Finite Group Databasebeta

← C3×#10593524 #13166890

Subgroups — C3×#10593524 #13166890

Order 1728.

630 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3805C211yesmaximal, characteristic
2C2×C3×#3776C211yesmaximal, characteristic, complemented
3C2×C3×#3775C211yesmaximal, characteristic, complemented
4C3×#63982C211yesmaximal, characteristic, complemented
5C3×C22⋊#622C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593524C311yesmaximal, characteristic, direct product factor
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C32×#67—21no—
11C2×C3×#627C2211yescharacteristic
12C2×C3×#627C2211yescharacteristic
13C2×C3×#625C2211yescharacteristic
14C3×#3805—22no—
15C3×#3805—22no—
16C2×C3×D3×Dic3—21no—
17C2×C3×#622C2211yescharacteristic
18C2×C3×#622C2211yescharacteristic
19C3×#3776—22no—
20C3×#3775—22no—
21C32×#217—21no—
22C3×#3724C2211yescharacteristic
23C3×Dic3⋊Dic3—21no—
24C2×C3×#237—61no—
25C2×#3805C2×C311yescharacteristic
26C2×#3776C2×C311yescharacteristic, complemented
27C2×#3775C2×C311yescharacteristic, complemented
28C3×#878—61no—
29#63982C2×C311yescharacteristic, complemented
30C22⋊#622C2×C311yescharacteristic, complemented
31#63740C2×C311yescharacteristic, complemented
32#63739C2×C311yescharacteristic, complemented
33C23×C33D411yescharacteristic
34C22×C32×D3—21no—
35C2×C3×C3⋊Dic3—21no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C3×C3⋊Dic3C2311yescharacteristic
39C32×#67—24no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C2×C32×Dic3—21no—
43C3×#627—21no—
44C3×#627—21no—
45C3×#627—21no—
46C3×#627D412yes—
47C3×#625—21no—
48C3×#625D412yes—
49C3×D3×Dic3—24no—
50C3×#622—21no—
51C3×#622—21no—
52C3×#452—91nomaximal
53C2×C32×D4—61no—
54C22×C3⋊Dic3C22×C311yescharacteristic
55C2×C3×#67—21no—
56C2×C3×#67—21no—
57C2×C3×#67—41no—
58C2×C3×#67—61no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61C22×C3×Dic3—61no—
62C3×#237—64no—
63C3×#237—64no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—61no—
66C2×C3×Dic6—61no—
67C2×#627C22×C311yescharacteristic
68C2×#627C22×C311yescharacteristic
69C2×#625C22×C311yescharacteristic
70#3805—22no—
71#3805—22no—
72C2×D3×Dic3—21no—
73C2×#622C22×C311yescharacteristic
74C2×#622C22×C311yescharacteristic
75#3776—22no—
76#3775—22no—
77C32×C22⋊C4—61no—
78C3×#217—21no—
79C3×#217—21no—
80C3×#217—41no—
81C3×C4⋊Dic3—61no—
82C3×Dic3⋊C4—61no—
83C3×Dic3⋊C4—61no—
84#3724C22×C311yescharacteristic
85Dic3⋊Dic3—21no—
86C22×C33—21no—
87C22×C33—21no—
88C22×C33—21no—
89C22×C33C2×D411yescharacteristic
90C2×C32×D3—22no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C3×C3⋊Dic3—21no—
94C3×C3⋊Dic3C2×D411yescharacteristic
95C3×C3⋊Dic3C2×D411yescharacteristic
96C32×Dic3—22no—
97C32×Dic3—41no—
98C32×Dic3—41no—
99C2×C3×#41—91no—
100C2×#237—61no—
101C2×C3×Q16—91no—
102C2×C3×SD16—91no—
103C3×C22⋊Q8—91no—
104#878—61no—
105C3×#103—91no—
106C3×#102—91no—
107C3×C22⋊C8—91no—
108C23×C32—21no—
109C23×C32—21no—
110C23×C32—41no—
111C23×C32—41no—
112C23×C32—61no—
113C23×C32C3×D411yescharacteristic
114C22×C3×D3—21no—
115C22×C3×D3—21no—
116C22×C3×D3—41no—
117C32×D4—64no—
118C2×C32×C4—61no—
119C2×C32×C4—61no—
120C2×C32×C4—61no—
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3C3×D411yescharacteristic
124C2×C3⋊Dic3C23×C311yescharacteristic
125C3×#67—24no—
126C3×#67—24no—
127C3×#67—44no—
128C3×#67—62no—
129C3×#67—62no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—121no—
147C2×C3×Dic3—121no—
148C3×C4×D3—64no—
149C3×Dic6—62no—
150C3×Dic6—62no—
151C3×Dic6—121no—
152C3×Dic6—121no—
153#627—21no—
154#627—21no—
155#627—21no—
156#627C3×D412yes—
157#625—21no—
158#625C3×D412yes—
159D3×Dic3—24no—
160#622—21no—
161#622—21no—
162#452—91noSylow
163C2×C33—41no—
164C2×C33#12011yescharacteristic
165C2×C33#13511yescharacteristic
166C2×C33#13711yescharacteristic
167C32×D3—42no—
168C3×#41—182no—
169C3×#41—182no—
170C2×C3×Q8—91no—
171C2×C3×Q8—91no—
172C2×C3×D4—61no—
173C2×C3×D4—61no—
174C2×C3×D4—91no—
175C2×C3×D4—121no—
176C22×C3×C4—91no—
177C22×C3×C4—181no—
178C2×#67—21no—
179C2×#67—61no—
180C22×Dic3—61no—
181C22×Dic3—61no—
182C22×Dic3—61no—
183#237—64no—
184#237—64no—
185C2×C4×D3—61no—
186C2×Dic6—61no—
187C2×Dic6—61no—
188C3×Q16—182no—
189C3×SD16—182no—
190C2×C3×C8—91no—
191C2×C3×C8—91no—
192C3×C4⋊C4—91no—
193C3×C4⋊C4—181no—
194C3×C22⋊C4—61no—
195C3×C22⋊C4—61no—
196C3×C22⋊C4—121no—
197#217—21no—
198C4⋊Dic3—61no—
199Dic3⋊C4—61no—
200Dic3⋊C4—61no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—42no—
215C22×C32—42no—
216C22×C32—62no—
217C22×C32—81no—
218C22×C32—81no—
219C22×C32—81no—
220C22×C32—81no—
221C22×C32—121no—
222C22×C32—121no—
223C22×C32C2×C3×D411yescharacteristic
224C2×C3×D3—22no—
225C2×C3×D3—22no—
226C2×C3×D3—42no—
227C2×C3×D3—41no—
228C2×C3×D3—41no—
229C2×C3×D3—41no—
230C2×C3×D3—41no—
231C2×C3×D3—81no—
232C2×C3×D3—81no—
233C32×C4—62no—
234C32×C4—121no—
235C32×C4—121no—
236C3⋊Dic3—21no—
237C3⋊Dic3C2×C3×D411yescharacteristic
238C3⋊Dic3C2×C3×D411yescharacteristic
239C3×Dic3—22no—
240C3×Dic3—22no—
241C3×Dic3—42no—
242C3×Dic3—41no—
243C3×Dic3—41no—
244C3×Dic3—41no—
245C3×Dic3—41no—
246C3×Dic3—61no—
247C3×Dic3—61no—
248C3×Dic3—61no—
249C3×Dic3—61no—
250C3×Dic3—62no—
251C3×Dic3—81no—
252C3×Dic3—81no—
253C3×Dic3—121no—
254C2×#41—91no—
255C2×Q16—91no—
256C2×SD16—91no—
257C22⋊Q8—91no—
258#103—91no—
259#102—91no—
260C22⋊C8—91no—
261C33#45211yescharacteristic, complemented, Sylow
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—61no—
267C23×C3—61no—
268C23×C3—121no—
269C23×C3GO+(4,𝔽2)11yescharacteristic
270C22×D3—21no—
271C3×Q8—92no—
272C3×Q8—181no—
273C3×Q8—181no—
274C3×Q8—181no—
275C3×D4—64no—
276C3×D4—64no—
277C3×D4—92no—
278C3×D4—124no—
279C3×D4—181no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—91no—
287C2×C3×C4—91no—
288C2×C3×C4—121no—
289C2×C3×C4—121no—
290C2×C3×C4—121no—
291C2×C3×C4—184no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294#67—24no—
295#67—62no—
296#67—62no—
297C2×Dic3—21no—
298C2×Dic3—21no—
299C2×Dic3—21no—
300C2×Dic3—61no—
301C2×Dic3—61no—
302C2×Dic3—61no—
303C2×Dic3—61no—
304C2×Dic3—64no—
305C2×Dic3—62no—
306C2×Dic3—62no—
307C2×Dic3—121no—
308C2×Dic3—121no—
309C4×D3—64no—
310Dic6—62no—
311Dic6—62no—
312Dic6—121no—
313Dic6—121no—
314C3×C8—181no—
315C3×C8—181no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32—42no—
330C2×C32—42no—
331C2×C32—82no—
332C2×C32—82no—
333C2×C32—122no—
334C2×C32C3×#12011yescharacteristic
335C2×C32C3×#13511yescharacteristic
336C2×C32C3×#13711yescharacteristic
337C3×D3—42no—
338C3×D3—42no—
339C3×D3—82no—
340#41—182no—
341#41—182no—
342C2×Q8—91no—
343C2×Q8—91no—
344C2×D4—61no—
345C2×D4—91no—
346C22×C4—91no—
347C22×C4—181no—
348Q16—182no—
349SD16—182no—
350C2×C8—91no—
351C2×C8—91no—
352C4⋊C4—91no—
353C4⋊C4—181no—
354C22⋊C4—61no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—41no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—62no—
371C22×C3—62no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3—122no—
377C22×C3—121no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3—121no—
381C22×C3—241no—
382C22×C3—241no—
383C22×C3C2×GO+(4,𝔽2)11yescharacteristic
384C2×D3—22no—
385C2×D3—41no—
386C2×D3—41no—
387C3×C4—62no—
388C3×C4—62no—
389C3×C4—91no—
390C3×C4—91no—
391C3×C4—122no—
392C3×C4—121no—
393C3×C4—121no—
394C3×C4—121no—
395C3×C4—121no—
396C3×C4—181no—
397C3×C4—241no—
398C3×C4—241no—
399Dic3—22no—
400Dic3—41no—
401Dic3—41no—
402Dic3—61no—
403Dic3—61no—
404Dic3—61no—
405Dic3—61no—
406Dic3—62no—
407Dic3—121no—
408C32—21no—
409C32—21no—
410C32—41no—
411C32—41no—
412C32C3×#45211yescharacteristic, complemented
413C23—61no—
414C23C3×GO+(4,𝔽2)11yescharacteristic
415Q8—92no—
416Q8—181no—
417Q8—181no—
418Q8—181no—
419D4—64no—
420D4—92no—
421D4—181no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—91no—
426C2×C4—91no—
427C2×C4—184no—
428C2×C4—181no—
429C2×C4—181no—
430C8—181no—
431C8—181no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—42no—
439C2×C3—42no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—41no—
445C2×C3—41no—
446C2×C3—41no—
447C2×C3—82no—
448C2×C3—82no—
449C2×C3—122no—
450C2×C3—122no—
451C2×C3—242no—
452C2×C3#6423711yescharacteristic
453C2×C3#6423911yescharacteristic
454C2×C3#6424111yescharacteristic
455D3—42no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—62no—
460C22—121no—
461C22—121no—
462C22C2×C3×GO+(4,𝔽2)11yescharacteristic
463C4—62no—
464C4—91no—
465C4—91no—
466C4—121no—
467C4—121no—
468C4—181no—
469C3—21no—
470C3—21no—
471C3—41no—
472C3—41no—
473C3#1059352411yescharacteristic, direct product factor
474C2—41no—
475C2—122no—
476C2C3×#6423711yescharacteristic
477C2C3×#6423911yescharacteristic
478C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×C3×#3805C2
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#452
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3776C2semidirect
3. C2×C3×#3775C2semidirect
4. C3×#63982C2semidirect
5. C3×C22⋊#622C2semidirect
6. C3×#63740C2semidirect
7. C3×#63739C2semidirect
8. #10593524C3direct (N × H)
26. C2×#3776C2×C3semidirect
27. C2×#3775C2×C3semidirect
29. #63982C2×C3semidirect
30. C22⋊#622C2×C3semidirect
31. #63740C2×C3semidirect
32. #63739C2×C3semidirect
261. C33#452semidirect
412. C32C3×#452semidirect
473. C3#10593524direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38052
C2×C3×#37762
C2×C3×#37752
C3×#639822
C3×C22⋊#6222
C3×#637402
C3×#637392
#105935243
C3×#4529