Finite Group Databasebeta

← C3×#18633 #10637240

Subgroups — C3×#18633 #10637240

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3503C211yesmaximal, characteristic
2C3×#3209C211yesmaximal, characteristic, complemented
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C3×#3079C211yesmaximal, characteristic
5C3×#1645C214yesmaximal, complemented
6C3×#1644C214yesmaximal, complemented
7C3×#1548C211yesmaximal, characteristic, complemented
8C3×#1493C211yesmaximal, characteristic, complemented
9C3×#1476C211yesmaximal, characteristic
10#18633C311yesmaximal, characteristic, direct product factor, Sylow
11C3×Q82C2212yes—
12C3×#557—21no—
13C3×#554—22no—
14C3×#554C2212yes—
15C3×D4×Q8—22no—
16C3×D4×Q8—21no—
17C3×#544—22no—
18C3×#544—21no—
19C3×#533—21no—
20C3×#533C2211yescharacteristic
21C2×C3×C4⋊Q8—21no—
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×C4⋊Q8C2211yescharacteristic
24C3×#520—21no—
25C3×C4×#41C2211yescharacteristic
26C3×#494—21no—
27C3×#487—22no—
28C3×#487—21no—
29C3×#487C2214yes—
30C3×#477—21no—
31C3×#476—21no—
32C3×#475C2212yes—
33C3×#474C2212yes—
34C3×#462C2214yes—
35C3×#451—22no—
36C3×#431C2211yescharacteristic
37C3×#428—21no—
38C3×#423C2211yescharacteristic
39C3×#417—21no—
40C2×C3×#103—22no—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#333C2214yes—
43C3×Q8⋊C8C2214yes—
44C3×Q8⋊C8C2212yes—
45C3×D4⋊C8C2212yes—
46#3503C2×C311yescharacteristic
47#3209C2×C311yescharacteristic, complemented
48C2×#487C2×C311yescharacteristic, complemented
49#3079C2×C311yescharacteristic
50#1645C2×C314yescomplemented
51#1644C2×C314yescomplemented
52#1548C2×C311yescharacteristic, complemented
53#1493C2×C311yescharacteristic, complemented
54#1476C2×C311yescharacteristic
55C2×C3×#41D411yescharacteristic
56C22×C3×Q8—22no—
57C22×C3×Q8—21no—
58C22×C3×Q8D412yes—
59C2×C3×Q16—24no—
60C2×C3×M16—21no—
61C22×C3×C8—21no—
62C3×C4⋊Q8—22no—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—24no—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8D412yes—
73C3×C4⋊Q8D411yescharacteristic
74C3×C4⋊Q8D411yescharacteristic
75C3×C4⋊Q8C2312yes—
76C3×C4⋊Q8C2312yes—
77C3×#125—22no—
78C3×#125—24no—
79C3×#125—22no—
80C3×#125—21no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—24no—
83C3×C22⋊Q8—24no—
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8D412yes—
88C3×C4×Q8—22no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C3×C4×Q8—21no—
92C3×C4×Q8D411yescharacteristic
93C3×C4×Q8C2311yescharacteristic
94C3×C4×Q8C2312yes—
95C3×C4×D4—24no—
96C3×C4×D4—22no—
97C3×C4×D4—21no—
98C3×C4×D4—21no—
99C3×C4×D4D411yescharacteristic
100C3×C4×D4C2311yescharacteristic
101C3×#117—22no—
102C3×#117—21no—
103C3×#117—21no—
104C3×#117D411yescharacteristic
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4—21no—
111C2×C3×C42—21no—
112C2×C3×C42C2311yescharacteristic
113C3×#107—22no—
114C3×#106—22no—
115C3×C4⋊C8—21no—
116C3×C4⋊C8C2312yes—
117C3×C4⋊C8C2312yes—
118C3×#103—28no—
119C3×#103—22no—
120C3×#103—24no—
121C3×#102—22no—
122C3×C22⋊C8—21no—
123C3×C4×C8—21no—
124C3×C4×C8—21no—
125C3×C4×C8C2312yes—
126Q82C22×C312yes—
127#557—21no—
128#554—22no—
129#554C22×C312yes—
130D4×Q8—22no—
131D4×Q8—21no—
132#544—22no—
133#544—21no—
134#533—21no—
135#533C22×C311yescharacteristic
136C2×C4⋊Q8—21no—
137C2×C4⋊Q8C22×C311yescharacteristic
138C2×C4⋊Q8C22×C311yescharacteristic
139#520—21no—
140C4×#41C22×C311yescharacteristic
141#494—21no—
142#487—22no—
143#487—21no—
144#487C22×C314yes—
145#477—21no—
146#476—21no—
147#475C22×C312yes—
148#474C22×C312yes—
149#462C22×C314yes—
150#451—22no—
151#431C22×C311yescharacteristic
152#428—21no—
153#423C22×C311yescharacteristic
154#417—21no—
155C2×#103—22no—
156C2×C4×C8C22×C311yescharacteristic
157#333C22×C314yes—
158Q8⋊C8C22×C314yes—
159Q8⋊C8C22×C312yes—
160D4⋊C8C22×C312yes—
161C3×#41—22no—
162C3×#41Q1614yescomplemented
163C2×C3×Q8—24no—
164C2×C3×Q8—22no—
165C2×C3×Q8—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—44no—
168C2×C3×Q8—41no—
169C2×C3×Q8—82no—
170C2×C3×Q8—84no—
171C2×C3×Q8C2×D412yes—
172C2×C3×Q8C2×D411yescharacteristic
173C2×C3×Q8C2×D414yes—
174C2×C3×D4—21no—
175C2×C3×D4C2×D411yescharacteristic
176C22×C3×C4—21no—
177C22×C3×C4—24no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—41no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C3×Q16—84no—
186C3×M16—42no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—42no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
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—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—41no—
209C3×C4⋊C4—41no—
210C3×C4⋊C4C2×D412yes—
211C3×C4⋊C4C2×D411yescharacteristic
212C3×C4⋊C4C2×D411yescharacteristic
213C3×C22⋊C4—24no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—21no—
217C3×C22⋊C4—21no—
218C3×C42—24no—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—22no—
223C3×C42C2×D411yescharacteristic
224C3×C42C2×D411yescharacteristic
225C3×C42C2×D411yescharacteristic
226C3×C42C2411yescharacteristic
227C2×#41C3×D411yescharacteristic
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8C3×D412yes—
231C2×Q16—24no—
232C2×M16—21no—
233C22×C8—21no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—24no—
237C4⋊Q8—22no—
238C4⋊Q8—22no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C3×D412yes—
245C4⋊Q8C3×D411yescharacteristic
246C4⋊Q8C3×D411yescharacteristic
247C4⋊Q8C23×C312yes—
248C4⋊Q8C23×C312yes—
249#125—22no—
250#125—24no—
251#125—22no—
252#125—21no—
253C22⋊Q8—22no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—22no—
257C22⋊Q8—24no—
258C22⋊Q8—21no—
259C22⋊Q8C3×D412yes—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—21no—
264C4×Q8C3×D411yescharacteristic
265C4×Q8C23×C311yescharacteristic
266C4×Q8C23×C312yes—
267C4×D4—24no—
268C4×D4—22no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C3×D411yescharacteristic
272C4×D4C23×C311yescharacteristic
273#117—22no—
274#117—21no—
275#117—21no—
276#117C3×D411yescharacteristic
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C42—21no—
284C2×C42C23×C311yescharacteristic
285#107—22no—
286#106—22no—
287C4⋊C8—21no—
288C4⋊C8C23×C312yes—
289C4⋊C8C23×C312yes—
290#103—28no—
291#103—22no—
292#103—24no—
293#102—22no—
294C22⋊C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8C23×C312yes—
298C23×C3—21no—
299C23×C3#12011yescharacteristic
300C3×Q8—21no—
301C3×Q8—24no—
302C3×Q8—44no—
303C3×Q8—44no—
304C3×Q8—42no—
305C3×Q8—44no—
306C3×Q8—82no—
307C3×Q8—84no—
308C3×Q8C2×Q1612yes—
309C3×D4—24no—
310C3×D4—21no—
311C3×D4C2×Q1612yes—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—82no—
325C2×C3×C4—84no—
326C2×C3×C4—82no—
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4C2×Q1612yes—
331C2×C3×C4#13712yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C22×D411yescharacteristic
335C3×C8—42no—
336C3×C8—42no—
337C3×C8—42no—
338#41—22no—
339#41C3×Q1614yescomplemented
340C2×Q8—24no—
341C2×Q8—22no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—44no—
345C2×Q8—41no—
346C2×Q8—82no—
347C2×Q8—84no—
348C2×Q8C2×C3×D412yes—
349C2×Q8C2×C3×D411yescharacteristic
350C2×Q8C2×C3×D414yes—
351C2×D4—21no—
352C2×D4C2×C3×D411yescharacteristic
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D411yescharacteristic
362Q16—84no—
363M16—42no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—42no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4C2×C3×D412yes—
388C4⋊C4C2×C3×D411yescharacteristic
389C4⋊C4C2×C3×D411yescharacteristic
390C22⋊C4—24no—
391C22⋊C4—22no—
392C22⋊C4—21no—
393C22⋊C4—21no—
394C22⋊C4—21no—
395C42—24no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—22no—
400C42C2×C3×D411yescharacteristic
401C42C2×C3×D411yescharacteristic
402C42C2×C3×D411yescharacteristic
403C42C24×C311yescharacteristic
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—22no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3#45112yes—
410C22×C3C2×#12011yescharacteristic
411C3×C4—22no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—41no—
415C3×C4—42no—
416C3×C4—41no—
417C3×C4—44no—
418C3×C4—42no—
419C3×C4—82no—
420C3×C4#45112yes—
421C3×C4#45512yes—
422C3×C4C22×Q1611yescharacteristic
423C3×C4C2×#13711yescharacteristic
424C3×C4#57712yes—
425C23—21no—
426C23C3×#12011yescharacteristic
427Q8—21no—
428Q8—24no—
429Q8—44no—
430Q8—44no—
431Q8—42no—
432Q8—44no—
433Q8—82no—
434Q8—84no—
435Q8C2×C3×Q1612yes—
436D4—24no—
437D4—21no—
438D4C2×C3×Q1612yes—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—82no—
452C2×C4—84no—
453C2×C4—82no—
454C2×C4C3×#12011yescharacteristic
455C2×C4C3×#12011yescharacteristic
456C2×C4C3×#12011yescharacteristic
457C2×C4C2×C3×Q1612yes—
458C2×C4C3×#13712yes—
459C2×C4C22×C3×D411yescharacteristic
460C2×C4C22×C3×D411yescharacteristic
461C2×C4C22×C3×D411yescharacteristic
462C8—42no—
463C8—42no—
464C8—42no—
465C2×C3—22no—
466C2×C3—42no—
467C2×C3C2×#45111yescharacteristic
468C2×C3#301011yescharacteristic
469C2×C3C2×#45511yescharacteristic
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22C3×#45112yes—
476C22C2×C3×#12011yescharacteristic
477C4—22no—
478C4—21no—
479C4—21no—
480C4—41no—
481C4—42no—
482C4—41no—
483C4—44no—
484C4—42no—
485C4—82no—
486C4C3×#45112yes—
487C4C3×#45512yes—
488C4C22×C3×Q1611yescharacteristic
489C4C2×C3×#13711yescharacteristic
490C4C3×#57712yes—
491C3#1863311yescharacteristic, direct product factor, Sylow
492C2—22no—
493C2—42no—
494C2C2×C3×#45111yescharacteristic
495C2C3×#301011yescharacteristic
496C2C2×C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC2×C3×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#3209C2semidirect
3. C2×C3×#487C2semidirect
5. C3×#1645C2semidirect
6. C3×#1644C2semidirect
7. C3×#1548C2semidirect
8. C3×#1493C2semidirect
10. #18633C3direct (N × H)
47. #3209C2×C3semidirect
48. C2×#487C2×C3semidirect
50. #1645C2×C3semidirect
51. #1644C2×C3semidirect
52. #1548C2×C3semidirect
53. #1493C2×C3semidirect
162. C3×#41Q16semidirect
339. #41C3×Q16semidirect
491. C3#18633direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35032
C3×#32092
C2×C3×#4872
C3×#30792
C3×#16452
C3×#16442
C3×#15482
C3×#14932
C3×#14762
#186333