Finite Group Databasebeta

← C3×#13932 #10632539

Subgroups — C3×#13932 #10632539

Order 768.

1226 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C4⋊C4C211yesmaximal, characteristic
2C3×#1443C212yesmaximal
3C3×#1441C212yesmaximal
4C3×#1436C212yesmaximal
5C3×#1436C212yesmaximal
6C3×C4⋊#95C211yesmaximal, characteristic
7C3×#1434C211yesmaximal, characteristic
8C3×#1434C211yesmaximal, characteristic
9C3×#1434C211yesmaximal, characteristic
10C3×#1434C211yesmaximal, characteristic
11C3×C4×#95C211yesmaximal, characteristic
12#13932C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C3×C42C2212yes—
18C22×C3×C42C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×C4×C4⋊C4C418yescomplemented
21C2×C3×#95C2211yescharacteristic
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×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2211yescharacteristic
37C2×C4×C4⋊C4C2×C311yescharacteristic
38#1443C2×C312yes—
39#1441C2×C312yes—
40#1436C2×C312yes—
41#1436C2×C312yes—
42C4⋊#95C2×C311yescharacteristic
43#1434C2×C311yescharacteristic
44#1434C2×C311yescharacteristic
45#1434C2×C311yescharacteristic
46#1434C2×C311yescharacteristic
47C4×#95C2×C311yescharacteristic
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—42no—
65C2×C3×C4⋊C4C2×C414yes—
66C2×C3×C4⋊C4C2×C414yes—
67C2×C3×C42—24no—
68C2×C3×C42—24no—
69C2×C3×C42—42no—
70C2×C3×C42—42no—
71C2×C3×C42—42no—
72C2×C3×C42—42no—
73C2×C3×C42C2×C414yes—
74C2×C3×C42C2×C414yes—
75C2×C3×C42C2×C414yes—
76C3×#95—24no—
77C3×#95—22no—
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—24no—
81C3×#95—24no—
82C3×#95—24no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—42no—
86C3×#95—42no—
87C3×#95—42no—
88C3×#95—42no—
89C3×#95C2×C414yes—
90C3×#95C2×C414yes—
91C3×#95Q814yes—
92C22×C4⋊C4C22×C312yes—
93C22×C4⋊C4C22×C311yescharacteristic
94C22×C4⋊C4C22×C311yescharacteristic
95C22×C42C22×C311yescharacteristic
96C22×C42C22×C312yes—
97C22×C42C22×C311yescharacteristic
98C22×C42C22×C311yescharacteristic
99C4×C4⋊C4C3×C418yescomplemented
100C2×#95C22×C311yescharacteristic
101C2×#95C22×C312yes—
102C2×#95C22×C311yescharacteristic
103C2×#95C22×C311yescharacteristic
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C311yescharacteristic
114C2×#95C22×C311yescharacteristic
115C2×#95C22×C311yescharacteristic
116C24×C3C2411yescharacteristic
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
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—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
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—21no—
150C22×C3×C4—21no—
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×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4C22×C412yes—
166C22×C3×C4C22×C412yes—
167C22×C3×C4C22×C412yes—
168C22×C3×C4C22×C412yes—
169C22×C3×C4C22×C412yes—
170C22×C3×C4C22×C412yes—
171C22×C3×C4C22×C412yes—
172C22×C3×C4C2×Q814yes—
173C22×C3×C4C2×Q812yes—
174C22×C3×C4#4114yes—
175C22×C3×C4#4114yes—
176C22×C3×C4#4112yes—
177C22×C3×C4#4114yes—
178C22×C3×C4#4112yes—
179C22×C3×C4#4112yes—
180C3×C4⋊C4—28no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—48no—
183C3×C42—28no—
184C3×C42—44no—
185C3×C42—44no—
186C3×C42—48no—
187C23×C4C23×C312yes—
188C23×C4C23×C312yes—
189C23×C4C23×C311yescharacteristic
190C23×C4C23×C311yescharacteristic
191C23×C4C23×C311yescharacteristic
192C23×C4C23×C312yes—
193C23×C4C23×C312yes—
194C23×C4C23×C311yescharacteristic
195C23×C4C23×C311yescharacteristic
196C23×C4C23×C311yescharacteristic
197C23×C4C23×C311yescharacteristic
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C2×C3×C414yes—
205C2×C4⋊C4C2×C3×C414yes—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C2×C3×C414yes—
213C2×C42C2×C3×C414yes—
214C2×C42C2×C3×C414yes—
215#95—24no—
216#95—22no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—22no—
223#95—22no—
224#95—42no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95C2×C3×C414yes—
229#95C2×C3×C414yes—
230#95C3×Q814yes—
231C23×C3C23×C411yescharacteristic
232C23×C3C22×Q811yescharacteristic
233C23×C3C2×#4112yes—
234C23×C3C2×#4111yescharacteristic
235C23×C3C2×#4112yes—
236C23×C3C2×#4111yescharacteristic
237C23×C3C2×#4111yescharacteristic
238C23×C3C2×#4112yes—
239C23×C3H(5,𝔽2)12yes—
240C23×C3H(5,𝔽2)11yescharacteristic
241C23×C32−1+411yescharacteristic
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
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—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—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4#11714yes—
302C2×C3×C4#11714yes—
303C2×C3×C4#11714yes—
304C2×C3×C4C4×Q814yes—
305C24C24×C311yescharacteristic
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
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—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4C22×C3×C412yes—
355C22×C4C22×C3×C412yes—
356C22×C4C22×C3×C412yes—
357C22×C4C22×C3×C412yes—
358C22×C4C22×C3×C412yes—
359C22×C4C22×C3×C412yes—
360C22×C4C22×C3×C412yes—
361C22×C4C2×C3×Q814yes—
362C22×C4C2×C3×Q812yes—
363C22×C4C3×#4114yes—
364C22×C4C3×#4114yes—
365C22×C4C3×#4112yes—
366C22×C4C3×#4114yes—
367C22×C4C3×#4112yes—
368C22×C4C3×#4112yes—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—48no—
372C42—28no—
373C42—44no—
374C42—44no—
375C42—48no—
376C22×C3C2×#11711yescharacteristic
377C22×C3C2×#11711yescharacteristic
378C22×C3C2×#11711yescharacteristic
379C22×C3C2×C4×Q811yescharacteristic
380C22×C3C4×#4111yescharacteristic
381C22×C3#51811yescharacteristic
382C22×C3#52011yescharacteristic
383C22×C3#52912yes—
384C22×C3#52912yes—
385C22×C3#52912yes—
386C22×C3#53312yes—
387C22×C3#53812yes—
388C22×C3#53912yes—
389C22×C3#55112yes—
390C22×C3#55312yes—
391C22×C3#55312yes—
392C22×C3#55312yes—
393C22×C3#55411yescharacteristic
394C22×C3#55412yes—
395C22×C3#55612yes—
396C22×C3#55611yescharacteristic
397C22×C3#55711yescharacteristic
398C22×C3#55911yescharacteristic
399C3×C4—24no—
400C3×C4—24no—
401C3×C4—24no—
402C3×C4—24no—
403C3×C4—42no—
404C3×C4—44no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—82no—
409C3×C4—82no—
410C23C23×C3×C411yescharacteristic
411C23C22×C3×Q811yescharacteristic
412C23C2×C3×#4111yescharacteristic
413C23C2×C3×#4111yescharacteristic
414C23C2×C3×#4112yes—
415C23C2×C3×#4112yes—
416C23C2×C3×#4111yescharacteristic
417C23C2×C3×#4112yes—
418C23C3×H(5,𝔽2)11yescharacteristic
419C23C3×H(5,𝔽2)12yes—
420C23C3×2−1+411yescharacteristic
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
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—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4C3×#11714yes—
481C2×C4C3×#11714yes—
482C2×C4C3×#11714yes—
483C2×C4C3×C4×Q814yes—
484C2×C3#228311yescharacteristic
485C2×C3#229511yescharacteristic
486C2×C3#231811yescharacteristic
487C2×C3#234511yescharacteristic
488C2×C3#234711yescharacteristic
489C2×C3#235311yescharacteristic
490C2×C3#235511yescharacteristic
491C2×C3#258712yes—
492C2×C3#258812yes—
493C2×C3#259412yes—
494C2×C3#259512yes—
495C22C2×C3×#11711yescharacteristic
496C22C2×C3×#11711yescharacteristic
497C22C2×C3×#11711yescharacteristic
498C22C2×C3×C4×Q811yescharacteristic
499C22C3×C4×#4111yescharacteristic
500C22C3×#51811yescharacteristic
501C22C3×#52011yescharacteristic
502C22C3×#52912yes—
503C22C3×#52912yes—
504C22C3×#52912yes—
505C22C3×#53312yes—
506C22C3×#53812yes—
507C22C3×#53912yes—
508C22C3×#55112yes—
509C22C3×#55312yes—
510C22C3×#55312yes—
511C22C3×#55312yes—
512C22C3×#55411yescharacteristic
513C22C3×#55412yes—
514C22C3×#55612yes—
515C22C3×#55611yescharacteristic
516C22C3×#55711yescharacteristic
517C22C3×#55911yescharacteristic
518C4—24no—
519C4—24no—
520C4—24no—
521C4—24no—
522C4—42no—
523C4—44no—
524C4—44no—
525C4—42no—
526C4—42no—
527C4—82no—
528C4—82no—
529C3#1393211yescharacteristic, direct product factor, Sylow
530C2C3×#228311yescharacteristic
531C2C3×#229511yescharacteristic
532C2C3×#231811yescharacteristic
533C2C3×#234511yescharacteristic
534C2C3×#234711yescharacteristic
535C2C3×#235311yescharacteristic
536C2C3×#235511yescharacteristic
537C2C3×#258712yes—
538C2C3×#258812yes—
539C2C3×#259412yes—
540C2C3×#259512yes—
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
12. #13932C3direct (N × H)
20. C3×C4×C4⋊C4C4semidirect
99. C4×C4⋊C4C3×C4semidirect
529. C3#13932direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C4⋊C42
C3×#14432
C3×#14412
C3×#14362
C3×C4⋊#952
C3×#14342
C3×C4×#952
#139323