Finite Group Databasebeta

← C3×#25696 #10644303

Subgroups — C3×#25696 #10644303

Order 768.

958 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3507C211yesmaximal, characteristic
2C3×#3165C211yesmaximal, characteristic, complemented
3C3×#3141C211yesmaximal, characteristic
4C3×#2568C211yesmaximal, characteristic, complemented
5C3×#2038C214yesmaximal, complemented
6C3×#1986C211yesmaximal, characteristic, complemented
7C3×#1980C211yesmaximal, characteristic
8C3×#1860C214yesmaximal, complemented
9C3×C4×#101C211yesmaximal, characteristic, complemented
10#25696C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#559—21no—
12C3×#557—21no—
13C3×#556—21no—
14C3×#556—22no—
15C3×#554—21no—
16C3×#552—22no—
17C3×#539—22no—
18C3×#536C2212yes—
19C3×#533C2211yescharacteristic
20C3×#532C2211yescharacteristic
21C2×C3×C4⋊Q8C2211yescharacteristic
22C3×#529—22no—
23C3×#520C2212yes—
24C2×C3×C4×Q8C2211yescharacteristic
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×#101C2212yes—
35C3×C4×M16C2211yescharacteristic
36C3×C4×M16C2211yescharacteristic
37C3×#400—22no—
38C3×#391—22no—
39C3×#391C2212yes—
40C3×#382C2212yes—
41C3×C4×C4⋊C4C2211yescharacteristic
42C3×#343C2212yes—
43C3×#339C2214yes—
44#3507C2×C311yescharacteristic
45#3165C2×C311yescharacteristic, complemented
46#3141C2×C311yescharacteristic
47#2568C2×C311yescharacteristic, complemented
48#2038C2×C314yescomplemented
49#1986C2×C311yescharacteristic, complemented
50#1980C2×C311yescharacteristic
51#1860C2×C314yescomplemented
52C4×#101C2×C311yescharacteristic, complemented
53C2×C3×#41C2312yes—
54C22×C3×Q8—21no—
55C22×C3×Q8C2311yescharacteristic
56C2×C3×M16C2312yes—
57C2×C3×M16C2312yes—
58C3×C4⋊Q8—22no—
59C3×C4⋊Q8—21no—
60C3×C4⋊Q8—21no—
61C3×C4⋊Q8—21no—
62C3×C4⋊Q8—21no—
63C3×C4⋊Q8D411yescharacteristic
64C3×C4⋊Q8D411yescharacteristic
65C3×C4⋊Q8D411yescharacteristic
66C3×#127D411yescharacteristic
67C3×#126—24no—
68C3×#126—24no—
69C3×#125—22no—
70C3×#125—24no—
71C3×#125D412yes—
72C3×#124—22no—
73C3×#124—22no—
74C3×#124D412yes—
75C3×#123—24no—
76C3×#123—24no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—21no—
79C3×C22⋊Q8—21no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×#121—21no—
83C3×#121—21no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8—21no—
87C3×C4×Q8—21no—
88C3×C4×Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×D4—22no—
93C3×C4×D4—22no—
94C3×C4×D4—24no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×#117—24no—
98C3×#117—22no—
99C3×#117C2312yes—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4D412yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C42D412yes—
107C2×C3×C42C2312yes—
108C2×C3×C42C2311yescharacteristic
109C3×#107—24no—
110C3×#106—24no—
111C3×#104—28no—
112C3×#103—24no—
113C3×#102—24no—
114C3×#101—24no—
115C3×#97—21no—
116C3×#97—21no—
117C3×C4×C8—21no—
118C3×C4×C8—21no—
119C3×#95—21no—
120C3×#95—24no—
121C3×#95—24no—
122#559—21no—
123#557—21no—
124#556—21no—
125#556—22no—
126#554—21no—
127#552—22no—
128#539—22no—
129#536C22×C312yes—
130#533C22×C311yescharacteristic
131#532C22×C311yescharacteristic
132C2×C4⋊Q8C22×C311yescharacteristic
133#529—22no—
134#520C22×C312yes—
135C2×C4×Q8C22×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×#101C22×C312yes—
146C4×M16C22×C311yescharacteristic
147C4×M16C22×C311yescharacteristic
148#400—22no—
149#391—22no—
150#391C22×C312yes—
151#382C22×C312yes—
152C4×C4⋊C4C22×C311yescharacteristic
153#343C22×C312yes—
154#339C22×C314yes—
155C3×#41—24no—
156C3×#41—24no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—44no—
164C2×C3×Q8—41no—
165C2×C3×Q8—41no—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—22no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4C2×D412yes—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C3×M16—22no—
181C3×M16—22no—
182C3×M16—22no—
183C3×M16#4114yes—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—42no—
202C3×C4⋊C4—41no—
203C3×C4⋊C4—41no—
204C3×C4⋊C4C2×D412yes—
205C3×C4⋊C4C2×D412yes—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—22no—
212C3×C42—24no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42—24no—
217C3×C42C2×D411yescharacteristic
218C3×C42C2×D411yescharacteristic
219C3×C42C2×D411yescharacteristic
220C3×C42C2×D411yescharacteristic
221C2×#41C23×C312yes—
222C22×Q8—21no—
223C22×Q8C23×C311yescharacteristic
224C2×M16C23×C312yes—
225C2×M16C23×C312yes—
226C4⋊Q8—22no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8C3×D411yescharacteristic
232C4⋊Q8C3×D411yescharacteristic
233C4⋊Q8C3×D411yescharacteristic
234#127C3×D411yescharacteristic
235#126—24no—
236#126—24no—
237#125—22no—
238#125—24no—
239#125C3×D412yes—
240#124—22no—
241#124—22no—
242#124C3×D412yes—
243#123—24no—
244#123—24no—
245C22⋊Q8—24no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250#121—21no—
251#121—21no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—24no—
263C4×D4—21no—
264C4×D4—21no—
265#117—24no—
266#117—22no—
267#117C23×C312yes—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C3×D412yes—
272C2×C4⋊C4C23×C312yes—
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C42C3×D412yes—
275C2×C42C23×C312yes—
276C2×C42C23×C311yescharacteristic
277#107—24no—
278#106—24no—
279#104—28no—
280#103—24no—
281#102—24no—
282#101—24no—
283#97—21no—
284#97—21no—
285C4×C8—21no—
286C4×C8—21no—
287#95—21no—
288#95—24no—
289#95—24no—
290C23×C3—22no—
291C23×C3C22×D411yescharacteristic
292C3×Q8—24no—
293C3×Q8—42no—
294C3×Q8—42no—
295C3×Q8—42no—
296C3×Q8—81no—
297C3×Q8—82no—
298C3×Q8—82no—
299C3×D4—24no—
300C3×D4—42no—
301C3×D4—42no—
302C3×D4—42no—
303C3×D4—81no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—41no—
313C2×C3×C4—44no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—44no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—84no—
321C2×C3×C4#12011yescharacteristic
322C2×C3×C4#12011yescharacteristic
323C2×C3×C4#12011yescharacteristic
324C2×C3×C4#12011yescharacteristic
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4112yes—
328C2×C3×C42−1+412yes—
329C3×C8—24no—
330C3×C8—42no—
331#41—24no—
332#41—24no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—44no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8C2×C3×D412yes—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4C2×C3×D412yes—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4C2×C3×D412yes—
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D412yes—
354C22×C4C2×C3×D411yescharacteristic
355C22×C4C24×C311yescharacteristic
356M16—22no—
357M16—22no—
358M16—22no—
359M16C3×#4114yes—
360C2×C8—22no—
361C2×C8—22no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4C2×C3×D412yes—
381C4⋊C4C2×C3×D412yes—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C42—24no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42C2×C3×D411yescharacteristic
394C42C2×C3×D411yescharacteristic
395C42C2×C3×D411yescharacteristic
396C42C2×C3×D411yescharacteristic
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—42no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3C2×#12011yescharacteristic
403C22×C3#53211yescharacteristic
404C22×C3#53611yescharacteristic
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—82no—
414C3×C4—82no—
415C3×C4#45412yes—
416C3×C4#54712yes—
417C3×C4#54812yes—
418C23—22no—
419C23C22×C3×D411yescharacteristic
420Q8—24no—
421Q8—42no—
422Q8—42no—
423Q8—42no—
424Q8—81no—
425Q8—82no—
426Q8—82no—
427D4—24no—
428D4—42no—
429D4—42no—
430D4—42no—
431D4—81no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—84no—
449C2×C4C3×#12011yescharacteristic
450C2×C4C3×#12011yescharacteristic
451C2×C4C3×#12011yescharacteristic
452C2×C4C3×#12011yescharacteristic
453C2×C4C22×C3×D411yescharacteristic
454C2×C4C22×C3×D411yescharacteristic
455C2×C4C2×C3×#4112yes—
456C2×C4C3×2−1+412yes—
457C8—24no—
458C8—42no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—82no—
462C2×C3#243211yescharacteristic
463C2×C3C2×#45411yescharacteristic
464C2×C3#302211yescharacteristic
465C22—21no—
466C22—21no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22C2×C3×#12011yescharacteristic
471C22C3×#53211yescharacteristic
472C22C3×#53611yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—82no—
482C4—82no—
483C4C3×#45412yes—
484C4C3×#54712yes—
485C4C3×#54812yes—
486C3#2569611yescharacteristic, direct product factor, Sylow
487C2—21no—
488C2—21no—
489C2—82no—
490C2C3×#243211yescharacteristic
491C2C2×C3×#45411yescharacteristic
492C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#532C22
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×#2568C2semidirect
5. C3×#2038C2semidirect
6. C3×#1986C2semidirect
8. C3×#1860C2semidirect
9. C3×C4×#101C2semidirect
10. #25696C3direct (N × H)
45. #3165C2×C3semidirect
47. #2568C2×C3semidirect
48. #2038C2×C3semidirect
49. #1986C2×C3semidirect
51. #1860C2×C3semidirect
52. C4×#101C2×C3semidirect
486. C3#25696direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35072
C3×#31652
C3×#31412
C3×#25682
C3×#20382
C3×#19862
C3×#19802
C3×#18602
C3×C4×#1012
#256963