Finite Group Databasebeta

← C3×#10591503 #13158518

Subgroups — C3×#10591503 #13158518

Order 1728.

1126 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1126 conjugacy classes of subgroups — too many to draw. Only its 148 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×D4×C3⋊D3C211yesmaximal, characteristic
2C3×#63818C212yesmaximal, complemented
3C32×#833C212yesmaximal, complemented
4C3×#63559C212yesmaximal, complemented
5C3×D3×#100—32nomaximal
6#10591503C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C412yes—
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638C412yes—
10C3×D4×C3⋊D3—22no—
11C3×D4×C3⋊D3—22no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C32×#208C2214yescomplemented
15C3×#3712—22no—
16C3×#3710—22no—
17C2×D4×C3⋊D3C2×C311yescharacteristic
18C2×C3×D3×D4—32no—
19C2×C3×D3×D4—61no—
20C3×D3×M16—34no—
21#63818C2×C312yescomplemented
22C32×#100—32no—
23C3×#833—22no—
24C3×#833D312yescomplemented
25C3×#833C2×C312yescomplemented
26C3×#823—34no—
27#63559C2×C312yescomplemented
28C23×C33C2×C412yes—
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—22no—
31C22×C3×C3⋊D3—44no—
32C22×C3×C3⋊D3C2×C411yescharacteristic
33C22×C3×C3⋊D3C2×C412yes—
34C33×D4—22no—
35C2×C33×C4C2311yescharacteristic
36C3×#638—24no—
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C3×C4⋊(C3⋊D3)—22no—
39C3×C4×C3⋊D3D412yes—
40C3×C4×C3⋊D3D412yes—
41C32×C3⋊C8—24no—
42C2×C3×#100—91no—
43D3×#100—32no—
44C23×C3⋊D3C3×C412yes—
45C23×C3×D3—34no—
46C23×C3×D3—62no—
47C2×C32×D4—21no—
48C2×C32×D4—22no—
49C2×C32×D4—41no—
50C2×C32×D4C2×D312yes—
51C2×C32×D4C22×C311yescharacteristic
52C2×#638C3×C412yes—
53D4×C3⋊D3—22no—
54D4×C3⋊D3—22no—
55C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×#67—34no—
58C2×C3×#67—62no—
59C3×D3×D4—64no—
60C3×D3×D4—64no—
61C3×D3×D4—64no—
62C3×D3×D4—64no—
63C2×C3×D12—32no—
64C2×C3×D12—61no—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67C32×M16—34no—
68C3×#208—24no—
69C3×#208C2×D314yescomplemented
70C3×#208C22×C314yescomplemented
71C3×ΓL(1,𝔽25)—64no—
72C3×D3×C8—64no—
73#3712—22no—
74#3710—22no—
75C22×C33—22no—
76C22×C33—42no—
77C22×C33C22×C411yescharacteristic
78C2×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—22no—
80C2×C3×C3⋊D3—44no—
81C2×C3×C3⋊D3—44no—
82C2×C3×C3⋊D3—42no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C22⋊C411yescharacteristic
85C33×C4C2×D412yes—
86C3×C3⋊Dic3C22⋊C412yes—
87C22×C3×D4—91no—
88C2×D3×D4—32no—
89C2×D3×D4—61no—
90C2×C3×M16—92no—
91D3×M16—34no—
92C3×#100—32no—
93C3×#100—32no—
94C3×#100—62no—
95C3×#100—92no—
96#833C3×D312yescomplemented
97#823—34no—
98C23×C32—22no—
99C23×C32—24no—
100C23×C32—42no—
101C23×C32C4×D314yes—
102C23×C32C2×C3×C412yes—
103C22×C3⋊D3—22no—
104C22×C3⋊D3—22no—
105C22×C3⋊D3—44no—
106C22×C3⋊D3C2×C3×C411yescharacteristic
107C22×C3⋊D3C2×C3×C412yes—
108C22×C3×D3—32no—
109C22×C3×D3—34no—
110C22×C3×D3—61no—
111C22×C3×D3—62no—
112C22×C3×D3—64no—
113C22×C3×D3—64no—
114C22×C3×D3—64no—
115C22×C3×D3—64no—
116C22×C3×D3—128no—
117C22×C3×D3—128no—
118C32×D4—22no—
119C32×D4—24no—
120C32×D4—24no—
121C32×D4—44no—
122C32×D4—44no—
123C2×C32×C4—21no—
124C2×C32×C4—22no—
125C2×C32×C4—41no—
126C2×C32×C4C22×D312yes—
127C2×C32×C4C23×C311yescharacteristic
128#638—24no—
129C2×C3⋊Dic3C2×C3×C411yescharacteristic
130C4⋊(C3⋊D3)—22no—
131C4×C3⋊D3C3×D412yes—
132C4×C3⋊D3C3×D412yes—
133C3×#67—68no—
134C3×#67—68no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—61no—
137C3×D12—64no—
138C3×D12—64no—
139C3×C4×D3—34no—
140C3×C4×D3—34no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C32×C8—64no—
144C3×C3⋊C8—24no—
145C3×C3⋊C8—24no—
146C3×C3⋊C8—44no—
147C2×#100—91noSylow
148C2×C33—21no—
149C2×C33—42no—
150C2×C33C2×C22⋊C411yescharacteristic
151C3×C3⋊D3—21no—
152C3×C3⋊D3—42no—
153C3×C3⋊D3#10012yescomplemented
154C24×C3—92no—
155C23×D3—34no—
156C23×D3—62no—
157C2×C3×D4—21no—
158C2×C3×D4—22no—
159C2×C3×D4—41no—
160C2×C3×D4—92no—
161C2×C3×D4—91no—
162C2×C3×D4—182no—
163C2×C3×D4—182no—
164C2×C3×D4D3211yescharacteristic
165C2×C3×D4C2×C3×D312yes—
166C22×C3×C4—91no—
167C2×#67—34no—
168C2×#67—62no—
169D3×D4—64no—
170D3×D4—64no—
171D3×D4—64no—
172D3×D4—64no—
173C2×D12—32no—
174C2×D12—61no—
175C2×C4×D3—32no—
176C2×C4×D3—61no—
177C3×M16—34no—
178C3×M16—34no—
179C3×M16—64no—
180C3×M16—182no—
181C2×C3×C8—182no—
182#208C2×C3×D314yescomplemented
183ΓL(1,𝔽25)—64no—
184D3×C8—64no—
185C22×C32—22no—
186C22×C32—24no—
187C22×C32—24no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—41no—
191C22×C32—44no—
192C22×C32—44no—
193C22×C32—42no—
194C22×C32—44no—
195C22×C32—44no—
196C22×C32—84no—
197C22×C32—84no—
198C22×C32C2×C4×D312yes—
199C22×C32C22×C3×C411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—22no—
202C2×C3⋊D3—44no—
203C2×C3⋊D3—44no—
204C2×C3⋊D3—42no—
205C2×C3⋊D3C3×C22⋊C411yescharacteristic
206C2×C3⋊D3C3×C22⋊C411yescharacteristic
207C2×C3×D3—32no—
208C2×C3×D3—32no—
209C2×C3×D3—61no—
210C2×C3×D3—61no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—64no—
214C2×C3×D3—64no—
215C2×C3×D3—128no—
216C2×C3×D3—128no—
217C2×C3×D3—128no—
218C2×C3×D3—128no—
219C2×C3×D3—124no—
220C2×C3×D3—124no—
221C32×C4—22no—
222C32×C4—24no—
223C32×C4—42no—
224C32×C4D3×D414yes—
225C32×C4C2×C3×D412yes—
226C3⋊Dic3C3×C22⋊C412yes—
227C3×Dic3—34no—
228C3×Dic3—62no—
229C22×D4—91no—
230C2×M16—92no—
231#100—32no—
232#100—92no—
233C33C2×#10011yescharacteristic, complemented, Sylow
234C23×C3—22no—
235C23×C3—24no—
236C23×C3—42no—
237C23×C3—91no—
238C23×C3—92no—
239C23×C3—182no—
240C23×C3—182no—
241C23×C3—364no—
242C23×C3#62412yes—
243C23×C3C3×C4×D314yes—
244C22×D3—32no—
245C22×D3—34no—
246C22×D3—61no—
247C22×D3—62no—
248C22×D3—64no—
249C22×D3—64no—
250C22×D3—64no—
251C22×D3—64no—
252C22×D3—128no—
253C22×D3—128no—
254C3×D4—24no—
255C3×D4—22no—
256C3×D4—24no—
257C3×D4—44no—
258C3×D4—44no—
259C3×D4—184no—
260C3×D4—182no—
261C2×C3×C4—22no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—91no—
265C2×C3×C4—92no—
266C2×C3×C4—92no—
267C2×C3×C4C2×D3211yescharacteristic
268C2×C3×C4C22×C3×D312yes—
269#67—68no—
270#67—68no—
271C2×Dic3—32no—
272C2×Dic3—61no—
273D12—64no—
274D12—64no—
275C4×D3—34no—
276C4×D3—34no—
277C4×D3—62no—
278C4×D3—62no—
279C3×C8—64no—
280C3×C8—64no—
281C3×C8—124no—
282C3⋊C8—24no—
283C2×C32—21no—
284C2×C32—22no—
285C2×C32—22no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—41no—
289C2×C32—42no—
290C2×C32—42no—
291C2×C32—42no—
292C2×C32—44no—
293C2×C32—44no—
294C2×C32—84no—
295C2×C32—84no—
296C2×C32D3×C22⋊C412yes—
297C2×C32C2×C3×C22⋊C411yescharacteristic
298C3⋊D3—21no—
299C3⋊D3—42no—
300C3⋊D3C3×#10012yescomplemented
301C3×D3—34no—
302C3×D3—62no—
303C3×D3—62no—
304C3×D3—62no—
305C3×D3—124no—
306C3×D3—124no—
307C24—92no—
308C2×D4—92no—
309C2×D4—91no—
310C2×D4—182no—
311C2×D4—182no—
312C2×D4C3×D3211yescharacteristic
313C22×C4—91no—
314M16—34no—
315M16—182no—
316C2×C8—182no—
317C22×C3—24no—
318C22×C3—24no—
319C22×C3—22no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—44no—
323C22×C3—44no—
324C22×C3—44no—
325C22×C3—42no—
326C22×C3—44no—
327C22×C3—41no—
328C22×C3—84no—
329C22×C3—84no—
330C22×C3—91no—
331C22×C3—91no—
332C22×C3—182no—
333C22×C3—182no—
334C22×C3—362no—
335C22×C3—364no—
336C22×C3—364no—
337C22×C3C2×#62411yescharacteristic
338C22×C3C2×C3×C4×D312yes—
339C2×D3—32no—
340C2×D3—32no—
341C2×D3—61no—
342C2×D3—61no—
343C2×D3—64no—
344C2×D3—64no—
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—128no—
348C2×D3—128no—
349C2×D3—128no—
350C2×D3—128no—
351C2×D3—124no—
352C2×D3—124no—
353C3×C4—24no—
354C3×C4—22no—
355C3×C4—42no—
356C3×C4—92no—
357C3×C4#381112yes—
358C3×C4C3×D3×D414yes—
359Dic3—34no—
360Dic3—62no—
361C32—21no—
362C32—22no—
363C32—41no—
364C32D3×#10012yescomplemented
365C32C2×C3×#10011yescharacteristic, complemented
366C23—91no—
367C23—92no—
368C23—182no—
369C23—182no—
370C23—364no—
371C23C3×#62412yes—
372D4—22no—
373D4—184no—
374D4—182no—
375C2×C4—91no—
376C2×C4—92no—
377C2×C4—92no—
378C2×C4C2×C3×D3211yescharacteristic
379C8—64no—
380C2×C3—22no—
381C2×C3—21no—
382C2×C3—22no—
383C2×C3—21no—
384C2×C3—22no—
385C2×C3—42no—
386C2×C3—42no—
387C2×C3—44no—
388C2×C3—42no—
389C2×C3—44no—
390C2×C3—41no—
391C2×C3—84no—
392C2×C3—84no—
393C2×C3—92no—
394C2×C3—181no—
395C2×C3—362no—
396C2×C3#6397411yescharacteristic
397C2×C3C3×D3×C22⋊C412yes—
398D3—34no—
399D3—62no—
400D3—62no—
401D3—62no—
402D3—124no—
403D3—124no—
404C22—22no—
405C22—42no—
406C22—91no—
407C22—91no—
408C22—182no—
409C22—182no—
410C22—362no—
411C22—364no—
412C22—364no—
413C22C2×C3×#62411yescharacteristic
414C4—92no—
415C4C3×#381112yes—
416C3—21no—
417C3—22no—
418C3—41no—
419C3#1059150311yescharacteristic, direct product factor
420C3C3×D3×#10012yescomplemented
421C2—21no—
422C2—42no—
423C2—92no—
424C2—181no—
425C2—362no—
426C2C3×#6397411yescharacteristic
subgroupG/H
centerC2×C3#63974
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D4D32
hypocenterC32C2×C3×#100
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63818C2semidirect
3. C32×#833C2semidirect
4. C3×#63559C2semidirect
6. #10591503C3direct (N × H)
14. C32×#208C22semidirect
21. #63818C2×C3semidirect
24. C3×#833D3semidirect
25. C3×#833C2×C3semidirect
27. #63559C2×C3semidirect
69. C3×#208C2×D3semidirect
70. C3×#208C22×C3semidirect
96. #833C3×D3semidirect
153. C3×C3⋊D3#100semidirect
182. #208C2×C3×D3semidirect
233. C33C2×#100semidirect
300. C3⋊D3C3×#100semidirect
364. C32D3×#100semidirect
365. C32C2×C3×#100semidirect
419. C3#10591503direct (N × H)
420. C3C3×D3×#100semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C3×#638182
C32×#8332
C3×#635592
C3×D3×#1003
#105915033