Finite Group Databasebeta

← C3×#18632 #10637239

Subgroups — C3×#18632 #10637239

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3503C211yesmaximal, characteristic
2C3×#3229C211yesmaximal, characteristic, complemented
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C3×#3039C211yesmaximal, characteristic
5C3×#1645C214yesmaximal, complemented
6C3×#1643C214yesmaximal, complemented
7C3×#1548C211yesmaximal, characteristic, complemented
8C3×#1493C211yesmaximal, characteristic, complemented
9C3×#1476C211yesmaximal, characteristic
10#18632C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#137—21no—
12C3×Q82C2212yes—
13C3×#557—21no—
14C3×#554—22no—
15C3×#554C2212yes—
16C3×D4×Q8—22no—
17C3×D4×Q8—21no—
18C3×#544—22no—
19C3×#544—21no—
20C3×#533—21no—
21C3×#533C2211yescharacteristic
22C2×C3×C4⋊Q8—21no—
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#520—21no—
26C3×C4×#41C2211yescharacteristic
27C3×#498—21no—
28C3×#487—22no—
29C3×#487—21no—
30C3×#487C2214yes—
31C3×#483—22no—
32C3×#475C2214yes—
33C3×#462—21no—
34C3×#462C2212yes—
35C3×#461—21no—
36C3×#461C2212yes—
37C3×#431C2211yescharacteristic
38C3×#423C2211yescharacteristic
39C3×#419—21no—
40C2×C3×#103—22no—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#333C2214yes—
43C3×Q8⋊C8C2214yes—
44C3×Q8⋊C8C2212yes—
45C3×D4⋊C8C2212yes—
46#3503C2×C311yescharacteristic
47#3229C2×C311yescharacteristic, complemented
48C2×#487C2×C311yescharacteristic, complemented
49#3039C2×C311yescharacteristic
50#1645C2×C314yescomplemented
51#1643C2×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—
59C3×#137—44no—
60C2×C3×Q16—22no—
61C2×C3×SD16—22no—
62C2×C3×M16—21no—
63C22×C3×C8—21no—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—24no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8D412yes—
75C3×C4⋊Q8D411yescharacteristic
76C3×C4⋊Q8D411yescharacteristic
77C3×C4⋊Q8C2312yes—
78C3×C4⋊Q8C2312yes—
79C3×#125—22no—
80C3×#125—24no—
81C3×#125—22no—
82C3×#125—21no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—24no—
86C3×C22⋊Q8—22no—
87C3×C22⋊Q8—24no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8D412yes—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—21no—
94C3×C4×Q8D411yescharacteristic
95C3×C4×Q8C2311yescharacteristic
96C3×C4×Q8C2312yes—
97C3×C4×D4—24no—
98C3×C4×D4—22no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4D411yescharacteristic
102C3×C4×D4C2311yescharacteristic
103C3×#117—22no—
104C3×#117—21no—
105C3×#117—21no—
106C3×#117D411yescharacteristic
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C3×#106—24no—
116C3×C4⋊C8—21no—
117C3×C4⋊C8C2312yes—
118C3×C4⋊C8C2312yes—
119C3×#103—28no—
120C3×#103—24no—
121C3×#103—22no—
122C3×#103—22no—
123C3×C22⋊C8—21no—
124C3×C4×C8—21no—
125C3×C4×C8—21no—
126C3×C4×C8C2312yes—
127C2×#137—21no—
128Q82C22×C312yes—
129#557—21no—
130#554—22no—
131#554C22×C312yes—
132D4×Q8—22no—
133D4×Q8—21no—
134#544—22no—
135#544—21no—
136#533—21no—
137#533C22×C311yescharacteristic
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8C22×C311yescharacteristic
140C2×C4⋊Q8C22×C311yescharacteristic
141#520—21no—
142C4×#41C22×C311yescharacteristic
143#498—21no—
144#487—22no—
145#487—21no—
146#487C22×C314yes—
147#483—22no—
148#475C22×C314yes—
149#462—21no—
150#462C22×C312yes—
151#461—21no—
152#461C22×C312yes—
153#431C22×C311yescharacteristic
154#423C22×C311yescharacteristic
155#419—21no—
156C2×#103—22no—
157C2×C4×C8C22×C311yescharacteristic
158#333C22×C314yes—
159Q8⋊C8C22×C314yes—
160Q8⋊C8C22×C312yes—
161D4⋊C8C22×C312yes—
162C3×#41—22no—
163C3×#41SD1614yes—
164C2×C3×Q8—24no—
165C2×C3×Q8—22no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—44no—
169C2×C3×Q8—41no—
170C2×C3×Q8—42no—
171C2×C3×Q8—42no—
172C2×C3×Q8—84no—
173C2×C3×Q8C2×D412yes—
174C2×C3×Q8C2×D411yescharacteristic
175C2×C3×Q8C2×D414yes—
176C2×C3×D4—21no—
177C2×C3×D4C2×D411yescharacteristic
178C22×C3×C4—21no—
179C22×C3×C4—24no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—41no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C3×Q16—44no—
188C3×SD16—44no—
189C3×M16—42no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—42no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
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×C4⋊C4C2×D412yes—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—22no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C42—24no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—22no—
226C3×C42C2×D411yescharacteristic
227C3×C42C2×D411yescharacteristic
228C3×C42C2×D411yescharacteristic
229C3×C42C2411yescharacteristic
230C2×#41C3×D411yescharacteristic
231C22×Q8—22no—
232C22×Q8—21no—
233C22×Q8C3×D412yes—
234#137—44no—
235C2×Q16—22no—
236C2×SD16—22no—
237C2×M16—21no—
238C22×C8—21no—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—24no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C3×D412yes—
250C4⋊Q8C3×D411yescharacteristic
251C4⋊Q8C3×D411yescharacteristic
252C4⋊Q8C23×C312yes—
253C4⋊Q8C23×C312yes—
254#125—22no—
255#125—24no—
256#125—22no—
257#125—21no—
258C22⋊Q8—22no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—22no—
262C22⋊Q8—24no—
263C22⋊Q8—21no—
264C22⋊Q8C3×D412yes—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—21no—
269C4×Q8C3×D411yescharacteristic
270C4×Q8C23×C311yescharacteristic
271C4×Q8C23×C312yes—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4C3×D411yescharacteristic
277C4×D4C23×C311yescharacteristic
278#117—22no—
279#117—21no—
280#117—21no—
281#117C3×D411yescharacteristic
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C42—21no—
289C2×C42C23×C311yescharacteristic
290#106—24no—
291C4⋊C8—21no—
292C4⋊C8C23×C312yes—
293C4⋊C8C23×C312yes—
294#103—28no—
295#103—24no—
296#103—22no—
297#103—22no—
298C22⋊C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8C23×C312yes—
302C23×C3—21no—
303C23×C3#12011yescharacteristic
304C3×Q8—21no—
305C3×Q8—24no—
306C3×Q8—44no—
307C3×Q8—44no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×Q8—42no—
311C3×Q8—84no—
312C3×Q8—82no—
313C3×Q8C2×SD1612yes—
314C3×D4—24no—
315C3×D4—21no—
316C3×D4C2×SD1612yes—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—82no—
330C2×C3×C4—84no—
331C2×C3×C4—82no—
332C2×C3×C4#12011yescharacteristic
333C2×C3×C4#12011yescharacteristic
334C2×C3×C4#12011yescharacteristic
335C2×C3×C4C2×SD1612yes—
336C2×C3×C4#13712yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C3×C8—42no—
341C3×C8—42no—
342C3×C8—42no—
343#41—22no—
344#41C3×SD1614yes—
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—44no—
350C2×Q8—41no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—84no—
354C2×Q8C2×C3×D412yes—
355C2×Q8C2×C3×D411yescharacteristic
356C2×Q8C2×C3×D414yes—
357C2×D4—21no—
358C2×D4C2×C3×D411yescharacteristic
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—41no—
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D411yescharacteristic
368Q16—44no—
369SD16—44no—
370M16—42no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—42no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4C2×C3×D411yescharacteristic
395C4⋊C4C2×C3×D411yescharacteristic
396C4⋊C4C2×C3×D412yes—
397C22⋊C4—24no—
398C22⋊C4—22no—
399C22⋊C4—21no—
400C22⋊C4—21no—
401C22⋊C4—21no—
402C42—24no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—22no—
407C42C2×C3×D411yescharacteristic
408C42C2×C3×D411yescharacteristic
409C42C2×C3×D411yescharacteristic
410C42C24×C311yescharacteristic
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—22no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3#44812yes—
417C22×C3C2×#12011yescharacteristic
418C3×C4—22no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—41no—
422C3×C4—42no—
423C3×C4—41no—
424C3×C4—44no—
425C3×C4—42no—
426C3×C4—82no—
427C3×C4#44812yes—
428C3×C4#45512yes—
429C3×C4C22×SD1611yescharacteristic
430C3×C4C2×#13711yescharacteristic
431C3×C4#57812yes—
432C23—21no—
433C23C3×#12011yescharacteristic
434Q8—21no—
435Q8—24no—
436Q8—44no—
437Q8—44no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—84no—
442Q8—82no—
443Q8C2×C3×SD1612yes—
444D4—24no—
445D4—21no—
446D4C2×C3×SD1612yes—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4C3×#12011yescharacteristic
463C2×C4C3×#12011yescharacteristic
464C2×C4C3×#12011yescharacteristic
465C2×C4C2×C3×SD1612yes—
466C2×C4C3×#13712yes—
467C2×C4C22×C3×D411yescharacteristic
468C2×C4C22×C3×D411yescharacteristic
469C2×C4C22×C3×D411yescharacteristic
470C8—42no—
471C8—42no—
472C8—42no—
473C2×C3—22no—
474C2×C3—42no—
475C2×C3C2×#44811yescharacteristic
476C2×C3#301111yescharacteristic
477C2×C3C2×#45511yescharacteristic
478C22—21no—
479C22—21no—
480C22—22no—
481C22—41no—
482C22—41no—
483C22C3×#44812yes—
484C22C2×C3×#12011yescharacteristic
485C4—22no—
486C4—21no—
487C4—21no—
488C4—41no—
489C4—42no—
490C4—41no—
491C4—44no—
492C4—42no—
493C4—82no—
494C4C3×#44812yes—
495C4C3×#45512yes—
496C4C22×C3×SD1611yescharacteristic
497C4C2×C3×#13711yescharacteristic
498C4C3×#57812yes—
499C3#1863211yescharacteristic, direct product factor, Sylow
500C2—22no—
501C2—42no—
502C2C2×C3×#44811yescharacteristic
503C2C3×#301111yescharacteristic
504C2C2×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×#3229C2semidirect
3. C2×C3×#487C2semidirect
5. C3×#1645C2semidirect
6. C3×#1643C2semidirect
7. C3×#1548C2semidirect
8. C3×#1493C2semidirect
10. #18632C3direct (N × H)
47. #3229C2×C3semidirect
48. C2×#487C2×C3semidirect
50. #1645C2×C3semidirect
51. #1643C2×C3semidirect
52. #1548C2×C3semidirect
53. #1493C2×C3semidirect
499. C3#18632direct (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×#32292
C2×C3×#4872
C3×#30392
C3×#16452
C3×#16432
C3×#15482
C3×#14932
C3×#14762
#186323