Finite Group Databasebeta

← C3×#10593351 #13166694

Subgroups — C3×#10593351 #13166694

Order 1728.

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

normal subgroups only

This group has 446 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×#63912C211yesmaximal, characteristic
2C3×#63874C211yesmaximal, characteristic, complemented
3C3×#63778C211yesmaximal, characteristic, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63739C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic, complemented
8#10593351C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#626—21no—
11C2×C3×#625C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C32×(C2×D3)⋊C4—21no—
15C32×C4×Dic3—21no—
16C3×#3724C2211yescharacteristic
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×#3722—21no—
20C3×#884—61no—
21C3×#872—61no—
22#63912C2×C311yescharacteristic
23#63874C2×C311yescharacteristic, complemented
24#63778C2×C311yescharacteristic, complemented
25#63743C2×C311yescharacteristic, complemented
26#63742C2×C311yescharacteristic, complemented
27#63739C2×C311yescharacteristic, complemented
28#63739C2×C311yescharacteristic, complemented
29C22×C3×C3⋊D3D411yescharacteristic
30C22×C32×D3—21no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3C2311yescharacteristic
33C3×C4×C3⋊D3—21no—
34C3×C4×C3⋊D3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C2×C32×Dic3—21no—
38C3×#626—42no—
39C3×#625—21no—
40C3×#625—21no—
41C3×#622—21no—
42C3×#622—21no—
43C3×#482—91nomaximal
44C2×C4×C3⋊D3C22×C311yescharacteristic
45C2×C3×#67—61no—
46C2×C3×#67—61no—
47C2×C3×D12—61no—
48C2×C3×C4×D3—61no—
49C2×C3×C4×D3—61no—
50C2×#626—21no—
51C2×#625C22×C311yescharacteristic
52C2×#622C22×C311yescharacteristic
53C2×#622C22×C311yescharacteristic
54C32×C22⋊C4—61no—
55C32×C42—61no—
56C3×(C2×D3)⋊C4—21no—
57C3×(C2×D3)⋊C4—21no—
58C3×(C2×D3)⋊C4—41no—
59C3×(C2×D3)⋊C4—61no—
60C3×(C2×D3)⋊C4—61no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×Dic3⋊C4—61no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3—21no—
66C3×C4×Dic3—41no—
67#3724C22×C311yescharacteristic
68#3724C22×C311yescharacteristic
69#3724C22×C311yescharacteristic
70#3722—21no—
71C22×C33C2×D411yescharacteristic
72C2×C3×C3⋊D3—21no—
73C2×C3×C3⋊D3—21no—
74C2×C3×C3⋊D3—21no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C33×C4—21no—
79C3×C3⋊Dic3#4111yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C32×Dic3—41no—
82C32×Dic3—41no—
83C32×Dic3—41no—
84C3×#121—91no—
85C3×#117—91no—
86#884—61no—
87#872—61no—
88C3×#107—91no—
89C3×#106—91no—
90C3×#102—91no—
91C3×#102—91no—
92C3×C22⋊C8—91no—
93C23×C32—61no—
94C22×C3⋊D3C3×D411yescharacteristic
95C22×C3×D3—21no—
96C22×C3×D3—21no—
97C22×C3×D3—41no—
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4—61no—
105C2×C32×C4—61no—
106C2×C32×C4—61no—
107C2×C32×C4C3×D411yescharacteristic
108C2×C3⋊Dic3C23×C311yescharacteristic
109C4×C3⋊D3—21no—
110C4×C3⋊D3—21no—
111C3×#67—121no—
112C3×#67—121no—
113C3×#67—122no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—21no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—21no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—41no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C3×D12—122no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—121no—
129C3×C4×D3—121no—
130#626—42no—
131#625—21no—
132#625—21no—
133#622—21no—
134#622—21no—
135#482—91noSylow
136C2×C33#12311yescharacteristic
137C2×C33#13511yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C3×C3⋊D3—41no—
140C32×D3—81no—
141C2×C3×D4—91no—
142C2×C3×D4—181no—
143C22×C3×C4—91no—
144C2×#67—61no—
145C2×#67—61no—
146C2×D12—61no—
147C2×C4×D3—61no—
148C2×C4×D3—61no—
149C2×C3×C8—91no—
150C2×C3×C8—91no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—91no—
153C3×C4⋊C4—91no—
154C3×C22⋊C4—61no—
155C3×C22⋊C4—61no—
156C3×C22⋊C4—121no—
157C3×C22⋊C4—181no—
158C3×C42—61no—
159C3×C42—61no—
160C3×C42—121no—
161(C2×D3)⋊C4—21no—
162(C2×D3)⋊C4—61no—
163(C2×D3)⋊C4—61no—
164Dic3⋊C4—61no—
165Dic3⋊C4—61no—
166Dic3⋊C4—61no—
167C4×Dic3—21no—
168C22×C32—21no—
169C22×C32—21no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32C2×C3×D411yescharacteristic
176C2×C3⋊D3—21no—
177C2×C3⋊D3—21no—
178C2×C3⋊D3—21no—
179C2×C3×D3—41no—
180C2×C3×D3—41no—
181C2×C3×D3—41no—
182C2×C3×D3—42no—
183C2×C3×D3—41no—
184C2×C3×D3—41no—
185C2×C3×D3—41no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C2×C3×D3—81no—
189C2×C3×D3—81no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C2×C3×D3—121no—
194C32×C4—22no—
195C32×C4—21no—
196C32×C4—41no—
197C32×C4—42no—
198C32×C4—81no—
199C32×C4—121no—
200C32×C4—121no—
201C32×C4—121no—
202C3⋊Dic3C3×#4111yescharacteristic
203C3⋊Dic3C3×#4111yescharacteristic
204C3×Dic3—41no—
205C3×Dic3—41no—
206C3×Dic3—41no—
207C3×Dic3—42no—
208C3×Dic3—42no—
209C3×Dic3—41no—
210C3×Dic3—41no—
211C3×Dic3—41no—
212C3×Dic3—61no—
213C3×Dic3—61no—
214C3×Dic3—61no—
215C3×Dic3—61no—
216C3×Dic3—81no—
217#121—91no—
218#117—91no—
219#107—91no—
220#106—91no—
221#102—91no—
222#102—91no—
223C22⋊C8—91no—
224C33#48211yescharacteristic, complemented, Sylow
225C23×C3—61no—
226C23×C3—61no—
227C23×C3—91no—
228C23×C3—121no—
229C22×D3—21no—
230C22×D3—61no—
231C22×D3—61no—
232C3×D4—181no—
233C3×D4—181no—
234C3×D4—362no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—61no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—91no—
246C2×C3×C4—121no—
247C2×C3×C4—121no—
248C2×C3×C4—121no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4GO+(4,𝔽2)11yescharacteristic
252#67—121no—
253#67—121no—
254#67—122no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—61no—
259C2×Dic3—61no—
260D12—122no—
261C4×D3—62no—
262C4×D3—62no—
263C4×D3—121no—
264C4×D3—121no—
265C3×C8—181no—
266C3×C8—181no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—41no—
277C2×C32—41no—
278C2×C32—41no—
279C2×C32—241no—
280C2×C32C3×#12311yescharacteristic
281C2×C32C3×#13511yescharacteristic
282C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
283C3⋊D3—41no—
284C3×D3—81no—
285C3×D3—82no—
286C3×D3—81no—
287C3×D3—122no—
288C3×D3—122no—
289C2×D4—91no—
290C2×D4—181no—
291C22×C4—91no—
292C2×C8—91no—
293C2×C8—91no—
294C4⋊C4—91no—
295C4⋊C4—91no—
296C4⋊C4—91no—
297C22⋊C4—61no—
298C22⋊C4—181no—
299C42—61no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—121no—
305C22×C3—121no—
306C22×C3—121no—
307C22×C3—122no—
308C22×C3—121no—
309C22×C3—121no—
310C22×C3—121no—
311C22×C3—181no—
312C22×C3—181no—
313C22×C3—181no—
314C22×C3—241no—
315C22×C3—241no—
316C22×C3C2×GO+(4,𝔽2)11yescharacteristic
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—62no—
321C2×D3—62no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—121no—
325C2×D3—121no—
326C3×C4—22no—
327C3×C4—21no—
328C3×C4—41no—
329C3×C4—42no—
330C3×C4—81no—
331C3×C4—91no—
332C3×C4—91no—
333C3×C4—121no—
334C3×C4—121no—
335C3×C4—121no—
336C3×C4—122no—
337C3×C4—122no—
338C3×C4—121no—
339C3×C4—121no—
340C3×C4—121no—
341C3×C4—241no—
342Dic3—41no—
343Dic3—41no—
344Dic3—41no—
345Dic3—61no—
346Dic3—61no—
347Dic3—61no—
348Dic3—61no—
349C32—21no—
350C32—21no—
351C32—41no—
352C32—41no—
353C32C3×#48211yescharacteristic, complemented
354C23—61no—
355C23—91no—
356D4—181no—
357D4—181no—
358D4—362no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—91no—
363C2×C4—181no—
364C2×C4—181no—
365C2×C4C3×GO+(4,𝔽2)11yescharacteristic
366C8—181no—
367C8—181no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3—41no—
380C2×C3—241no—
381C2×C3—242no—
382C2×C3—241no—
383C2×C3—361no—
384C2×C3#6422311yescharacteristic
385C2×C3#6422411yescharacteristic
386C2×C3#6423311yescharacteristic
387D3—81no—
388D3—122no—
389D3—122no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—181no—
394C22—181no—
395C22—181no—
396C22C2×C3×GO+(4,𝔽2)11yescharacteristic
397C4—21no—
398C4—91no—
399C4—91no—
400C4—121no—
401C4—121no—
402C4—121no—
403C3—21no—
404C3—21no—
405C3—41no—
406C3—41no—
407C3#1059335111yescharacteristic, direct product factor
408C2—241no—
409C2—361no—
410C2C3×#6422311yescharacteristic
411C2C3×#6422411yescharacteristic
412C2C3×#6423311yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC3×#63912C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#482
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. C3×#63874C2semidirect
3. C3×#63778C2semidirect
4. C3×#63743C2semidirect
5. C3×#63742C2semidirect
6. C3×#63739C2semidirect
7. C3×#63739C2semidirect
8. #10593351C3direct (N × H)
23. #63874C2×C3semidirect
24. #63778C2×C3semidirect
25. #63743C2×C3semidirect
26. #63742C2×C3semidirect
27. #63739C2×C3semidirect
28. #63739C2×C3semidirect
224. C33#482semidirect
353. C32C3×#482semidirect
407. C3#10593351direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#639122
C3×#638742
C3×#637782
C3×#637432
C3×#637422
C3×#637392
#105933513
C3×#4829