Finite Group Databasebeta

← #13161973

Subgroups — #13161973

Order 1728.

950 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 143 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×#3830C211yesmaximal, characteristic
2#11723667C211yesmaximal, characteristic, complemented
3C2×#88808C211yesmaximal, characteristic, complemented
4C3×#63659C212yesmaximal, complemented
5#11722076C212yesmaximal, complemented
6#10591424—31nomaximal
7#10591420—32nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—21no—
10C22×C3×C3⋊Dic3—21no—
11C3×#3830—22no—
12C3×#3830C414yescomplemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17#88814—21no—
18#88813—21no—
19#88808—22no—
20C2×#3830D311yescharacteristic
21C2×C3×#237—32no—
22C2×C3×#237—61no—
23#63849—31no—
24#63841—32no—
25C2×#3710—31no—
26C2×D3×C3⋊C8—32no—
27#63659D312yescomplemented
28C3×#832—32no—
29C3×#832—32no—
30C3×#832—62no—
31#63567—32no—
32#63566—32no—
33#63565—32no—
34C23×C33—21no—
35C22×C3×C3⋊D3D411yescharacteristic
36C33×D4—21no—
37C33×D4C2×C412yes—
38C2×C33×C4C2311yescharacteristic
39C3×#638—24no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—24no—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4×C3⋊D3C2×C412yes—
45C3×C4×C3⋊D3D412yes—
46C3×#634—21no—
47C3×#634C2×C412yes—
48C3×C3⋊(C3⋊C8)—22no—
49#5233—91no—
50#4772—92no—
51C2×C32×D4—21no—
52C2×C32×D4—22no—
53C2×C32×D4—41no—
54C2×C32×D4C2×D312yes—
55C2×C32×D4C2×D311yescharacteristic
56C2×#638—21no—
57C22×C3⋊Dic3—21no—
58#3830—22no—
59#3830Dic314yescomplemented
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×#634C2×D311yescharacteristic
62C2×C3×#67—62no—
63C2×C3×#67—62no—
64C22×C3×Dic3—62no—
65C22×C3×Dic3—62no—
66C3×#237—38no—
67C3×#237—64no—
68C3×#237—64no—
69C3×#237—64no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—61no—
72C2×C3×Dic6—32no—
73C2×C3×Dic6—61no—
74C4⋊(C3⋊Dic3)C2×D312yes—
75C2×C3⋊(C3⋊C8)C2×D312yes—
76C3×C4⋊Dic3—32no—
77C3×C4⋊Dic3—32no—
78C3×C4⋊Dic3—62no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—32no—
81C2×C3×C3⋊C8—62no—
82#3722—61no—
83(C2×D3)⋊Dic3—62no—
84Dic32—62no—
85Dic32—61no—
86#3710—62no—
87D3×C3⋊C8—64no—
88C22×C33—22no—
89C22×C33—42no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—22no—
92C2×C3×C3⋊D3C22⋊C412yes—
93C33×C4C22×C411yescharacteristic
94C33×C4C2×D411yescharacteristic
95C3×C3⋊Dic3—22no—
96C3×C3⋊Dic3—42no—
97C3×C3⋊Dic3C22⋊C412yes—
98C2×C3×#41—91no—
99C2×#237—32no—
100C2×#237—61no—
101#926—91no—
102C22×C3⋊C8—91no—
103C2×D3×C8—92no—
104#892—92no—
105C3×#102—92no—
106#835—92no—
107#832—32no—
108#832—32no—
109#832—62no—
110#816—92no—
111#810—92no—
112C23×C32—21no—
113C23×C32—22no—
114C23×C32—22no—
115C23×C32—42no—
116C23×C32—42no—
117C22×C3⋊D3#6711yescharacteristic
118C22×C3×D3—32no—
119C22×C3×D3—61no—
120C32×D4—21no—
121C32×D4—22no—
122C32×D4—22no—
123C32×D4—24no—
124C32×D4—22no—
125C32×D4—42no—
126C32×D4—42no—
127C32×D4—42no—
128C32×D4C4×D314yes—
129C32×D4C2×Dic312yes—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4C22×D312yes—
134C2×C32×C4C22×D311yescharacteristic
135#638—24no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—24no—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3#6711yescharacteristic
140C4×C3⋊D3C2×Dic312yes—
141C4×C3⋊D3#6712yes—
142#634—21no—
143#634C2×Dic312yes—
144C3×#67—68no—
145C3×#67—68no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—68no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—68no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—122no—
155C3×C4×D3—34no—
156C3×C4×D3—34no—
157C3×C4×D3—62no—
158C3×C4×D3—62no—
159C3×Dic6—34no—
160C3×Dic6—62no—
161C3×Dic6—62no—
162C3×Dic6—62no—
163C3⋊(C3⋊C8)—22no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—62no—
166C3×C3⋊C8—122no—
167#416—271noSylow
168C2×C33—42no—
169C2×C33C2×C22⋊C411yescharacteristic
170C2×C33#13512yes—
171C3×C3⋊D3—22no—
172C3×#41—94no—
173C3×#41—182no—
174C2×C3×Q8—91no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4—41no—
178C2×C3×D4—181no—
179C2×C3×D4D3212yes—
180C2×C3×D4D3211yescharacteristic
181C22×C3×C4—91no—
182C22×C3×C4—181no—
183C2×#67—62no—
184C2×#67—62no—
185C22×Dic3—62no—
186C22×Dic3—62no—
187#237—38no—
188#237—64no—
189#237—64no—
190#237—64no—
191C2×C4×D3—32no—
192C2×C4×D3—61no—
193C2×Dic6—32no—
194C2×Dic6—61no—
195C2×C3×C8—92no—
196C3×C4⋊C4—92no—
197#217—181no—
198(C2×D3)⋊C4—182no—
199C4⋊Dic3—32no—
200C4⋊Dic3—32no—
201C4⋊Dic3—62no—
202C4×Dic3—182no—
203C4×Dic3—181no—
204C2×C3⋊C8—32no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—62no—
207C2×C3⋊C8—182no—
208D3×C8—184no—
209C22×C32—22no—
210C22×C32—24no—
211C22×C32—24no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—41no—
215C22×C32—44no—
216C22×C32—44no—
217C22×C32—42no—
218C22×C32—44no—
219C22×C32—42no—
220C22×C32—42no—
221C22×C32—82no—
222C22×C32—82no—
223C22×C32—84no—
224C22×C32D3×D412yes—
225C22×C32C2×#6711yescharacteristic
226C2×C3⋊D3—22no—
227C2×C3⋊D3#21712yes—
228C2×C3×D3—34no—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C2×C3×D3—122no—
233C32×C4—22no—
234C32×C4—21no—
235C32×C4—22no—
236C32×C4—21no—
237C32×C4—41no—
238C32×C4—41no—
239C32×C4C2×C4×D312yes—
240C32×C4D3×D412yes—
241C32×C4C22×Dic311yescharacteristic
242C32×C4C2×#6711yescharacteristic
243C3⋊Dic3—22no—
244C3⋊Dic3—42no—
245C3⋊Dic3#21712yes—
246C3×Dic3—34no—
247C3×Dic3—62no—
248C3×Dic3—64no—
249C3×Dic3—64no—
250C3×Dic3—122no—
251C3×Dic3—122no—
252C3×Dic3—242no—
253C2×#41—91no—
254C22×C8—271no—
255#117—271no—
256#103—272no—
257#102—92no—
258C33#41611yescharacteristic, complemented, Sylow
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—21no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—91no—
265C22×D3—32no—
266C22×D3—61no—
267C3×Q8—92no—
268C3×Q8—181no—
269C3×D4—22no—
270C3×D4—22no—
271C3×D4—21no—
272C3×D4—22no—
273C3×D4—24no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—184no—
278C3×D4D3×Dic314yes—
279C3×D4#62412yes—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—41no—
283C2×C3×C4—91no—
284C2×C3×C4—92no—
285C2×C3×C4—92no—
286C2×C3×C4—184no—
287C2×C3×C4—181no—
288C2×C3×C4—182no—
289C2×C3×C4C2×D3212yes—
290C2×C3×C4C2×D3211yescharacteristic
291#67—68no—
292#67—68no—
293C2×Dic3—32no—
294C2×Dic3—61no—
295C2×Dic3—68no—
296C2×Dic3—62no—
297C2×Dic3—68no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—62no—
301C2×Dic3—122no—
302C4×D3—34no—
303C4×D3—34no—
304C4×D3—62no—
305C4×D3—62no—
306Dic6—34no—
307Dic6—62no—
308Dic6—62no—
309Dic6—62no—
310C3×C8—182no—
311C3⋊C8—62no—
312C3⋊C8—62no—
313C3⋊C8—122no—
314C2×C32—22no—
315C2×C32—22no—
316C2×C32—22no—
317C2×C32—22no—
318C2×C32—21no—
319C2×C32—41no—
320C2×C32—42no—
321C2×C32—42no—
322C2×C32—44no—
323C2×C32—44no—
324C2×C32—84no—
325C2×C32—84no—
326C2×C32D3×C22⋊C412yes—
327C2×C32#91212yes—
328C2×C32#91612yes—
329C2×C32#95012yes—
330C2×C32C2×#21711yescharacteristic
331C3⋊D3—22no—
332C3×D3—64no—
333C3×D3—122no—
334#41—94no—
335#41—182no—
336C2×Q8—91no—
337C2×D4—181no—
338C2×D4#111311yescharacteristic
339C22×C4—91no—
340C22×C4—181no—
341C2×C8—92no—
342C2×C8—542no—
343C4⋊C4—92no—
344C22⋊C4—541no—
345C42—541no—
346C22×C3—24no—
347C22×C3—24no—
348C22×C3—22no—
349C22×C3—21no—
350C22×C3—22no—
351C22×C3—44no—
352C22×C3—44no—
353C22×C3—44no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—41no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—84no—
361C22×C3—92no—
362C22×C3—182no—
363C22×C3D3×#6712yes—
364C22×C3#381111yescharacteristic
365C2×D3—34no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—62no—
369C2×D3—122no—
370C3×C4—21no—
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—21no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—92no—
377C3×C4—182no—
378C3×C4—362no—
379C3×C4C2×D3×Dic312yes—
380C3×C4C2×#62411yescharacteristic
381C3×C4D3×#6712yes—
382C3×C4#381111yescharacteristic
383Dic3—34no—
384Dic3—62no—
385Dic3—64no—
386Dic3—64no—
387Dic3—122no—
388Dic3—122no—
389Dic3—242no—
390C32—22no—
391C32—21no—
392C32—41no—
393C32#477212yescomplemented
394C32#523311yescharacteristic, complemented
395C23—21no—
396C23—91no—
397Q8—92no—
398Q8—181no—
399D4—21no—
400D4—184no—
401D4#629412yes—
402C2×C4—91no—
403C2×C4—92no—
404C2×C4—92no—
405C2×C4—184no—
406C2×C4—181no—
407C2×C4—182no—
408C2×C4C2×#111311yescharacteristic
409C8—182no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—22no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—44no—
416C2×C3—44no—
417C2×C3—42no—
418C2×C3—42no—
419C2×C3—41no—
420C2×C3—84no—
421C2×C3—84no—
422C2×C3—182no—
423C2×C3#6393312yes—
424C2×C3#6393512yes—
425C2×C3#6393612yes—
426C2×C3D3×#21712yes—
427C2×C3#6397411yescharacteristic
428D3—64no—
429D3—122no—
430C22—22no—
431C22—42no—
432C22—92no—
433C22—182no—
434C22#8904911yescharacteristic
435C4—92no—
436C4—182no—
437C4—362no—
438C4C2×#629411yescharacteristic
439C4#8904911yescharacteristic
440C3—21no—
441C3—22no—
442C3—41no—
443C3#1059142012yescomplemented
444C3#1059142411yescharacteristic, complemented
445C2—42no—
446C2—182no—
447C2#1172370712yes—
448C2#1172372311yescharacteristic
subgroupG/H
centerC22#89049
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
cosocleC2×C33×C4C23
hypercenterC2×D4#1113
hypocenterC33#416
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11723667C2semidirect
3. C2×#88808C2semidirect
4. C3×#63659C2semidirect
5. #11722076C2semidirect
12. C3×#3830C4semidirect
27. #63659D3semidirect
59. #3830Dic3semidirect
258. C33#416semidirect
393. C32#4772semidirect
394. C32#5233semidirect
443. C3#10591420semidirect
444. C3#10591424semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
#117236672
C2×#888082
C3×#636592
#117220762
#105914243
#105914203