Finite Group Databasebeta

← C3×#10593403 #13166764

Subgroups — C3×#10593403 #13166764

Order 1728.

876 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D32C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×D32⋊C4C212yesmaximal
4C3×#63900C211yesmaximal, characteristic, complemented
5C3×#63737C212yesmaximal, complemented
6#10593403C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2211yescharacteristic
8C22×C3×#157C2212yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×C4×D3—21no—
11C2×C3×#624C2212yes—
12C2×C3×#624C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C3×C4×D32—22no—
15C3×C4×D32C414yescomplemented
16C3×C4⋊#157—22no—
17C3×D32⋊C4—24no—
18C32×C4⋊Dic3—21no—
19C3×#3724—21no—
20C22×C3×C4×D3—61no—
21C2×C4×D32C2×C311yescharacteristic
22C2×C4⋊#157C2×C311yescharacteristic, complemented
23C2×D32⋊C4C2×C312yes—
24C3×D3×C4⋊C4—61no—
25#63900C2×C311yescharacteristic, complemented
26#63737C2×C312yescomplemented
27C22×C3×C3⋊D3C2311yescharacteristic
28C22×C32×D3—21no—
29C2×C3×D32—21no—
30C2×C3×D32—21no—
31C2×C3×D32—21no—
32C2×C3×D32C2×C412yes—
33C2×C3×D32D412yes—
34C2×C3×D32Q812yes—
35C2×C3×#157—22no—
36C2×C3×#157—22no—
37C2×C3×#157—22no—
38C2×C33×C4D411yescharacteristic
39C2×C3×C3⋊Dic3D411yescharacteristic
40C3×C4×C3⋊D3C2×C412yes—
41C3×C4×C3⋊D3D412yes—
42C2×C32×Dic3—21no—
43C2×C32×Dic3—22no—
44C32×C4×D3—24no—
45C3×#624—21no—
46C3×#624—22no—
47C3×#624—22no—
48C3×#624C2×C412yes—
49C3×D3×Dic3—24no—
50C3×#380—91nomaximal
51C23×C3×D3—61no—
52C22×D32C22×C311yescharacteristic
53C22×#157C22×C312yes—
54C22×C32×C4—61no—
55C2×C4×C3⋊D3C22×C311yescharacteristic
56C22×C3×Dic3—61no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—41no—
60C2×C3×C4×D3—64no—
61C2×C3×C4×D3—64no—
62C2×C3×C4×D3—61no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C2×C3×C4×D3—61no—
66C2×C3×C4×D3—62no—
67C2×#624C22×C312yes—
68C2×#624C22×C311yescharacteristic
69C2×D3×Dic3—21no—
70C4×D32—22no—
71C4×D32C3×C414yescomplemented
72C4⋊#157—22no—
73D32⋊C4—24no—
74C32×C4⋊C4—61no—
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—41no—
78C3×Dic3⋊C4—62no—
79#3724—21no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3C22×C411yescharacteristic
82C2×C3×C3⋊D3C2×D411yescharacteristic
83C2×C3×C3⋊D3C2×D411yescharacteristic
84C2×C3×C3⋊D3C2×Q811yescharacteristic
85C2×C3×C3⋊D3#4111yescharacteristic
86C2×C3×C3⋊D3#4111yescharacteristic
87C2×C32×D3—22no—
88C2×C32×D3—22no—
89C2×C32×D3—22no—
90C3×D32—22no—
91C3×D32—22no—
92C3×D32—22no—
93C3×D32C4⋊C414yescomplemented
94C3×#157—42no—
95C33×C4C22⋊C412yescomplemented
96C3×C3⋊Dic3C22⋊C412yescomplemented
97C32×Dic3—22no—
98C32×Dic3—42no—
99C23×C3×C4—91no—
100C22×C4×D3—61no—
101C2×C3×C4⋊C4—91no—
102C2×C3×C4⋊C4—91no—
103C2×C3×C22⋊C4—92no—
104D3×C4⋊C4—61no—
105C3×#95—92no—
106C23×C32—61no—
107C22×C3⋊D3C23×C311yescharacteristic
108C22×C3×D3—21no—
109C22×C3×D3—21no—
110C22×C3×D3—41no—
111C22×C3×D3—62no—
112C22×C3×D3—62no—
113C22×C3×D3—62no—
114C22×C3×D3—61no—
115C22×C3×D3—61no—
116C22×C3×D3—62no—
117C22×C3×D3—62no—
118C22×C3×D3—62no—
119C2×D32—21no—
120C2×D32—21no—
121C2×D32—21no—
122C2×D32C2×C3×C412yes—
123C2×D32C3×D412yes—
124C2×D32C3×Q812yes—
125C2×#157—22no—
126C2×#157—22no—
127C2×#157—22no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4—64no—
133C2×C32×C4—61no—
134C2×C32×C4—62no—
135C2×C32×C4C3×D411yescharacteristic
136C2×C3⋊Dic3C3×D411yescharacteristic
137C4×C3⋊D3C2×C3×C412yes—
138C4×C3⋊D3C3×D412yes—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—42no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—64no—
148C3×C4×D3—24no—
149C3×C4×D3—24no—
150C3×C4×D3—44no—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×C4×D3—62no—
157C3×C4×D3—64no—
158C3×C4×D3—122no—
159C3×C4×D3—122no—
160#624—21no—
161#624—22no—
162#624—22no—
163#624C2×C3×C412yes—
164D3×Dic3—24no—
165#380—91noSylow
166C2×C33C2×C22⋊C411yescharacteristic
167C2×C33#12111yescharacteristic
168C2×C33C22⋊Q811yescharacteristic
169C3×C3⋊D3C2×C4⋊C411yescharacteristic
170C3×C3⋊D3#11711yescharacteristic
171C3×C3⋊D3#12111yescharacteristic
172C3×C3⋊D3C22⋊Q811yescharacteristic
173C32×D3—24no—
174C24×C3—91no—
175C23×D3—61no—
176C22×C3×C4—61no—
177C22×C3×C4—61no—
178C22×C3×C4—91no—
179C22×C3×C4—94no—
180C22×C3×C4—92no—
181C22×C3×C4—91no—
182C22×C3×C4—92no—
183C22×C3×C4—121no—
184C22×C3×C4—182no—
185C22×C3×C4—181no—
186C22×Dic3—61no—
187C2×C4×D3—21no—
188C2×C4×D3—64no—
189C2×C4×D3—61no—
190C2×C4×D3—61no—
191C2×C4×D3—61no—
192C2×C4×D3—62no—
193C2×C4×D3—64no—
194C2×C4×D3—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—121no—
198C3×C4⋊C4—181no—
199C3×C4⋊C4—182no—
200C3×C22⋊C4—184no—
201C4⋊Dic3—21no—
202Dic3⋊C4—62no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—62no—
208C22×C32—62no—
209C22×C32—62no—
210C22×C32C2×C3×D411yescharacteristic
211C2×C3⋊D3C22×C3×C411yescharacteristic
212C2×C3⋊D3C2×C3×D411yescharacteristic
213C2×C3⋊D3C2×C3×D411yescharacteristic
214C2×C3⋊D3C2×C3×Q811yescharacteristic
215C2×C3⋊D3C3×#4111yescharacteristic
216C2×C3⋊D3C3×#4111yescharacteristic
217C2×C3×D3—22no—
218C2×C3×D3—22no—
219C2×C3×D3—22no—
220C2×C3×D3—22no—
221C2×C3×D3—22no—
222C2×C3×D3—22no—
223C2×C3×D3—42no—
224C2×C3×D3—42no—
225C2×C3×D3—42no—
226C2×C3×D3—64no—
227C2×C3×D3—61no—
228C2×C3×D3—61no—
229C2×C3×D3—61no—
230C2×C3×D3—61no—
231C2×C3×D3—61no—
232C2×C3×D3—61no—
233C2×C3×D3—61no—
234C2×C3×D3—61no—
235C2×C3×D3—61no—
236C2×C3×D3—61no—
237C2×C3×D3—61no—
238C2×C3×D3—61no—
239C2×C3×D3—64no—
240C2×C3×D3—64no—
241C2×C3×D3—64no—
242D32—22no—
243D32—22no—
244D32—22no—
245D32C3×C4⋊C414yescomplemented
246#157—42no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—42no—
250C32×C4—42no—
251C32×C4—62no—
252C32×C4—122no—
253C32×C4C3×C22⋊C412yescomplemented
254C3⋊Dic3C3×C22⋊C412yescomplemented
255C3×Dic3—22no—
256C3×Dic3—22no—
257C3×Dic3—42no—
258C3×Dic3—42no—
259C3×Dic3—42no—
260C3×Dic3—62no—
261C3×Dic3—62no—
262C3×Dic3—82no—
263C23×C4—91no—
264C2×C4⋊C4—91no—
265C2×C4⋊C4—91no—
266C2×C22⋊C4—92no—
267#95—92no—
268C33#38011yescharacteristic, complemented, Sylow
269C23×C3—61no—
270C23×C3—61no—
271C23×C3—92no—
272C23×C3—92no—
273C23×C3—92no—
274C23×C3—91no—
275C23×C3—121no—
276C23×C3—181no—
277C23×C3—181no—
278C23×C3—181no—
279C22×D3—21no—
280C22×D3—62no—
281C22×D3—62no—
282C22×D3—62no—
283C22×D3—61no—
284C22×D3—61no—
285C22×D3—62no—
286C22×D3—62no—
287C22×D3—62no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—64no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—64no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4—92no—
299C2×C3×C4—91no—
300C2×C3×C4—92no—
301C2×C3×C4—92no—
302C2×C3×C4—124no—
303C2×C3×C4—121no—
304C2×C3×C4—122no—
305C2×C3×C4—181no—
306C2×C3×C4—184no—
307C2×C3×C4—182no—
308C2×C3×C4—182no—
309C2×C3×C4—182no—
310C2×C3×C4—182no—
311C2×C3×C4—182no—
312C2×C3×C4GO+(4,𝔽2)11yescharacteristic
313C2×Dic3—21no—
314C2×Dic3—22no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—64no—
318C4×D3—24no—
319C4×D3—62no—
320C4×D3—62no—
321C4×D3—62no—
322C4×D3—62no—
323C4×D3—62no—
324C4×D3—62no—
325C4×D3—64no—
326C4×D3—122no—
327C4×D3—122no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—64no—
341C2×C32C2×C3×C22⋊C411yescharacteristic
342C2×C32C3×#12111yescharacteristic
343C2×C32C3×C22⋊Q811yescharacteristic
344C3⋊D3C2×C3×C4⋊C411yescharacteristic
345C3⋊D3C3×#11711yescharacteristic
346C3⋊D3C3×#12111yescharacteristic
347C3⋊D3C3×C22⋊Q811yescharacteristic
348C3×D3—24no—
349C3×D3—24no—
350C3×D3—44no—
351C3×D3—61no—
352C3×D3—61no—
353C3×D3—61no—
354C3×D3—61no—
355C3×D3—61no—
356C3×D3—61no—
357C3×D3—61no—
358C3×D3—61no—
359C24—91no—
360C22×C4—61no—
361C22×C4—91no—
362C22×C4—94no—
363C22×C4—92no—
364C22×C4—91no—
365C22×C4—92no—
366C22×C4—182no—
367C22×C4—181no—
368C4⋊C4—61no—
369C4⋊C4—181no—
370C4⋊C4—182no—
371C22⋊C4—184no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—62no—
377C22×C3—62no—
378C22×C3—62no—
379C22×C3—62no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3—94no—
383C22×C3—91no—
384C22×C3—91no—
385C22×C3—91no—
386C22×C3—91no—
387C22×C3—91no—
388C22×C3—91no—
389C22×C3—122no—
390C22×C3—122no—
391C22×C3—122no—
392C22×C3—182no—
393C22×C3—182no—
394C22×C3—182no—
395C22×C3C2×GO+(4,𝔽2)11yescharacteristic
396C2×D3—22no—
397C2×D3—22no—
398C2×D3—22no—
399C2×D3—64no—
400C2×D3—61no—
401C2×D3—61no—
402C2×D3—61no—
403C2×D3—61no—
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—61no—
408C2×D3—61no—
409C2×D3—61no—
410C2×D3—61no—
411C2×D3—61no—
412C2×D3—64no—
413C2×D3—64no—
414C2×D3—64no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—62no—
420C3×C4—62no—
421C3×C4—92no—
422C3×C4—122no—
423C3×C4—122no—
424C3×C4—122no—
425C3×C4—242no—
426C3×C4—362no—
427C3×C4D32⋊C412yescomplemented
428Dic3—22no—
429Dic3—42no—
430Dic3—62no—
431Dic3—62no—
432C32—21no—
433C32—21no—
434C32—41no—
435C32—41no—
436C32C3×#38011yescharacteristic, complemented
437C23—61no—
438C23—92no—
439C23—92no—
440C23—92no—
441C23—91no—
442C23—181no—
443C23—181no—
444C23—181no—
445C2×C4—61no—
446C2×C4—64no—
447C2×C4—62no—
448C2×C4—92no—
449C2×C4—91no—
450C2×C4—92no—
451C2×C4—92no—
452C2×C4—181no—
453C2×C4—184no—
454C2×C4—182no—
455C2×C4—182no—
456C2×C4—182no—
457C2×C4—182no—
458C2×C4—182no—
459C2×C4C3×GO+(4,𝔽2)11yescharacteristic
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—64no—
473C2×C3—64no—
474C2×C3—91no—
475C2×C3—91no—
476C2×C3—91no—
477C2×C3—91no—
478C2×C3—124no—
479C2×C3D32⋊Q811yescharacteristic
480C2×C3#6423011yescharacteristic
481C2×C3C2×D32⋊C411yescharacteristic
482D3—24no—
483D3—61no—
484D3—61no—
485D3—61no—
486D3—61no—
487D3—61no—
488D3—61no—
489D3—61no—
490D3—61no—
491C22—62no—
492C22—62no—
493C22—62no—
494C22—94no—
495C22—91no—
496C22—91no—
497C22—91no—
498C22—91no—
499C22—91no—
500C22—91no—
501C22—182no—
502C22—182no—
503C22—182no—
504C22C2×C3×GO+(4,𝔽2)11yescharacteristic
505C4—62no—
506C4—92no—
507C4—122no—
508C4—362no—
509C4C3×D32⋊C412yescomplemented
510C3—21no—
511C3—21no—
512C3—41no—
513C3—41no—
514C3#1059340311yescharacteristic, direct product factor
515C2—64no—
516C2—91no—
517C2—91no—
518C2—91no—
519C2—91no—
520C2C3×D32⋊Q811yescharacteristic
521C2C3×#6423011yescharacteristic
522C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×D32C22
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#380
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊#157C2semidirect
4. C3×#63900C2semidirect
5. C3×#63737C2semidirect
6. #10593403C3direct (N × H)
15. C3×C4×D32C4semidirect
22. C2×C4⋊#157C2×C3semidirect
25. #63900C2×C3semidirect
26. #63737C2×C3semidirect
71. C4×D32C3×C4semidirect
93. C3×D32C4⋊C4semidirect
95. C33×C4C22⋊C4semidirect
96. C3×C3⋊Dic3C22⋊C4semidirect
245. D32C3×C4⋊C4semidirect
253. C32×C4C3×C22⋊C4semidirect
254. C3⋊Dic3C3×C22⋊C4semidirect
268. C33#380semidirect
427. C3×C4D32⋊C4semidirect
436. C32C3×#380semidirect
509. C4C3×D32⋊C4semidirect
514. C3#10593403direct (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×D322
C2×C3×C4⋊#1572
C2×C3×D32⋊C42
C3×#639002
C3×#637372
#105934033
C3×#3809