Finite Group Databasebeta

← C3×#10593451 #13166820

Subgroups — C3×#10593451 #13166820

Order 1728.

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

normal subgroups only

This group has 588 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×#64298C211yesmaximal, characteristic, complemented
2C3×#64236C212yesmaximal, complemented
3C3×#63913C211yesmaximal, characteristic, complemented
4C3×#63780C211yesmaximal, characteristic, complemented
5C3×#63741C212yesmaximal, complemented
6#10593451C311yesmaximal, characteristic, direct product factor
7C2×C3×#634C2211yescharacteristic
8C32×#235—21no—
9C2×C3×#625C2211yescharacteristic
10C3×#3805—21no—
11C3×#3805C2212yes—
12C3×#3795—22no—
13C3×D3×Dic6—22no—
14C3×#3792C2212yes—
15C3×#3775—22no—
16C3×#3774—22no—
17C32×(C2×D3)⋊C4—21no—
18C3×(C2×D3)⋊Dic3—21no—
19C3×Dic32C2211yescharacteristic
20C3×#1010—61no—
21#64298C2×C311yescharacteristic, complemented
22#64236C2×C312yescomplemented
23C3×#885—61no—
24#63913C2×C311yescharacteristic, complemented
25#63780C2×C311yescharacteristic, complemented
26#63741C2×C312yescomplemented
27C22×C32×D3—21no—
28C2×C33×C4D411yescharacteristic
29C2×C3×C3⋊Dic3D411yescharacteristic
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3×#634—22no—
32C32×#67—22no—
33C2×C32×Dic3—21no—
34C32×D12—21no—
35C32×C4×D3—22no—
36C32×Dic6—21no—
37C3×#627—21no—
38C3×#627D412yes—
39C3×#625—21no—
40C3×#625—22no—
41C3×#625D412yes—
42C3×D3×Dic3—22no—
43C3×D3×Dic3—22no—
44C3×#622—22no—
45C3×#454—91nomaximal
46C32×#41—61no—
47C2×#634C22×C311yescharacteristic
48C2×C3×#67—61no—
49C3×D3×Q8—62no—
50C3×#237—62no—
51C3×#237—62no—
52C3×#237—62no—
53C3×#235—21no—
54C3×#235—21no—
55C3×#235—41no—
56C3×#235—62no—
57C2×C3×Dic6—62no—
58C2×C3×Dic6—61no—
59C2×C3×Dic6—61no—
60C2×#625C22×C311yescharacteristic
61#3805—21no—
62#3805C22×C312yes—
63#3795—22no—
64D3×Dic6—22no—
65#3792C22×C312yes—
66#3775—22no—
67#3774—22no—
68C32×C22⋊C4—61no—
69C3×#217—61no—
70C3×(C2×D3)⋊C4—21no—
71C3×(C2×D3)⋊C4—21no—
72C3×(C2×D3)⋊C4—41no—
73C3×(C2×D3)⋊C4—61no—
74C3×C4×Dic3—61no—
75(C2×D3)⋊Dic3—21no—
76Dic32C22×C311yescharacteristic
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—22no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C2×C32×D3—41no—
82C33×C4—21no—
83C3×C3⋊Dic3—21no—
84C3×C3⋊Dic3C2×D412yes—
85C32×Dic3—22no—
86C32×Dic3—41no—
87C3×2−1+4—91no—
88#1010—61no—
89C3×AGL(1,ℤ/8)—92no—
90C3×#124—91no—
91#885—61no—
92C3×#104—92no—
93C3×#101—91no—
94C23×C32—61no—
95C22×C3×D3—21no—
96C22×C3×D3—21no—
97C22×C3×D3—41no—
98C32×Q8—61no—
99C32×D4—61no—
100C32×D4—62no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—41no—
104C2×C32×C4—41no—
105C2×C32×C4—62no—
106C2×C32×C4—61no—
107C2×C32×C4C3×D411yescharacteristic
108C2×C3⋊Dic3C3×D411yescharacteristic
109C2×C3⋊Dic3C23×C311yescharacteristic
110#634—22no—
111C3×#67—22no—
112C3×#67—22no—
113C3×#67—42no—
114C3×#67—62no—
115C3×#67—62no—
116C3×#67—122no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C3×D12—21no—
127C3×D12—21no—
128C3×D12—41no—
129C3×C4×D3—22no—
130C3×C4×D3—22no—
131C3×C4×D3—42no—
132C3×C4×D3—62no—
133C3×C4×D3—62no—
134C3×Dic6—21no—
135C3×Dic6—21no—
136C3×Dic6—41no—
137C3×Dic6—62no—
138C3×Dic6—62no—
139C3×Dic6—62no—
140C3×Dic6—62no—
141C3×Dic6—122no—
142#627—21no—
143#627C3×D412yes—
144#625—21no—
145#625—22no—
146#625C3×D412yes—
147D3×Dic3—22no—
148D3×Dic3—22no—
149#622—22no—
150#454—91noSylow
151C2×C33—21no—
152C2×C33#12011yescharacteristic
153C32×D3—42no—
154C32×D3—81no—
155C3×#41—61no—
156C3×#41—61no—
157C3×#41—92no—
158C3×#41—121no—
159C3×#41—182no—
160C3×#41—181no—
161C2×C3×Q8—91no—
162C2×C3×Q8—182no—
163C2×C3×D4—91no—
164C2×#67—61no—
165D3×Q8—62no—
166#237—62no—
167#237—62no—
168#237—62no—
169#235—21no—
170#235—62no—
171C2×Dic6—61no—
172C2×Dic6—61no—
173C2×Dic6—62no—
174C3×SD16—182no—
175C3×D8—182no—
176C3×M16—92no—
177C3×C22⋊C4—61no—
178C3×C22⋊C4—61no—
179C3×C22⋊C4—121no—
180C3×C22⋊C4—181no—
181C3×C42—91no—
182#217—61no—
183(C2×D3)⋊C4—21no—
184(C2×D3)⋊C4—61no—
185C4×Dic3—61no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—62no—
191C22×C32—121no—
192C22×C32—121no—
193C22×C32—121no—
194C22×C32C2×C3×D411yescharacteristic
195C2×C3×D3—22no—
196C2×C3×D3—22no—
197C2×C3×D3—42no—
198C2×C3×D3—41no—
199C2×C3×D3—41no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C2×C3×D3—41no—
203C2×C3×D3—42no—
204C2×C3×D3—41no—
205C2×C3×D3—81no—
206C2×C3×D3—81no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—41no—
213C32×C4—62no—
214C32×C4—81no—
215C32×C4—121no—
216C3⋊Dic3—21no—
217C3⋊Dic3C2×C3×D412yes—
218C3×Dic3—22no—
219C3×Dic3—22no—
220C3×Dic3—42no—
221C3×Dic3—41no—
222C3×Dic3—41no—
223C3×Dic3—62no—
224C3×Dic3—62no—
225C3×Dic3—62no—
226C3×Dic3—81no—
227C3×Dic3—121no—
2282−1+4—91no—
229AGL(1,ℤ/8)—92no—
230#124—91no—
231#104—92no—
232#101—91no—
233C33#45411yescharacteristic, complemented, Sylow
234C23×C3—61no—
235C23×C3—61no—
236C23×C3—121no—
237C22×D3—21no—
238C3×Q8—61no—
239C3×Q8—61no—
240C3×Q8—92no—
241C3×Q8—121no—
242C3×Q8—182no—
243C3×Q8—181no—
244C3×D4—61no—
245C3×D4—61no—
246C3×D4—62no—
247C3×D4—62no—
248C3×D4—92no—
249C3×D4—121no—
250C3×D4—122no—
251C3×D4—182no—
252C3×D4—181no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—62no—
258C2×C3×C4—62no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—91no—
262C2×C3×C4—91no—
263C2×C3×C4—122no—
264C2×C3×C4—121no—
265C2×C3×C4—182no—
266C2×C3×C4—182no—
267C2×C3×C4GO+(4,𝔽2)11yescharacteristic
268#67—22no—
269#67—62no—
270#67—62no—
271#67—122no—
272C2×Dic3—21no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—62no—
278C2×Dic3—62no—
279D12—21no—
280C4×D3—22no—
281C4×D3—62no—
282C4×D3—62no—
283Dic6—21no—
284Dic6—62no—
285Dic6—62no—
286Dic6—62no—
287Dic6—62no—
288Dic6—122no—
289C3×C8—182no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—41no—
300C2×C32—81no—
301C2×C32—122no—
302C2×C32—241no—
303C2×C32C3×#12011yescharacteristic
304C3×D3—42no—
305C3×D3—42no—
306C3×D3—82no—
307C3×D3—81no—
308C3×D3—82no—
309C3×D3—81no—
310#41—61no—
311#41—92no—
312#41—182no—
313#41—181no—
314C2×Q8—91no—
315C2×Q8—182no—
316C2×D4—91no—
317SD16—182no—
318D8—182no—
319M16—92no—
320C22⋊C4—61no—
321C22⋊C4—181no—
322C42—91no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—62no—
328C22×C3—62no—
329C22×C3—122no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3—121no—
333C22×C3—121no—
334C22×C3—121no—
335C22×C3—122no—
336C22×C3—121no—
337C22×C3—241no—
338C22×C3—241no—
339C22×C3C2×GO+(4,𝔽2)11yescharacteristic
340C2×D3—22no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—62no—
351C3×C4—62no—
352C3×C4—81no—
353C3×C4—92no—
354C3×C4—122no—
355C3×C4—121no—
356C3×C4—121no—
357C3×C4—181no—
358C3×C4—241no—
359Dic3—22no—
360Dic3—41no—
361Dic3—62no—
362Dic3—62no—
363Dic3—62no—
364Dic3—121no—
365C32—21no—
366C32—21no—
367C32—41no—
368C32—41no—
369C32C3×#45411yescharacteristic, complemented
370C23—61no—
371Q8—61no—
372Q8—92no—
373Q8—182no—
374Q8—181no—
375D4—61no—
376D4—62no—
377D4—92no—
378D4—182no—
379D4—181no—
380C2×C4—62no—
381C2×C4—61no—
382C2×C4—91no—
383C2×C4—91no—
384C2×C4—182no—
385C2×C4—182no—
386C2×C4C3×GO+(4,𝔽2)11yescharacteristic
387C8—182no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—41no—
394C2×C3—41no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—81no—
399C2×C3—122no—
400C2×C3—122no—
401C2×C3—242no—
402C2×C3—241no—
403C2×C3—242no—
404C2×C3—241no—
405C2×C3#6424111yescharacteristic
406D3—42no—
407D3—81no—
408C22—62no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22C2×C3×GO+(4,𝔽2)11yescharacteristic
413C4—21no—
414C4—62no—
415C4—92no—
416C4—121no—
417C4—181no—
418C3—21no—
419C3—21no—
420C3—41no—
421C3—41no—
422C3#1059345111yescharacteristic, direct product factor
423C2—21no—
424C2—122no—
425C2—241no—
426C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33#120
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#454
LCSG ▷ C2×C3⋊Dic3 ▷ 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×#64298C2semidirect
2. C3×#64236C2semidirect
3. C3×#63913C2semidirect
4. C3×#63780C2semidirect
5. C3×#63741C2semidirect
6. #10593451C3direct (N × H)
21. #64298C2×C3semidirect
22. #64236C2×C3semidirect
24. #63913C2×C3semidirect
25. #63780C2×C3semidirect
26. #63741C2×C3semidirect
233. C33#454semidirect
369. C32C3×#454semidirect
422. C3#10593451direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#642982
C3×#642362
C3×#639132
C3×#637802
C3×#637412
#105934513
C3×#4549