Finite Group Databasebeta

← C3×#10593455 #13166824

Subgroups — C3×#10593455 #13166824

Order 1728.

1100 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1100 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C22×C3×GO+(4,𝔽2)C212yesmaximal, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C2×C3×D32⋊C4C212yesmaximal, complemented
4C3×#63921C211yesmaximal, characteristic, complemented
5C3×#63781C211yesmaximal, characteristic, complemented
6#10593455C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2211yescharacteristic
8C22×C3×D32C2212yes—
9C22×C3×#157C2212yes—
10C2×C3×GO+(4,𝔽2)—24no—
11C2×C3×GO+(4,𝔽2)—24no—
12C2×C3×GO+(4,𝔽2)—24no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×D12—21no—
15C2×C3×#625—21no—
16C2×C3×#624C2211yescharacteristic
17C3×#3802—24no—
18C3×D32⋊C4—24no—
19C32×(C2×D3)⋊C4—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C22×GO+(4,𝔽2)C2×C312yescomplemented
22C2×C3×D3×D4—61no—
23C2×#3802C2×C311yescharacteristic, complemented
24C2×D32⋊C4C2×C312yescomplemented
25C3×D3×C22⋊C4—61no—
26#63921C2×C311yescharacteristic, complemented
27#63781C2×C311yescharacteristic, complemented
28C22×C3×C3⋊D3C2311yescharacteristic
29C22×C32×D3—22no—
30C22×C32×D3—21no—
31C2×C3×D32—22no—
32C2×C3×D32—22no—
33C2×C3×D32—22no—
34C2×C3×D32—22no—
35C2×C3×D32—22no—
36C2×C3×D32—22no—
37C2×C3×D32—22no—
38C2×C3×D32—21no—
39C2×C3×D32D412yescomplemented
40C2×C3×D32D414yescomplemented
41C2×C3×#157—22no—
42C2×C3×#157—22no—
43C2×C3×#157D414yescomplemented
44C3×GO+(4,𝔽2)—216no—
45C2×C33×C4D411yescharacteristic, complemented
46C2×C3×C3⋊Dic3D411yescharacteristic, complemented
47C3×C4×C3⋊D3—22no—
48C2×C32×Dic3—21no—
49C32×D12—24no—
50C3×#625—24no—
51C3×#624—22no—
52C3×#392—91nomaximal
53C23×C3×D3—61no—
54C23×C3×D3—62no—
55C22×D32C22×C312yes—
56C22×D32C22×C311yescharacteristic
57C22×#157C22×C312yes—
58C2×GO+(4,𝔽2)—24no—
59C2×GO+(4,𝔽2)—24no—
60C2×GO+(4,𝔽2)—24no—
61C2×C32×D4—61no—
62C2×C4×C3⋊D3C22×C311yescharacteristic
63C2×C3×#67—62no—
64C3×D3×D4—68no—
65C2×C3×D12—21no—
66C2×C3×D12—21no—
67C2×C3×D12—41no—
68C2×C3×C4×D3—61no—
69C2×C3×C4×D3—61no—
70C2×C3×C4×D3—61no—
71C2×#625—21no—
72C2×#624C22×C311yescharacteristic
73#3802—24no—
74D32⋊C4—24no—
75C32×C22⋊C4—61no—
76C3×#217—61no—
77C3×(C2×D3)⋊C4—21no—
78C3×(C2×D3)⋊C4—21no—
79C3×(C2×D3)⋊C4—41no—
80C3×(C2×D3)⋊C4—61no—
81(C2×D3)⋊Dic3—21no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3C2×D411yescharacteristic
84C2×C3×C3⋊D3C2×D412yes—
85C2×C3×C3⋊D3C2×D412yes—
86C2×C3×C3⋊D3#4111yescharacteristic
87C2×C32×D3—24no—
88C2×C32×D3—22no—
89C2×C32×D3—41no—
90C2×C32×D3—42no—
91C2×C32×D3—41no—
92C2×C32×D3—42no—
93C3×D32—24no—
94C3×D32—24no—
95C3×D32—24no—
96C3×D32—42no—
97C3×D32—42no—
98C3×D32—42no—
99C3×#157—24no—
100C33×C4—21no—
101C3×C3⋊Dic3—21no—
102C32×Dic3—41no—
103C22×C3×D4—91no—
104C22×C3×D4—92no—
105C2×D3×D4—61no—
106C2×C3×C22⋊C4—91no—
107C2×C3×C22⋊C4—92no—
108D3×C22⋊C4—61no—
109C3×#95—91no—
110C23×C32—61no—
111C23×C32—62no—
112C22×C3⋊D3C23×C311yescharacteristic
113C22×C3×D3—21no—
114C22×C3×D3—22no—
115C22×C3×D3—21no—
116C22×C3×D3—22no—
117C22×C3×D3—41no—
118C22×C3×D3—42no—
119C22×C3×D3—62no—
120C22×C3×D3—64no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C22×C3×D3—62no—
124C22×C3×D3—64no—
125C22×C3×D3—122no—
126C22×C3×D3—122no—
127C22×C3×D3—122no—
128C22×C3×D3—122no—
129C22×C3×D3—122no—
130C22×C3×D3—122no—
131C2×D32—22no—
132C2×D32—21no—
133C2×D32—22no—
134C2×D32—22no—
135C2×D32—22no—
136C2×D32—22no—
137C2×D32—22no—
138C2×D32—22no—
139C2×D32C3×D412yescomplemented
140C2×D32C3×D414yescomplemented
141C2×#157—22no—
142C2×#157—22no—
143C2×#157C3×D414yescomplemented
144GO+(4,𝔽2)—216no—
145C32×D4—64no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—41no—
149C2×C32×C4—41no—
150C2×C32×C4—61no—
151C2×C32×C4C3×D411yescharacteristic, complemented
152C2×C3⋊Dic3C3×D411yescharacteristic, complemented
153C4×C3⋊D3—22no—
154C3×#67—68no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—41no—
158C2×C3×Dic3—61no—
159C2×C3×Dic3—61no—
160C3×D12—24no—
161C3×D12—24no—
162C3×D12—44no—
163C3×C4×D3—64no—
164C3×C4×D3—122no—
165C3×C4×D3—122no—
166#625—24no—
167#624—22no—
168#392—91noSylow
169C2×C33#12011yescharacteristic
170C2×C33#12111yescharacteristic
171C2×C33#12711yescharacteristic
172C3×C3⋊D3#12012yes—
173C3×C3⋊D3#12112yes—
174C32×D3—42no—
175C32×D3—44no—
176C24×C3—91no—
177C24×C3—92no—
178C23×D3—61no—
179C23×D3—62no—
180C2×C3×D4—61no—
181C2×C3×D4—61no—
182C2×C3×D4—121no—
183C2×C3×D4—181no—
184C2×C3×D4—184no—
185C2×C3×D4—184no—
186C2×C3×D4—184no—
187C2×C3×D4—184no—
188C22×C3×C4—91no—
189C22×C3×C4—92no—
190C22×C3×C4—91no—
191C2×#67—62no—
192D3×D4—68no—
193C2×D12—21no—
194C2×C4×D3—61no—
195C2×C4×D3—61no—
196C2×C4×D3—61no—
197C3×C22⋊C4—61no—
198C3×C22⋊C4—61no—
199C3×C22⋊C4—121no—
200C3×C22⋊C4—181no—
201C3×C22⋊C4—184no—
202#217—61no—
203(C2×D3)⋊C4—21no—
204(C2×D3)⋊C4—61no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—64no—
210C22×C32—62no—
211C22×C32—122no—
212C22×C32—121no—
213C22×C32—122no—
214C22×C32—121no—
215C22×C32C2×C3×D411yescharacteristic
216C2×C3⋊D3C2×C3×D411yescharacteristic
217C2×C3⋊D3C2×C3×D412yes—
218C2×C3⋊D3C2×C3×D412yes—
219C2×C3⋊D3C3×#4111yescharacteristic
220C2×C3×D3—24no—
221C2×C3×D3—22no—
222C2×C3×D3—24no—
223C2×C3×D3—22no—
224C2×C3×D3—44no—
225C2×C3×D3—42no—
226C2×C3×D3—42no—
227C2×C3×D3—41no—
228C2×C3×D3—42no—
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—42no—
232C2×C3×D3—41no—
233C2×C3×D3—42no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C2×C3×D3—62no—
237C2×C3×D3—61no—
238C2×C3×D3—61no—
239C2×C3×D3—62no—
240C2×C3×D3—61no—
241C2×C3×D3—61no—
242C2×C3×D3—81no—
243C2×C3×D3—82no—
244C2×C3×D3—81no—
245C2×C3×D3—82no—
246C2×C3×D3—124no—
247C2×C3×D3—124no—
248C2×C3×D3—124no—
249C2×C3×D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3—124no—
252D32—24no—
253D32—24no—
254D32—24no—
255D32—42no—
256D32—42no—
257D32—42no—
258#157—24no—
259C32×C4—22no—
260C32×C4—21no—
261C32×C4—41no—
262C32×C4—42no—
263C32×C4—81no—
264C32×C4—121no—
265C3⋊Dic3—21no—
266C3×Dic3—41no—
267C3×Dic3—41no—
268C3×Dic3—62no—
269C3×Dic3—81no—
270C3×Dic3—121no—
271C22×D4—91no—
272C22×D4—92no—
273C2×C22⋊C4—91no—
274C2×C22⋊C4—92no—
275#95—91no—
276C33#39211yescharacteristic, complemented, Sylow
277C23×C3—61no—
278C23×C3—62no—
279C23×C3—61no—
280C23×C3—62no—
281C23×C3—92no—
282C23×C3—94no—
283C23×C3—91no—
284C23×C3—121no—
285C23×C3—122no—
286C23×C3—182no—
287C23×C3—181no—
288C23×C3—182no—
289C23×C3—182no—
290C23×C3—182no—
291C23×C3—182no—
292C23×C3—182no—
293C23×C3—182no—
294C22×D3—21no—
295C22×D3—22no—
296C22×D3—64no—
297C22×D3—62no—
298C22×D3—61no—
299C22×D3—61no—
300C22×D3—64no—
301C22×D3—62no—
302C22×D3—122no—
303C22×D3—122no—
304C22×D3—122no—
305C22×D3—122no—
306C22×D3—122no—
307C22×D3—122no—
308C3×D4—64no—
309C3×D4—64no—
310C3×D4—124no—
311C3×D4—184no—
312C3×D4—1816no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—94no—
320C2×C3×C4—91no—
321C2×C3×C4—121no—
322C2×C3×C4—182no—
323C2×C3×C4—182no—
324C2×C3×C4—182no—
325C2×C3×C4—182no—
326C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
327#67—68no—
328C2×Dic3—21no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331D12—24no—
332C4×D3—64no—
333C4×D3—122no—
334C4×D3—122no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32—41no—
347C2×C32—124no—
348C2×C32—122no—
349C2×C32C3×#12011yescharacteristic
350C2×C32C3×#12111yescharacteristic
351C2×C32C3×#12711yescharacteristic
352C3⋊D3C3×#12012yes—
353C3⋊D3C3×#12112yes—
354C3×D3—44no—
355C3×D3—42no—
356C3×D3—42no—
357C3×D3—44no—
358C3×D3—62no—
359C3×D3—62no—
360C3×D3—62no—
361C3×D3—62no—
362C3×D3—84no—
363C3×D3—82no—
364C24—91no—
365C24—92no—
366C2×D4—61no—
367C2×D4—181no—
368C2×D4—184no—
369C2×D4—184no—
370C2×D4—184no—
371C2×D4—184no—
372C22×C4—91no—
373C22×C4—92no—
374C22×C4—91no—
375C22⋊C4—61no—
376C22⋊C4—181no—
377C22⋊C4—184no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—64no—
383C22×C3—62no—
384C22×C3—62no—
385C22×C3—64no—
386C22×C3—91no—
387C22×C3—91no—
388C22×C3—92no—
389C22×C3—92no—
390C22×C3—124no—
391C22×C3—122no—
392C22×C3—121no—
393C22×C3—122no—
394C22×C3—121no—
395C22×C3—122no—
396C22×C3—122no—
397C22×C3—121no—
398C22×C3—122no—
399C22×C3—121no—
400C22×C3—184no—
401C22×C3—184no—
402C22×C3—184no—
403C22×C3—241no—
404C22×C3—242no—
405C22×C3—241no—
406C22×C3—242no—
407C22×C3—362no—
408C22×C3—362no—
409C22×C3—362no—
410C22×C3C2×GO+(4,𝔽2)11yescharacteristic
411C2×D3—24no—
412C2×D3—22no—
413C2×D3—41no—
414C2×D3—42no—
415C2×D3—41no—
416C2×D3—42no—
417C2×D3—62no—
418C2×D3—62no—
419C2×D3—62no—
420C2×D3—61no—
421C2×D3—61no—
422C2×D3—61no—
423C2×D3—62no—
424C2×D3—61no—
425C2×D3—124no—
426C2×D3—124no—
427C2×D3—124no—
428C2×D3—124no—
429C2×D3—124no—
430C2×D3—124no—
431C3×C4—22no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—42no—
435C3×C4—81no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—184no—
439C3×C4—181no—
440C3×C4—241no—
441Dic3—41no—
442Dic3—62no—
443Dic3—121no—
444C32—21no—
445C32—21no—
446C32—41no—
447C32—41no—
448C32C3×#39211yescharacteristic, complemented
449C23—61no—
450C23—62no—
451C23—94no—
452C23—92no—
453C23—91no—
454C23—182no—
455C23—181no—
456C23—182no—
457C23—182no—
458C23—182no—
459C23—182no—
460C23—182no—
461C23—182no—
462D4—64no—
463D4—184no—
464D4—1816no—
465C2×C4—61no—
466C2×C4—94no—
467C2×C4—91no—
468C2×C4—182no—
469C2×C4—182no—
470C2×C4—182no—
471C2×C4—182no—
472C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—92no—
486C2×C3—92no—
487C2×C3—124no—
488C2×C3—122no—
489C2×C3—122no—
490C2×C3—124no—
491C2×C3—244no—
492C2×C3—242no—
493C2×C3#6423011yescharacteristic
494C2×C3#6423111yescharacteristic
495C2×C3#6424111yescharacteristic
496D3—44no—
497D3—42no—
498D3—62no—
499D3—62no—
500D3—62no—
501D3—62no—
502C22—64no—
503C22—62no—
504C22—92no—
505C22—91no—
506C22—91no—
507C22—92no—
508C22—121no—
509C22—122no—
510C22—121no—
511C22—122no—
512C22—184no—
513C22—184no—
514C22—184no—
515C22—362no—
516C22—362no—
517C22—362no—
518C22C2×C3×GO+(4,𝔽2)11yescharacteristic
519C4—21no—
520C4—121no—
521C4—184no—
522C4—181no—
523C3—21no—
524C3—21no—
525C3—41no—
526C3—41no—
527C3#1059345511yescharacteristic, direct product factor
528C2—92no—
529C2—92no—
530C2—124no—
531C2—122no—
532C2C3×#6423011yescharacteristic
533C2C3×#6423111yescharacteristic
534C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#392
LCSG ▷ C22×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×GO+(4,𝔽2)C2semidirect
2. C2×C3×#3802C2semidirect
3. C2×C3×D32⋊C4C2semidirect
4. C3×#63921C2semidirect
5. C3×#63781C2semidirect
6. #10593455C3direct (N × H)
21. C22×GO+(4,𝔽2)C2×C3semidirect
23. C2×#3802C2×C3semidirect
24. C2×D32⋊C4C2×C3semidirect
26. #63921C2×C3semidirect
27. #63781C2×C3semidirect
39. C2×C3×D32D4semidirect
40. C2×C3×D32D4semidirect
43. C2×C3×#157D4semidirect
45. C2×C33×C4D4semidirect
46. C2×C3×C3⋊Dic3D4semidirect
139. C2×D32C3×D4semidirect
140. C2×D32C3×D4semidirect
143. C2×#157C3×D4semidirect
151. C2×C32×C4C3×D4semidirect
152. C2×C3⋊Dic3C3×D4semidirect
276. C33#392semidirect
326. C2×C3×C4GO+(4,𝔽2)semidirect
448. C32C3×#392semidirect
472. C2×C4C3×GO+(4,𝔽2)semidirect
527. C3#10593455direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C2×C3×#38022
C2×C3×D32⋊C42
C3×#639212
C3×#637812
#105934553
C3×#3929