Finite Group Databasebeta

← C3×#10593405 #13166766

Subgroups — C3×#10593405 #13166766

Order 1728.

944 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3802C211yesmaximal, characteristic
2C2×C3×C4⋊#157C211yesmaximal, characteristic, complemented
3C2×C3×D32⋊C4C214yesmaximal, complemented
4C3×C4×#624C211yesmaximal, characteristic, complemented
5#10593405C311yesmaximal, characteristic, direct product factor
6C22×C3×D32C2212yes—
7C22×C3×#157C2212yes—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12—21no—
10C2×C3×#625—21no—
11C2×C3×#624C2212yes—
12C3×#3802—22no—
13C3×#3802C414yescomplemented
14C3×C4⋊#157—22no—
15C3×D32⋊C4—28no—
16C32×C4×Dic3—21no—
17C3×Dic32—21no—
18C2×C3×D3×D4—61no—
19C2×#3802C2×C311yescharacteristic
20C2×C4⋊#157C2×C311yescharacteristic, complemented
21C2×D32⋊C4C2×C314yescomplemented
22C3×C42×D3—61no—
23C4×#624C2×C311yescharacteristic, complemented
24C22×C3×C3⋊D3C2311yescharacteristic
25C22×C32×D3—22no—
26C2×C3×D32—24no—
27C2×C3×D32—24no—
28C2×C3×D32—22no—
29C2×C3×D32C2×C414yes—
30C2×C3×#157—24no—
31C2×C3×#157—22no—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4×C3⋊D3C2×C412yes—
35C3×C4×C3⋊D3D412yes—
36C2×C32×Dic3—22no—
37C32×D12—24no—
38C3×#625—24no—
39C3×#624—22no—
40C3×#624D414yescomplemented
41C3×#390—91nomaximal
42C23×C3×D3—62no—
43C22×D32C22×C312yes—
44C22×#157C22×C312yes—
45C2×C32×D4—61no—
46C2×C4×C3⋊D3C22×C311yescharacteristic
47C2×C3×#67—62no—
48C3×D3×D4—64no—
49C3×D3×D4—64no—
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—41no—
53C2×C3×C4×D3—62no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×#625—21no—
57C2×#624C22×C312yes—
58#3802—22no—
59#3802C3×C414yescomplemented
60C4⋊#157—22no—
61D32⋊C4—28no—
62C32×C42—61no—
63C3×C4×Dic3—21no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3—41no—
66C3×C4×Dic3—62no—
67Dic32—21no—
68C22×C33C2×D411yescharacteristic
69C2×C3×C3⋊D3C22×C411yescharacteristic
70C2×C3×C3⋊D3C2×D412yes—
71C2×C3×C3⋊D3C2×D411yescharacteristic
72C2×C3×C3⋊D3#4112yes—
73C2×C32×D3—24no—
74C2×C32×D3—44no—
75C3×D32—28no—
76C3×D32—42no—
77C3×D32—42no—
78C3×#157—42no—
79C33×C4C22⋊C412yescomplemented
80C3×C3⋊Dic3C22⋊C412yescomplemented
81C32×Dic3—24no—
82C22×C3×D4—91no—
83C2×D3×D4—61no—
84C2×C3×C4⋊C4—91no—
85C2×C3×C22⋊C4—94no—
86C2×C3×C42—91no—
87C42×D3—61no—
88C23×C32—62no—
89C22×C3⋊D3C23×C311yescharacteristic
90C22×C3×D3—22no—
91C22×C3×D3—22no—
92C22×C3×D3—42no—
93C22×C3×D3—64no—
94C22×C3×D3—61no—
95C22×C3×D3—61no—
96C22×C3×D3—64no—
97C22×C3×D3—124no—
98C22×C3×D3—124no—
99C2×D32—22no—
100C2×D32—24no—
101C2×D32—24no—
102C2×D32C2×C3×C414yes—
103C2×#157—24no—
104C2×#157—22no—
105C32×D4—64no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—62no—
111C2×C32×C4C3×D411yescharacteristic
112C2×C3⋊Dic3C3×D411yescharacteristic
113C4×C3⋊D3C2×C3×C412yes—
114C4×C3⋊D3C3×D412yes—
115C3×#67—68no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—42no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C3×D12—24no—
122C3×D12—24no—
123C3×D12—44no—
124C3×C4×D3—64no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—64no—
130#625—24no—
131#624—22no—
132#624C3×D414yescomplemented
133#390—91noSylow
134C2×C33C2×C22⋊C411yescharacteristic
135C2×C33#12112yes—
136C3×C3⋊D3C4×D412yes—
137C3×C3⋊D3#12411yescharacteristic
138C3×C3⋊D3#12711yescharacteristic
139C32×D3—44no—
140C24×C3—92no—
141C23×D3—62no—
142C2×C3×D4—61no—
143C2×C3×D4—61no—
144C2×C3×D4—94no—
145C2×C3×D4—121no—
146C2×C3×D4—182no—
147C2×C3×D4—181no—
148C22×C3×C4—92no—
149C22×C3×C4—91no—
150C22×C3×C4—92no—
151C2×#67—62no—
152D3×D4—64no—
153D3×D4—64no—
154C2×D12—21no—
155C2×C4×D3—62no—
156C2×C4×D3—61no—
157C2×C4×D3—61no—
158C3×C4⋊C4—182no—
159C3×C22⋊C4—188no—
160C3×C42—61no—
161C3×C42—61no—
162C3×C42—121no—
163C3×C42—181no—
164C4×Dic3—21no—
165C4×Dic3—62no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—64no—
171C22×C32—124no—
172C22×C32C2×C3×D411yescharacteristic
173C2×C3⋊D3C22×C3×C411yescharacteristic
174C2×C3⋊D3C2×C3×D412yes—
175C2×C3⋊D3C2×C3×D411yescharacteristic
176C2×C3⋊D3C3×#4112yes—
177C2×C3×D3—24no—
178C2×C3×D3—24no—
179C2×C3×D3—44no—
180C2×C3×D3—44no—
181C2×C3×D3—44no—
182C2×C3×D3—61no—
183C2×C3×D3—61no—
184C2×C3×D3—62no—
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—62no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C2×C3×D3—84no—
191C2×C3×D3—128no—
192C2×C3×D3—124no—
193C2×C3×D3—124no—
194D32—28no—
195D32—42no—
196D32—42no—
197#157—42no—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—42no—
201C32×C4—42no—
202C32×C4—64no—
203C32×C4C3×C22⋊C412yescomplemented
204C3⋊Dic3C3×C22⋊C412yescomplemented
205C3×Dic3—24no—
206C3×Dic3—24no—
207C3×Dic3—44no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C22×D4—91no—
211C2×C4⋊C4—91no—
212C2×C22⋊C4—94no—
213C2×C42—91no—
214C33#39011yescharacteristic, complemented, Sylow
215C23×C3—62no—
216C23×C3—62no—
217C23×C3—94no—
218C23×C3—91no—
219C23×C3—122no—
220C23×C3—182no—
221C23×C3—184no—
222C23×C3—184no—
223C22×D3—22no—
224C22×D3—64no—
225C22×D3—61no—
226C22×D3—61no—
227C22×D3—64no—
228C22×D3—124no—
229C22×D3—124no—
230C3×D4—64no—
231C3×D4—64no—
232C3×D4—124no—
233C3×D4—184no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—94no—
241C2×C3×C4—91no—
242C2×C3×C4—92no—
243C2×C3×C4—92no—
244C2×C3×C4—122no—
245C2×C3×C4—182no—
246C2×C3×C4—184no—
247C2×C3×C4—182no—
248C2×C3×C4GO+(4,𝔽2)11yescharacteristic
249#67—68no—
250C2×Dic3—22no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253D12—24no—
254C4×D3—64no—
255C4×D3—62no—
256C4×D3—62no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—64no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32—124no—
269C2×C32C2×C3×C22⋊C411yescharacteristic
270C2×C32C3×#12112yes—
271C3⋊D3C3×C4×D412yes—
272C3⋊D3C3×#12411yescharacteristic
273C3⋊D3C3×#12711yescharacteristic
274C3×D3—44no—
275C3×D3—44no—
276C3×D3—62no—
277C3×D3—62no—
278C3×D3—61no—
279C3×D3—61no—
280C3×D3—61no—
281C3×D3—61no—
282C3×D3—84no—
283C24—92no—
284C2×D4—61no—
285C2×D4—94no—
286C2×D4—182no—
287C2×D4—181no—
288C22×C4—92no—
289C22×C4—91no—
290C22×C4—92no—
291C4⋊C4—182no—
292C22⋊C4—188no—
293C42—61no—
294C42—181no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—64no—
300C22×C3—64no—
301C22×C3—92no—
302C22×C3—92no—
303C22×C3—91no—
304C22×C3—91no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3—124no—
308C22×C3—188no—
309C22×C3—244no—
310C22×C3—362no—
311C22×C3—362no—
312C22×C3C2×GO+(4,𝔽2)11yescharacteristic
313C2×D3—24no—
314C2×D3—44no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—61no—
318C2×D3—62no—
319C2×D3—61no—
320C2×D3—62no—
321C2×D3—62no—
322C2×D3—62no—
323C2×D3—128no—
324C2×D3—124no—
325C2×D3—124no—
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—42no—
329C3×C4—42no—
330C3×C4—64no—
331C3×C4—64no—
332C3×C4—92no—
333C3×C4—124no—
334C3×C4—362no—
335C3×C4D32⋊C412yescomplemented
336Dic3—24no—
337Dic3—62no—
338Dic3—62no—
339C32—21no—
340C32—21no—
341C32—41no—
342C32—41no—
343C32C3×#39011yescharacteristic, complemented
344C23—62no—
345C23—94no—
346C23—91no—
347C23—182no—
348C23—184no—
349C23—184no—
350D4—64no—
351D4—184no—
352C2×C4—62no—
353C2×C4—94no—
354C2×C4—91no—
355C2×C4—92no—
356C2×C4—92no—
357C2×C4—182no—
358C2×C4—184no—
359C2×C4—182no—
360C2×C4C3×GO+(4,𝔽2)11yescharacteristic
361C2×C3—22no—
362C2×C3—22no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—42no—
366C2×C3—42no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3—91no—
370C2×C3—92no—
371C2×C3—91no—
372C2×C3—124no—
373C2×C3—124no—
374C2×C3—244no—
375C2×C3#6423012yes—
376C2×C3C2×D32⋊C411yescharacteristic
377D3—44no—
378D3—62no—
379D3—61no—
380D3—61no—
381D3—62no—
382D3—61no—
383D3—61no—
384C22—64no—
385C22—92no—
386C22—92no—
387C22—91no—
388C22—91no—
389C22—124no—
390C22—188no—
391C22—362no—
392C22—362no—
393C22C2×C3×GO+(4,𝔽2)11yescharacteristic
394C4—64no—
395C4—92no—
396C4—362no—
397C4C3×D32⋊C412yescomplemented
398C3—21no—
399C3—21no—
400C3—41no—
401C3—41no—
402C3#1059340511yescharacteristic, direct product factor
403C2—91no—
404C2—92no—
405C2—91no—
406C2—124no—
407C2C3×#6423012yes—
408C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#390
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊#157C2semidirect
3. C2×C3×D32⋊C4C2semidirect
4. C3×C4×#624C2semidirect
5. #10593405C3direct (N × H)
13. C3×#3802C4semidirect
20. C2×C4⋊#157C2×C3semidirect
21. C2×D32⋊C4C2×C3semidirect
23. C4×#624C2×C3semidirect
40. C3×#624D4semidirect
59. #3802C3×C4semidirect
79. C33×C4C22⋊C4semidirect
80. C3×C3⋊Dic3C22⋊C4semidirect
132. #624C3×D4semidirect
203. C32×C4C3×C22⋊C4semidirect
204. C3⋊Dic3C3×C22⋊C4semidirect
214. C33#390semidirect
335. C3×C4D32⋊C4semidirect
343. C32C3×#390semidirect
397. C4C3×D32⋊C4semidirect
402. C3#10593405direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×C3×C4⋊#1572
C2×C3×D32⋊C42
C3×C4×#6242
#105934053
C3×#3909