Finite Group Databasebeta

← C3×#25693 #10644300

Subgroups — C3×#25693 #10644300

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3C3×#3141C211yesmaximal, characteristic, complemented
4C3×#2563C211yesmaximal, characteristic, complemented
5C3×#2037C214yesmaximal, complemented
6C3×#1986C211yesmaximal, characteristic, complemented
7C3×#1980C211yesmaximal, characteristic, complemented
8C3×#1860C214yesmaximal, complemented
9C3×C4×#100C211yesmaximal, characteristic, complemented
10#25693C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#559—22no—
12C3×#556—21no—
13C3×#555—21no—
14C3×#553—22no—
15C3×#550—21no—
16C3×#547—21no—
17C3×#538—22no—
18C3×#535C2212yes—
19C3×#533C2211yescharacteristic
20C3×#532C2211yescharacteristic
21C2×C3×C4⋊Q8C2211yescharacteristic
22C3×#525—22no—
23C3×#520C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#501—22no—
26C3×#500—21no—
27C3×#498—21no—
28C3×#489—22no—
29C3×#487—21no—
30C3×#486—21no—
31C3×#428C2214yes—
32C2×C3×#104C2214yes—
33C3×#417C2214yes—
34C2×C3×#100C2212yes—
35C3×C4×M16C2211yescharacteristic
36C3×C4×M16C2211yescharacteristic
37C3×#393—22no—
38C3×#390C2212yes—
39C3×#382C2212yes—
40C3×#380—22no—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#343C2212yes—
43C3×#339C2214yes—
44#3506C2×C311yescharacteristic
45#3165C2×C311yescharacteristic, complemented
46#3141C2×C311yescharacteristic, complemented
47#2563C2×C311yescharacteristic, complemented
48#2037C2×C314yescomplemented
49#1986C2×C311yescharacteristic, complemented
50#1980C2×C311yescharacteristic, complemented
51#1860C2×C314yescomplemented
52C4×#100C2×C311yescharacteristic, complemented
53C2×C3×#41—22no—
54C2×C3×#41C2312yes—
55C22×C3×Q8—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4D412yes—
58C2×C3×M16C2312yes—
59C2×C3×M16C2312yes—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8—21no—
64C3×C4⋊Q8D411yescharacteristic
65C3×C4⋊Q8D411yescharacteristic
66C3×C4⋊Q8D411yescharacteristic
67C3×#127—22no—
68C3×#127D411yescharacteristic
69C3×#126—24no—
70C3×#125D412yes—
71C3×#124—24no—
72C3×#124—22no—
73C3×#124D412yes—
74C3×#123—24no—
75C3×#123—24no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—21no—
78C3×C22⋊Q8—21no—
79C3×#121—22no—
80C3×#121—24no—
81C3×#121—21no—
82C3×#121—21no—
83C3×#121—22no—
84C3×#121—22no—
85C3×#120—24no—
86C3×C4×Q8—22no—
87C3×C4×Q8—21no—
88C3×C4×Q8—21no—
89C3×C4×D4—24no—
90C3×C4×D4—22no—
91C3×C4×D4—24no—
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×#117—22no—
99C3×#117—22no—
100C3×#117C2312yes—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C22⋊C4—24no—
107C2×C3×C22⋊C4D412yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C3×#107—24no—
111C3×#106—24no—
112C3×#104—28no—
113C3×#103—24no—
114C3×#102—24no—
115C3×#100—24no—
116C3×#97—21no—
117C3×#97—21no—
118C3×C4×C8—21no—
119C3×C4×C8—21no—
120C3×#95—21no—
121C3×#95—24no—
122#559—22no—
123#556—21no—
124#555—21no—
125#553—22no—
126#550—21no—
127#547—21no—
128#538—22no—
129#535C22×C312yes—
130#533C22×C311yescharacteristic
131#532C22×C311yescharacteristic
132C2×C4⋊Q8C22×C311yescharacteristic
133#525—22no—
134#520C22×C312yes—
135C2×C4×D4C22×C311yescharacteristic
136#501—22no—
137#500—21no—
138#498—21no—
139#489—22no—
140#487—21no—
141#486—21no—
142#428C22×C314yes—
143C2×#104C22×C314yes—
144#417C22×C314yes—
145C2×#100C22×C312yes—
146C4×M16C22×C311yescharacteristic
147C4×M16C22×C311yescharacteristic
148#393—22no—
149#390C22×C312yes—
150#382C22×C312yes—
151#380—22no—
152C4×C4⋊C4C22×C311yescharacteristic
153#343C22×C312yes—
154#339C22×C314yes—
155C24×C3C2×D412yes—
156C3×#41—24no—
157C3×#41—24no—
158C3×#41—84no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—41no—
162C2×C3×Q8—41no—
163C2×C3×Q8—82no—
164C2×C3×Q8C2×D412yes—
165C2×C3×D4—22no—
166C2×C3×D4—24no—
167C2×C3×D4—24no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4C2×D412yes—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—44no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16#4114yescomplemented
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—41no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4C2×D412yes—
207C3×C4⋊C4C2×D412yes—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—84no—
217C3×C42—24no—
218C3×C42—22no—
219C3×C42—24no—
220C3×C42C2×D411yescharacteristic
221C3×C42C2×D411yescharacteristic
222C3×C42C2×D411yescharacteristic
223C3×C42C2×D411yescharacteristic
224C2×#41—22no—
225C2×#41C23×C312yes—
226C22×Q8—21no—
227C22×D4C23×C311yescharacteristic
228C23×C4C3×D412yes—
229C2×M16C23×C312yes—
230C2×M16C23×C312yes—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8C3×D411yescharacteristic
236C4⋊Q8C3×D411yescharacteristic
237C4⋊Q8C3×D411yescharacteristic
238#127—22no—
239#127C3×D411yescharacteristic
240#126—24no—
241#125C3×D412yes—
242#124—24no—
243#124—22no—
244#124C3×D412yes—
245#123—24no—
246#123—24no—
247C22⋊Q8—22no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250#121—22no—
251#121—24no—
252#121—21no—
253#121—21no—
254#121—22no—
255#121—22no—
256#120—24no—
257C4×Q8—22no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—24no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117—22no—
270#117—22no—
271#117C23×C312yes—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4C23×C312yes—
276C2×C4⋊C4C23×C311yescharacteristic
277C2×C22⋊C4—24no—
278C2×C22⋊C4C3×D412yes—
279C2×C42C23×C312yes—
280C2×C42C23×C311yescharacteristic
281#107—24no—
282#106—24no—
283#104—28no—
284#103—24no—
285#102—24no—
286#100—24no—
287#97—21no—
288#97—21no—
289C4×C8—21no—
290C4×C8—21no—
291#95—21no—
292#95—24no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3C22×D411yescharacteristic
300C3×Q8—24no—
301C3×Q8—42no—
302C3×Q8—81no—
303C3×Q8—82no—
304C3×Q8—82no—
305C3×D4—24no—
306C3×D4—42no—
307C3×D4—42no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—42no—
311C3×D4—81no—
312C3×D4—84no—
313C3×D4—84no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—41no—
320C2×C3×C4—44no—
321C2×C3×C4—44no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—82no—
329C2×C3×C4—84no—
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4#12011yescharacteristic
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C2×#4112yes—
337C2×C3×C42−1+412yes—
338C3×C8—24no—
339C3×C8—42no—
340C24C2×C3×D412yes—
341#41—24no—
342#41—24no—
343#41—84no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—82no—
349C2×Q8C2×C3×D412yes—
350C2×D4—22no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4C2×C3×D412yes—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—44no—
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D412yes—
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C24×C311yescharacteristic
371M16—22no—
372M16—22no—
373M16—22no—
374M16C3×#4114yescomplemented
375C2×C8—22no—
376C2×C8—22no—
377C4⋊C4—24no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4C2×C3×D412yes—
392C4⋊C4C2×C3×D412yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—84no—
402C42—24no—
403C42—22no—
404C42—24no—
405C42C2×C3×D411yescharacteristic
406C42C2×C3×D411yescharacteristic
407C42C2×C3×D411yescharacteristic
408C42C2×C3×D411yescharacteristic
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—44no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—82no—
419C22×C3C2×#12011yescharacteristic
420C22×C3#53211yescharacteristic
421C22×C3#53611yescharacteristic
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—42no—
429C3×C4—82no—
430C3×C4—82no—
431C3×C4#45512yes—
432C3×C4#54712yes—
433C3×C4#54812yes—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23C22×C3×D411yescharacteristic
441Q8—24no—
442Q8—42no—
443Q8—81no—
444Q8—82no—
445Q8—82no—
446D4—24no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—81no—
453D4—84no—
454D4—84no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—82no—
470C2×C4—84no—
471C2×C4C3×#12011yescharacteristic
472C2×C4C3×#12011yescharacteristic
473C2×C4C3×#12011yescharacteristic
474C2×C4C3×#12011yescharacteristic
475C2×C4C22×C3×D411yescharacteristic
476C2×C4C22×C3×D411yescharacteristic
477C2×C4C2×C3×#4112yes—
478C2×C4C3×2−1+412yes—
479C8—24no—
480C8—42no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—82no—
484C2×C3—82no—
485C2×C3#243211yescharacteristic
486C2×C3C2×#45511yescharacteristic
487C2×C3#302111yescharacteristic
488C22—21no—
489C22—21no—
490C22—44no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—82no—
498C22C2×C3×#12011yescharacteristic
499C22C3×#53211yescharacteristic
500C22C3×#53611yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—82no—
510C4C3×#45512yes—
511C4C3×#54712yes—
512C4C3×#54812yes—
513C3#2569311yescharacteristic, direct product factor, Sylow
514C2—21no—
515C2—21no—
516C2—82no—
517C2—82no—
518C2C3×#243211yescharacteristic
519C2C2×C3×#45511yescharacteristic
520C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3506C2
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
3. C3×#3141C2semidirect
4. C3×#2563C2semidirect
5. C3×#2037C2semidirect
6. C3×#1986C2semidirect
7. C3×#1980C2semidirect
8. C3×#1860C2semidirect
9. C3×C4×#100C2semidirect
10. #25693C3direct (N × H)
45. #3165C2×C3semidirect
46. #3141C2×C3semidirect
47. #2563C2×C3semidirect
48. #2037C2×C3semidirect
49. #1986C2×C3semidirect
50. #1980C2×C3semidirect
51. #1860C2×C3semidirect
52. C4×#100C2×C3semidirect
189. C3×M16#41semidirect
374. M16C3×#41semidirect
513. C3#25693direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35062
C3×#31652
C3×#31412
C3×#25632
C3×#20372
C3×#19862
C3×#19802
C3×#18602
C3×C4×#1002
#256933