Finite Group Databasebeta

← C3×#10593338 #13166771

Subgroups — C3×#10593338 #13166771

Order 1728.

564 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3773C211yesmaximal, characteristic
2C3×#63883C211yesmaximal, characteristic
3C3×C4×#624C211yesmaximal, characteristic
4C3×#63781C211yesmaximal, characteristic
5C3×#63737C211yesmaximal, characteristic
6C3×#63737C211yesmaximal, characteristic
7C3×#63737C211yesmaximal, characteristic
8#10593338C311yesmaximal, characteristic, direct product factor
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#624C2211yescharacteristic
13C2×C3×#624C2211yescharacteristic
14C2×C3×#624C2211yescharacteristic
15C2×C3×#624C2211yescharacteristic
16C3×#3773C414yescomplemented
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3—21no—
19C3×Dic3⋊Dic3—21no—
20C3×Dic32—21no—
21C2×#3773C2×C311yescharacteristic
22C3×D3×C4⋊C4—61no—
23C3×C42×D3—61no—
24#63883C2×C311yescharacteristic
25C4×#624C2×C311yescharacteristic
26#63781C2×C311yescharacteristic
27#63737C2×C311yescharacteristic
28#63737C2×C311yescharacteristic
29#63737C2×C311yescharacteristic
30C22×C3×C3⋊D3C2311yescharacteristic
31C2×C3×#157—21no—
32C2×C3×#157—21no—
33C2×C3×#157—21no—
34C2×C3×#157—21no—
35C2×C3×#157—21no—
36C2×C3×#157C2×C412yes—
37C2×C33×C4D411yescharacteristic
38C2×C3×C3⋊Dic3D411yescharacteristic
39C3×C4×C3⋊D3—21no—
40C3×C4×C3⋊D3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3—21no—
45C3×#624—21no—
46C3×#624—21no—
47C3×#624—21no—
48C3×#624—21no—
49C3×#624—21no—
50C3×#624C2×C412yes—
51C3×#624C2×C412yes—
52C3×#624Q812yes—
53C3×#387—91nomaximal
54C22×#157C22×C311yescharacteristic
55C22×#157C22×C311yescharacteristic
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×C4×D3—61no—
58C2×C3×C4×D3—61no—
59C2×C3×C4×D3—61no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—61no—
62C2×C3×C4×D3—61no—
63C2×#624C22×C311yescharacteristic
64C2×#624C22×C311yescharacteristic
65C2×#624C22×C311yescharacteristic
66C2×#624C22×C311yescharacteristic
67#3773C3×C414yescomplemented
68C32×C4⋊C4—61no—
69C32×C42—61no—
70C3×C4⋊Dic3—61no—
71C3×Dic3⋊C4—21no—
72C3×Dic3⋊C4—21no—
73C3×Dic3⋊C4—41no—
74C3×Dic3⋊C4—61no—
75C3×C4×Dic3—21no—
76C3×C4×Dic3—21no—
77C3×C4×Dic3—41no—
78C3×C4×Dic3—61no—
79C3×C4×Dic3—61no—
80Dic3⋊Dic3—21no—
81Dic32—21no—
82C22×C33C2×D411yescharacteristic
83C2×C3×C3⋊D3C22×C411yescharacteristic
84C2×C3×C3⋊D3C2×Q811yescharacteristic
85C2×C3×C3⋊D3#4111yescharacteristic
86C2×C3×C3⋊D3#4111yescharacteristic
87C2×C3×C3⋊D3#4111yescharacteristic
88C2×C3×C3⋊D3#4111yescharacteristic
89C3×#157—22no—
90C3×#157—41no—
91C33×C4—21no—
92C3×C3⋊Dic3—21no—
93C32×Dic3—22no—
94C32×Dic3—22no—
95C32×Dic3—22no—
96C32×Dic3—41no—
97C2×C3×C4⋊C4—91no—
98C2×C3×C4⋊C4—91no—
99C2×C3×C42—91no—
100D3×C4⋊C4—61no—
101C42×D3—61no—
102C3×#95—91no—
103C3×#95—91no—
104C3×#95—91no—
105C3×#95—91no—
106C22×C3⋊D3C23×C311yescharacteristic
107C22×C3×D3—61no—
108C22×C3×D3—61no—
109C2×#157—21no—
110C2×#157—21no—
111C2×#157—21no—
112C2×#157—21no—
113C2×#157—21no—
114C2×#157C2×C3×C412yes—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4—61no—
120C2×C32×C4—61no—
121C2×C32×C4—61no—
122C2×C32×C4—61no—
123C2×C32×C4C3×D411yescharacteristic
124C2×C3⋊Dic3C3×D411yescharacteristic
125C4×C3⋊D3—21no—
126C4×C3⋊D3—21no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—62no—
144C3×C4×D3—62no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×C4×D3—62no—
148C3×C4×D3—62no—
149C3×C4×D3—121no—
150C3×C4×D3—121no—
151C3×C4×D3—121no—
152C3×C4×D3—121no—
153#624—21no—
154#624—21no—
155#624—21no—
156#624—21no—
157#624—21no—
158#624C2×C3×C412yes—
159#624C2×C3×C412yes—
160#624C3×Q812yes—
161#387—91noSylow
162C2×C33C4×D411yescharacteristic
163C2×C33C22⋊Q811yescharacteristic
164C2×C33#12311yescharacteristic
165C3×C3⋊D3#11711yescharacteristic
166C3×C3⋊D3C4×Q811yescharacteristic
167C3×C3⋊D3#12511yescharacteristic
168C3×C3⋊D3#12611yescharacteristic
169C22×C3×C4—91no—
170C22×C3×C4—91no—
171C22×C3×C4—91no—
172C22×C3×C4—91no—
173C22×C3×C4—91no—
174C22×C3×C4—91no—
175C22×C3×C4—91no—
176C2×C4×D3—61no—
177C2×C4×D3—61no—
178C2×C4×D3—61no—
179C2×C4×D3—61no—
180C2×C4×D3—61no—
181C2×C4×D3—61no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4—61no—
184C3×C4⋊C4—94no—
185C3×C4⋊C4—121no—
186C3×C4⋊C4—181no—
187C3×C42—61no—
188C3×C42—61no—
189C3×C42—121no—
190C3×C42—181no—
191C4⋊Dic3—61no—
192Dic3⋊C4—21no—
193Dic3⋊C4—61no—
194C4×Dic3—21no—
195C4×Dic3—61no—
196C4×Dic3—61no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32C2×C3×D411yescharacteristic
202C2×C3⋊D3C22×C3×C411yescharacteristic
203C2×C3⋊D3C2×C3×Q811yescharacteristic
204C2×C3⋊D3C3×#4111yescharacteristic
205C2×C3⋊D3C3×#4111yescharacteristic
206C2×C3⋊D3C3×#4111yescharacteristic
207C2×C3⋊D3C3×#4111yescharacteristic
208C2×C3×D3—61no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—61no—
215C2×C3×D3—61no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220#157—22no—
221#157—41no—
222C32×C4—22no—
223C32×C4—21no—
224C32×C4—41no—
225C32×C4—42no—
226C32×C4—62no—
227C32×C4—62no—
228C32×C4—62no—
229C32×C4—81no—
230C32×C4—121no—
231C3⋊Dic3—21no—
232C3×Dic3—22no—
233C3×Dic3—22no—
234C3×Dic3—22no—
235C3×Dic3—22no—
236C3×Dic3—22no—
237C3×Dic3—22no—
238C3×Dic3—42no—
239C3×Dic3—42no—
240C3×Dic3—42no—
241C3×Dic3—41no—
242C3×Dic3—41no—
243C3×Dic3—62no—
244C3×Dic3—81no—
245C3×Dic3—121no—
246C2×C4⋊C4—91no—
247C2×C4⋊C4—91no—
248C2×C42—91no—
249#95—91no—
250#95—91no—
251#95—91no—
252#95—91no—
253C33#38711yescharacteristic, complemented, Sylow
254C23×C3—91no—
255C22×D3—61no—
256C22×D3—61no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—92no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—91no—
274C2×C3×C4—121no—
275C2×C3×C4—121no—
276C2×C3×C4—121no—
277C2×C3×C4—121no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×C3×C4—181no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4—181no—
287C2×C3×C4—181no—
288C2×C3×C4—181no—
289C2×C3×C4—181no—
290C2×C3×C4GO+(4,𝔽2)11yescharacteristic
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—61no—
296C2×Dic3—61no—
297C4×D3—62no—
298C4×D3—62no—
299C4×D3—62no—
300C4×D3—62no—
301C4×D3—62no—
302C4×D3—62no—
303C4×D3—62no—
304C4×D3—62no—
305C4×D3—121no—
306C4×D3—121no—
307C4×D3—121no—
308C4×D3—121no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32C3×C4×D411yescharacteristic
322C2×C32C3×C22⋊Q811yescharacteristic
323C2×C32C3×#12311yescharacteristic
324C3⋊D3C3×#11711yescharacteristic
325C3⋊D3C3×C4×Q811yescharacteristic
326C3⋊D3C3×#12511yescharacteristic
327C3⋊D3C3×#12611yescharacteristic
328C3×D3—61no—
329C3×D3—61no—
330C3×D3—61no—
331C3×D3—61no—
332C3×D3—61no—
333C3×D3—61no—
334C3×D3—61no—
335C3×D3—61no—
336C22×C4—91no—
337C22×C4—91no—
338C22×C4—91no—
339C22×C4—91no—
340C22×C4—91no—
341C22×C4—91no—
342C22×C4—91no—
343C4⋊C4—61no—
344C4⋊C4—94no—
345C4⋊C4—181no—
346C42—61no—
347C42—181no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—91no—
353C22×C3—91no—
354C22×C3—91no—
355C22×C3—91no—
356C22×C3—91no—
357C22×C3—91no—
358C22×C3C2×GO+(4,𝔽2)11yescharacteristic
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—61no—
363C2×D3—61no—
364C2×D3—61no—
365C2×D3—61no—
366C2×D3—61no—
367C2×D3—61no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—61no—
371C3×C4—22no—
372C3×C4—21no—
373C3×C4—41no—
374C3×C4—42no—
375C3×C4—62no—
376C3×C4—62no—
377C3×C4—62no—
378C3×C4—62no—
379C3×C4—62no—
380C3×C4—62no—
381C3×C4—81no—
382C3×C4—122no—
383C3×C4—122no—
384C3×C4—122no—
385C3×C4—121no—
386C3×C4—121no—
387C3×C4—182no—
388C3×C4—181no—
389C3×C4—241no—
390C3×C4—361no—
391Dic3—22no—
392Dic3—22no—
393Dic3—22no—
394Dic3—41no—
395Dic3—62no—
396Dic3—121no—
397C32—21no—
398C32—21no—
399C32—41no—
400C32—41no—
401C32C3×#38711yescharacteristic, complemented
402C23—91no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—92no—
408C2×C4—92no—
409C2×C4—92no—
410C2×C4—92no—
411C2×C4—91no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—181no—
415C2×C4—181no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4—181no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—181no—
423C2×C4—181no—
424C2×C4C3×GO+(4,𝔽2)11yescharacteristic
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—41no—
437C2×C3—91no—
438C2×C3—91no—
439C2×C3—91no—
440C2×C3—91no—
441C2×C3D32⋊Q811yescharacteristic
442C2×C3C4×GO+(4,𝔽2)11yescharacteristic
443C2×C3#6423311yescharacteristic
444D3—61no—
445D3—61no—
446D3—61no—
447D3—61no—
448D3—61no—
449D3—61no—
450D3—61no—
451D3—61no—
452C22—91no—
453C22—91no—
454C22—91no—
455C22—91no—
456C22—91no—
457C22—91no—
458C22C2×C3×GO+(4,𝔽2)11yescharacteristic
459C4—21no—
460C4—62no—
461C4—62no—
462C4—62no—
463C4—121no—
464C4—182no—
465C4—181no—
466C4—361no—
467C3—21no—
468C3—21no—
469C3—41no—
470C3—41no—
471C3#1059333811yescharacteristic, direct product factor
472C2—91no—
473C2—91no—
474C2—91no—
475C2—91no—
476C2C3×D32⋊Q811yescharacteristic
477C2C3×C4×GO+(4,𝔽2)11yescharacteristic
478C2C3×#6423311yescharacteristic
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×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#387
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
8. #10593338C3direct (N × H)
16. C3×#3773C4semidirect
67. #3773C3×C4semidirect
253. C33#387semidirect
401. C32C3×#387semidirect
471. C3#10593338direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37732
C3×#638832
C3×C4×#6242
C3×#637812
C3×#637372
#105933383
C3×#3879