Finite Group Databasebeta

← C3×#10593434 #13166798

Subgroups — C3×#10593434 #13166798

Order 1728.

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

normal subgroups only

This group has 622 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×#3798C211yesmaximal, characteristic
2C2×C3×#3776C211yesmaximal, characteristic, complemented
3C2×C3×#3775C211yesmaximal, characteristic, complemented
4C3×#63865C211yesmaximal, characteristic, complemented
5C3×#63778C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593434C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×C4×D3—21no—
11C2×C3×#627C2211yescharacteristic
12C2×C3×#627C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C3×#625C2211yescharacteristic
15C3×#3798—22no—
16C3×#3798—22no—
17C2×C3×#622C2211yescharacteristic
18C2×C3×#622C2211yescharacteristic
19C3×#3776—22no—
20C3×#3775—22no—
21C32×Dic3⋊C4—21no—
22C3×#3724C2211yescharacteristic
23C3×#3722—21no—
24C2×C3×#235—61no—
25C2×#3798C2×C311yescharacteristic
26C2×#3776C2×C311yescharacteristic, complemented
27C2×#3775C2×C311yescharacteristic, complemented
28C3×#896—61no—
29#63865C2×C311yescharacteristic, complemented
30#63778C2×C311yescharacteristic, complemented
31#63740C2×C311yescharacteristic, complemented
32#63739C2×C311yescharacteristic, complemented
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C32×D3—21no—
35C2×C33×C4D411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C3×C4×C3⋊D3—21no—
38C3×C4×C3⋊D3—21no—
39C2×C32×Dic3—21no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C32×C4×D3—24no—
43C3×#627—21no—
44C3×#627—21no—
45C3×#627—21no—
46C3×#627D412yes—
47C3×#626—24no—
48C3×#625—21no—
49C3×#625D412yes—
50C3×#622—21no—
51C3×#622—21no—
52C3×#452—91nomaximal
53C22×C32×C4—61no—
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C2×C3×#67—61no—
56C2×C3×#67—61no—
57C3×#235—64no—
58C3×#235—64no—
59C2×C3×D12—61no—
60C2×C3×C4×D3—21no—
61C2×C3×C4×D3—21no—
62C2×C3×C4×D3—41no—
63C2×C3×C4×D3—61no—
64C2×C3×C4×D3—61no—
65C2×C3×Dic6—61no—
66C2×C3×Dic6—61no—
67C2×#627C22×C311yescharacteristic
68C2×#627C22×C311yescharacteristic
69C2×#626—21no—
70C2×#625C22×C311yescharacteristic
71#3798—22no—
72#3798—22no—
73C2×#622C22×C311yescharacteristic
74C2×#622C22×C311yescharacteristic
75#3776—22no—
76#3775—22no—
77C32×C4⋊C4—61no—
78C3×(C2×D3)⋊C4—61no—
79C3×(C2×D3)⋊C4—61no—
80C3×Dic3⋊C4—21no—
81C3×Dic3⋊C4—21no—
82C3×Dic3⋊C4—41no—
83C3×Dic3⋊C4—61no—
84#3724C22×C311yescharacteristic
85#3722—21no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—21no—
90C2×C32×D3—22no—
91C2×C32×D3—41no—
92C2×C32×D3—41no—
93C33×C4—21no—
94C3×C3⋊Dic3C2×D411yescharacteristic
95C3×C3⋊Dic3C2×D411yescharacteristic
96C32×Dic3—22no—
97C32×Dic3—41no—
98C32×Dic3—41no—
99C2×C3×#41—91no—
100C2×#235—61no—
101C2×C3×Q16—91no—
102C2×C3×SD16—91no—
103C3×C22⋊Q8—91no—
104#896—61no—
105C3×#103—91no—
106C3×#102—91no—
107C3×C22⋊C8—91no—
108C23×C32—61no—
109C22×C3⋊D3C3×D411yescharacteristic
110C22×C3×D3—21no—
111C22×C3×D3—21no—
112C22×C3×D3—41no—
113C22×C3×D3—61no—
114C22×C3×D3—61no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4—64no—
120C2×C32×C4—61no—
121C2×C32×C4—61no—
122C2×C32×C4—61no—
123C2×C32×C4C3×D411yescharacteristic
124C2×C3⋊Dic3C23×C311yescharacteristic
125C4×C3⋊D3—21no—
126C4×C3⋊D3—21no—
127C3×#67—62no—
128C3×#67—64no—
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—
141C3×D12—64no—
142C3×C4×D3—24no—
143C3×C4×D3—24no—
144C3×C4×D3—44no—
145C3×C4×D3—62no—
146C3×C4×D3—62no—
147C3×C4×D3—121no—
148C3×C4×D3—121no—
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#626—24no—
158#625—21no—
159#625C3×D412yes—
160#622—21no—
161#622—21no—
162#452—91noSylow
163C2×C33#12011yescharacteristic
164C2×C33#13511yescharacteristic
165C2×C33#13711yescharacteristic
166C3×C3⋊D3—41no—
167C32×D3—42no—
168C3×#41—182no—
169C3×#41—182no—
170C2×C3×Q8—91no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C22×C3×C4—61no—
175C22×C3×C4—61no—
176C22×C3×C4—91no—
177C22×C3×C4—121no—
178C2×#67—61no—
179C2×#67—61no—
180#235—64no—
181#235—64no—
182C2×D12—61no—
183C2×C4×D3—21no—
184C2×C4×D3—61no—
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—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—91no—
195C3×C4⋊C4—121no—
196C3×C22⋊C4—181no—
197(C2×D3)⋊C4—61no—
198(C2×D3)⋊C4—61no—
199Dic3⋊C4—21no—
200Dic3⋊C4—61no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—62no—
206C22×C32—121no—
207C22×C32—121no—
208C22×C32C2×C3×D411yescharacteristic
209C2×C3⋊D3—21no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3×D3—22no—
213C2×C3×D3—22no—
214C2×C3×D3—42no—
215C2×C3×D3—41no—
216C2×C3×D3—41no—
217C2×C3×D3—42no—
218C2×C3×D3—42no—
219C2×C3×D3—41no—
220C2×C3×D3—41no—
221C2×C3×D3—62no—
222C2×C3×D3—62no—
223C2×C3×D3—121no—
224C2×C3×D3—121no—
225C2×C3×D3—121no—
226C2×C3×D3—121no—
227C32×C4—22no—
228C32×C4—21no—
229C32×C4—41no—
230C32×C4—42no—
231C32×C4—62no—
232C32×C4—81no—
233C32×C4—121no—
234C32×C4—121no—
235C3⋊Dic3C2×C3×D411yescharacteristic
236C3⋊Dic3C2×C3×D411yescharacteristic
237C3×Dic3—22no—
238C3×Dic3—22no—
239C3×Dic3—42no—
240C3×Dic3—41no—
241C3×Dic3—41no—
242C3×Dic3—42no—
243C3×Dic3—41no—
244C3×Dic3—41no—
245C3×Dic3—61no—
246C3×Dic3—61no—
247C3×Dic3—61no—
248C3×Dic3—61no—
249C3×Dic3—81no—
250C2×#41—91no—
251C2×Q16—91no—
252C2×SD16—91no—
253C22⋊Q8—91no—
254#103—91no—
255#102—91no—
256C22⋊C8—91no—
257C33#45211yescharacteristic, complemented, Sylow
258C23×C3—61no—
259C23×C3—61no—
260C23×C3—91no—
261C23×C3—121no—
262C22×D3—21no—
263C22×D3—61no—
264C22×D3—61no—
265C3×Q8—92no—
266C3×Q8—181no—
267C3×Q8—181no—
268C3×Q8—181no—
269C3×D4—92no—
270C3×D4—184no—
271C3×D4—181no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—64no—
277C2×C3×C4—64no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—91no—
285C2×C3×C4—124no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4—121no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4GO+(4,𝔽2)11yescharacteristic
292#67—62no—
293#67—64no—
294#67—62no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—21no—
298C2×Dic3—61no—
299C2×Dic3—61no—
300D12—64no—
301C4×D3—24no—
302C4×D3—62no—
303C4×D3—62no—
304C4×D3—121no—
305C4×D3—121no—
306Dic6—62no—
307Dic6—62no—
308Dic6—121no—
309Dic6—121no—
310C3×C8—181no—
311C3×C8—181no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—122no—
325C2×C32C3×#12011yescharacteristic
326C2×C32C3×#13511yescharacteristic
327C2×C32C3×#13711yescharacteristic
328C3⋊D3—41no—
329C3×D3—42no—
330C3×D3—44no—
331C3×D3—42no—
332C3×D3—122no—
333C3×D3—122no—
334#41—182no—
335#41—182no—
336C2×Q8—91no—
337C2×Q8—91no—
338C2×D4—91no—
339C2×D4—181no—
340C22×C4—61no—
341C22×C4—91no—
342Q16—182no—
343SD16—182no—
344C2×C8—91no—
345C2×C8—91no—
346C4⋊C4—61no—
347C4⋊C4—91no—
348C22⋊C4—181no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—62no—
354C22×C3—62no—
355C22×C3—122no—
356C22×C3—121no—
357C22×C3—121no—
358C22×C3—122no—
359C22×C3—122no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—181no—
363C22×C3—181no—
364C22×C3—181no—
365C22×C3C2×GO+(4,𝔽2)11yescharacteristic
366C2×D3—22no—
367C2×D3—41no—
368C2×D3—41no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—121no—
375C3×C4—22no—
376C3×C4—21no—
377C3×C4—41no—
378C3×C4—42no—
379C3×C4—62no—
380C3×C4—62no—
381C3×C4—81no—
382C3×C4—91no—
383C3×C4—91no—
384C3×C4—122no—
385C3×C4—121no—
386C3×C4—121no—
387C3×C4—122no—
388C3×C4—121no—
389C3×C4—121no—
390C3×C4—241no—
391Dic3—22no—
392Dic3—41no—
393Dic3—41no—
394Dic3—61no—
395Dic3—61no—
396Dic3—61no—
397Dic3—61no—
398C32—21no—
399C32—21no—
400C32—41no—
401C32—41no—
402C32C3×#45211yescharacteristic, complemented
403C23—61no—
404C23—91no—
405Q8—92no—
406Q8—181no—
407Q8—181no—
408Q8—181no—
409D4—92no—
410D4—184no—
411D4—181no—
412C2×C4—64no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—91no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4C3×GO+(4,𝔽2)11yescharacteristic
420C8—181no—
421C8—181no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—41no—
429C2×C3—41no—
430C2×C3—41no—
431C2×C3—41no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—122no—
435C2×C3—124no—
436C2×C3—122no—
437C2×C3—361no—
438C2×C3#6422311yescharacteristic
439C2×C3#6422611yescharacteristic
440C2×C3#6424111yescharacteristic
441D3—42no—
442D3—122no—
443D3—122no—
444C22—62no—
445C22—121no—
446C22—121no—
447C22—181no—
448C22—181no—
449C22—181no—
450C22C2×C3×GO+(4,𝔽2)11yescharacteristic
451C4—21no—
452C4—62no—
453C4—91no—
454C4—91no—
455C4—121no—
456C4—121no—
457C3—21no—
458C3—21no—
459C3—41no—
460C3—41no—
461C3#1059343411yescharacteristic, direct product factor
462C2—122no—
463C2—361no—
464C2C3×#6422311yescharacteristic
465C2C3×#6422611yescharacteristic
466C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3798C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#452
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3776C2semidirect
3. C2×C3×#3775C2semidirect
4. C3×#63865C2semidirect
5. C3×#63778C2semidirect
6. C3×#63740C2semidirect
7. C3×#63739C2semidirect
8. #10593434C3direct (N × H)
26. C2×#3776C2×C3semidirect
27. C2×#3775C2×C3semidirect
29. #63865C2×C3semidirect
30. #63778C2×C3semidirect
31. #63740C2×C3semidirect
32. #63739C2×C3semidirect
257. C33#452semidirect
402. C32C3×#452semidirect
461. C3#10593434direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C2×C3×#37762
C2×C3×#37752
C3×#638652
C3×#637782
C3×#637402
C3×#637392
#105934343
C3×#4529