Finite Group Databasebeta

← C3×#24191 #10642798

Subgroups — C3×#24191 #10642798

Order 768.

1050 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic
3C2×C3×C4×Q16C211yesmaximal, characteristic
4C3×#2582C211yesmaximal, characteristic
5C3×#2043C212yesmaximal
6C3×#2016C212yesmaximal
7C3×#1984C211yesmaximal, characteristic
8C3×#1956C211yesmaximal, characteristic
9C3×#1878C212yesmaximal
10C3×#1863C212yesmaximal
11C3×#1803C211yesmaximal, characteristic
12#24191C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C3×#557—24no—
16C3×#557—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125—21no—
20C2×C3×#125—22no—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×Q8C2211yescharacteristic
26C3×#462—24no—
27C3×C4×Q16—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#103C2211yescharacteristic
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×C4×C8C2211yescharacteristic
37C3×#398—22no—
38C3×#391—21no—
39C3×#391C2212yes—
40C3×#389—21no—
41C3×#389C2212yes—
42C3×#384C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#557C2×C311yescharacteristic
47C2×#462C2×C311yescharacteristic
48C2×C4×Q16C2×C311yescharacteristic
49#2582C2×C311yescharacteristic
50#2043C2×C312yes—
51#2016C2×C312yes—
52#1984C2×C311yescharacteristic
53#1956C2×C311yescharacteristic
54#1878C2×C312yes—
55#1863C2×C312yes—
56#1803C2×C311yescharacteristic
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C2×C3×Q16—24no—
64C2×C3×Q16—24no—
65C2×C3×Q16—22no—
66C2×C3×Q16—42no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×C4⋊Q8—24no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8D414yes—
74C3×#125—216no—
75C3×#125—28no—
76C3×C4×Q8—28no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8D414yes—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
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⋊C4D412yes—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C42—22no—
101C2×C3×C42—21no—
102C2×C3×C42—21no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×#107—22no—
106C3×C4⋊C8—22no—
107C3×#103—22no—
108C3×#103—24no—
109C3×#103—24no—
110C3×#103—22no—
111C3×#103—42no—
112C3×#103—42no—
113C3×C4×C8—22no—
114C3×#95—22no—
115C3×#95—22no—
116C3×#95—22no—
117C22×Q16C22×C311yescharacteristic
118C22×Q16C22×C312yes—
119#557—24no—
120#557—24no—
121C2×C4⋊Q8C22×C312yes—
122C2×C4⋊Q8C22×C311yescharacteristic
123C2×#125—21no—
124C2×#125—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8C22×C311yescharacteristic
127C2×C4×Q8C22×C312yes—
128C2×C4×Q8C22×C311yescharacteristic
129C2×C4×Q8C22×C311yescharacteristic
130#462—24no—
131C4×Q16—24no—
132C2×#107C22×C311yescharacteristic
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C311yescharacteristic
137C2×#103C22×C311yescharacteristic
138C2×#103C22×C312yes—
139C2×#103C22×C312yes—
140C2×C4×C8C22×C311yescharacteristic
141#398—22no—
142#391—21no—
143#391C22×C312yes—
144#389—21no—
145#389C22×C312yes—
146#384C22×C311yescharacteristic
147#340C22×C312yes—
148#336C22×C312yes—
149#336C22×C312yes—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—22no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8Q1614yescomplemented
168C2×C3×Q8C2×D414yes—
169C2×C3×Q8C2×D412yes—
170C2×C3×Q8#4112yes—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C3×Q16—28no—
183C3×Q16—48no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8#4112yes—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
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—42no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4—81no—
209C3×C4⋊C4—82no—
210C3×C4⋊C4C2×D412yes—
211C3×C42—28no—
212C3×C42—24no—
213C3×C42—22no—
214C3×C42—22no—
215C3×C42—42no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218C22×Q8C3×D411yescharacteristic
219C22×Q8C23×C312yes—
220C22×Q8C23×C311yescharacteristic
221C22×Q8C23×C311yescharacteristic
222C2×Q16—22no—
223C2×Q16—22no—
224C2×Q16—24no—
225C2×Q16—24no—
226C2×Q16—22no—
227C2×Q16—42no—
228C22×C8C23×C312yes—
229C22×C8C23×C311yescharacteristic
230C22×C8C23×C311yescharacteristic
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—22no—
234C4⋊Q8C3×D414yes—
235#125—216no—
236#125—28no—
237C4×Q8—28no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8C3×D414yes—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C3×D411yescharacteristic
257C2×C4⋊C4C3×D412yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C312yes—
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42C23×C312yes—
265C2×C42C23×C311yescharacteristic
266#107—22no—
267C4⋊C8—22no—
268#103—22no—
269#103—24no—
270#103—24no—
271#103—22no—
272#103—42no—
273#103—42no—
274C4×C8—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278C23×C3C22×D411yescharacteristic
279C3×Q8—28no—
280C3×Q8—22no—
281C3×Q8—22no—
282C3×Q8—22no—
283C3×Q8—22no—
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—42no—
287C3×Q8—42no—
288C3×Q8—41no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×Q8—41no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4#12012yes—
318C2×C3×C4#12012yes—
319C2×C3×C4C2×Q1614yes—
320C2×C3×C4C2×Q1612yes—
321C2×C3×C4#13712yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C2×C3×C42−1+411yescharacteristic
328C3×C8—22no—
329C3×C8—41no—
330C3×C8—42no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8C3×Q1614yescomplemented
349C2×Q8C2×C3×D414yes—
350C2×Q8C2×C3×D412yes—
351C2×Q8C3×#4112yes—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D412yes—
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C24×C311yescharacteristic
363Q16—28no—
364Q16—48no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8C3×#4112yes—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—81no—
390C4⋊C4—82no—
391C4⋊C4C2×C3×D412yes—
392C42—28no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42—42no—
397C42C2×C3×D412yes—
398C42C2×C3×D412yes—
399C22×C3C2×#12011yescharacteristic
400C22×C3#54811yescharacteristic
401C22×C3#55011yescharacteristic
402C22×C3C22×Q1611yescharacteristic
403C22×C3C2×#13711yescharacteristic
404C22×C3#57711yescharacteristic
405C22×C3#57811yescharacteristic
406C3×C4—24no—
407C3×C4—22no—
408C3×C4—42no—
409C3×C4—44no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—82no—
414C3×C4—81no—
415C3×C4#45114yes—
416C3×C4#52511yescharacteristic
417C3×C4#53711yescharacteristic
418C3×C4#54611yescharacteristic
419C3×C4#54711yescharacteristic
420C23C22×C3×D411yescharacteristic
421Q8—28no—
422Q8—22no—
423Q8—22no—
424Q8—22no—
425Q8—22no—
426Q8—24no—
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—41no—
431Q8—41no—
432Q8—41no—
433Q8—41no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4C3×#12012yes—
460C2×C4C3×#12012yes—
461C2×C4C2×C3×Q1614yes—
462C2×C4C2×C3×Q1612yes—
463C2×C4C3×#13712yes—
464C2×C4C22×C3×D411yescharacteristic
465C2×C4C22×C3×D411yescharacteristic
466C2×C4C2×C3×#4111yescharacteristic
467C2×C4C2×C3×#4111yescharacteristic
468C2×C4C3×H(5,𝔽2)11yescharacteristic
469C2×C4C3×2−1+411yescharacteristic
470C8—22no—
471C8—41no—
472C8—42no—
473C2×C3#240711yescharacteristic
474C2×C3C2×#45111yescharacteristic
475C2×C3#301311yescharacteristic
476C2×C3#336311yescharacteristic
477C2×C3#337411yescharacteristic
478C2×C3#339511yescharacteristic
479C2×C3#340011yescharacteristic
480C22C2×C3×#12011yescharacteristic
481C22C3×#54811yescharacteristic
482C22C3×#55011yescharacteristic
483C22C22×C3×Q1611yescharacteristic
484C22C2×C3×#13711yescharacteristic
485C22C3×#57711yescharacteristic
486C22C3×#57811yescharacteristic
487C4—24no—
488C4—22no—
489C4—42no—
490C4—44no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—82no—
495C4—81no—
496C4C3×#45114yes—
497C4C3×#52511yescharacteristic
498C4C3×#53711yescharacteristic
499C4C3×#54611yescharacteristic
500C4C3×#54711yescharacteristic
501C3#2419111yescharacteristic, direct product factor, Sylow
502C2C3×#240711yescharacteristic
503C2C2×C3×#45111yescharacteristic
504C2C3×#301311yescharacteristic
505C2C3×#336311yescharacteristic
506C2C3×#337411yescharacteristic
507C2C3×#339511yescharacteristic
508C2C3×#340011yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
12. #24191C3direct (N × H)
167. C2×C3×Q8Q16semidirect
348. C2×Q8C3×Q16semidirect
501. C3#24191direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#4622
C2×C3×C4×Q162
C3×#25822
C3×#20432
C3×#20162
C3×#19842
C3×#19562
C3×#18782
C3×#18632
C3×#18032
#241913