Finite Group Databasebeta

← C3×#10593407 #13166768

Subgroups — C3×#10593407 #13166768

Order 1728.

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

normal subgroups only

This group has 924 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×#63900C211yesmaximal, characteristic, complemented
5#10593407C311yesmaximal, 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×#3724—21no—
18C2×C3×D3×D4—61no—
19C2×#3802C2×C311yescharacteristic
20C2×C4×#157C2×C311yescharacteristic, complemented
21C2×D32⋊C4C2×C314yescomplemented
22C3×D3×C4⋊C4—61no—
23#63900C2×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—22no—
31C2×C3×#157—22no—
32C2×C3×#157D414yescomplemented
33C2×C33×C4D411yescharacteristic
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×C4×C3⋊D3C2×C412yes—
36C3×C4×C3⋊D3D412yes—
37C2×C32×Dic3—22no—
38C32×D12—24no—
39C3×#625—24no—
40C3×#624—24no—
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—61no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—62no—
56C2×#625—21no—
57C2×#624C22×C312yes—
58#3802—22no—
59#3802C3×C414yescomplemented
60C4×#157—22no—
61D32⋊C4—28no—
62C32×C4⋊C4—61no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3—41no—
66C3×Dic3⋊C4—62no—
67#3724—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—42no—
75C2×C32×D3—42no—
76C3×D32—28no—
77C3×D32—44no—
78C3×#157—24no—
79C33×C4C22⋊C412yescomplemented
80C3×C3⋊Dic3C22⋊C412yescomplemented
81C32×Dic3—42no—
82C22×C3×D4—91no—
83C2×D3×D4—61no—
84C2×C3×C4⋊C4—91no—
85C2×C3×C22⋊C4—94no—
86C2×C3×C42—91no—
87D3×C4⋊C4—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—22no—
104C2×#157—22no—
105C2×#157C3×D414yescomplemented
106C32×D4—64no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—62no—
112C2×C32×C4C3×D411yescharacteristic
113C2×C3⋊Dic3C3×D411yescharacteristic
114C4×C3⋊D3C2×C3×C412yes—
115C4×C3⋊D3C3×D412yes—
116C3×#67—68no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—42no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C3×D12—24no—
123C3×D12—24no—
124C3×D12—44no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—124no—
130#625—24no—
131#624—24no—
132#390—91noSylow
133C2×C33C2×C22⋊C411yescharacteristic
134C2×C33#12411yescharacteristic
135C2×C33#12711yescharacteristic
136C3×C3⋊D3C4×D412yes—
137C3×C3⋊D3#12112yes—
138C32×D3—44no—
139C24×C3—92no—
140C23×D3—62no—
141C2×C3×D4—61no—
142C2×C3×D4—61no—
143C2×C3×D4—94no—
144C2×C3×D4—121no—
145C2×C3×D4—182no—
146C2×C3×D4—181no—
147C22×C3×C4—92no—
148C22×C3×C4—91no—
149C22×C3×C4—92no—
150C2×#67—62no—
151D3×D4—64no—
152D3×D4—64no—
153C2×D12—21no—
154C2×C4×D3—61no—
155C2×C4×D3—61no—
156C2×C4×D3—62no—
157C3×C4⋊C4—61no—
158C3×C4⋊C4—61no—
159C3×C4⋊C4—121no—
160C3×C4⋊C4—181no—
161C3×C22⋊C4—188no—
162C3×C42—182no—
163C4⋊Dic3—21no—
164Dic3⋊C4—62no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—41no—
168C22×C32—41no—
169C22×C32—64no—
170C22×C32—122no—
171C22×C32—122no—
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—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C2×C3×D3—42no—
184C2×C3×D3—61no—
185C2×C3×D3—61no—
186C2×C3×D3—61no—
187C2×C3×D3—61no—
188C2×C3×D3—62no—
189C2×C3×D3—62no—
190C2×C3×D3—62no—
191C2×C3×D3—62no—
192C2×C3×D3—82no—
193C2×C3×D3—82no—
194C2×C3×D3—128no—
195C2×C3×D3—128no—
196D32—28no—
197D32—44no—
198#157—24no—
199C32×C4—22no—
200C32×C4—22no—
201C32×C4—42no—
202C32×C4—42no—
203C32×C4—122no—
204C32×C4C3×C22⋊C412yescomplemented
205C3⋊Dic3C3×C22⋊C412yescomplemented
206C3×Dic3—42no—
207C3×Dic3—42no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C3×Dic3—82no—
211C22×D4—91no—
212C2×C4⋊C4—91no—
213C2×C22⋊C4—94no—
214C2×C42—91no—
215C33#39011yescharacteristic, complemented, Sylow
216C23×C3—62no—
217C23×C3—62no—
218C23×C3—94no—
219C23×C3—91no—
220C23×C3—122no—
221C23×C3—182no—
222C23×C3—184no—
223C23×C3—184no—
224C22×D3—22no—
225C22×D3—64no—
226C22×D3—61no—
227C22×D3—61no—
228C22×D3—64no—
229C22×D3—124no—
230C22×D3—124no—
231C3×D4—64no—
232C3×D4—64no—
233C3×D4—124no—
234C3×D4—184no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—91no—
242C2×C3×C4—92no—
243C2×C3×C4—92no—
244C2×C3×C4—94no—
245C2×C3×C4—122no—
246C2×C3×C4—184no—
247C2×C3×C4—182no—
248C2×C3×C4—182no—
249C2×C3×C4GO+(4,𝔽2)11yescharacteristic
250#67—68no—
251C2×Dic3—22no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254D12—24no—
255C4×D3—62no—
256C4×D3—62no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—124no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—124no—
273C2×C32C2×C3×C22⋊C411yescharacteristic
274C2×C32C3×#12411yescharacteristic
275C2×C32C3×#12711yescharacteristic
276C3⋊D3C3×C4×D412yes—
277C3⋊D3C3×#12112yes—
278C3×D3—44no—
279C3×D3—44no—
280C3×D3—62no—
281C3×D3—62no—
282C3×D3—62no—
283C3×D3—62no—
284C3×D3—84no—
285C24—92no—
286C2×D4—61no—
287C2×D4—94no—
288C2×D4—182no—
289C2×D4—181no—
290C22×C4—92no—
291C22×C4—91no—
292C22×C4—92no—
293C4⋊C4—61no—
294C4⋊C4—181no—
295C22⋊C4—188no—
296C42—182no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—64no—
302C22×C3—64no—
303C22×C3—92no—
304C22×C3—92no—
305C22×C3—91no—
306C22×C3—91no—
307C22×C3—124no—
308C22×C3—122no—
309C22×C3—122no—
310C22×C3—122no—
311C22×C3—122no—
312C22×C3—188no—
313C22×C3—242no—
314C22×C3—242no—
315C22×C3—364no—
316C22×C3C2×GO+(4,𝔽2)11yescharacteristic
317C2×D3—24no—
318C2×D3—42no—
319C2×D3—42no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—61no—
323C2×D3—61no—
324C2×D3—62no—
325C2×D3—62no—
326C2×D3—62no—
327C2×D3—62no—
328C2×D3—128no—
329C2×D3—128no—
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—92no—
335C3×C4—122no—
336C3×C4—122no—
337C3×C4—184no—
338C3×C4—242no—
339C3×C4D32⋊C412yescomplemented
340Dic3—42no—
341Dic3—62no—
342Dic3—62no—
343C32—21no—
344C32—21no—
345C32—41no—
346C32—41no—
347C32C3×#39011yescharacteristic, complemented
348C23—62no—
349C23—94no—
350C23—91no—
351C23—182no—
352C23—184no—
353C23—184no—
354D4—64no—
355D4—184no—
356C2×C4—62no—
357C2×C4—91no—
358C2×C4—92no—
359C2×C4—92no—
360C2×C4—94no—
361C2×C4—184no—
362C2×C4—182no—
363C2×C4—182no—
364C2×C4C3×GO+(4,𝔽2)11yescharacteristic
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—41no—
375C2×C3—41no—
376C2×C3—41no—
377C2×C3—92no—
378C2×C3—92no—
379C2×C3—124no—
380C2×C3—124no—
381C2×C3—244no—
382C2×C3#6422111yescharacteristic
383C2×C3#6423111yescharacteristic
384C2×C3C2×D32⋊C411yescharacteristic
385D3—44no—
386D3—62no—
387D3—62no—
388D3—62no—
389D3—62no—
390C22—64no—
391C22—92no—
392C22—92no—
393C22—91no—
394C22—91no—
395C22—122no—
396C22—122no—
397C22—188no—
398C22—364no—
399C22C2×C3×GO+(4,𝔽2)11yescharacteristic
400C4—92no—
401C4—122no—
402C4—184no—
403C4C3×D32⋊C412yescomplemented
404C3—21no—
405C3—21no—
406C3—41no—
407C3—41no—
408C3#1059340711yescharacteristic, direct product factor
409C2—92no—
410C2—92no—
411C2—124no—
412C2C3×#6422111yescharacteristic
413C2C3×#6423111yescharacteristic
414C2C2×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×#63900C2semidirect
5. #10593407C3direct (N × H)
13. C3×#3802C4semidirect
20. C2×C4×#157C2×C3semidirect
21. C2×D32⋊C4C2×C3semidirect
23. #63900C2×C3semidirect
32. C2×C3×#157D4semidirect
59. #3802C3×C4semidirect
79. C33×C4C22⋊C4semidirect
80. C3×C3⋊Dic3C22⋊C4semidirect
105. C2×#157C3×D4semidirect
204. C32×C4C3×C22⋊C4semidirect
205. C3⋊Dic3C3×C22⋊C4semidirect
215. C33#390semidirect
339. C3×C4D32⋊C4semidirect
347. C32C3×#390semidirect
403. C4C3×D32⋊C4semidirect
408. C3#10593407direct (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×#639002
#105934073
C3×#3909