Finite Group Databasebeta

← C3×#15778 #10634385

Subgroups — C3×#15778 #10634385

Order 768.

1318 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×#398C212yesmaximal
2C2×C3×#389C212yesmaximal
3C2×C3×#387C212yesmaximal
4C3×#1444C211yesmaximal, characteristic
5C3×#1442C211yesmaximal, characteristic
6C3×#1442C211yesmaximal, characteristic
7C3×#1441C212yesmaximal
8C3×#1441C212yesmaximal
9C3×#1434C212yesmaximal
10#15778C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C42C2212yes—
19C3×#398—28no—
20C3×#389—28no—
21C3×#387—28no—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2211yescharacteristic
35C2×#398C2×C312yes—
36C2×#389C2×C312yes—
37C2×#387C2×C312yes—
38#1444C2×C311yescharacteristic
39#1442C2×C311yescharacteristic
40#1442C2×C311yescharacteristic
41#1441C2×C312yes—
42#1441C2×C312yes—
43#1434C2×C312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—41no—
67C2×C3×C4⋊C4—41no—
68C2×C3×C4⋊C4—41no—
69C2×C3×C4⋊C4—42no—
70C2×C3×C4⋊C4D418yes—
71C2×C3×C42—24no—
72C2×C3×C42—24no—
73C2×C3×C42—24no—
74C3×#95—24no—
75C3×#95—24no—
76C3×#95—24no—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—24no—
81C3×#95—24no—
82C3×#95—24no—
83C3×#95—22no—
84C3×#95—41no—
85C3×#95D414yes—
86C3×#95Q814yes—
87C22×C4⋊C4C22×C312yes—
88C22×C4⋊C4C22×C312yes—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C312yes—
91C22×C4⋊C4C22×C311yescharacteristic
92C22×C4⋊C4C22×C311yescharacteristic
93C22×C4⋊C4C22×C312yes—
94C22×C42C22×C312yes—
95#398—28no—
96#389—28no—
97#387—28no—
98C2×#95C22×C312yes—
99C2×#95C22×C312yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C311yescharacteristic
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C311yescharacteristic
110C2×#95C22×C311yescharacteristic
111C24×C3C2411yescharacteristic
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
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×C4C2×D412yes—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D414yes—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D414yes—
164C22×C3×C4C2×Q814yes—
165C22×C3×C4C2×Q812yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4114yes—
168C22×C3×C4#4114yes—
169C3×C4⋊C4—216no—
170C3×C4⋊C4—216no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4—48no—
173C3×C4⋊C4—44no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4—48no—
176C3×C42—48no—
177C23×C4C23×C312yes—
178C23×C4C23×C312yes—
179C23×C4C23×C312yes—
180C23×C4C23×C311yescharacteristic
181C23×C4C23×C311yescharacteristic
182C23×C4C23×C312yes—
183C23×C4C23×C312yes—
184C23×C4C23×C311yescharacteristic
185C23×C4C23×C312yes—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C3×D418yes—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—41no—
218#95C3×D414yes—
219#95C3×Q814yes—
220C23×C3C22×D411yescharacteristic
221C23×C3C22×D412yes—
222C23×C3C22×Q811yescharacteristic
223C23×C3C2×#4112yes—
224C23×C3C2×#4111yescharacteristic
225C23×C3C2×#4112yes—
226C23×C3H(5,𝔽2)11yescharacteristic
227C23×C32−1+411yescharacteristic
228C23×C32−1+412yes—
229C23×C32−1+411yescharacteristic
230C23×C32−1+411yescharacteristic
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
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#12118yes—
288C24C24×C311yescharacteristic
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
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×C4C2×C3×D412yes—
337C22×C4C2×C3×D414yes—
338C22×C4C2×C3×D414yes—
339C22×C4C2×C3×D414yes—
340C22×C4C2×C3×D414yes—
341C22×C4C2×C3×Q814yes—
342C22×C4C2×C3×Q812yes—
343C22×C4C3×#4112yes—
344C22×C4C3×#4114yes—
345C22×C4C3×#4114yes—
346C4⋊C4—216no—
347C4⋊C4—216no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C42—48no—
354C22×C3C2×#12112yes—
355C22×C3#52512yes—
356C22×C3#53312yes—
357C22×C3#53612yes—
358C22×C3#53711yescharacteristic
359C22×C3#53911yescharacteristic
360C22×C3#54112yes—
361C22×C3#54112yes—
362C22×C3#54411yescharacteristic
363C22×C3D4211yescharacteristic
364C22×C3#54711yescharacteristic
365C22×C3#54712yes—
366C22×C3#54712yes—
367C22×C3#54812yes—
368C22×C3D4×Q812yes—
369C22×C3D4×Q811yescharacteristic
370C22×C3#55111yescharacteristic
371C22×C3#55212yes—
372C22×C3#55212yes—
373C22×C3#55511yescharacteristic
374C22×C3#55711yescharacteristic
375C22×C3#56311yescharacteristic
376C22×C3#56411yescharacteristic
377C3×C4—28no—
378C3×C4—42no—
379C3×C4—44no—
380C3×C4—42no—
381C3×C4—44no—
382C3×C4—42no—
383C3×C4—44no—
384C3×C4—44no—
385C3×C4—82no—
386C23C22×C3×D411yescharacteristic
387C23C22×C3×D412yes—
388C23C22×C3×Q811yescharacteristic
389C23C2×C3×#4112yes—
390C23C2×C3×#4111yescharacteristic
391C23C2×C3×#4112yes—
392C23C3×H(5,𝔽2)11yescharacteristic
393C23C3×2−1+412yes—
394C23C3×2−1+411yescharacteristic
395C23C3×2−1+411yescharacteristic
396C23C3×2−1+411yescharacteristic
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
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—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4C3×#12118yes—
454C2×C3#241312yes—
455C2×C3#242912yes—
456C2×C3#246212yes—
457C2×C3#246311yescharacteristic
458C2×C3#253711yescharacteristic
459C2×C3#261211yescharacteristic
460C2×C3#267711yescharacteristic
461C2×C3#271912yes—
462C2×C3#272211yescharacteristic
463C2×C3#272611yescharacteristic
464C2×C3#276911yescharacteristic
465C22C2×C3×#12112yes—
466C22C3×#52512yes—
467C22C3×#53312yes—
468C22C3×#53612yes—
469C22C3×#53711yescharacteristic
470C22C3×#53911yescharacteristic
471C22C3×#54112yes—
472C22C3×#54112yes—
473C22C3×#54411yescharacteristic
474C22C3×D4211yescharacteristic
475C22C3×#54711yescharacteristic
476C22C3×#54712yes—
477C22C3×#54712yes—
478C22C3×#54812yes—
479C22C3×D4×Q812yes—
480C22C3×D4×Q811yescharacteristic
481C22C3×#55111yescharacteristic
482C22C3×#55212yes—
483C22C3×#55212yes—
484C22C3×#55511yescharacteristic
485C22C3×#55711yescharacteristic
486C22C3×#56311yescharacteristic
487C22C3×#56411yescharacteristic
488C4—28no—
489C4—42no—
490C4—44no—
491C4—42no—
492C4—44no—
493C4—42no—
494C4—44no—
495C4—44no—
496C4—82no—
497C3#1577811yescharacteristic, direct product factor, Sylow
498C2C3×#241312yes—
499C2C3×#242912yes—
500C2C3×#246212yes—
501C2C3×#246311yescharacteristic
502C2C3×#253711yescharacteristic
503C2C3×#261211yescharacteristic
504C2C3×#267711yescharacteristic
505C2C3×#271912yes—
506C2C3×#272211yescharacteristic
507C2C3×#272611yescharacteristic
508C2C3×#276911yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
10. #15778C3direct (N × H)
497. C3#15778direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3982
C2×C3×#3892
C2×C3×#3872
C3×#14442
C3×#14422
C3×#14412
C3×#14342
#157783