Finite Group Databasebeta

← C3×#14156 #10632763

Subgroups — C3×#14156 #10632763

Order 768.

1310 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×C3×#387C212yesmaximal
3C3×#1445C212yesmaximal
4C3×#1441C212yesmaximal
5C3×#1436C212yesmaximal
6C3×#1436C212yesmaximal
7C3×#1434C212yesmaximal
8C3×C4×#95C211yesmaximal, characteristic
9C3×C4×#95C211yesmaximal, characteristic
10#14156C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2212yes—
16C22×C3×C42C2212yes—
17C22×C3×C42C2211yescharacteristic
18C3×#389C418yescomplemented
19C3×#387—28no—
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2211yescharacteristic
34C2×#389C2×C311yescharacteristic
35C2×#387C2×C312yes—
36#1445C2×C312yes—
37#1441C2×C312yes—
38#1436C2×C312yes—
39#1436C2×C312yes—
40#1434C2×C312yes—
41C4×#95C2×C311yescharacteristic
42C4×#95C2×C311yescharacteristic
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2312yes—
47C23×C3×C4C2312yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—42no—
58C2×C3×C4⋊C4—42no—
59C2×C3×C4⋊C4C2×C414yes—
60C2×C3×C4⋊C4C2×C418yes—
61C2×C3×C4⋊C4C2×C414yes—
62C2×C3×C42—24no—
63C2×C3×C42—24no—
64C2×C3×C42—24no—
65C2×C3×C42—24no—
66C2×C3×C42—42no—
67C2×C3×C42—42no—
68C2×C3×C42—42no—
69C2×C3×C42C2×C414yes—
70C3×#95—24no—
71C3×#95—24no—
72C3×#95—24no—
73C3×#95—24no—
74C3×#95—24no—
75C3×#95—24no—
76C3×#95—24no—
77C3×#95—24no—
78C3×#95—22no—
79C3×#95—22no—
80C3×#95—42no—
81C3×#95C2×C418yes—
82C3×#95D414yes—
83C3×#95Q814yes—
84C22×C4⋊C4C22×C312yes—
85C22×C4⋊C4C22×C312yes—
86C22×C4⋊C4C22×C311yescharacteristic
87C22×C4⋊C4C22×C311yescharacteristic
88C22×C42C22×C312yes—
89C22×C42C22×C312yes—
90C22×C42C22×C311yescharacteristic
91#389C3×C418yescomplemented
92#387—28no—
93C2×#95C22×C311yescharacteristic
94C2×#95C22×C311yescharacteristic
95C2×#95C22×C312yes—
96C2×#95C22×C312yes—
97C2×#95C22×C312yes—
98C2×#95C22×C312yes—
99C2×#95C22×C312yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C312yes—
105C2×#95C22×C311yescharacteristic
106C2×#95C22×C311yescharacteristic
107C24×C3C2411yescharacteristic
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
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—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—22no—
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×C4C22×C414yes—
152C22×C3×C4C22×C414yes—
153C22×C3×C4C22×C412yes—
154C22×C3×C4C22×C412yes—
155C22×C3×C4C22×C412yes—
156C22×C3×C4C2×D414yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×Q814yes—
159C22×C3×C4C2×Q812yes—
160C22×C3×C4#4114yes—
161C22×C3×C4#4114yes—
162C22×C3×C4#4114yes—
163C22×C3×C4#4112yes—
164C22×C3×C4#4112yes—
165C3×C4⋊C4—28no—
166C3×C4⋊C4—216no—
167C3×C4⋊C4—44no—
168C3×C4⋊C4—48no—
169C3×C42—44no—
170C3×C42—48no—
171C3×C42—48no—
172C23×C4C23×C312yes—
173C23×C4C23×C312yes—
174C23×C4C23×C312yes—
175C23×C4C23×C312yes—
176C23×C4C23×C312yes—
177C23×C4C23×C312yes—
178C23×C4C23×C311yescharacteristic
179C23×C4C23×C311yescharacteristic
180C23×C4C23×C311yescharacteristic
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C2×C3×C414yes—
189C2×C4⋊C4C2×C3×C418yes—
190C2×C4⋊C4C2×C3×C414yes—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C2×C3×C414yes—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—22no—
208#95—22no—
209#95—42no—
210#95C2×C3×C418yes—
211#95C3×D414yes—
212#95C3×Q814yes—
213C23×C3C23×C411yescharacteristic
214C23×C3C22×D411yescharacteristic
215C23×C3C22×Q811yescharacteristic
216C23×C3C2×#4112yes—
217C23×C3C2×#4111yescharacteristic
218C23×C3C2×#4111yescharacteristic
219C23×C3C2×#4112yes—
220C23×C3C2×#4112yes—
221C23×C3H(5,𝔽2)12yes—
222C23×C32−1+411yescharacteristic
223C23×C32−1+411yescharacteristic
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—24no—
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—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—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—42no—
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—42no—
266C2×C3×C4—42no—
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#11714yes—
279C2×C3×C4C4×D414yes—
280C2×C3×C4C4×Q814yes—
281C24C24×C311yescharacteristic
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
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—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×C4C22×C3×C414yes—
326C22×C4C22×C3×C414yes—
327C22×C4C22×C3×C412yes—
328C22×C4C22×C3×C412yes—
329C22×C4C22×C3×C412yes—
330C22×C4C2×C3×D414yes—
331C22×C4C2×C3×D412yes—
332C22×C4C2×C3×Q814yes—
333C22×C4C2×C3×Q812yes—
334C22×C4C3×#4114yes—
335C22×C4C3×#4114yes—
336C22×C4C3×#4114yes—
337C22×C4C3×#4112yes—
338C22×C4C3×#4112yes—
339C4⋊C4—28no—
340C4⋊C4—216no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C42—44no—
344C42—48no—
345C42—48no—
346C22×C3C2×#11711yescharacteristic
347C22×C3C2×C4×D411yescharacteristic
348C22×C3C2×C4×Q811yescharacteristic
349C22×C3C4×#4111yescharacteristic
350C22×C3C4×#4111yescharacteristic
351C22×C3#51811yescharacteristic
352C22×C3#51911yescharacteristic
353C22×C3#52912yes—
354C22×C3#52912yes—
355C22×C3#53212yes—
356C22×C3#53312yes—
357C22×C3#53912yes—
358C22×C3#53912yes—
359C22×C3#54612yes—
360C22×C3#54711yescharacteristic
361C22×C3D4×Q811yescharacteristic
362C22×C3#55112yes—
363C22×C3#55212yes—
364C22×C3#55212yes—
365C22×C3#55312yes—
366C22×C3#55412yes—
367C22×C3#55611yescharacteristic
368C22×C3#55711yescharacteristic
369C3×C4—24no—
370C3×C4—24no—
371C3×C4—24no—
372C3×C4—44no—
373C3×C4—44no—
374C3×C4—44no—
375C3×C4—44no—
376C3×C4—44no—
377C3×C4—44no—
378C23C23×C3×C411yescharacteristic
379C23C22×C3×D411yescharacteristic
380C23C22×C3×Q811yescharacteristic
381C23C2×C3×#4112yes—
382C23C2×C3×#4111yescharacteristic
383C23C2×C3×#4111yescharacteristic
384C23C2×C3×#4112yes—
385C23C2×C3×#4112yes—
386C23C3×H(5,𝔽2)12yes—
387C23C3×2−1+411yescharacteristic
388C23C3×2−1+411yescharacteristic
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
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—24no—
409C2×C4—24no—
410C2×C4—24no—
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—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
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—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4C3×#11714yes—
444C2×C4C3×C4×D414yes—
445C2×C4C3×C4×Q814yes—
446C2×C3#229311yescharacteristic
447C2×C3#229511yescharacteristic
448C2×C3#231311yescharacteristic
449C2×C3#236411yescharacteristic
450C2×C3#236511yescharacteristic
451C2×C3#236811yescharacteristic
452C2×C3#237111yescharacteristic
453C2×C3#255312yes—
454C2×C3#255612yes—
455C2×C3#257312yes—
456C2×C3#258512yes—
457C22C2×C3×#11711yescharacteristic
458C22C2×C3×C4×D411yescharacteristic
459C22C2×C3×C4×Q811yescharacteristic
460C22C3×C4×#4111yescharacteristic
461C22C3×C4×#4111yescharacteristic
462C22C3×#51811yescharacteristic
463C22C3×#51911yescharacteristic
464C22C3×#52912yes—
465C22C3×#52912yes—
466C22C3×#53212yes—
467C22C3×#53312yes—
468C22C3×#53912yes—
469C22C3×#53912yes—
470C22C3×#54612yes—
471C22C3×#54711yescharacteristic
472C22C3×D4×Q811yescharacteristic
473C22C3×#55112yes—
474C22C3×#55212yes—
475C22C3×#55212yes—
476C22C3×#55312yes—
477C22C3×#55412yes—
478C22C3×#55611yescharacteristic
479C22C3×#55711yescharacteristic
480C4—24no—
481C4—24no—
482C4—24no—
483C4—44no—
484C4—44no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4—44no—
489C3#1415611yescharacteristic, direct product factor, Sylow
490C2C3×#229311yescharacteristic
491C2C3×#229511yescharacteristic
492C2C3×#231311yescharacteristic
493C2C3×#236411yescharacteristic
494C2C3×#236511yescharacteristic
495C2C3×#236811yescharacteristic
496C2C3×#237111yescharacteristic
497C2C3×#255312yes—
498C2C3×#255612yes—
499C2C3×#257312yes—
500C2C3×#258512yes—
subgroupG/H
center, socle, prime radicalC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
10. #14156C3direct (N × H)
18. C3×#389C4semidirect
91. #389C3×C4semidirect
489. C3#14156direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×C3×#3872
C3×#14452
C3×#14412
C3×#14362
C3×#14342
C3×C4×#952
#141563