Finite Group Databasebeta

← C3×#33046 #10651653

Subgroups — C3×#33046 #10651653

Order 768.

830 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#487C211yesmaximal, characteristic, complemented
2C2×C3×#405C211yesmaximal, characteristic
3C3×#2628C211yesmaximal, characteristic
4C3×#2621C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6C3×#1960C211yesmaximal, characteristic, complemented
7C3×#1935C212yesmaximal, complemented
8C3×#1933C212yesmaximal, complemented
9C3×#1894C212yesmaximal
10C3×#1892C212yesmaximal
11C3×#1835C211yesmaximal, characteristic, complemented
12#33046C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×#117C2211yescharacteristic
19C3×#487—24no—
20C2×C3×#103C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C3×#405—24no—
25C2×C3×#97C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C3×#400—21no—
30C3×#399—21no—
31C3×#398—21no—
32C3×#398—21no—
33C3×#397—21no—
34C3×#397—21no—
35C3×#395—21no—
36C3×#393—21no—
37C3×#391C2212yes—
38C3×#389C2212yes—
39C3×#382C2211yescharacteristic
40C3×#382C2211yescharacteristic
41C3×#380C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#340C2212yes—
46C2×#487C2×C311yescharacteristic, complemented
47C2×#405C2×C311yescharacteristic
48#2628C2×C311yescharacteristic
49#2621C2×C311yescharacteristic
50#1993C2×C311yescharacteristic, complemented
51#1960C2×C311yescharacteristic, complemented
52#1935C2×C312yescomplemented
53#1933C2×C312yescomplemented
54#1894C2×C312yes—
55#1892C2×C312yes—
56#1835C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C23×C3×C4C2311yescharacteristic
60C2×C3×M16—22no—
61C2×C3×M16—22no—
62C2×C3×M16—22no—
63C22×C3×C8—22no—
64C22×C3×C8—22no—
65C22×C3×C8—22no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8—22no—
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—24no—
76C3×#117—22no—
77C3×#117—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D411yescharacteristic
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×#103—24no—
95C3×#103—24no—
96C3×#103—24no—
97C3×#103—24no—
98C3×#97—22no—
99C3×#97—22no—
100C3×C4×C8—22no—
101C3×C4×C8D414yes—
102C3×#95—22no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C3×#95—21no—
107C3×#95—22no—
108C3×#95—22no—
109C3×#95—22no—
110C22×M16C22×C311yescharacteristic
111C23×C8C22×C311yescharacteristic
112C2×C4⋊Q8C22×C311yescharacteristic
113C2×C4⋊Q8C22×C311yescharacteristic
114C2×C22⋊Q8C22×C312yes—
115C2×#117C22×C311yescharacteristic
116#487—24no—
117C2×#103C22×C312yes—
118C2×#103C22×C312yes—
119C2×#103C22×C312yes—
120C2×#103C22×C312yes—
121#405—24no—
122C2×#97C22×C311yescharacteristic
123C2×#97C22×C311yescharacteristic
124C2×C4×C8C22×C311yescharacteristic
125C2×C4×C8C22×C311yescharacteristic
126#400—21no—
127#399—21no—
128#398—21no—
129#398—21no—
130#397—21no—
131#397—21no—
132#395—21no—
133#393—21no—
134#391C22×C312yes—
135#389C22×C312yes—
136#382C22×C311yescharacteristic
137#382C22×C311yescharacteristic
138#380C22×C312yes—
139#340C22×C312yes—
140#340C22×C312yes—
141#340C22×C312yes—
142#340C22×C312yes—
143C24×C3C2×D411yescharacteristic
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—42no—
151C2×C3×Q8—82no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×C3×C4C2411yescharacteristic
168C3×M16—28no—
169C2×C3×C8—28no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8C2×D412yes—
179C2×C3×C8C2×D412yes—
180C2×C3×C8#4112yes—
181C2×C3×C8#4112yes—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—44no—
194C3×C4⋊C4—82no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—82no—
199C3×C42—21no—
200C3×C42—21no—
201C3×C42C2×D412yes—
202C3×C42#4112yes—
203C22×Q8—21no—
204C22×Q8C23×C312yes—
205C23×C4C23×C311yescharacteristic
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8C23×C311yescharacteristic
213C22×C8C23×C311yescharacteristic
214C22×C8C23×C311yescharacteristic
215C22×C8C23×C311yescharacteristic
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222#117—22no—
223#117—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4C3×D411yescharacteristic
230C2×C4⋊C4C3×D411yescharacteristic
231C2×C4⋊C4C23×C312yes—
232C2×C4⋊C4C23×C312yes—
233C2×C4⋊C4C23×C312yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4C3×D411yescharacteristic
237C2×C22⋊C4C3×D411yescharacteristic
238C2×C42C23×C311yescharacteristic
239C2×C42C23×C311yescharacteristic
240#103—24no—
241#103—24no—
242#103—24no—
243#103—24no—
244#97—22no—
245#97—22no—
246C4×C8—22no—
247C4×C8C3×D414yes—
248#95—22no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95—22no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3#12412yes—
261C23×C3C22×D411yescharacteristic
262C3×Q8—42no—
263C3×Q8—42no—
264C3×Q8—42no—
265C3×Q8—42no—
266C3×Q8—84no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4#12112yes—
292C2×C3×C4#12112yes—
293C2×C3×C4#12412yes—
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C2×#4111yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C3×C8—22no—
301C3×C8—22no—
302C3×C8—22no—
303C3×C8—22no—
304C24C2×C3×D411yescharacteristic
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—42no—
312C2×Q8—82no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C3×#4112yes—
327C22×C4C3×#4112yes—
328C22×C4C24×C311yescharacteristic
329M16—28no—
330C2×C8—28no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8C2×C3×D412yes—
340C2×C8C2×C3×D412yes—
341C2×C8C3×#4112yes—
342C2×C8C3×#4112yes—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—44no—
355C4⋊C4—82no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—44no—
359C22⋊C4—82no—
360C42—21no—
361C42—21no—
362C42C2×C3×D412yes—
363C42C3×#4112yes—
364C22×C3—22no—
365C22×C3—24no—
366C22×C3—42no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3C2×#12111yescharacteristic
370C22×C3C2×#12411yescharacteristic
371C22×C3#53211yescharacteristic
372C22×C3#57712yes—
373C22×C3#57812yes—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—82no—
380C3×C4—82no—
381C3×C4—82no—
382C3×C4—82no—
383C3×C4#52511yescharacteristic
384C3×C4#52511yescharacteristic
385C3×C4#52911yescharacteristic
386C3×C4#52911yescharacteristic
387C23—22no—
388C23—22no—
389C23—21no—
390C23—21no—
391C23C3×#12412yes—
392C23C22×C3×D411yescharacteristic
393Q8—42no—
394Q8—42no—
395Q8—42no—
396Q8—42no—
397Q8—84no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4C3×#12112yes—
423C2×C4C3×#12112yes—
424C2×C4C3×#12412yes—
425C2×C4C22×C3×D411yescharacteristic
426C2×C4C2×C3×#4111yescharacteristic
427C2×C4C2×C3×#4111yescharacteristic
428C2×C4C2×C3×#4111yescharacteristic
429C2×C4C2×C3×#4111yescharacteristic
430C2×C4C2×C3×#4111yescharacteristic
431C8—22no—
432C8—22no—
433C8—22no—
434C8—22no—
435C2×C3—42no—
436C2×C3#285011yescharacteristic
437C2×C3#305711yescharacteristic
438C2×C3#305911yescharacteristic
439C2×C3#313712yes—
440C2×C3#317812yes—
441C22—24no—
442C22—22no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22C2×C3×#12111yescharacteristic
447C22C2×C3×#12411yescharacteristic
448C22C3×#53211yescharacteristic
449C22C3×#57712yes—
450C22C3×#57812yes—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—82no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4C3×#52511yescharacteristic
461C4C3×#52511yescharacteristic
462C4C3×#52911yescharacteristic
463C4C3×#52911yescharacteristic
464C3#3304611yescharacteristic, direct product factor, Sylow
465C2—42no—
466C2C3×#285011yescharacteristic
467C2C3×#305711yescharacteristic
468C2C3×#305911yescharacteristic
469C2C3×#313712yes—
470C2C3×#317812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#487C2semidirect
5. C3×#1993C2semidirect
6. C3×#1960C2semidirect
7. C3×#1935C2semidirect
8. C3×#1933C2semidirect
11. C3×#1835C2semidirect
12. #33046C3direct (N × H)
46. C2×#487C2×C3semidirect
50. #1993C2×C3semidirect
51. #1960C2×C3semidirect
52. #1935C2×C3semidirect
53. #1933C2×C3semidirect
56. #1835C2×C3semidirect
464. C3#33046direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4872
C2×C3×#4052
C3×#26282
C3×#26212
C3×#19932
C3×#19602
C3×#19352
C3×#19332
C3×#18942
C3×#18922
C3×#18352
#330463