Finite Group Databasebeta

← C3×#10591167 #13158346

Subgroups — C3×#10591167 #13158346

Order 1728.

830 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 830 conjugacy classes of subgroups — too many to draw. Only its 132 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×#3748C211yesmaximal, characteristic, complemented
2C32×#930C212yesmaximal, complemented
3C2×C3×#3713C211yesmaximal, characteristic, complemented
4C3×#63574C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6C3×#5039—32nomaximal
7#10591167C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C2×C32×#67—22no—
10C2×C32×D12C2212yes—
11C3×#3748—22no—
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C32×(C2×D3)⋊C4—22no—
15C32×C4⋊Dic3C2212yes—
16C3×#3713—22no—
17C32×#121—32no—
18C2×#3748C2×C311yescharacteristic, complemented
19C2×C3×#213—32no—
20C3×#930—22no—
21C3×#930D312yescomplemented
22C3×#930C2×C312yescomplemented
23C2×C3×#208—32no—
24C2×C3×#208—61no—
25C2×#3713C2×C311yescharacteristic, complemented
26C3×#832—32no—
27C3×#809—32no—
28C3×#808—32no—
29#63574C2×C311yescharacteristic, complemented
30#63562C2×C312yescomplemented
31C23×C33D411yescharacteristic
32C22×C32×D3—22no—
33C2×C33×C4—21no—
34C2×C33×C4—21no—
35C2×C33×C4D411yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C32×#67—44no—
38C2×C32×Dic3—22no—
39C32×D12—22no—
40C32×D12—22no—
41C3×C3⋊(C3⋊C8)—21no—
42C3×C3⋊(C3⋊C8)D412yescomplemented
43C3×#469—91no—
44#5039—32no—
45C2×C32×D4—32no—
46C2×C32×D4—62no—
47C22×C32×C4—21no—
48C22×C32×C4—22no—
49C22×C32×C4—41no—
50C22×C32×C4C2×D312yes—
51C22×C32×C4C22×C311yescharacteristic
52C2×C3×#67—24no—
53C2×C3×#67—22no—
54C2×C3×#67—22no—
55C2×C3×D12—22no—
56C2×C3×D12C2×D312yes—
57C2×C3×D12C22×C312yes—
58C32×C4⋊C4—32no—
59C32×C22⋊C4—62no—
60#3748—22no—
61C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
62C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
63C3×#213—64no—
64C3×(C2×D3)⋊C4—24no—
65C3×(C2×D3)⋊C4—22no—
66C3×(C2×D3)⋊C4—22no—
67C3×C4⋊Dic3—22no—
68C3×C4⋊Dic3C2×D312yes—
69C3×C4⋊Dic3C22×C312yes—
70C3×#208—64no—
71C3×#208—64no—
72C2×C3×C3⋊C8—32no—
73C2×C3×C3⋊C8—32no—
74C2×C3×C3⋊C8—61no—
75C2×C3×C3⋊C8—61no—
76#3713—22no—
77C22×C33—21no—
78C22×C33—22no—
79C22×C33C2×D411yescharacteristic
80C2×C32×D3—42no—
81C2×C32×D3—42no—
82C2×C32×D3—42no—
83C33×C4—21no—
84C33×C4C2×D411yescharacteristic
85C33×C4#4111yescharacteristic
86C32×Dic3—42no—
87C2×C3×D8—91no—
88C2×C3×M16—91no—
89C3×#121—32no—
90C3×#121—32no—
91C3×#121—62no—
92C2×#213—32no—
93#930C3×D312yescomplemented
94C2×#208—32no—
95C2×#208—61no—
96C3×#106—91no—
97C3×#102—92no—
98#832—32no—
99#809—32no—
100#808—32no—
101C23×C32—21no—
102C23×C32—22no—
103C23×C32—41no—
104C23×C32—62no—
105C23×C32#6712yes—
106C23×C32C3×D411yescharacteristic
107C22×C3×D3—24no—
108C22×C3×D3—22no—
109C22×C3×D3—22no—
110C32×D4—62no—
111C32×D4—62no—
112C32×D4—124no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4—42no—
126C2×C32×C4—42no—
127C2×C32×C4—42no—
128C2×C32×C4—42no—
129C2×C32×C4—62no—
130C2×C32×C4#6712yes—
131C2×C32×C4C3×D411yescharacteristic
132C2×C32×C4C22×D312yes—
133C2×C32×C4C23×C311yescharacteristic
134C3×#67—48no—
135C3×#67—44no—
136C3×#67—44no—
137C2×C3×Dic3—24no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C3×D12—22no—
141C3×D12—22no—
142C3×D12—22no—
143C3×D12—22no—
144C3×D12—42no—
145C3×D12—42no—
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)C3×D412yescomplemented
148C3×C3⋊C8—34no—
149C3×C3⋊C8—62no—
150C3×C3⋊C8—62no—
151C3×C3⋊C8—62no—
152#469—91noSylow
153C2×C33—41no—
154C2×C33#12111yescharacteristic
155C2×C33AGL(1,ℤ/8)12yes—
156C32×D3—82no—
157C2×C3×D4—32no—
158C2×C3×D4—32no—
159C2×C3×D4—62no—
160C2×C3×D4—64no—
161C2×C3×D4—62no—
162C2×C3×D4—62no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—41no—
166C22×C3×C4D3211yescharacteristic
167C22×C3×C4C2×C3×D312yes—
168C2×#67—22no—
169C2×D12C2×C3×D312yes—
170C3×D8—182no—
171C3×M16—182no—
172C2×C3×C8—91no—
173C2×C3×C8—91no—
174C3×C4⋊C4—32no—
175C3×C4⋊C4—32no—
176C3×C4⋊C4—62no—
177C3×C22⋊C4—64no—
178C3×C22⋊C4—62no—
179C3×C22⋊C4—62no—
180#213—64no—
181(C2×D3)⋊C4—22no—
182C4⋊Dic3C2×C3×D312yes—
183#208—64no—
184#208—64no—
185C2×C3⋊C8—32no—
186C2×C3⋊C8—32no—
187C2×C3⋊C8—61no—
188C2×C3⋊C8—61no—
189C22×C32—22no—
190C22×C32—21no—
191C22×C32—24no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—22no—
198C22×C32—41no—
199C22×C32—42no—
200C22×C32—42no—
201C22×C32—42no—
202C22×C32—42no—
203C22×C32—44no—
204C22×C32—122no—
205C22×C32—122no—
206C22×C32—122no—
207C22×C32C2×#6712yes—
208C22×C32C2×C3×D411yescharacteristic
209C2×C3×D3—44no—
210C2×C3×D3—44no—
211C2×C3×D3—44no—
212C2×C3×D3—42no—
213C2×C3×D3—42no—
214C2×C3×D3—42no—
215C2×C3×D3—42no—
216C2×C3×D3—42no—
217C2×C3×D3—42no—
218C32×C4—21no—
219C32×C4—22no—
220C32×C4—22no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4—42no—
228C32×C4—42no—
229C32×C4—122no—
230C32×C4D3×D412yes—
231C32×C4#23712yes—
232C32×C4C2×C3×D411yescharacteristic
233C32×C4C3×#4111yescharacteristic
234C3×Dic3—44no—
235C3×Dic3—42no—
236C3×Dic3—42no—
237C2×D8—91no—
238C2×M16—91no—
239#121—32no—
240#106—91no—
241#102—92no—
242C33#46911yescharacteristic, complemented, Sylow
243C23×C3—22no—
244C23×C3—21no—
245C23×C3—41no—
246C23×C3—64no—
247C23×C3—62no—
248C23×C3—62no—
249C23×C3#62511yescharacteristic
250C23×C3C3×#6712yes—
251C22×D3—22no—
252C3×D4—62no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—122no—
257C3×D4—122no—
258C3×D4—128no—
259C3×D4—124no—
260C3×D4—124no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—64no—
278C2×C3×C4—62no—
279C2×C3×C4—62no—
280C2×C3×C4#62511yescharacteristic
281C2×C3×C4C3×#6712yes—
282C2×C3×C4C2×D3211yescharacteristic
283C2×C3×C4C22×C3×D312yes—
284#67—44no—
285C2×Dic3—22no—
286D12—22no—
287D12—22no—
288C3×C8—92no—
289C3×C8—181no—
290C3⋊C8—34no—
291C3⋊C8—62no—
292C3⋊C8—62no—
293C3⋊C8—62no—
294C2×C32—22no—
295C2×C32—22no—
296C2×C32—22no—
297C2×C32—22no—
298C2×C32—21no—
299C2×C32—41no—
300C2×C32—42no—
301C2×C32—44no—
302C2×C32—42no—
303C2×C32—44no—
304C2×C32—42no—
305C2×C32—41no—
306C2×C32—242no—
307C2×C32#93212yes—
308C2×C32#93912yes—
309C2×C32#94912yes—
310C2×C32C3×#12111yescharacteristic
311C2×C32C3×AGL(1,ℤ/8)12yes—
312C3×D3—84no—
313C3×D3—82no—
314C3×D3—82no—
315C2×D4—32no—
316C2×D4—62no—
317C22×C4C3×D3211yescharacteristic
318D8—182no—
319M16—182no—
320C2×C8—91no—
321C2×C8—91no—
322C4⋊C4—32no—
323C22⋊C4—62no—
324C22×C3—22no—
325C22×C3—24no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—21no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—44no—
338C22×C3—41no—
339C22×C3—124no—
340C22×C3—124no—
341C22×C3—124no—
342C22×C3—122no—
343C22×C3—122no—
344C22×C3—122no—
345C22×C3—122no—
346C22×C3—122no—
347C22×C3—122no—
348C22×C3C2×#62511yescharacteristic
349C22×C3C2×C3×#6712yes—
350C2×D3—42no—
351C2×D3—42no—
352C2×D3—42no—
353C3×C4—22no—
354C3×C4—22no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—22no—
358C3×C4—21no—
359C3×C4—22no—
360C3×C4—42no—
361C3×C4—42no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—124no—
365C3×C4—122no—
366C3×C4—122no—
367C3×C4#380511yescharacteristic
368C3×C4#381111yescharacteristic
369C3×C4C3×D3×D412yes—
370C3×C4C3×#23712yes—
371Dic3—42no—
372C32—22no—
373C32—21no—
374C32—41no—
375C32#503912yescomplemented
376C32C3×#46911yescharacteristic, complemented
377C23—62no—
378C23C3×#62511yescharacteristic
379D4—62no—
380D4—62no—
381D4—124no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—62no—
385C2×C4C3×#62511yescharacteristic
386C2×C4C2×C3×D3211yescharacteristic
387C8—92no—
388C8—181no—
389C2×C3—21no—
390C2×C3—22no—
391C2×C3—22no—
392C2×C3—22no—
393C2×C3—22no—
394C2×C3—44no—
395C2×C3—42no—
396C2×C3—44no—
397C2×C3—42no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—42no—
401C2×C3—244no—
402C2×C3—242no—
403C2×C3—242no—
404C2×C3#6382312yes—
405C2×C3#6398011yescharacteristic
406C2×C3C3×#93212yes—
407C2×C3C3×#93912yes—
408C2×C3C3×#94912yes—
409D3—82no—
410C22—21no—
411C22—22no—
412C22—122no—
413C22—122no—
414C22—122no—
415C22C2×C3×#62511yescharacteristic
416C4—21no—
417C4—122no—
418C4C3×#380511yescharacteristic
419C4C3×#381111yescharacteristic
420C3—22no—
421C3—21no—
422C3—41no—
423C3#1059116711yescharacteristic, direct product factor
424C3C3×#503912yescomplemented
425C2—41no—
426C2—242no—
427C2C3×#6382312yes—
428C2C3×#6398011yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C2×D4
hypercenterC22×C3×C4D32
hypocenterC32C3×#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3748C2semidirect
2. C32×#930C2semidirect
3. C2×C3×#3713C2semidirect
4. C3×#63574C2semidirect
5. C3×#63562C2semidirect
7. #10591167C3direct (N × H)
18. C2×#3748C2×C3semidirect
21. C3×#930D3semidirect
22. C3×#930C2×C3semidirect
25. C2×#3713C2×C3semidirect
29. #63574C2×C3semidirect
30. #63562C2×C3semidirect
42. C3×C3⋊(C3⋊C8)D4semidirect
93. #930C3×D3semidirect
147. C3⋊(C3⋊C8)C3×D4semidirect
242. C33#469semidirect
375. C32#5039semidirect
376. C32C3×#469semidirect
423. C3#10591167direct (N × H)
424. C3C3×#5039semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
C32×#9302
C2×C3×#37132
C3×#635742
C3×#635622
C3×#50393
#105911673