Finite Group Databasebeta

← #13155800

Subgroups — #13155800

Order 1728.

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

normal subgroups only

This group has 836 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×#3797C211yesmaximal, characteristic, complemented
2#11723522C211yesmaximal, characteristic
3C2×C3×#3715C212yesmaximal, complemented
4#11722072C211yesmaximal, characteristic, complemented
5#11722038C212yesmaximal, complemented
6#10591583—32nomaximal
7#10591518—31nomaximal
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12C2211yescharacteristic
10C2×C32×Dic6C2212yes—
11C2×#6290—21no—
12C2×C3×#627—21no—
13C2×C3×#624—21no—
14C3×#3797—24no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17(C2×D3)⋊(C3⋊Dic3)—21no—
18C3×#3715—24no—
19C2×C3×D3×Q8—32no—
20C2×#3797D311yescharacteristic, complemented
21#64147—31no—
22C2×C3×#215—32no—
23C2×C3×#214—32no—
24#63908—32no—
25#63908—61no—
26C2×#3715D312yescomplemented
27#63662—32no—
28#63566—32no—
29#63566—32no—
30#63566—62no—
31#63557—31no—
32(C2×D3)⋊(C3⋊C8)—32no—
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C32×D3—21no—
35C2×C33×C4C2311yescharacteristic
36C2×C3⋊(C3⋊Dic3)—21no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3—22no—
39C2×C32×Dic3—22no—
40C32×D12—22no—
41C32×Dic6—22no—
42C32×Dic6D414yescomplemented
43#6290—42no—
44C3×#627—24no—
45C3×#624—24no—
46C3×C3⋊(C3⋊C8)—22no—
47#5227—91no—
48#5170—92no—
49C2×C32×Q8—32no—
50C2×C32×D4—31no—
51C2×#638—61no—
52C2×C4×C3⋊D3C2×D311yescharacteristic
53C3×D3×Q8—64no—
54C3×D3×Q8—64no—
55C2×C3×D12—21no—
56C2×C3×D12C2×D312yes—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—61no—
60C2×C3×Dic6—22no—
61C2×C3×Dic6—62no—
62C2×C3×Dic6C2×D312yes—
63C2×C3×Dic6C2×D312yes—
64C2×#627—21no—
65C2×#625—62no—
66C2×#625—62no—
67C2×#624—21no—
68#3797—24no—
69#3757—61no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—32no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)—62no—
74C4⋊(C3⋊Dic3)—121no—
75C2×C3⋊(C3⋊C8)C2×D312yes—
76C3×#215—64no—
77C3×#214—64no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—62no—
81(C2×D3)⋊Dic3—62no—
82(C2×D3)⋊Dic3—62no—
83#3715—24no—
84C22×C33C2×D411yescharacteristic
85C2×C3×C3⋊D3—21no—
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3SD1612yescomplemented
88C2×C32×D3—41no—
89C2×C32×D3—41no—
90C2×C32×D3—41no—
91C33×C4C2×D412yes—
92C3⋊(C3⋊Dic3)—41no—
93C3×C3⋊Dic3—21no—
94C32×Dic3—24no—
95C22×C3×Q8—91no—
96C2×D3×Q8—32no—
97C2×C3×SD16—92no—
98C2×#215—32no—
99#938—92no—
100#938—181no—
101C2×#214—32no—
102#930—91no—
103#835—92no—
104#835—92no—
105#835—182no—
106#830—91no—
107(C2×D3)⋊C8—92no—
108#816—92no—
109C23×C32—61no—
110C22×C3⋊D3#6711yescharacteristic
111C22×C3×D3—22no—
112C22×C3×D3—22no—
113C22×C3×D3—32no—
114C22×C3×D3—61no—
115C32×Q8—34no—
116C32×Q8—62no—
117C32×D4—62no—
118C2×C32×C4—22no—
119C2×C32×C4—21no—
120C2×C32×C4—41no—
121C2×C32×C4—62no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D311yescharacteristic
124#638—122no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—62no—
127C2×C3⋊Dic3—62no—
128C2×C3⋊Dic3—122no—
129C2×C3⋊Dic3—122no—
130C2×C3⋊Dic3#6711yescharacteristic
131C4×C3⋊D3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—32no—
135C2×C3×Dic3—42no—
136C2×C3×Dic3—61no—
137C3×D12—22no—
138C3×D12—22no—
139C3×D12—42no—
140C3×C4×D3—68no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—122no—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—24no—
147C3×Dic6—42no—
148C3×Dic6—68no—
149C3×Dic6#6714yescomplemented
150C3×Dic6#6714yescomplemented
151#627—24no—
152#625—124no—
153#625—124no—
154#624—24no—
155C3⋊(C3⋊C8)—22no—
156C3×C3⋊C8—62no—
157C3×C3⋊C8—62no—
158C3×C3⋊C8—122no—
159#448—271noSylow
160C2×C33#12011yescharacteristic
161C2×C33C2×SD1611yescharacteristic
162C2×C33#13711yescharacteristic
163C3×C3⋊D3—22no—
164C32×D3—81no—
165C2×C3×Q8—32no—
166C2×C3×Q8—32no—
167C2×C3×Q8—62no—
168C2×C3×Q8—181no—
169C2×C3×Q8—184no—
170C2×C3×D4—32no—
171C2×C3×D4—61no—
172C22×C3×C4—91no—
173C22×C3×C4—181no—
174C2×#67—181no—
175C2×#67—182no—
176C2×#67—182no—
177D3×Q8—64no—
178D3×Q8—64no—
179C2×D12D3211yescharacteristic
180C2×C4×D3—32no—
181C2×C4×D3—61no—
182C2×C4×D3—62no—
183C2×Dic6—62no—
184C2×Dic6D3212yes—
185C3×SD16—184no—
186C2×C3×C8—92no—
187#217—182no—
188#217—182no—
189#215—64no—
190#214—64no—
191(C2×D3)⋊C4—181no—
192C4⋊Dic3—92no—
193C4⋊Dic3—91no—
194C4⋊Dic3—181no—
195C4⋊Dic3—182no—
196C4⋊Dic3—361no—
197C2×C3⋊C8—32no—
198C2×C3⋊C8—32no—
199C2×C3⋊C8—62no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—41no—
203C22×C32—121no—
204C22×C32—121no—
205C22×C32—121no—
206C22×C32D3×D412yes—
207C22×C32C2×#6711yescharacteristic
208C2×C3⋊D3—21no—
209C2×C3⋊D3—21no—
210C2×C3⋊D3#21512yescomplemented
211C2×C3×D3—34no—
212C2×C3×D3—42no—
213C2×C3×D3—42no—
214C2×C3×D3—42no—
215C2×C3×D3—42no—
216C2×C3×D3—42no—
217C2×C3×D3—42no—
218C2×C3×D3—62no—
219C2×C3×D3—62no—
220C2×C3×D3—62no—
221C2×C3×D3—121no—
222C2×C3×D3—121no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—22no—
226C32×C4—42no—
227C32×C4—64no—
228C32×C4D3×D412yes—
229C32×C4C2×#6712yes—
230C32×C4C2×#6712yes—
231C3⋊Dic3—21no—
232C3⋊Dic3—121no—
233C3⋊Dic3—122no—
234C3⋊Dic3—122no—
235C3⋊Dic3—242no—
236C3⋊Dic3—242no—
237C3×Dic3—24no—
238C3×Dic3—24no—
239C3×Dic3—44no—
240C3×Dic3—62no—
241C3×Dic3—121no—
242C22×Q8—91no—
243C2×SD16—92no—
244#121—271no—
245#103—272no—
246C22⋊C8—271no—
247C33#44811yescharacteristic, complemented, Sylow
248C23×C3—62no—
249C23×C3—62no—
250C23×C3—91no—
251C22×D3—21no—
252C22×D3—32no—
253C22×D3—61no—
254C3×Q8—34no—
255C3×Q8—34no—
256C3×Q8—64no—
257C3×Q8—62no—
258C3×Q8—62no—
259C3×Q8—122no—
260C3×Q8—184no—
261C3×D4—62no—
262C3×D4—62no—
263C3×D4—122no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—91no—
270C2×C3×C4—122no—
271C2×C3×C4—184no—
272C2×C3×C4—182no—
273C2×C3×C4C2×D3212yes—
274C2×C3×C4C2×D3211yescharacteristic
275#67—364no—
276#67—362no—
277#67—364no—
278C2×Dic3—22no—
279C2×Dic3—32no—
280C2×Dic3—61no—
281C2×Dic3—182no—
282C2×Dic3—182no—
283C2×Dic3—181no—
284C2×Dic3—362no—
285C2×Dic3—362no—
286D12—22no—
287C4×D3—68no—
288C4×D3—62no—
289C4×D3—62no—
290C4×D3—122no—
291Dic6—22no—
292Dic6—68no—
293Dic6#62514yescomplemented
294C3×C8—182no—
295C3⋊C8—62no—
296C3⋊C8—62no—
297C3⋊C8—122no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—22no—
301C2×C32—22no—
302C2×C32—22no—
303C2×C32—21no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—41no—
307C2×C32—241no—
308C2×C32D3×SD1612yes—
309C2×C32#91512yes—
310C2×C32#93712yes—
311C2×C32C2×#21511yescharacteristic
312C2×C32#94211yescharacteristic
313C2×C32#95311yescharacteristic
314C3⋊D3—22no—
315C3×D3—64no—
316C3×D3—82no—
317C3×D3—82no—
318C3×D3—122no—
319C2×Q8—32no—
320C2×Q8—181no—
321C2×Q8—184no—
322C2×D4—31no—
323C2×D4—541no—
324C22×C4—91no—
325C22×C4—181no—
326SD16—184no—
327C2×C8—92no—
328C4⋊C4—271no—
329C22⋊C4—541no—
330C22×C3—21no—
331C22×C3—22no—
332C22×C3—41no—
333C22×C3—92no—
334C22×C3—122no—
335C22×C3—122no—
336C22×C3—122no—
337C22×C3—122no—
338C22×C3—122no—
339C22×C3—122no—
340C22×C3—181no—
341C22×C3—181no—
342C22×C3D3×#6712yes—
343C22×C3#381111yescharacteristic
344C2×D3—34no—
345C2×D3—41no—
346C2×D3—41no—
347C2×D3—41no—
348C2×D3—62no—
349C2×D3—62no—
350C2×D3—62no—
351C2×D3—121no—
352C2×D3—121no—
353C3×C4—22no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—42no—
357C3×C4—64no—
358C3×C4—64no—
359C3×C4—124no—
360C3×C4—181no—
361C3×C4C2×#62512yes—
362C3×C4D3×#6712yes—
363C3×C4D3×#6712yes—
364Dic3—24no—
365Dic3—62no—
366Dic3—121no—
367Dic3—362no—
368Dic3—361no—
369Dic3—362no—
370Dic3—722no—
371Dic3—722no—
372C32—21no—
373C32—22no—
374C32—41no—
375C32#517012yescomplemented
376C32#522711yescharacteristic, complemented
377C23—61no—
378C23—91no—
379Q8—34no—
380Q8—62no—
381Q8—184no—
382D4—62no—
383D4—1082no—
384C2×C4—62no—
385C2×C4—91no—
386C2×C4—184no—
387C2×C4—182no—
388C2×C4—541no—
389C2×C4D3311yescharacteristic
390C8—182no—
391C2×C3—21no—
392C2×C3—22no—
393C2×C3—21no—
394C2×C3—22no—
395C2×C3—21no—
396C2×C3—22no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—182no—
401C2×C3—242no—
402C2×C3—242no—
403C2×C3#6393011yescharacteristic
404C2×C3#6393111yescharacteristic
405C2×C3D3×#21512yes—
406C2×C3#6394612yes—
407C2×C3#6397612yes—
408C2×C3#6398311yescharacteristic
409D3—64no—
410D3—81no—
411D3—122no—
412C22—92no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22—181no—
417C22—181no—
418C22#8895211yescharacteristic
419C4—64no—
420C4—181no—
421C4—1081no—
422C4D3×#62512yes—
423C3—21no—
424C3—22no—
425C3—41no—
426C3#1059151811yescharacteristic, complemented
427C3#1059158312yescomplemented
428C2—182no—
429C2—241no—
430C2#1172283811yescharacteristic
431C2#1172288911yescharacteristic
432C2#1172294611yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723522C2
hypocenterC33#448
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
1. C2×C3×#3797C2semidirect
3. C2×C3×#3715C2semidirect
4. #11722072C2semidirect
5. #11722038C2semidirect
20. C2×#3797D3semidirect
26. C2×#3715D3semidirect
42. C32×Dic6D4semidirect
87. C2×C3×C3⋊D3SD16semidirect
149. C3×Dic6#67semidirect
150. C3×Dic6#67semidirect
210. C2×C3⋊D3#215semidirect
247. C33#448semidirect
293. Dic6#625semidirect
375. C32#5170semidirect
376. C32#5227semidirect
426. C3#10591518semidirect
427. C3#10591583semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
#117235222
C2×C3×#37152
#117220722
#117220382
#105915833
#105915183