Finite Group Databasebeta

← C3×#15779 #10634386

Subgroups — C3×#15779 #10634386

Order 768.

1370 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#395C212yesmaximal
2C2×C3×#389C212yesmaximal
3C3×#1446C211yesmaximal, characteristic
4C3×#1444C211yesmaximal, characteristic
5C3×#1444C211yesmaximal, characteristic
6C3×#1442C212yesmaximal
7C3×#1441C212yesmaximal
8C3×#1436C212yesmaximal
9C3×#1436C212yesmaximal
10#15779C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C42C2212yes—
19C3×#395—28no—
20C3×#389—28no—
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×#395C2×C312yes—
35C2×#389C2×C312yes—
36#1446C2×C311yescharacteristic
37#1444C2×C311yescharacteristic
38#1444C2×C311yescharacteristic
39#1442C2×C312yes—
40#1441C2×C312yes—
41#1436C2×C312yes—
42#1436C2×C312yes—
43C22×C3×Q8—24no—
44C22×C3×Q8—22no—
45C22×C3×Q8—41no—
46C22×C3×Q8—42no—
47C22×C3×Q8—41no—
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
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—42no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4—41no—
70C2×C3×C4⋊C4—41no—
71C2×C3×C4⋊C4—42no—
72C2×C3×C4⋊C4D418yes—
73C2×C3×C42—24no—
74C2×C3×C42—42no—
75C2×C3×C42—42no—
76C3×#95—24no—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—24no—
81C3×#95—24no—
82C3×#95—22no—
83C3×#95—42no—
84C3×#95—42no—
85C3×#95—41no—
86C3×#95—41no—
87C3×#95Q818yescomplemented
88C3×#95Q814yescomplemented
89C23×Q8C22×C311yescharacteristic
90C22×C4⋊C4C22×C312yes—
91C22×C4⋊C4C22×C312yes—
92C22×C4⋊C4C22×C312yes—
93C22×C4⋊C4C22×C311yescharacteristic
94C22×C4⋊C4C22×C311yescharacteristic
95C22×C4⋊C4C22×C312yes—
96C22×C42C22×C312yes—
97#395—28no—
98#389—28no—
99C2×#95C22×C312yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C311yescharacteristic
105C2×#95C22×C312yes—
106C2×#95C22×C311yescharacteristic
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C311yescharacteristic
110C2×#95C22×C312yes—
111C2×#95C22×C311yescharacteristic
112C24×C3C2411yescharacteristic
113C2×C3×Q8—44no—
114C2×C3×Q8—48no—
115C2×C3×Q8—48no—
116C2×C3×Q8—44no—
117C2×C3×Q8—44no—
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—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
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—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4C2×D414yes—
165C22×C3×C4C2×D414yes—
166C22×C3×C4C2×D414yes—
167C22×C3×C4C2×Q814yes—
168C22×C3×C4C2×Q814yes—
169C22×C3×C4C2×Q812yes—
170C22×C3×C4C2×Q814yes—
171C22×C3×C4C2×Q814yes—
172C22×C3×C4#4112yes—
173C22×C3×C4#4114yes—
174C22×C3×C4#4112yes—
175C3×C4⋊C4—216no—
176C3×C4⋊C4—48no—
177C3×C4⋊C4—44no—
178C3×C4⋊C4—48no—
179C3×C4⋊C4—48no—
180C3×C4⋊C4—44no—
181C3×C42—48no—
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—41no—
185C22×Q8—42no—
186C22×Q8—41no—
187C23×C4C23×C312yes—
188C23×C4C23×C312yes—
189C23×C4C23×C312yes—
190C23×C4C23×C311yescharacteristic
191C23×C4C23×C312yes—
192C23×C4C23×C312yes—
193C23×C4C23×C311yescharacteristic
194C23×C4C23×C311yescharacteristic
195C23×C4C23×C312yes—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C3×D418yes—
212C2×C42—24no—
213C2×C42—42no—
214C2×C42—42no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—22no—
222#95—42no—
223#95—42no—
224#95—41no—
225#95—41no—
226#95C3×Q818yescomplemented
227#95C3×Q814yescomplemented
228C23×C3C22×D412yes—
229C23×C3C22×Q811yescharacteristic
230C23×C3C22×Q812yes—
231C23×C3C2×#4111yescharacteristic
232C23×C3C2×#4112yes—
233C23×C3C2×#4111yescharacteristic
234C23×C3H(5,𝔽2)11yescharacteristic
235C23×C3H(5,𝔽2)12yes—
236C23×C3H(5,𝔽2)11yescharacteristic
237C23×C32−1+411yescharacteristic
238C23×C32−1+411yescharacteristic
239C3×Q8—416no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
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—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—22no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—42no—
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—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
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—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4C4⋊Q818yes—
297C24C24×C311yescharacteristic
298C2×Q8—44no—
299C2×Q8—48no—
300C2×Q8—48no—
301C2×Q8—44no—
302C2×Q8—44no—
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—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4C2×C3×D414yes—
350C22×C4C2×C3×D414yes—
351C22×C4C2×C3×D414yes—
352C22×C4C2×C3×Q814yes—
353C22×C4C2×C3×Q814yes—
354C22×C4C2×C3×Q812yes—
355C22×C4C2×C3×Q814yes—
356C22×C4C2×C3×Q814yes—
357C22×C4C3×#4112yes—
358C22×C4C3×#4114yes—
359C22×C4C3×#4112yes—
360C4⋊C4—216no—
361C4⋊C4—48no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C42—48no—
367C22×C3#52512yes—
368C22×C3C2×C4⋊Q812yes—
369C22×C3#53312yes—
370C22×C3#53412yes—
371C22×C3#53811yescharacteristic
372C22×C3#53911yescharacteristic
373C22×C3#54212yes—
374C22×C3#54412yes—
375C22×C3#54411yescharacteristic
376C22×C3D4211yescharacteristic
377C22×C3#54612yes—
378C22×C3#54812yes—
379C22×C3D4×Q812yes—
380C22×C3#55111yescharacteristic
381C22×C3#55411yescharacteristic
382C22×C3#55412yes—
383C22×C3#55412yes—
384C22×C3#55412yes—
385C22×C3#55611yescharacteristic
386C22×C3#55711yescharacteristic
387C22×C3Q8211yescharacteristic
388C22×C3#56011yescharacteristic
389C22×C3#56311yescharacteristic
390C3×C4—28no—
391C3×C4—44no—
392C3×C4—44no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—44no—
396C3×C4—44no—
397C3×C4—44no—
398C3×C4—42no—
399C23C22×C3×D412yes—
400C23C22×C3×Q812yes—
401C23C22×C3×Q811yescharacteristic
402C23C2×C3×#4111yescharacteristic
403C23C2×C3×#4112yes—
404C23C2×C3×#4111yescharacteristic
405C23C3×H(5,𝔽2)11yescharacteristic
406C23C3×H(5,𝔽2)12yes—
407C23C3×H(5,𝔽2)11yescharacteristic
408C23C3×2−1+411yescharacteristic
409C23C3×2−1+411yescharacteristic
410Q8—416no—
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—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
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×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4C3×C4⋊Q818yes—
468C2×C3#246311yescharacteristic
469C2×C3#246512yes—
470C2×C3#255012yes—
471C2×C3#255111yescharacteristic
472C2×C3#258312yes—
473C2×C3#258611yescharacteristic
474C2×C3#266811yescharacteristic
475C2×C3#272912yes—
476C2×C3#274211yescharacteristic
477C2×C3#279211yescharacteristic
478C2×C3#280511yescharacteristic
479C22C3×#52512yes—
480C22C2×C3×C4⋊Q812yes—
481C22C3×#53312yes—
482C22C3×#53412yes—
483C22C3×#53811yescharacteristic
484C22C3×#53911yescharacteristic
485C22C3×#54212yes—
486C22C3×#54412yes—
487C22C3×#54411yescharacteristic
488C22C3×D4211yescharacteristic
489C22C3×#54612yes—
490C22C3×#54812yes—
491C22C3×D4×Q812yes—
492C22C3×#55111yescharacteristic
493C22C3×#55412yes—
494C22C3×#55412yes—
495C22C3×#55412yes—
496C22C3×#55411yescharacteristic
497C22C3×#55611yescharacteristic
498C22C3×#55711yescharacteristic
499C22C3×Q8211yescharacteristic
500C22C3×#56011yescharacteristic
501C22C3×#56311yescharacteristic
502C4—28no—
503C4—44no—
504C4—42no—
505C4—44no—
506C4—42no—
507C4—44no—
508C4—44no—
509C4—44no—
510C4—42no—
511C3#1577911yescharacteristic, direct product factor, Sylow
512C2C3×#246311yescharacteristic
513C2C3×#246512yes—
514C2C3×#255012yes—
515C2C3×#255111yescharacteristic
516C2C3×#258312yes—
517C2C3×#258611yescharacteristic
518C2C3×#266811yescharacteristic
519C2C3×#272912yes—
520C2C3×#274211yescharacteristic
521C2C3×#279211yescharacteristic
522C2C3×#280511yescharacteristic
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. #15779C3direct (N × H)
87. C3×#95Q8semidirect
88. C3×#95Q8semidirect
226. #95C3×Q8semidirect
227. #95C3×Q8semidirect
511. C3#15779direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3952
C2×C3×#3892
C3×#14462
C3×#14442
C3×#14422
C3×#14412
C3×#14362
#157793