Finite Group Databasebeta

← C3×#29699 #10648306

Subgroups — C3×#29699 #10648306

Order 768.

948 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic, complemented
3C2×C3×#475C211yesmaximal, characteristic, complemented
4C3×#2628C211yesmaximal, characteristic
5C3×#2059C212yesmaximal, complemented
6C3×#2043C212yesmaximal, complemented
7C3×#2036C212yesmaximal
8C3×#2035C212yesmaximal
9C3×#1805C211yesmaximal, characteristic, complemented
10C3×#1805C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic
12#29699C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#541—24no—
14C3×#541—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#126—21no—
18C2×C3×#126—21no—
19C2×C3×#125—21no—
20C2×C3×#125—21no—
21C2×C3×#125C2211yescharacteristic
22C2×C3×C22⋊Q8C2211yescharacteristic
23C2×C3×C22⋊Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic
27C3×#477—24no—
28C3×#475—24no—
29C2×C3×#107C2212yes—
30C2×C3×#106C2212yes—
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×C22⋊C8C2212yes—
38C3×#400—21no—
39C3×#399—21no—
40C3×#395—21no—
41C3×#393—21no—
42C3×#389C2212yes—
43C3×#382C2211yescharacteristic
44C3×#380C2212yes—
45C3×#340C2212yes—
46C3×#340C2212yes—
47C3×#336C2212yes—
48C2×#541C2×C311yescharacteristic
49C2×#477C2×C311yescharacteristic, complemented
50C2×#475C2×C311yescharacteristic, complemented
51#2628C2×C311yescharacteristic
52#2059C2×C312yescomplemented
53#2043C2×C312yescomplemented
54#2036C2×C312yes—
55#2035C2×C312yes—
56#1805C2×C311yescharacteristic, complemented
57#1805C2×C311yescharacteristic, complemented
58#1798C2×C311yescharacteristic
59C22×C3×Q8—21no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C23×C3×C4C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×#126—28no—
70C3×#126—28no—
71C3×#125—28no—
72C3×#125—28no—
73C3×#125—22no—
74C3×#125—22no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8D414yes—
82C3×C4×Q8D414yes—
83C3×#117—22no—
84C3×#117—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4D411yescharacteristic
96C2×C3×C4⋊C4D411yescharacteristic
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C22⋊C4—21no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4D411yescharacteristic
105C2×C3×C22⋊C4D411yescharacteristic
106C2×C3×C42—21no—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C3×#107—24no—
111C3×#106—24no—
112C3×C4⋊C8—22no—
113C3×C4⋊C8—22no—
114C3×#103—24no—
115C3×#103—24no—
116C3×#103—42no—
117C3×#103—42no—
118C3×C22⋊C8—42no—
119C3×#95—22no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—22no—
123#541—24no—
124#541—24no—
125C2×C4⋊Q8C22×C311yescharacteristic
126C2×C4⋊Q8C22×C311yescharacteristic
127C2×#126—21no—
128C2×#126—21no—
129C2×#125—21no—
130C2×#125—21no—
131C2×#125C22×C311yescharacteristic
132C2×C22⋊Q8C22×C311yescharacteristic
133C2×C22⋊Q8C22×C311yescharacteristic
134C2×C4×Q8C22×C311yescharacteristic
135C2×C4×Q8C22×C311yescharacteristic
136C2×#117C22×C311yescharacteristic
137#477—24no—
138#475—24no—
139C2×#107C22×C312yes—
140C2×#106C22×C312yes—
141C2×C4⋊C8C22×C311yescharacteristic
142C2×C4⋊C8C22×C311yescharacteristic
143C2×#103C22×C312yes—
144C2×#103C22×C312yes—
145C2×#103C22×C312yes—
146C2×#103C22×C312yes—
147C2×C22⋊C8C22×C312yes—
148#400—21no—
149#399—21no—
150#395—21no—
151#393—21no—
152#389C22×C312yes—
153#382C22×C311yescharacteristic
154#380C22×C312yes—
155#340C22×C312yes—
156#340C22×C312yes—
157#336C22×C312yes—
158C24×C3C2×D411yescharacteristic
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—42no—
170C2×C3×Q8—82no—
171C2×C3×Q8C2×D412yes—
172C2×C3×Q8C2×D412yes—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—82no—
215C3×C4⋊C4C2×D412yes—
216C3×C4⋊C4C2×D412yes—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—82no—
222C3×C42—24no—
223C3×C42—24no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42C2×D412yes—
227C3×C42C2×D412yes—
228C22×Q8—21no—
229C22×Q8C23×C311yescharacteristic
230C22×Q8C23×C311yescharacteristic
231C23×C4C23×C311yescharacteristic
232C22×C8C23×C312yes—
233C22×C8C23×C312yes—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238#126—28no—
239#126—28no—
240#125—28no—
241#125—28no—
242#125—22no—
243#125—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×Q8C3×D414yes—
251C4×Q8C3×D414yes—
252#117—22no—
253#117—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4C3×D411yescharacteristic
265C2×C4⋊C4C3×D411yescharacteristic
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C312yes—
268C2×C4⋊C4C23×C311yescharacteristic
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4C3×D411yescharacteristic
274C2×C22⋊C4C3×D411yescharacteristic
275C2×C42—21no—
276C2×C42—21no—
277C2×C42C23×C311yescharacteristic
278C2×C42C23×C311yescharacteristic
279#107—24no—
280#106—24no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283#103—24no—
284#103—24no—
285#103—42no—
286#103—42no—
287C22⋊C8—42no—
288#95—22no—
289#95—21no—
290#95—21no—
291#95—22no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3C22×D411yescharacteristic
298C3×Q8—22no—
299C3×Q8—22no—
300C3×Q8—22no—
301C3×Q8—22no—
302C3×Q8—41no—
303C3×Q8—41no—
304C3×Q8—41no—
305C3×Q8—41no—
306C3×Q8—84no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4#12012yes—
337C2×C3×C4#12012yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4H(5,𝔽2)11yescharacteristic
341C2×C3×C42−1+411yescharacteristic
342C2×C3×C42−1+411yescharacteristic
343C2×C3×C42−1+411yescharacteristic
344C3×C8—42no—
345C3×C8—42no—
346C24C2×C3×D411yescharacteristic
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×Q8—82no—
359C2×Q8C2×C3×D412yes—
360C2×Q8C2×C3×D412yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4C2×C3×D411yescharacteristic
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C2×C3×D411yescharacteristic
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C24×C311yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—42no—
402C4⋊C4—82no—
403C4⋊C4C2×C3×D412yes—
404C4⋊C4C2×C3×D412yes—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—82no—
410C42—24no—
411C42—24no—
412C42—21no—
413C42—21no—
414C42C2×C3×D412yes—
415C42C2×C3×D412yes—
416C22×C3—41no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3C2×#12011yescharacteristic
422C22×C3#53611yescharacteristic
423C22×C3#53711yescharacteristic
424C22×C3#57712yes—
425C22×C3#57812yes—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—41no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—41no—
432C3×C4—42no—
433C3×C4—41no—
434C3×C4—42no—
435C3×C4—82no—
436C3×C4—82no—
437C3×C4#53611yescharacteristic
438C3×C4#53611yescharacteristic
439C3×C4#53711yescharacteristic
440C3×C4#53711yescharacteristic
441C23—22no—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23C22×C3×D411yescharacteristic
447Q8—22no—
448Q8—22no—
449Q8—22no—
450Q8—22no—
451Q8—41no—
452Q8—41no—
453Q8—41no—
454Q8—41no—
455Q8—84no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4C3×#12012yes—
486C2×C4C3×#12012yes—
487C2×C4C22×C3×D411yescharacteristic
488C2×C4C22×C3×D411yescharacteristic
489C2×C4C3×H(5,𝔽2)11yescharacteristic
490C2×C4C3×2−1+411yescharacteristic
491C2×C4C3×2−1+411yescharacteristic
492C2×C4C3×2−1+411yescharacteristic
493C8—42no—
494C8—42no—
495C2×C3—81no—
496C2×C3#261611yescharacteristic
497C2×C3#301011yescharacteristic
498C2×C3#301111yescharacteristic
499C2×C3#324212yes—
500C2×C3#325812yes—
501C22—41no—
502C22—42no—
503C22—42no—
504C22—41no—
505C22—41no—
506C22C2×C3×#12011yescharacteristic
507C22C3×#53611yescharacteristic
508C22C3×#53711yescharacteristic
509C22C3×#57712yes—
510C22C3×#57812yes—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—82no—
521C4—82no—
522C4C3×#53611yescharacteristic
523C4C3×#53611yescharacteristic
524C4C3×#53711yescharacteristic
525C4C3×#53711yescharacteristic
526C3#2969911yescharacteristic, direct product factor, Sylow
527C2—81no—
528C2C3×#261611yescharacteristic
529C2C3×#301011yescharacteristic
530C2C3×#301111yescharacteristic
531C2C3×#324212yes—
532C2C3×#325812yes—
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
2. C2×C3×#477C2semidirect
3. C2×C3×#475C2semidirect
5. C3×#2059C2semidirect
6. C3×#2043C2semidirect
9. C3×#1805C2semidirect
10. C3×#1805C2semidirect
12. #29699C3direct (N × H)
49. C2×#477C2×C3semidirect
50. C2×#475C2×C3semidirect
52. #2059C2×C3semidirect
53. #2043C2×C3semidirect
56. #1805C2×C3semidirect
57. #1805C2×C3semidirect
526. C3#29699direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×C3×#4772
C2×C3×#4752
C3×#26282
C3×#20592
C3×#20432
C3×#20362
C3×#20352
C3×#18052
C3×#17982
#296993