Finite Group Databasebeta

← C3×#10593517 #13166883

Subgroups — C3×#10593517 #13166883

Order 1728.

800 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C3×#64330C211yesmaximal, characteristic, complemented
2C3×#64241C212yesmaximal, complemented
3C3×#63983C211yesmaximal, characteristic, complemented
4C3×#63786C211yesmaximal, characteristic, complemented
5C3×#63738C212yesmaximal, complemented
6#10593517C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2211yescharacteristic, complemented
8C2×C3×GO+(4,𝔽2)—22no—
9C2×C3×#638C2211yescharacteristic
10C2×C32×#67—21no—
11C2×C32×#67—21no—
12C3×#3811C2212yes—
13C3×D3×#67—22no—
14C3×#3805—21no—
15C3×#3804—22no—
16C3×C22⋊#157C2212yescomplemented
17C3×D32⋊C4—22no—
18C3×#3722C2211yescharacteristic, complemented
19C3×(C2×D3)⋊Dic3—21no—
20C3×#1004—61no—
21#64330C2×C311yescharacteristic, complemented
22#64241C2×C312yescomplemented
23C3×#937—61no—
24#63983C2×C311yescharacteristic, complemented
25#63786C2×C311yescharacteristic, complemented
26#63738C2×C312yescomplemented
27C23×C33D411yescharacteristic, complemented
28C22×C3×C3⋊D3C2311yescharacteristic
29C22×C32×D3—21no—
30C22×C32×D3—21no—
31C2×C3×D32—22no—
32C2×C3×D32—42no—
33C2×C3×D32D412yescomplemented
34C2×C3×#157—22no—
35C3×GO+(4,𝔽2)—44no—
36C3×#638—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic, complemented
38C32×#67—22no—
39C32×#67—22no—
40C32×#67—42no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—21no—
43C3×#627—21no—
44C3×#626—22no—
45C3×#625—21no—
46C3×#624D412yescomplemented
47C3×D3×Dic3—22no—
48C3×B(4,𝔽2)—91nomaximal
49C23×C3×D3—61no—
50C22×D32C22×C311yescharacteristic, complemented
51C2×GO+(4,𝔽2)—22no—
52C2×C32×D4—61no—
53C2×C32×D4—61no—
54C2×#638C22×C311yescharacteristic
55C2×C3×#67—21no—
56C2×C3×#67—21no—
57C2×C3×#67—21no—
58C2×C3×#67—21no—
59C2×C3×#67—41no—
60C2×C3×#67—41no—
61C2×C3×#67—62no—
62C2×C3×#67—61no—
63C2×C3×#67—61no—
64C3×#237—62no—
65C3×#237—62no—
66C3×D3×D4—62no—
67C3×D3×D4—62no—
68C3×#235—62no—
69#3811C22×C312yes—
70D3×#67—22no—
71#3805—21no—
72#3804—22no—
73C22⋊#157C22×C312yescomplemented
74D32⋊C4—22no—
75C3×#217—61no—
76C3×(C2×D3)⋊C4—61no—
77C3×(C2×D3)⋊C4—61no—
78#3722C22×C311yescharacteristic, complemented
79(C2×D3)⋊Dic3—21no—
80C22×C33—21no—
81C22×C33—41no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3C2×D412yes—
85C2×C32×D3—22no—
86C2×C32×D3—22no—
87C2×C32×D3—41no—
88C2×C32×D3—41no—
89C2×C32×D3—81no—
90C3×D32—24no—
91C3×D32—42no—
92C3×D32—42no—
93C3×#157—42no—
94C3×C3⋊Dic3—21no—
95C32×Dic3—22no—
96C32×Dic3—41no—
97C3×H(5,𝔽2)—91no—
98#1004—61no—
99C3×#120—91no—
100C3×#120—92no—
101#937—61no—
102C3×#99—91no—
103C3×#99—92no—
104C23×C32—21no—
105C23×C32—21no—
106C23×C32—41no—
107C23×C32—41no—
108C23×C32—61no—
109C23×C32—61no—
110C23×C32C3×D411yescharacteristic, complemented
111C22×C3⋊D3C23×C311yescharacteristic
112C22×C3×D3—21no—
113C22×C3×D3—21no—
114C22×C3×D3—21no—
115C22×C3×D3—21no—
116C22×C3×D3—41no—
117C22×C3×D3—41no—
118C22×C3×D3—62no—
119C22×C3×D3—61no—
120C22×C3×D3—61no—
121C22×C3×D3—122no—
122C22×C3×D3—122no—
123C22×C3×D3—122no—
124C2×D32—22no—
125C2×D32—42no—
126C2×D32C3×D412yescomplemented
127C2×#157—22no—
128GO+(4,𝔽2)—44no—
129C32×D4—62no—
130C32×D4—62no—
131C32×D4—122no—
132C2×C32×C4—61no—
133C2×C32×C4—61no—
134#638—22no—
135C2×C3⋊Dic3C3×D411yescharacteristic, complemented
136C3×#67—22no—
137C3×#67—22no—
138C3×#67—22no—
139C3×#67—22no—
140C3×#67—42no—
141C3×#67—42no—
142C3×#67—42no—
143C3×#67—42no—
144C3×#67—44no—
145C3×#67—62no—
146C3×#67—62no—
147C3×#67—62no—
148C3×#67—62no—
149C3×#67—122no—
150C2×C3×Dic3—21no—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—41no—
155C2×C3×Dic3—41no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3—62no—
159C3×D12—62no—
160C3×C4×D3—62no—
161C3×C4×D3—62no—
162C3×Dic6—62no—
163#627—21no—
164#626—22no—
165#625—21no—
166#624C3×D412yescomplemented
167D3×Dic3—22no—
168B(4,𝔽2)—91noSylow
169C2×C33—21no—
170C2×C33—41no—
171C2×C33#12011yescharacteristic
172C3×C3⋊D3—22no—
173C32×D3—42no—
174C32×D3—42no—
175C24×C3—91no—
176C23×D3—61no—
177C3×#41—182no—
178C3×#41—181no—
179C2×C3×D4—61no—
180C2×C3×D4—61no—
181C2×C3×D4—61no—
182C2×C3×D4—61no—
183C2×C3×D4—92no—
184C2×C3×D4—91no—
185C2×C3×D4—121no—
186C2×C3×D4—121no—
187C2×C3×D4—182no—
188C2×C3×D4—182no—
189C2×#67—21no—
190C2×#67—21no—
191C2×#67—61no—
192C2×#67—61no—
193C2×#67—62no—
194#237—62no—
195#237—62no—
196D3×D4—62no—
197D3×D4—62no—
198#235—62no—
199C3×C22⋊C4—91no—
200C3×C22⋊C4—92no—
201C3×C22⋊C4—181no—
202C3×C22⋊C4—182no—
203#217—61no—
204(C2×D3)⋊C4—61no—
205(C2×D3)⋊C4—61no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—42no—
220C22×C32—62no—
221C22×C32—62no—
222C22×C32—81no—
223C22×C32—81no—
224C22×C32—81no—
225C22×C32—82no—
226C22×C32—121no—
227C22×C32—121no—
228C22×C32—241no—
229C22×C32C2×C3×D411yescharacteristic
230C2×C3⋊D3—41no—
231C2×C3⋊D3C2×C3×D412yes—
232C2×C3×D3—22no—
233C2×C3×D3—22no—
234C2×C3×D3—22no—
235C2×C3×D3—22no—
236C2×C3×D3—42no—
237C2×C3×D3—42no—
238C2×C3×D3—41no—
239C2×C3×D3—41no—
240C2×C3×D3—41no—
241C2×C3×D3—41no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—81no—
245C2×C3×D3—81no—
246C2×C3×D3—81no—
247C2×C3×D3—81no—
248C2×C3×D3—82no—
249C2×C3×D3—124no—
250C2×C3×D3—124no—
251C2×C3×D3—121no—
252C2×C3×D3—121no—
253C2×C3×D3—242no—
254C2×C3×D3—241no—
255D32—24no—
256D32—42no—
257D32—42no—
258#157—42no—
259C32×C4—62no—
260C32×C4—121no—
261C3⋊Dic3—21no—
262C3×Dic3—22no—
263C3×Dic3—22no—
264C3×Dic3—42no—
265C3×Dic3—41no—
266C3×Dic3—41no—
267C3×Dic3—42no—
268C3×Dic3—62no—
269C3×Dic3—121no—
270H(5,𝔽2)—91no—
271#120—91no—
272#120—92no—
273#99—91no—
274#99—92no—
275C33B(4,𝔽2)11yescharacteristic, complemented, Sylow
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—61no—
281C23×C3—61no—
282C23×C3—61no—
283C23×C3—61no—
284C23×C3—92no—
285C23×C3—91no—
286C23×C3—121no—
287C23×C3—121no—
288C23×C3—182no—
289C23×C3—362no—
290C23×C3GO+(4,𝔽2)11yescharacteristic, complemented
291C22×D3—21no—
292C22×D3—21no—
293C22×D3—62no—
294C22×D3—61no—
295C22×D3—61no—
296C22×D3—122no—
297C22×D3—122no—
298C22×D3—122no—
299C3×Q8—181no—
300C3×D4—62no—
301C3×D4—62no—
302C3×D4—62no—
303C3×D4—62no—
304C3×D4—122no—
305C3×D4—122no—
306C3×D4—122no—
307C3×D4—122no—
308C3×D4—124no—
309C3×D4—182no—
310C3×D4—182no—
311C3×D4—181no—
312C3×D4—364no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—92no—
318C2×C3×C4—91no—
319C2×C3×C4—121no—
320C2×C3×C4—121no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323#67—22no—
324#67—22no—
325#67—42no—
326#67—62no—
327#67—62no—
328#67—62no—
329#67—62no—
330#67—122no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—62no—
336D12—62no—
337C4×D3—62no—
338C4×D3—62no—
339Dic6—62no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—41no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—42no—
353C2×C32—41no—
354C2×C32—81no—
355C2×C32—81no—
356C2×C32—81no—
357C2×C32—82no—
358C2×C32—122no—
359C2×C32—122no—
360C2×C32C3×#12011yescharacteristic
361C3⋊D3—22no—
362C3×D3—42no—
363C3×D3—42no—
364C3×D3—42no—
365C3×D3—42no—
366C3×D3—82no—
367C3×D3—82no—
368C3×D3—122no—
369C3×D3—122no—
370C24—91no—
371#41—182no—
372#41—181no—
373C2×D4—61no—
374C2×D4—61no—
375C2×D4—92no—
376C2×D4—91no—
377C2×D4—182no—
378C2×D4—182no—
379C22⋊C4—91no—
380C22⋊C4—92no—
381C22⋊C4—181no—
382C22⋊C4—182no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—42no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—62no—
398C22×C3—62no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3—82no—
405C22×C3—92no—
406C22×C3—122no—
407C22×C3—122no—
408C22×C3—121no—
409C22×C3—121no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3—184no—
413C22×C3—241no—
414C22×C3—241no—
415C22×C3—241no—
416C22×C3—241no—
417C22×C3—242no—
418C22×C3—361no—
419C22×C3—362no—
420C22×C3—362no—
421C22×C3C2×GO+(4,𝔽2)11yescharacteristic
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—41no—
425C2×D3—41no—
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—81no—
429C2×D3—124no—
430C2×D3—124no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—242no—
434C2×D3—241no—
435C3×C4—62no—
436C3×C4—62no—
437C3×C4—122no—
438C3×C4—121no—
439C3×C4—121no—
440C3×C4—122no—
441C3×C4—181no—
442C3×C4—362no—
443Dic3—22no—
444Dic3—41no—
445Dic3—62no—
446Dic3—121no—
447C32—21no—
448C32—21no—
449C32—41no—
450C32—41no—
451C32C3×B(4,𝔽2)11yescharacteristic, complemented
452C23—61no—
453C23—61no—
454C23—92no—
455C23—91no—
456C23—182no—
457C23—362no—
458C23C3×GO+(4,𝔽2)11yescharacteristic, complemented
459Q8—181no—
460D4—62no—
461D4—62no—
462D4—122no—
463D4—182no—
464D4—182no—
465D4—181no—
466D4—364no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—92no—
470C2×C4—91no—
471C2×C4—182no—
472C2×C4—182no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3—42no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—41no—
487C2×C3—81no—
488C2×C3—81no—
489C2×C3—81no—
490C2×C3—82no—
491C2×C3—122no—
492C2×C3—122no—
493C2×C3—122no—
494C2×C3—122no—
495C2×C3—182no—
496C2×C3—242no—
497C2×C3—242no—
498C2×C3#6424111yescharacteristic
499D3—42no—
500D3—42no—
501D3—122no—
502D3—122no—
503C22—21no—
504C22—41no—
505C22—62no—
506C22—62no—
507C22—92no—
508C22—121no—
509C22—121no—
510C22—184no—
511C22—241no—
512C22—361no—
513C22—362no—
514C22—362no—
515C22C2×C3×GO+(4,𝔽2)11yescharacteristic
516C4—62no—
517C4—121no—
518C4—181no—
519C4—362no—
520C3—21no—
521C3—21no—
522C3—41no—
523C3—41no—
524C3#1059351711yescharacteristic, direct product factor
525C2—21no—
526C2—41no—
527C2—122no—
528C2—122no—
529C2—182no—
530C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC2×C33#120
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×B(4,𝔽2)
LCSG ▷ C22×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C3×#64330C2semidirect
2. C3×#64241C2semidirect
3. C3×#63983C2semidirect
4. C3×#63786C2semidirect
5. C3×#63738C2semidirect
6. #10593517C3direct (N × H)
7. C22×C3×D32C22semidirect
16. C3×C22⋊#157C22semidirect
18. C3×#3722C22semidirect
21. #64330C2×C3semidirect
22. #64241C2×C3semidirect
24. #63983C2×C3semidirect
25. #63786C2×C3semidirect
26. #63738C2×C3semidirect
27. C23×C33D4semidirect
33. C2×C3×D32D4semidirect
37. C2×C3×C3⋊Dic3D4semidirect
46. C3×#624D4semidirect
50. C22×D32C22×C3semidirect
73. C22⋊#157C22×C3semidirect
78. #3722C22×C3semidirect
110. C23×C32C3×D4semidirect
126. C2×D32C3×D4semidirect
135. C2×C3⋊Dic3C3×D4semidirect
166. #624C3×D4semidirect
275. C33B(4,𝔽2)semidirect
290. C23×C3GO+(4,𝔽2)semidirect
451. C32C3×B(4,𝔽2)semidirect
458. C23C3×GO+(4,𝔽2)semidirect
524. C3#10593517direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643302
C3×#642412
C3×#639832
C3×#637862
C3×#637382
#105935173
C3×B(4,𝔽2)9