Finite Group Databasebeta

← C3×#10593427 #13166792

Subgroups — C3×#10593427 #13166792

Order 1728.

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

normal subgroups only

This group has 560 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×#3775C212yesmaximal, complemented
2C2×C3×#3774C212yesmaximal, complemented
3C3×#63915C211yesmaximal, characteristic, complemented
4C3×#63914C211yesmaximal, characteristic, complemented
5C3×#63777C211yesmaximal, characteristic, complemented
6#10593427C311yesmaximal, characteristic, direct product factor
7C2×C32×D12—21no—
8C2×C3×#627C2211yescharacteristic
9C2×C3×#625—21no—
10C2×C3×#625C2211yescharacteristic
11C2×C3×#625C2212yes—
12C2×C3×#622C2212yes—
13C3×#3775—24no—
14C3×#3774—24no—
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×(C2×D3)⋊C4—21no—
17C3×(C2×D3)⋊Dic3—21no—
18C2×#3775C2×C312yescomplemented
19C2×#3774C2×C312yescomplemented
20C3×#940—61no—
21C3×#885—61no—
22#63915C2×C311yescharacteristic, complemented
23#63914C2×C311yescharacteristic, complemented
24#63777C2×C311yescharacteristic, complemented
25C22×C32×D3—22no—
26C22×C32×D3—21no—
27C2×C33×C4D411yescharacteristic
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30C2×C32×Dic3—21no—
31C32×D12—42no—
32C3×#627—22no—
33C3×#625—22no—
34C3×#625—22no—
35C3×#625—22no—
36C3×#625—42no—
37C3×#622D414yescomplemented
38C3×#496—91nomaximal
39C2×C32×D4—61no—
40C2×C3×#67—61no—
41C2×C3×#67—62no—
42C2×C3×#67—62no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C2×C3×D12—41no—
46C2×C3×Dic6—61no—
47C2×#627C22×C311yescharacteristic
48C2×#625—21no—
49C2×#625C22×C312yes—
50C2×#625C22×C311yescharacteristic
51C2×#622C22×C312yes—
52#3775—24no—
53#3774—24no—
54C32×C22⋊C4—61no—
55C4×C3⋊Dic3C22×C311yescharacteristic
56C3×#217—61no—
57C3×(C2×D3)⋊C4—21no—
58C3×(C2×D3)⋊C4—21no—
59C3×(C2×D3)⋊C4—41no—
60C3×(C2×D3)⋊C4—61no—
61C3×C4×Dic3—61no—
62C3×C4×Dic3—61no—
63(C2×D3)⋊Dic3—21no—
64C22×C33C2×D411yescharacteristic
65C2×C32×D3—41no—
66C2×C32×D3—42no—
67C2×C32×D3—42no—
68C2×C32×D3—41no—
69C2×C32×D3—42no—
70C2×C32×D3—41no—
71C33×C4—21no—
72C3×C3⋊Dic3—21no—
73C3×C3⋊Dic3C2×D412yes—
74C32×Dic3—41no—
75C2×C3×SD16—92no—
76C2×C3×D8—92no—
77C3×#127—91no—
78C3×#124—91no—
79#940—61no—
80#885—61no—
81C3×#97—91no—
82C23×C32—61no—
83C23×C32—62no—
84C22×C3×D3—21no—
85C22×C3×D3—22no—
86C22×C3×D3—21no—
87C22×C3×D3—22no—
88C22×C3×D3—41no—
89C22×C3×D3—42no—
90C32×D4—122no—
91C2×C32×C4—21no—
92C2×C32×C4—21no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4—61no—
96C2×C32×C4C3×D411yescharacteristic
97C2×C3⋊Dic3C3×D411yescharacteristic
98C2×C3⋊Dic3C23×C311yescharacteristic
99C3×#67—122no—
100C3×#67—122no—
101C3×#67—122no—
102C3×#67—124no—
103C2×C3×Dic3—21no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—61no—
107C2×C3×Dic3—61no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—61no—
110C3×D12—42no—
111C3×D12—42no—
112C3×D12—44no—
113C3×Dic6—122no—
114#627—22no—
115#625—22no—
116#625—22no—
117#625—22no—
118#625—42no—
119#622C3×D414yescomplemented
120#496—91noSylow
121C2×C33#12711yescharacteristic
122C2×C33AGL(1,ℤ/8)11yescharacteristic
123C2×C33AGL(1,ℤ/8)11yescharacteristic
124C32×D3—82no—
125C32×D3—81no—
126C2×C3×Q8—91no—
127C2×C3×D4—61no—
128C2×C3×D4—61no—
129C2×C3×D4—91no—
130C2×C3×D4—92no—
131C2×C3×D4—121no—
132C2×C3×D4—181no—
133C2×#67—61no—
134C2×#67—62no—
135C2×#67—62no—
136C2×D12—21no—
137C2×Dic6—61no—
138C3×SD16—184no—
139C3×D8—184no—
140C2×C3×C8—92no—
141C3×C22⋊C4—61no—
142C3×C22⋊C4—61no—
143C3×C22⋊C4—121no—
144C3×C22⋊C4—181no—
145C3×C42—91no—
146#217—61no—
147(C2×D3)⋊C4—21no—
148(C2×D3)⋊C4—61no—
149C4×Dic3—61no—
150C4×Dic3—61no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32—122no—
156C22×C32—121no—
157C22×C32—122no—
158C22×C32—121no—
159C22×C32—122no—
160C22×C32—121no—
161C22×C32C2×C3×D411yescharacteristic
162C2×C3×D3—42no—
163C2×C3×D3—42no—
164C2×C3×D3—41no—
165C2×C3×D3—42no—
166C2×C3×D3—41no—
167C2×C3×D3—41no—
168C2×C3×D3—44no—
169C2×C3×D3—42no—
170C2×C3×D3—41no—
171C2×C3×D3—42no—
172C2×C3×D3—42no—
173C2×C3×D3—42no—
174C2×C3×D3—41no—
175C2×C3×D3—41no—
176C2×C3×D3—81no—
177C2×C3×D3—82no—
178C2×C3×D3—82no—
179C2×C3×D3—81no—
180C32×C4—22no—
181C32×C4—21no—
182C32×C4—41no—
183C32×C4—42no—
184C32×C4—81no—
185C32×C4—121no—
186C3⋊Dic3—21no—
187C3⋊Dic3C2×C3×D412yes—
188C3×Dic3—41no—
189C3×Dic3—41no—
190C3×Dic3—62no—
191C3×Dic3—62no—
192C3×Dic3—62no—
193C3×Dic3—81no—
194C3×Dic3—121no—
195C2×SD16—92no—
196C2×D8—92no—
197#127—91no—
198#124—91no—
199#97—91no—
200C33#49611yescharacteristic, complemented, Sylow
201C23×C3—61no—
202C23×C3—62no—
203C23×C3—61no—
204C23×C3—62no—
205C23×C3—121no—
206C23×C3—122no—
207C22×D3—21no—
208C22×D3—22no—
209C3×Q8—182no—
210C3×D4—122no—
211C3×D4—122no—
212C3×D4—124no—
213C3×D4—182no—
214C3×D4—182no—
215C3×D4—182no—
216C3×D4—362no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—121no—
226C2×C3×C4GO+(4,𝔽2)11yescharacteristic
227#67—122no—
228#67—122no—
229#67—122no—
230#67—124no—
231C2×Dic3—21no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C2×Dic3—61no—
236D12—42no—
237Dic6—122no—
238C3×C8—94no—
239C2×C32—21no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—41no—
249C2×C32—41no—
250C2×C32—41no—
251C2×C32—242no—
252C2×C32—241no—
253C2×C32C3×#12711yescharacteristic
254C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
255C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
256C3×D3—82no—
257C3×D3—81no—
258C3×D3—84no—
259C3×D3—82no—
260C3×D3—82no—
261C3×D3—81no—
262C2×Q8—91no—
263C2×D4—61no—
264C2×D4—91no—
265C2×D4—92no—
266C2×D4—181no—
267SD16—184no—
268D8—184no—
269C2×C8—92no—
270C22⋊C4—61no—
271C22⋊C4—181no—
272C42—91no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—122no—
278C22×C3—121no—
279C22×C3—122no—
280C22×C3—122no—
281C22×C3—121no—
282C22×C3—121no—
283C22×C3—124no—
284C22×C3—122no—
285C22×C3—122no—
286C22×C3—121no—
287C22×C3—121no—
288C22×C3—122no—
289C22×C3—122no—
290C22×C3—121no—
291C22×C3—241no—
292C22×C3—242no—
293C22×C3—241no—
294C22×C3—242no—
295C22×C3C2×GO+(4,𝔽2)11yescharacteristic
296C2×D3—42no—
297C2×D3—42no—
298C2×D3—41no—
299C2×D3—42no—
300C2×D3—41no—
301C2×D3—41no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—42no—
306C3×C4—81no—
307C3×C4—92no—
308C3×C4—121no—
309C3×C4—121no—
310C3×C4—181no—
311C3×C4—241no—
312Dic3—41no—
313Dic3—62no—
314Dic3—62no—
315Dic3—62no—
316Dic3—121no—
317C32—21no—
318C32—21no—
319C32—41no—
320C32—41no—
321C32C3×#49611yescharacteristic, complemented
322C23—61no—
323C23—62no—
324Q8—182no—
325D4—122no—
326D4—182no—
327D4—182no—
328D4—182no—
329D4—362no—
330C2×C4—61no—
331C2×C4—91no—
332C2×C4—91no—
333C2×C4C3×GO+(4,𝔽2)11yescharacteristic
334C8—94no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—21no—
339C2×C3—21no—
340C2×C3—21no—
341C2×C3—41no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—242no—
348C2×C3—241no—
349C2×C3—244no—
350C2×C3—242no—
351C2×C3—242no—
352C2×C3—241no—
353C2×C3#6422411yescharacteristic
354C2×C3#6423111yescharacteristic
355C2×C3#6423611yescharacteristic
356D3—82no—
357D3—81no—
358C22—122no—
359C22—121no—
360C22—122no—
361C22—121no—
362C22—121no—
363C22—122no—
364C22C2×C3×GO+(4,𝔽2)11yescharacteristic
365C4—21no—
366C4—92no—
367C4—121no—
368C4—181no—
369C3—21no—
370C3—21no—
371C3—41no—
372C3—41no—
373C3#1059342711yescharacteristic, direct product factor
374C2—242no—
375C2—241no—
376C2C3×#6422411yescharacteristic
377C2C3×#6423111yescharacteristic
378C2C3×#6423611yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#496
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
1. C2×C3×#3775C2semidirect
2. C2×C3×#3774C2semidirect
3. C3×#63915C2semidirect
4. C3×#63914C2semidirect
5. C3×#63777C2semidirect
6. #10593427C3direct (N × H)
18. C2×#3775C2×C3semidirect
19. C2×#3774C2×C3semidirect
22. #63915C2×C3semidirect
23. #63914C2×C3semidirect
24. #63777C2×C3semidirect
37. C3×#622D4semidirect
119. #622C3×D4semidirect
200. C33#496semidirect
321. C32C3×#496semidirect
373. C3#10593427direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37752
C2×C3×#37742
C3×#639152
C3×#639142
C3×#637772
#105934273
C3×#4969