Finite Group Databasebeta

← C3×#10593414 #13166778

Subgroups — C3×#10593414 #13166778

Order 1728.

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

normal subgroups only

This group has 766 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×#64308C211yesmaximal, characteristic, complemented
2C3×#64241C212yesmaximal, complemented
3C3×#63923C211yesmaximal, characteristic, complemented
4C3×#63774C211yesmaximal, characteristic, complemented
5C3×#63738C212yesmaximal, complemented
6#10593414C311yesmaximal, characteristic, direct product factor
7C22×C3×D32C2211yescharacteristic, complemented
8C2×C3×GO+(4,𝔽2)—22no—
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C32×#235—21no—
11C3×#3811—21no—
12C3×#3811C2212yes—
13C2×C3×#626—21no—
14C3×D3×D12—22no—
15C3×#3799—22no—
16C3×C22⋊#157C2212yescomplemented
17C3×D32⋊C4—22no—
18C32×(C2×D3)⋊C4—21no—
19C3×#3722C2211yescharacteristic, complemented
20C3×#1009—61no—
21#64308C2×C311yescharacteristic, complemented
22#64241C2×C312yescomplemented
23C3×#882—61no—
24#63923C2×C311yescharacteristic, complemented
25#63774C2×C311yescharacteristic, complemented
26#63738C2×C312yescomplemented
27C22×C3×C3⋊D3D411yescharacteristic, complemented
28C22×C3×C3⋊D3C2311yescharacteristic
29C22×C32×D3—21no—
30C2×C3×D32—22no—
31C2×C3×D32—21no—
32C2×C3×D32—42no—
33C2×C3×D32D412yescomplemented
34C2×C3×#157—22no—
35C3×GO+(4,𝔽2)—44no—
36C2×C33×C4D411yescharacteristic, complemented
37C3×C4⋊(C3⋊D3)—22no—
38C32×#67—22no—
39C2×C32×Dic3—21no—
40C32×D12—21no—
41C32×C4×D3—22no—
42C32×Dic6—21no—
43C3×#626—22no—
44C3×#626—22no—
45C3×#626—42no—
46C3×#624—21no—
47C3×#624D412yescomplemented
48C3×B(4,𝔽2)—91nomaximal
49C23×C3×D3—61no—
50C22×D32C22×C311yescharacteristic, complemented
51C2×GO+(4,𝔽2)—22no—
52C32×#41—61no—
53C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
54C2×C3×#67—61no—
55C3×#239—62no—
56C3×D3×D4—62no—
57C3×D3×D4—62no—
58C3×D3×D4—62no—
59C3×#235—21no—
60C3×#235—21no—
61C3×#235—41no—
62C3×#235—62no—
63C2×C3×D12—62no—
64C2×C3×D12—61no—
65C2×C3×D12—61no—
66C2×C3×D12—61no—
67#3811—21no—
68#3811C22×C312yes—
69C2×#626—21no—
70D3×D12—22no—
71#3799—22no—
72C22⋊#157C22×C312yescomplemented
73D32⋊C4—22no—
74C32×C22⋊C4—61no—
75C3×(C2×D3)⋊C4—21no—
76C3×(C2×D3)⋊C4—21no—
77C3×(C2×D3)⋊C4—41no—
78C3×(C2×D3)⋊C4—61no—
79#3722C22×C311yescharacteristic, complemented
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—21no—
82C2×C3×C3⋊D3—41no—
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—
89C3×D32—24no—
90C3×D32—42no—
91C3×D32—42no—
92C3×D32—81no—
93C3×#157—42no—
94C33×C4—21no—
95C32×Dic3—22no—
96C32×Dic3—41no—
97C3×H(5,𝔽2)—91no—
98#1009—61no—
99C3×#120—91no—
100C3×#120—92no—
101#882—61no—
102C3×#99—91no—
103C3×#99—92no—
104C23×C32—61no—
105C22×C3⋊D3C3×D411yescharacteristic, complemented
106C22×C3⋊D3C23×C311yescharacteristic
107C22×C3×D3—21no—
108C22×C3×D3—21no—
109C22×C3×D3—41no—
110C22×C3×D3—62no—
111C22×C3×D3—61no—
112C22×C3×D3—61no—
113C22×C3×D3—61no—
114C22×C3×D3—61no—
115C22×C3×D3—62no—
116C22×C3×D3—122no—
117C22×C3×D3—122no—
118C22×C3×D3—122no—
119C2×D32—21no—
120C2×D32—22no—
121C2×D32—42no—
122C2×D32C3×D412yescomplemented
123C2×#157—22no—
124GO+(4,𝔽2)—44no—
125C32×Q8—61no—
126C32×D4—61no—
127C32×D4—62no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4—62no—
133C2×C32×C4—61no—
134C2×C32×C4C3×D411yescharacteristic, complemented
135C4⋊(C3⋊D3)—22no—
136C3×#67—22no—
137C3×#67—22no—
138C3×#67—42no—
139C3×#67—62no—
140C3×#67—62no—
141C3×#67—122no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—41no—
145C3×D12—21no—
146C3×D12—21no—
147C3×D12—41no—
148C3×D12—62no—
149C3×D12—62no—
150C3×D12—62no—
151C3×D12—62no—
152C3×D12—122no—
153C3×D12—122no—
154C3×C4×D3—22no—
155C3×C4×D3—22no—
156C3×C4×D3—42no—
157C3×C4×D3—62no—
158C3×C4×D3—62no—
159C3×Dic6—21no—
160C3×Dic6—21no—
161C3×Dic6—41no—
162#626—22no—
163#626—22no—
164#626—42no—
165#624—21no—
166#624C3×D412yescomplemented
167B(4,𝔽2)—91noSylow
168C2×C33—21no—
169C2×C33#12011yescharacteristic
170C3×C3⋊D3—22no—
171C3×C3⋊D3—41no—
172C32×D3—42no—
173C32×D3—42no—
174C24×C3—91no—
175C23×D3—61no—
176C3×#41—61no—
177C3×#41—61no—
178C3×#41—121no—
179C3×#41—182no—
180C2×C3×D4—92no—
181C2×C3×D4—91no—
182C2×C3×D4—182no—
183C2×C3×D4—181no—
184C2×C3×D4—181no—
185C2×C3×D4—182no—
186C2×#67—61no—
187#239—62no—
188D3×D4—62no—
189D3×D4—62no—
190D3×D4—62no—
191#235—21no—
192#235—62no—
193C2×D12—61no—
194C2×D12—61no—
195C2×D12—62no—
196C2×D12—61no—
197C3×C22⋊C4—61no—
198C3×C22⋊C4—61no—
199C3×C22⋊C4—91no—
200C3×C22⋊C4—92no—
201C3×C22⋊C4—121no—
202C3×C22⋊C4—182no—
203(C2×D3)⋊C4—21no—
204(C2×D3)⋊C4—61no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—62no—
210C22×C32—62no—
211C22×C32—121no—
212C22×C32—121no—
213C22×C32C2×C3×D411yescharacteristic
214C2×C3⋊D3—21no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3—41no—
217C2×C3⋊D3C2×C3×D412yes—
218C2×C3×D3—22no—
219C2×C3×D3—22no—
220C2×C3×D3—22no—
221C2×C3×D3—22no—
222C2×C3×D3—42no—
223C2×C3×D3—42no—
224C2×C3×D3—41no—
225C2×C3×D3—41no—
226C2×C3×D3—41no—
227C2×C3×D3—41no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—81no—
232C2×C3×D3—81no—
233C2×C3×D3—124no—
234C2×C3×D3—121no—
235C2×C3×D3—124no—
236C2×C3×D3—121no—
237C2×C3×D3—121no—
238C2×C3×D3—121no—
239C2×C3×D3—121no—
240C2×C3×D3—242no—
241C2×C3×D3—241no—
242C2×C3×D3—241no—
243C2×C3×D3—242no—
244D32—24no—
245D32—42no—
246D32—42no—
247D32—81no—
248#157—42no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—41no—
253C32×C4—41no—
254C32×C4—41no—
255C32×C4—62no—
256C32×C4—81no—
257C32×C4—121no—
258C3×Dic3—22no—
259C3×Dic3—22no—
260C3×Dic3—42no—
261C3×Dic3—41no—
262C3×Dic3—41no—
263C3×Dic3—81no—
264H(5,𝔽2)—91no—
265#120—91no—
266#120—92no—
267#99—91no—
268#99—92no—
269C33B(4,𝔽2)11yescharacteristic, complemented, Sylow
270C23×C3—61no—
271C23×C3—61no—
272C23×C3—92no—
273C23×C3—91no—
274C23×C3—91no—
275C23×C3—121no—
276C23×C3—181no—
277C23×C3—182no—
278C23×C3—362no—
279C22×D3—21no—
280C22×D3—62no—
281C22×D3—61no—
282C22×D3—61no—
283C22×D3—61no—
284C22×D3—61no—
285C22×D3—62no—
286C22×D3—122no—
287C22×D3—122no—
288C22×D3—122no—
289C3×Q8—61no—
290C3×Q8—61no—
291C3×Q8—121no—
292C3×D4—61no—
293C3×D4—61no—
294C3×D4—62no—
295C3×D4—62no—
296C3×D4—121no—
297C3×D4—122no—
298C3×D4—182no—
299C3×D4—182no—
300C3×D4—182no—
301C3×D4—362no—
302C3×D4—364no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—92no—
312C2×C3×C4—122no—
313C2×C3×C4—121no—
314C2×C3×C4—181no—
315C2×C3×C4—182no—
316C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
317#67—22no—
318#67—62no—
319#67—62no—
320#67—122no—
321C2×Dic3—21no—
322D12—21no—
323D12—62no—
324D12—62no—
325D12—62no—
326D12—62no—
327D12—122no—
328D12—122no—
329C4×D3—22no—
330C4×D3—62no—
331C4×D3—62no—
332Dic6—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—81no—
344C2×C32—122no—
345C2×C32—122no—
346C2×C32C3×#12011yescharacteristic
347C3⋊D3—22no—
348C3⋊D3—41no—
349C3×D3—42no—
350C3×D3—42no—
351C3×D3—42no—
352C3×D3—42no—
353C3×D3—82no—
354C3×D3—82no—
355C3×D3—122no—
356C3×D3—122no—
357C3×D3—122no—
358C3×D3—241no—
359C24—91no—
360#41—61no—
361#41—182no—
362C2×D4—92no—
363C2×D4—91no—
364C2×D4—182no—
365C2×D4—181no—
366C2×D4—181no—
367C2×D4—182no—
368C22⋊C4—61no—
369C22⋊C4—91no—
370C22⋊C4—92no—
371C22⋊C4—182no—
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—92no—
381C22×C3—122no—
382C22×C3—122no—
383C22×C3—121no—
384C22×C3—121no—
385C22×C3—121no—
386C22×C3—121no—
387C22×C3—184no—
388C22×C3—181no—
389C22×C3—241no—
390C22×C3—241no—
391C22×C3—361no—
392C22×C3—361no—
393C22×C3—362no—
394C22×C3—362no—
395C22×C3—721no—
396C22×C3C2×GO+(4,𝔽2)11yescharacteristic
397C2×D3—22no—
398C2×D3—22no—
399C2×D3—41no—
400C2×D3—41no—
401C2×D3—62no—
402C2×D3—62no—
403C2×D3—62no—
404C2×D3—124no—
405C2×D3—121no—
406C2×D3—124no—
407C2×D3—121no—
408C2×D3—121no—
409C2×D3—121no—
410C2×D3—121no—
411C2×D3—242no—
412C2×D3—241no—
413C2×D3—241no—
414C2×D3—242no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—62no—
422C3×C4—62no—
423C3×C4—81no—
424C3×C4—122no—
425C3×C4—121no—
426C3×C4—121no—
427C3×C4—241no—
428C3×C4—362no—
429Dic3—22no—
430Dic3—41no—
431C32—21no—
432C32—21no—
433C32—41no—
434C32—41no—
435C32C3×B(4,𝔽2)11yescharacteristic, complemented
436C23—61no—
437C23—92no—
438C23—91no—
439C23—91no—
440C23—181no—
441C23—182no—
442C23—362no—
443Q8—61no—
444D4—61no—
445D4—62no—
446D4—182no—
447D4—182no—
448D4—182no—
449D4—362no—
450D4—364no—
451C2×C4—62no—
452C2×C4—61no—
453C2×C4—92no—
454C2×C4—181no—
455C2×C4—182no—
456C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—81no—
468C2×C3—122no—
469C2×C3—122no—
470C2×C3—122no—
471C2×C3—122no—
472C2×C3—182no—
473C2×C3—242no—
474C2×C3—242no—
475C2×C3—361no—
476C2×C3#6424111yescharacteristic
477D3—42no—
478D3—42no—
479D3—122no—
480D3—122no—
481D3—122no—
482D3—241no—
483C22—62no—
484C22—62no—
485C22—92no—
486C22—121no—
487C22—121no—
488C22—184no—
489C22—181no—
490C22—361no—
491C22—361no—
492C22—362no—
493C22—362no—
494C22—721no—
495C22C2×C3×GO+(4,𝔽2)11yescharacteristic
496C4—21no—
497C4—62no—
498C4—121no—
499C4—362no—
500C3—21no—
501C3—21no—
502C3—41no—
503C3—41no—
504C3#1059341411yescharacteristic, direct product factor
505C2—21no—
506C2—122no—
507C2—122no—
508C2—182no—
509C2—361no—
510C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC22×C3⋊D3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33#120
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×B(4,𝔽2)
LCSG ▷ C22×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64308C2semidirect
2. C3×#64241C2semidirect
3. C3×#63923C2semidirect
4. C3×#63774C2semidirect
5. C3×#63738C2semidirect
6. #10593414C3direct (N × H)
7. C22×C3×D32C22semidirect
16. C3×C22⋊#157C22semidirect
19. C3×#3722C22semidirect
21. #64308C2×C3semidirect
22. #64241C2×C3semidirect
24. #63923C2×C3semidirect
25. #63774C2×C3semidirect
26. #63738C2×C3semidirect
27. C22×C3×C3⋊D3D4semidirect
33. C2×C3×D32D4semidirect
36. C2×C33×C4D4semidirect
47. C3×#624D4semidirect
50. C22×D32C22×C3semidirect
72. C22⋊#157C22×C3semidirect
79. #3722C22×C3semidirect
105. C22×C3⋊D3C3×D4semidirect
122. C2×D32C3×D4semidirect
134. C2×C32×C4C3×D4semidirect
166. #624C3×D4semidirect
269. C33B(4,𝔽2)semidirect
316. C2×C3×C4GO+(4,𝔽2)semidirect
435. C32C3×B(4,𝔽2)semidirect
456. C2×C4C3×GO+(4,𝔽2)semidirect
504. C3#10593414direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643082
C3×#642412
C3×#639232
C3×#637742
C3×#637382
#105934143
C3×B(4,𝔽2)9