteristic
234C22×Q8—21no—
235C22×Q8C23×C312yes—
236C2×M16C23×C312yes—
237C2×M16C23×C312yes—
238C4⋊Q8—22no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8C3×D411yescharacteristic
248C4⋊Q8C3×D411yescharacteristic
249#126—24no—
250#126—24no—
251#125—22no—
252#125—22no—
253#125—24no—
254#125—24no—
255#125C3×D412yes—
256#124—22no—
257#123—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—21no—
261C22⋊Q8—21no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8C3×D411yescharacteristic
273C4×Q8C3×D411yescharacteristic
274C4×D4—21no—
275C4×D4—21no—
276C4×D4C3×D411yescharacteristic
277C4×D4C3×D411yescharacteristic
278#117—22no—
279#117—22no—
280#117—24no—
281#117—21no—
282#117C23×C311yescharacteristic
283#117C23×C312yes—
284C2×C4⋊C4—21no—
285C2×C4⋊C4C3×D412yes—
286C2×C4⋊C4C3×D412yes—
287C2×C4⋊C4C23×C312yes—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C23×C312yes—
292C2×C42C23×C311yescharacteristic
293#107—22no—
294#107—24no—
295#106—22no—
296#106—24no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299#104—24no—
300#103—22no—
301#102—22no—
302#101—28no—
303#97—21no—
304C4×C8—21no—
305#95—28no—
306C23×C3—21no—
307C23×C3C22×D411yescharacteristic
308C3×Q8—21no—
309C3×Q8—21no—
310C3×Q8—42no—
311C3×Q8—42no—
312C3×Q8—44no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×Q8—82no—
316C3×Q8—82no—
317C3×D4—21no—
318C3×D4—21no—
319C3×D4—42no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—21no—
329C2×C3×C4—28no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—44no—
336C2×C3×C4—42no—
337C2×C3×C4#12011yescharacteristic
338C2×C3×C4#12011yescharacteristic
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C2×#4112yes—
344C2×C3×C42−1+412yes—
345C3×C8—24no—
346C3×C8—42no—
347#41—22no—
348#41—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—44no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8C2×C3×D411yescharacteristic
359C2×D4—21no—
360C2×D4C2×C3×D411yescharacteristic
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4C2×C3×D414yes—
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C24×C311yescharacteristic
370M16—22no—
371M16—24no—
372M16C3×#4114yes—
373C2×C8—22no—
374C2×C8—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4C2×C3×D411yescharacteristic
394C4⋊C4C2×C3×D412yes—
395C4⋊C4C2×C3×D411yescharacteristic
396C4⋊C4C2×C3×D412yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—21no—
400C22⋊C4—22no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—42no—
413C42C2×C3×D411yescharacteristic
414C42C2×C3×D411yescharacteristic
415C42C2×C3×D411yescharacteristic
416C42C2×C3×D411yescharacteristic
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3C2×#12011yescharacteristic
423C22×C3#54712yes—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—44no—
427C3×C4—44no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—42no—
431C3×C4—41no—
432C3×C4—42no—
433C3×C4—82no—
434C3×C4#45412yes—
435C3×C4#45612yes—
436C3×C4#53211yescharacteristic
437C3×C4#53611yescharacteristic
438C3×C4#54812yes—
439C23—21no—
440C23C22×C3×D411yescharacteristic
441Q8—21no—
442Q8—21no—
443Q8—42no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—44no—
448Q8—82no—
449Q8—82no—
450D4—21no—
451D4—21no—
452D4—42no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—28no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4C3×#12011yescharacteristic
471C2×C4C3×#12011yescharacteristic
472C2×C4C3×#12011yescharacteristic
473C2×C4C3×#12011yescharacteristic
474C2×C4C22×C3×D411yescharacteristic
475C2×C4C22×C3×D411yescharacteristic
476C2×C4C2×C3×#4112yes—
477C2×C4C3×2−1+412yes—
478C8—24no—
479C8—42no—
480C2×C3—22no—
481C2×C3—81no—
482C2×C3#243211yescharacteristic
483C2×C3C2×#45411yescharacteristic
484C2×C3C2×#45611yescharacteristic
485C22—21no—
486C22—21no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22C2×C3×#12011yescharacteristic
491C22C3×#54712yes—
492C4—21no—
493C4—21no—
494C4—44no—
495C4—44no—
496C4—41no—
497C4—41no—
498C4—42no—
499C4—41no—
500C4—42no—
501C4—82no—
502C4C3×#45412yes—
503C4C3×#45612yes—
504C4C3×#53211yescharacteristic
505C4C3×#53611yescharacteristic
506C4C3×#54812yes—
507C3#2568811yescharacteristic, direct product factor, Sylow
508C2—22no—
509C2—81no—
510C2C3×#243211yescharacteristic
511C2C2×C3×#45411yescharacteristic
512C2C2×C3×#45611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#41C23
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×#3165C2semidirect
4. C3×#2862C2semidirect
6. C3×#1994C2semidirect
7. C3×#1966C2semidirect
8. C3×#1857C2semidirect
10. #25688C3direct (N × H)
47. #3165C2×C3semidirect
49. #2862C2×C3semidirect
51. #1994C2×C3semidirect
52. #1966C2×C3semidirect
53. #1857C2×C3semidirect
507. C3#25688direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35092
C3×#31652
C3×#30792
C3×#28622
C3×#20382
C3×#19942
C3×#19662
C3×#18572
C3×C4×#1042
#256883