Finite Group Databasebeta

← C3×#10593439 #13166803

Subgroups — C3×#10593439 #13166803

Order 1728.

518 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 74 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
2C3×C4×#627C211yesmaximal, characteristic, complemented
3C3×#63914C211yesmaximal, characteristic
4C3×C4⋊#622C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593439C311yesmaximal, characteristic, direct product factor
7C2×C32×D12—21no—
8C2×C3×#627C2211yescharacteristic
9C2×C3×#625—21no—
10C2×C3×#625C2212yes—
11C2×C3×#622C2212yes—
12C3×#3775—24no—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C32×C4×Dic3—21no—
15C3×#3724C2211yescharacteristic
16C3×Dic3⋊Dic3—21no—
17C2×#3775C2×C312yescomplemented
18C3×#940—61no—
19C3×C4×Dic6—61no—
20C4×#627C2×C311yescharacteristic, complemented
21#63914C2×C311yescharacteristic
22C4⋊#622C2×C311yescharacteristic, complemented
23#63739C2×C312yescomplemented
24C22×C32×D3—22no—
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—21no—
29C2×C32×Dic3—21no—
30C32×D12—42no—
31C3×#627—21no—
32C3×#627D412yescomplemented
33C3×#625—22no—
34C3×#625—22no—
35C3×#625—42no—
36C3×#622—22no—
37C3×#460—91nomaximal
38C2×C32×D4—61no—
39C2×C3×#67—62no—
40C2×C3×#67—62no—
41C2×C3×D12—21no—
42C2×C3×D12—21no—
43C2×C3×D12—41no—
44C2×C3×Dic6—61no—
45C2×#627C22×C311yescharacteristic
46C2×#625—21no—
47C2×#625C22×C312yes—
48C2×#622C22×C312yes—
49#3775—24no—
50C32×C42—61no—
51C4×C3⋊Dic3C22×C311yescharacteristic
52C3×C4⋊Dic3—61no—
53C3×Dic3⋊C4—61no—
54C3×Dic3⋊C4—61no—
55C3×C4×Dic3—21no—
56C3×C4×Dic3—21no—
57C3×C4×Dic3—41no—
58C3×C4×Dic3—61no—
59C3×C4×Dic3—61no—
60#3724C22×C311yescharacteristic
61Dic3⋊Dic3—21no—
62C22×C33C2×D411yescharacteristic
63C2×C32×D3—42no—
64C2×C32×D3—42no—
65C2×C32×D3—42no—
66C33×C4SD1612yescomplemented
67C3×C3⋊Dic3—21no—
68C3×C3⋊Dic3C2×D411yescharacteristic
69C3×C3⋊Dic3#4111yescharacteristic
70C32×Dic3—22no—
71C32×Dic3—41no—
72C2×C3×SD16—92no—
73C3×#127—91no—
74C3×C4×Q8—91no—
75#940—61no—
76C4×Dic6—61no—
77C3×C4⋊C8—91no—
78C3×#102—92no—
79C23×C32—62no—
80C22×C3×D3—22no—
81C22×C3×D3—22no—
82C22×C3×D3—42no—
83C32×D4—122no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—41no—
87C2×C32×C4—41no—
88C2×C32×C4—61no—
89C2×C32×C4—61no—
90C2×C32×C4C3×D411yescharacteristic
91C2×C3⋊Dic3C3×D411yescharacteristic
92C2×C3⋊Dic3C23×C311yescharacteristic
93C3×#67—122no—
94C3×#67—122no—
95C3×#67—124no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—21no—
98C2×C3×Dic3—21no—
99C2×C3×Dic3—21no—
100C2×C3×Dic3—41no—
101C2×C3×Dic3—41no—
102C2×C3×Dic3—61no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C3×D12—42no—
107C3×D12—44no—
108C3×D12—42no—
109C3×Dic6—62no—
110C3×Dic6—62no—
111#627—21no—
112#627C3×D412yescomplemented
113#625—22no—
114#625—22no—
115#625—42no—
116#622—22no—
117#460—91noSylow
118C2×C33#12111yescharacteristic
119C2×C33C2×SD1611yescharacteristic
120C2×C33AGL(1,ℤ/8)11yescharacteristic
121C32×D3—82no—
122C2×C3×Q8—91no—
123C2×C3×D4—61no—
124C2×C3×D4—61no—
125C2×C3×D4—92no—
126C2×C3×D4—121no—
127C2×C3×D4—181no—
128C2×#67—62no—
129C2×#67—62no—
130C2×D12—21no—
131C2×Dic6—61no—
132C3×SD16—184no—
133C2×C3×C8—92no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4—181no—
136C3×C42—61no—
137C3×C42—61no—
138C3×C42—91no—
139C3×C42—121no—
140C4⋊Dic3—61no—
141Dic3⋊C4—61no—
142Dic3⋊C4—61no—
143C4×Dic3—21no—
144C4×Dic3—61no—
145C4×Dic3—61no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—41no—
149C22×C32—41no—
150C22×C32—122no—
151C22×C32—122no—
152C22×C32—122no—
153C22×C32C2×C3×D411yescharacteristic
154C2×C3×D3—42no—
155C2×C3×D3—42no—
156C2×C3×D3—42no—
157C2×C3×D3—44no—
158C2×C3×D3—42no—
159C2×C3×D3—42no—
160C2×C3×D3—42no—
161C2×C3×D3—82no—
162C2×C3×D3—82no—
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—42no—
166C32×C4—42no—
167C32×C4—62no—
168C32×C4—121no—
169C32×C4C3×SD1612yescomplemented
170C3⋊Dic3—21no—
171C3⋊Dic3C2×C3×D411yescharacteristic
172C3⋊Dic3C3×#4111yescharacteristic
173C3×Dic3—22no—
174C3×Dic3—22no—
175C3×Dic3—42no—
176C3×Dic3—41no—
177C3×Dic3—42no—
178C3×Dic3—41no—
179C3×Dic3—61no—
180C3×Dic3—61no—
181C3×Dic3—61no—
182C3×Dic3—61no—
183C3×Dic3—62no—
184C3×Dic3—121no—
185C2×SD16—92no—
186#127—91no—
187C4×Q8—91no—
188C4⋊C8—91no—
189#102—92no—
190C33#46011yescharacteristic, complemented, Sylow
191C23×C3—62no—
192C23×C3—62no—
193C23×C3—122no—
194C22×D3—22no—
195C3×Q8—92no—
196C3×Q8—181no—
197C3×D4—122no—
198C3×D4—124no—
199C3×D4—122no—
200C3×D4—182no—
201C3×D4—182no—
202C3×D4—362no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—121no—
214C2×C3×C4—121no—
215C2×C3×C4GO+(4,𝔽2)11yescharacteristic
216#67—122no—
217#67—122no—
218#67—124no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—61no—
222C2×Dic3—61no—
223C2×Dic3—61no—
224C2×Dic3—61no—
225D12—42no—
226Dic6—62no—
227Dic6—62no—
228C3×C8—182no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—41no—
241C2×C32—242no—
242C2×C32C3×#12111yescharacteristic
243C2×C32C2×C3×SD1611yescharacteristic
244C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
245C3×D3—82no—
246C3×D3—84no—
247C3×D3—82no—
248C2×Q8—91no—
249C2×D4—61no—
250C2×D4—92no—
251C2×D4—181no—
252SD16—184no—
253C2×C8—92no—
254C4⋊C4—91no—
255C4⋊C4—181no—
256C42—61no—
257C42—91no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—122no—
263C22×C3—122no—
264C22×C3—122no—
265C22×C3—124no—
266C22×C3—122no—
267C22×C3—122no—
268C22×C3—122no—
269C22×C3—242no—
270C22×C3—242no—
271C22×C3C2×GO+(4,𝔽2)11yescharacteristic
272C2×D3—42no—
273C2×D3—42no—
274C2×D3—42no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—42no—
278C3×C4—42no—
279C3×C4—62no—
280C3×C4—62no—
281C3×C4—91no—
282C3×C4—91no—
283C3×C4—122no—
284C3×C4—121no—
285C3×C4—122no—
286C3×C4—121no—
287C3×C4—181no—
288C3×C4#377512yescomplemented
289Dic3—22no—
290Dic3—41no—
291Dic3—61no—
292Dic3—61no—
293Dic3—61no—
294Dic3—61no—
295Dic3—62no—
296Dic3—121no—
297C32—21no—
298C32—21no—
299C32—41no—
300C32—41no—
301C32C3×#46011yescharacteristic, complemented
302C23—62no—
303Q8—92no—
304Q8—181no—
305D4—122no—
306D4—182no—
307D4—182no—
308D4—362no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4C3×GO+(4,𝔽2)11yescharacteristic
314C8—182no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—242no—
328C2×C3—244no—
329C2×C3—242no—
330C2×C3#6422411yescharacteristic
331C2×C3#6423011yescharacteristic
332C2×C3C2×#377511yescharacteristic
333D3—82no—
334C22—122no—
335C22—122no—
336C22—122no—
337C22C2×C3×GO+(4,𝔽2)11yescharacteristic
338C4—62no—
339C4—91no—
340C4—91no—
341C4—121no—
342C4—181no—
343C4C3×#377512yescomplemented
344C3—21no—
345C3—21no—
346C3—41no—
347C3—41no—
348C3#1059343911yescharacteristic, direct product factor
349C2—242no—
350C2C3×#6422411yescharacteristic
351C2C3×#6423011yescharacteristic
352C2C2×C3×#377511yescharacteristic
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×#63914C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#460
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. C3×C4×#627C2semidirect
4. C3×C4⋊#622C2semidirect
5. C3×#63739C2semidirect
6. #10593439C3direct (N × H)
17. C2×#3775C2×C3semidirect
20. C4×#627C2×C3semidirect
22. C4⋊#622C2×C3semidirect
23. #63739C2×C3semidirect
32. C3×#627D4semidirect
66. C33×C4SD16semidirect
112. #627C3×D4semidirect
169. C32×C4C3×SD16semidirect
190. C33#460semidirect
288. C3×C4#3775semidirect
301. C32C3×#460semidirect
343. C4C3×#3775semidirect
348. C3#10593439direct (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
C3×C4×#6272
C3×#639142
C3×C4⋊#6222
C3×#637392
#105934393
C3×#4609