Finite Group Databasebeta

← C3×#29746 #10648353

Subgroups — C3×#29746 #10648353

Order 768.

966 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
1C2×C3×#470C211yesmaximal, characteristic, complemented
2C2×C3×#408C211yesmaximal, characteristic
3C3×#2616C212yesmaximal
4C3×#2077C212yesmaximal, complemented
5C3×#2036C212yesmaximal, complemented
6C3×#2035C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#1892C212yesmaximal, complemented
9C3×#1809C211yesmaximal, characteristic, complemented
10#29746C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×C3×Q16C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C22⋊Q8C2212yes—
18C3×#470—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C3×#408—24no—
26C2×C3×C22⋊C8C2212yes—
27C2×C3×C22⋊C8C2212yes—
28C3×#395—22no—
29C3×#394—22no—
30C3×#394—22no—
31C3×#394—22no—
32C3×#380C2212yes—
33C3×#380C2212yes—
34C3×#380C2212yes—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#340C2212yes—
38C2×#470C2×C311yescharacteristic, complemented
39C2×#408C2×C311yescharacteristic
40#2616C2×C312yes—
41#2077C2×C312yescomplemented
42#2036C2×C312yescomplemented
43#2035C2×C312yescomplemented
44#2001C2×C312yescomplemented
45#1892C2×C312yescomplemented
46#1809C2×C311yescharacteristic, complemented
47C2×C3×#41—22no—
48C2×C3×#41—22no—
49C2×C3×#41—41no—
50C2×C3×#41—41no—
51C2×C3×#41—41no—
52C2×C3×#41—41no—
53C22×C3×Q8D411yescharacteristic
54C22×C3×Q8C2312yes—
55C22×C3×D4D412yes—
56C22×C3×D4D411yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C2×C3×Q16—22no—
60C2×C3×Q16—41no—
61C2×C3×Q16—41no—
62C2×C3×M16—41no—
63C2×C3×M16—41no—
64C2×C3×M16D414yes—
65C22×C3×C8—22no—
66C22×C3×C8—41no—
67C22×C3×C8—41no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—42no—
74C3×C22⋊Q8—42no—
75C3×C22⋊Q8—42no—
76C3×C22⋊Q8—42no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C3×#107—41no—
90C3×#106—22no—
91C3×#106—41no—
92C3×#103—24no—
93C3×#103—42no—
94C3×#103—42no—
95C3×C22⋊C8—24no—
96C3×C22⋊C8—24no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C22×#41C22×C311yescharacteristic
101C22×Q16C22×C311yescharacteristic
102C22×M16C22×C311yescharacteristic
103C23×C8C22×C311yescharacteristic
104C2×C22⋊Q8C22×C312yes—
105C2×C22⋊Q8C22×C312yes—
106C2×C22⋊Q8C22×C312yes—
107#470—24no—
108C2×#107C22×C311yescharacteristic
109C2×#106C22×C311yescharacteristic
110C2×#106C22×C311yescharacteristic
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×#103C22×C312yes—
114#408—24no—
115C2×C22⋊C8C22×C312yes—
116C2×C22⋊C8C22×C312yes—
117#395—22no—
118#394—22no—
119#394—22no—
120#394—22no—
121#380C22×C312yes—
122#380C22×C312yes—
123#380C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#340C22×C312yes—
127C24×C3C2×D412yes—
128C24×C3C2×D411yescharacteristic
129C3×#41—44no—
130C3×#41—44no—
131C3×#41—44no—
132C3×#41—44no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—41no—
141C2×C3×Q8—41no—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4—41no—
145C2×C3×D4—41no—
146C2×C3×D4—42no—
147C2×C3×D4—42no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4#4112yes—
166C22×C3×C4C2411yescharacteristic
167C3×Q16—44no—
168C3×M16—44no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—44no—
177C2×C3×C8C2×D412yes—
178C2×C3×C8C2×D412yes—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—44no—
186C3×C4⋊C4—44no—
187C3×C4⋊C4—44no—
188C3×C22⋊C4—44no—
189C3×C22⋊C4—44no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—82no—
192C3×C22⋊C4—82no—
193C3×C22⋊C4—82no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—41no—
197C2×#41—41no—
198C2×#41—41no—
199C2×#41—41no—
200C22×Q8C3×D411yescharacteristic
201C22×Q8C23×C312yes—
202C22×D4C3×D412yes—
203C22×D4C3×D411yescharacteristic
204C23×C4C23×C312yes—
205C23×C4C23×C311yescharacteristic
206C2×Q16—22no—
207C2×Q16—41no—
208C2×Q16—41no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16C3×D414yes—
212C22×C8—22no—
213C22×C8—41no—
214C22×C8—41no—
215C22×C8C23×C312yes—
216C22×C8C23×C311yescharacteristic
217C22×C8C23×C311yescharacteristic
218C22⋊Q8—24no—
219C22⋊Q8—24no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223C22⋊Q8—42no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4C23×C312yes—
228C2×C4⋊C4C23×C312yes—
229C2×C4⋊C4C23×C312yes—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236#107—41no—
237#106—22no—
238#106—41no—
239#103—24no—
240#103—42no—
241#103—42no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244#95—22no—
245#95—22no—
246#95—22no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—22no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3#12112yes—
259C23×C3C22×D411yescharacteristic
260C3×Q8—44no—
261C3×Q8—42no—
262C3×Q8—42no—
263C3×Q8—42no—
264C3×Q8—42no—
265C3×D4—44no—
266C3×D4—48no—
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—22no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
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—42no—
292C2×C3×C4#12112yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4H(5,𝔽2)11yescharacteristic
296C2×C3×C4H(5,𝔽2)11yescharacteristic
297C2×C3×C4H(5,𝔽2)11yescharacteristic
298C2×C3×C4H(5,𝔽2)11yescharacteristic
299C3×C8—22no—
300C3×C8—22no—
301C3×C8—42no—
302C24C2×C3×D412yes—
303C24C2×C3×D411yescharacteristic
304#41—44no—
305#41—44no—
306#41—44no—
307#41—44no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D412yes—
340C22×C4C3×#4112yes—
341C22×C4C24×C311yescharacteristic
342Q16—44no—
343M16—44no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—44no—
352C2×C8C2×C3×D412yes—
353C2×C8C2×C3×D412yes—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22×C3—22no—
370C22×C3—24no—
371C22×C3—42no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3C2×#12111yescharacteristic
382C22×C3#53411yescharacteristic
383C22×C3#53411yescharacteristic
384C22×C3#57712yes—
385C22×C3#57812yes—
386C3×C4—22no—
387C3×C4—42no—
388C3×C4—82no—
389C3×C4—82no—
390C3×C4—82no—
391C3×C4—82no—
392C3×C4#53411yescharacteristic
393C3×C4#53511yescharacteristic
394C3×C4#53811yescharacteristic
395C3×C4#53811yescharacteristic
396C23—22no—
397C23—22no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23C3×#12112yes—
408C23C22×C3×D411yescharacteristic
409Q8—44no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414D4—44no—
415D4—48no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4C3×#12112yes—
442C2×C4C22×C3×D411yescharacteristic
443C2×C4C2×C3×#4111yescharacteristic
444C2×C4C3×H(5,𝔽2)11yescharacteristic
445C2×C4C3×H(5,𝔽2)11yescharacteristic
446C2×C4C3×H(5,𝔽2)11yescharacteristic
447C2×C4C3×H(5,𝔽2)11yescharacteristic
448C8—22no—
449C8—22no—
450C8—42no—
451C2×C3—42no—
452C2×C3—82no—
453C2×C3#261711yescharacteristic
454C2×C3#306312yes—
455C2×C3#320112yes—
456C2×C3#320211yescharacteristic
457C2×C3#320411yescharacteristic
458C22—22no—
459C22—24no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22C2×C3×#12111yescharacteristic
471C22C3×#53411yescharacteristic
472C22C3×#53411yescharacteristic
473C22C3×#57712yes—
474C22C3×#57812yes—
475C4—22no—
476C4—42no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—82no—
481C4C3×#53411yescharacteristic
482C4C3×#53511yescharacteristic
483C4C3×#53811yescharacteristic
484C4C3×#53811yescharacteristic
485C3#2974611yescharacteristic, direct product factor, Sylow
486C2—42no—
487C2—82no—
488C2C3×#261711yescharacteristic
489C2C3×#306312yes—
490C2C3×#320112yes—
491C2C3×#320211yescharacteristic
492C2C3×#320411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#470C2semidirect
4. C3×#2077C2semidirect
5. C3×#2036C2semidirect
6. C3×#2035C2semidirect
7. C3×#2001C2semidirect
8. C3×#1892C2semidirect
9. C3×#1809C2semidirect
10. #29746C3direct (N × H)
38. C2×#470C2×C3semidirect
41. #2077C2×C3semidirect
42. #2036C2×C3semidirect
43. #2035C2×C3semidirect
44. #2001C2×C3semidirect
45. #1892C2×C3semidirect
46. #1809C2×C3semidirect
485. C3#29746direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4702
C2×C3×#4082
C3×#26162
C3×#20772
C3×#20362
C3×#20352
C3×#20012
C3×#18922
C3×#18092
#297463