Finite Group Databasebeta

← #13155844

Subgroups — #13155844

Order 1728.

952 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 118 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
2#11723435C211yesmaximal, characteristic, complemented
3C2×C3×#3715C212yesmaximal, complemented
4#11722072C211yesmaximal, characteristic, complemented
5#11722032C212yesmaximal, complemented
6#10591652—31nomaximal
7#10591458—32nomaximal
8C22×C3×D32—21no—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12C2212yes—
11C2×C32×Dic6C2211yescharacteristic, complemented
12C2×C3×#625—21no—
13C3×#3802—24no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
15C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
16#88806—21no—
17#88803—21no—
18C3×#3715—24no—
19C2×C3×D3×D4—32no—
20C2×#3802D311yescharacteristic
21#64155—31no—
22C2×C3×#215—32no—
23C2×C3×#214—32no—
24#63856—32no—
25#63856—61no—
26C2×#3715D312yescomplemented
27#63659—32no—
28#63563—32no—
29#63563—32no—
30#63563—62no—
31#63557—31no—
32(C2×D3)⋊(C3⋊C8)—32no—
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C32×D3—22no—
35C2×C3×D32—24no—
36C2×C3×D32—42no—
37C2×C3×D32—42no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4×C3⋊D3—22no—
42C2×C32×Dic3—21no—
43C32×D12—22no—
44C32×D12D414yes—
45C32×Dic6—22no—
46C3×#625—24no—
47C3×C3⋊(C3⋊C8)—22no—
48#5218—91no—
49#5169—92no—
50C23×C3×D3—62no—
51C22×D32—21no—
52C2×C32×Q8—31no—
53C2×C32×D4—32no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×C3×#67—62no—
56C3×D3×D4—64no—
57C3×D3×D4—64no—
58C2×C3×D12—22no—
59C2×C3×D12C2×D312yes—
60C2×C3×D12C2×D312yes—
61C2×C3×C4×D3—32no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6C2×D312yescomplemented
65C2×#625—21no—
66#3802—24no—
67#3752—61no—
68C4⋊(C3⋊Dic3)—31no—
69C4⋊(C3⋊Dic3)—32no—
70C4⋊(C3⋊Dic3)—61no—
71C4⋊(C3⋊Dic3)—62no—
72C4⋊(C3⋊Dic3)—121no—
73#3750—61no—
74C2×C3⋊(C3⋊C8)C2×D312yescomplemented
75C3×#215—64no—
76C3×#214—64no—
77C2×C3×C3⋊C8—32no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—62no—
80#3724—62no—
81#3724—62no—
82(C2×D3)⋊Dic3—62no—
83(C2×D3)⋊Dic3—62no—
84#3715—24no—
85C22×C33C2×D411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3SD1612yescomplemented
89C2×C32×D3—24no—
90C2×C32×D3—42no—
91C2×C32×D3—42no—
92C3×D32—48no—
93C33×C4C2×D412yes—
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—21no—
96C32×Dic3—41no—
97C22×C3×D4—91no—
98C2×D3×D4—32no—
99C2×C3×SD16—92no—
100#946—92no—
101#946—181no—
102C2×#215—32no—
103C2×#214—32no—
104#924—91no—
105#832—92no—
106#832—92no—
107#832—182no—
108#830—91no—
109#821—92no—
110(C2×D3)⋊C8—92no—
111C23×C32—62no—
112C22×C3⋊D3#6711yescharacteristic
113C22×C3×D3—22no—
114C22×C3×D3—22no—
115C22×C3×D3—32no—
116C22×C3×D3—42no—
117C22×C3×D3—61no—
118C22×C3×D3—68no—
119C22×C3×D3—124no—
120C22×C3×D3—124no—
121C2×D32—24no—
122C2×D32—42no—
123C2×D32—42no—
124C32×Q8—62no—
125C32×D4—34no—
126C32×D4—62no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4—61no—
131C2×C32×C4C22×D312yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—62no—
135C2×C3⋊Dic3—62no—
136C2×C3⋊Dic3—122no—
137C2×C3⋊Dic3—122no—
138C2×C3⋊Dic3#6711yescharacteristic
139C4×C3⋊D3—22no—
140C3×#67—68no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—61no—
145C3×D12—22no—
146C3×D12—22no—
147C3×D12—24no—
148C3×D12—42no—
149C3×D12#6714yes—
150C3×D12#6714yes—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×C4×D3—122no—
154C3×Dic6—22no—
155C3×Dic6—22no—
156C3×Dic6—42no—
157#625—24no—
158C3⋊(C3⋊C8)—22no—
159C3×C3⋊C8—62no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—122no—
162#450—271noSylow
163C2×C33#12011yescharacteristic
164C2×C33C2×SD1611yescharacteristic
165C2×C33AGL(1,ℤ/8)11yescharacteristic
166C3×C3⋊D3—22no—
167C32×D3—44no—
168C24×C3—181no—
169C23×D3—62no—
170C2×C3×Q8—32no—
171C2×C3×Q8—61no—
172C2×C3×D4—32no—
173C2×C3×D4—32no—
174C2×C3×D4—62no—
175C2×C3×D4—181no—
176C2×C3×D4—184no—
177C22×C3×C4—91no—
178C2×#67—62no—
179D3×D4—64no—
180D3×D4—64no—
181C2×D12D3212yes—
182C2×C4×D3—32no—
183C2×C4×D3—61no—
184C2×Dic6D3211yescharacteristic, complemented
185C3×SD16—184no—
186C2×C3×C8—92no—
187#217—181no—
188#215—64no—
189#214—64no—
190(C2×D3)⋊C4—182no—
191(C2×D3)⋊C4—182no—
192C4⋊Dic3—92no—
193C4⋊Dic3—91no—
194C4⋊Dic3—181no—
195C4⋊Dic3—182no—
196C4⋊Dic3—361no—
197Dic3⋊C4—181no—
198Dic3⋊C4—182no—
199Dic3⋊C4—182no—
200C2×C3⋊C8—32no—
201C2×C3⋊C8—32no—
202C2×C3⋊C8—62no—
203C22×C32—22no—
204C22×C32—21no—
205C22×C32—41no—
206C22×C32—64no—
207C22×C32—122no—
208C22×C32—122no—
209C22×C32D3×D412yes—
210C22×C32C2×#6711yescharacteristic
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3#21412yescomplemented
214C2×C3×D3—24no—
215C2×C3×D3—24no—
216C2×C3×D3—34no—
217C2×C3×D3—44no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—42no—
221C2×C3×D3—42no—
222C2×C3×D3—62no—
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—82no—
226C2×C3×D3—82no—
227C2×C3×D3—121no—
228C2×C3×D3—121no—
229C2×C3×D3—1216no—
230D32—48no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—22no—
234C32×C4—42no—
235C32×C4—121no—
236C32×C4D3×D412yes—
237C32×C4C2×#6712yes—
238C32×C4C2×#6712yes—
239C3⋊Dic3—21no—
240C3⋊Dic3—121no—
241C3⋊Dic3—122no—
242C3⋊Dic3—122no—
243C3⋊Dic3—242no—
244C3⋊Dic3—242no—
245C3×Dic3—42no—
246C3×Dic3—42no—
247C3×Dic3—62no—
248C3×Dic3—121no—
249C22×D4—91no—
250C2×SD16—92no—
251C22⋊Q8—271no—
252#102—272no—
253C22⋊C8—271no—
254C33#45011yescharacteristic, complemented, Sylow
255C23×C3—62no—
256C23×C3—62no—
257C23×C3—91no—
258C23×C3—122no—
259C23×C3—184no—
260C23×C3—362no—
261C23×C3—362no—
262C22×D3—22no—
263C22×D3—32no—
264C22×D3—61no—
265C22×D3—68no—
266C22×D3—124no—
267C22×D3—124no—
268C3×Q8—62no—
269C3×Q8—62no—
270C3×Q8—122no—
271C3×D4—34no—
272C3×D4—34no—
273C3×D4—64no—
274C3×D4—62no—
275C3×D4—62no—
276C3×D4—122no—
277C3×D4—184no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—41no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—91no—
284C2×C3×C4—182no—
285C2×C3×C4C2×D3212yes—
286C2×C3×C4C2×D3211yescharacteristic
287#67—68no—
288C2×Dic3—21no—
289C2×Dic3—32no—
290C2×Dic3—61no—
291C2×Dic3—182no—
292C2×Dic3—182no—
293C2×Dic3—181no—
294C2×Dic3—362no—
295C2×Dic3—362no—
296D12—22no—
297D12#62514yes—
298C4×D3—62no—
299C4×D3—62no—
300C4×D3—122no—
301Dic6—22no—
302C3×C8—182no—
303C3⋊C8—62no—
304C3⋊C8—62no—
305C3⋊C8—122no—
306C2×C32—22no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—22no—
311C2×C32—21no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—124no—
316C2×C32D3×SD1612yes—
317C2×C32#91412yes—
318C2×C32#93211yescharacteristic
319C2×C32C2×#21411yescharacteristic
320C2×C32#93712yes—
321C2×C32#95311yescharacteristic
322C3⋊D3—22no—
323C3×D3—44no—
324C3×D3—44no—
325C3×D3—64no—
326C3×D3—84no—
327C3×D3—122no—
328C24—181no—
329C2×Q8—31no—
330C2×D4—32no—
331C2×D4—181no—
332C2×D4—184no—
333C22×C4—91no—
334SD16—184no—
335C2×C8—92no—
336C4⋊C4—271no—
337C4⋊C4—541no—
338C22⋊C4—541no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—41no—
342C22×C3—64no—
343C22×C3—64no—
344C22×C3—92no—
345C22×C3—124no—
346C22×C3—122no—
347C22×C3—122no—
348C22×C3—122no—
349C22×C3—122no—
350C22×C3—181no—
351C22×C3—181no—
352C22×C3—242no—
353C22×C3—242no—
354C22×C3—368no—
355C22×C3D3×#6712yes—
356C22×C3#381111yescharacteristic
357C2×D3—24no—
358C2×D3—34no—
359C2×D3—42no—
360C2×D3—42no—
361C2×D3—62no—
362C2×D3—62no—
363C2×D3—62no—
364C2×D3—121no—
365C2×D3—121no—
366C2×D3—1216no—
367C3×C4—22no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—42no—
371C3×C4—122no—
372C3×C4—122no—
373C3×C4—181no—
374C3×C4C2×#62512yes—
375C3×C4D3×#6712yes—
376C3×C4D3×#6712yes—
377Dic3—41no—
378Dic3—62no—
379Dic3—121no—
380Dic3—362no—
381Dic3—362no—
382Dic3—361no—
383Dic3—722no—
384Dic3—722no—
385C32—21no—
386C32—22no—
387C32—41no—
388C32#516912yescomplemented
389C32#521811yescharacteristic, complemented
390C23—62no—
391C23—91no—
392C23—184no—
393C23—362no—
394C23—362no—
395Q8—62no—
396D4—34no—
397D4—62no—
398D4—184no—
399C2×C4—61no—
400C2×C4—91no—
401C2×C4—182no—
402C2×C4—541no—
403C2×C4D3311yescharacteristic
404C8—182no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—21no—
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—124no—
415C2×C3—124no—
416C2×C3—182no—
417C2×C3—244no—
418C2×C3D3×#21412yes—
419C2×C3#6393412yes—
420C2×C3#6394011yescharacteristic
421C2×C3#6394111yescharacteristic
422C2×C3#6397612yes—
423C2×C3#6398311yescharacteristic
424D3—44no—
425D3—64no—
426D3—122no—
427C22—64no—
428C22—92no—
429C22—122no—
430C22—122no—
431C22—181no—
432C22—181no—
433C22—368no—
434C22#8895211yescharacteristic
435C4—121no—
436C4—181no—
437C4—1081no—
438C4D3×#62512yes—
439C3—21no—
440C3—22no—
441C3—41no—
442C3#1059145812yescomplemented
443C3#1059165211yescharacteristic, complemented
444C2—124no—
445C2—182no—
446C2#1172283711yescharacteristic
447C2#1172287311yescharacteristic
448C2#1172294611yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3802C2
hypocenterC33#450
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723435C2semidirect
3. C2×C3×#3715C2semidirect
4. #11722072C2semidirect
5. #11722032C2semidirect
11. C2×C32×Dic6C22semidirect
14. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
15. C2×C3×C3⋊(C3⋊C8)C22semidirect
26. C2×#3715D3semidirect
64. C2×C3×Dic6C2×D3semidirect
74. C2×C3⋊(C3⋊C8)C2×D3semidirect
88. C2×C3×C3⋊D3SD16semidirect
184. C2×Dic6D32semidirect
213. C2×C3⋊D3#214semidirect
254. C33#450semidirect
388. C32#5169semidirect
389. C32#5218semidirect
442. C3#10591458semidirect
443. C3#10591652semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
#117234352
C2×C3×#37152
#117220722
#117220322
#105916523
#105914583