Finite Group Databasebeta

← #13167387

Subgroups — #13167387

Order 1728.

758 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 79 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×C4×#1113C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic, complemented
3#11723656C211yesmaximal, characteristic, complemented
4#11722780C214yesmaximal, complemented
5#10593307—31nomaximal
6C22×#1113C411yescharacteristic
7C2×C3×C4×C3⋊D3—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×#6294—21no—
10C2×#6294C411yescharacteristic
11C4×#1113—21no—
12C4×#1113—21no—
13C4×#1113C412yes—
14C4×#1113C2212yes—
15C3×#3792—21no—
16C3×#3788—21no—
17C3×#3787C2212yescomplemented
18C3×#3748—21no—
19#88809—21no—
20#88808C2212yescomplemented
21C2×C4×D32—31no—
22C2×C4×D32—61no—
23#64283D311yescharacteristic, complemented
24#63818—31no—
25D3×#208—61no—
26D32⋊C8—34no—
27C22×C3×C3⋊D3—21no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C2×#1113—21no—
30C2×#1113—24no—
31C2×#1113C2×C412yes—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3—22no—
37C3×C4×C3⋊D3C2×C411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3C2311yescharacteristic
41#6294—21no—
42#6294—22no—
43#6294—22no—
44#6294C2×C412yes—
45C3×#622—22no—
46C3×C3⋊(C3⋊C8)—22no—
47#5126—91nomaximal
48C22×D32—31no—
49C22×D32—61no—
50C2×C4×C3⋊D3—21no—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—61no—
54C2×C3×C4×D3—61no—
55C2×C3×C4×D3—61no—
56C2×C3×C4×D3—121no—
57C2×#624—31no—
58C2×#624—61no—
59C2×D3×Dic3—61no—
60C2×D3×Dic3—61no—
61C2×D3×Dic3—61no—
62C4×D32—32no—
63C4×D32—32no—
64C4×D32—62no—
65C4×D32—62no—
66C4×D32—62no—
67C4×D32—62no—
68C4×D32—61no—
69C4×D32—61no—
70#3792—21no—
71#3788—21no—
72#3787C2×D312yescomplemented
73#3748—21no—
74C3×#208—61no—
75C3×#208—61no—
76C3×#208—121no—
77#3712—61no—
78#3711—62no—
79#3710—32no—
80D3×C3⋊C8—62no—
81C22×C33C22⋊C411yescharacteristic
82C2×C3×C3⋊D3—24no—
83C2×C3×C3⋊D3—22no—
84C2×C3×C3⋊D3—22no—
85C2×C3×C3⋊D3C22⋊C411yescharacteristic
86C2×C3×C3⋊D3C22×C411yescharacteristic
87#1113—24no—
88#1113—22no—
89#1113M1614yescomplemented
90C33×C4C22⋊C411yescharacteristic
91C33×C4C2×D411yescharacteristic
92C3×C3⋊Dic3—22no—
93C3×C3⋊Dic3C22⋊C411yescharacteristic
94C3×C3⋊Dic3C2×D411yescharacteristic
95C22×C4×D3—91no—
96C22×C4×D3—181no—
97C2×C3×M16—91no—
98C2×#208—91no—
99D3×M16—181no—
100(C2×D3)⋊C8—94no—
101C22×C3⋊D3—21no—
102C22×C3⋊D3C4×D311yescharacteristic
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C22×C3×D3—61no—
106C22×C3×D3—61no—
107C22×C3×D3—121no—
108C2×D32—32no—
109C2×D32—62no—
110C2×D32—64no—
111C2×D32—68no—
112C2×D32—62no—
113C2×D32—61no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4D1211yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3#6711yescharacteristic
121C4×C3⋊D3—22no—
122C4×C3⋊D3—22no—
123C4×C3⋊D3C4×D311yescharacteristic
124C4×C3⋊D3D1211yescharacteristic
125C4×C3⋊D3#6711yescharacteristic
126C4×C3⋊D3C22×D311yescharacteristic
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—121no—
132C3×C4×D3—61no—
133C3×C4×D3—61no—
134C3×C4×D3—61no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—62no—
141C3×C4×D3—62no—
142C3×C4×D3—62no—
143C3×C4×D3—62no—
144C3×C4×D3—122no—
145C3×C4×D3—122no—
146#624—32no—
147#624—62no—
148#624—62no—
149#624—61no—
150D3×Dic3—62no—
151D3×Dic3—62no—
152D3×Dic3—62no—
153D3×Dic3—62no—
154D3×Dic3—62no—
155D3×Dic3—62no—
156#622—22no—
157C3⋊(C3⋊C8)—22no—
158C3×C3⋊C8—62no—
159C3×C3⋊C8—62no—
160C3×C3⋊C8—122no—
161#407—271noSylow
162C2×C33—21no—
163C2×C33C2×C22⋊C411yescharacteristic
164C3×C3⋊D3—21no—
165C3×C3⋊D3—24no—
166C3×C3⋊D3C2×M1612yes—
167C23×D3—91no—
168C23×D3—181no—
169C22×C3×C4—91no—
170C22×C3×C4—181no—
171C22×Dic3—91no—
172C22×Dic3—181no—
173C2×C4×D3—61no—
174C2×C4×D3—61no—
175C2×C4×D3—61no—
176C2×C4×D3—61no—
177C2×C4×D3—92no—
178C2×C4×D3—92no—
179C2×C4×D3—121no—
180C2×C4×D3—182no—
181C2×C4×D3—182no—
182C2×C4×D3—181no—
183C2×C4×D3—181no—
184C2×C4×D3—181no—
185C2×C4×D3—182no—
186C2×C4×D3—182no—
187C2×C4×D3—181no—
188C2×C4×D3—181no—
189C3×M16—181no—
190C3×M16—181no—
191C3×M16—181no—
192C2×C3×C8—92no—
193#208—61no—
194#208—61no—
195#208—121no—
196#208—181no—
197C2×C3⋊C8—92no—
198ΓL(1,𝔽25)—182no—
199D3×C8—182no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32(C2×D3)⋊C411yescharacteristic
205C2×C3⋊D3—22no—
206C2×C3⋊D3—24no—
207C2×C3⋊D3—22no—
208C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
209C2×C3⋊D3C2×C4×D311yescharacteristic
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—62no—
215C2×C3×D3—64no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—62no—
219C2×C3×D3—122no—
220C2×C3×D3—122no—
221C2×C3×D3—124no—
222D32—34no—
223D32—64no—
224D32—64no—
225D32—68no—
226D32—64no—
227D32—62no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—41no—
233C32×C4—41no—
234C32×C4—41no—
235C32×C4—41no—
236C32×C4(C2×D3)⋊C411yescharacteristic
237C32×C4C2×D1211yescharacteristic
238C3⋊Dic3—22no—
239C3⋊Dic3(C2×D3)⋊C411yescharacteristic
240C3⋊Dic3C2×#6711yescharacteristic
241C3×Dic3—61no—
242C3×Dic3—61no—
243C3×Dic3—61no—
244C3×Dic3—61no—
245C3×Dic3—62no—
246C3×Dic3—62no—
247C3×Dic3—122no—
248C23×C4—271no—
249C2×M16—91no—
250C2×M16—271no—
251C22⋊C8—274no—
252C33#40711yescharacteristic, complemented, Sylow
253C23×C3—91no—
254C23×C3—181no—
255C22×D3—61no—
256C22×D3—61no—
257C22×D3—61no—
258C22×D3—61no—
259C22×D3—92no—
260C22×D3—121no—
261C22×D3—182no—
262C22×D3—184no—
263C22×D3—181no—
264C22×D3—182no—
265C22×D3—188no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—91no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—182no—
276C2×C3×C4—182no—
277C2×C3×C4—181no—
278C2×C3×C4GO+(4,𝔽2)11yescharacteristic
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—92no—
284C2×Dic3—121no—
285C2×Dic3—181no—
286C2×Dic3—182no—
287C2×Dic3—182no—
288C4×D3—61no—
289C4×D3—61no—
290C4×D3—61no—
291C4×D3—61no—
292C4×D3—61no—
293C4×D3—61no—
294C4×D3—61no—
295C4×D3—61no—
296C4×D3—62no—
297C4×D3—62no—
298C4×D3—62no—
299C4×D3—62no—
300C4×D3—122no—
301C4×D3—122no—
302C4×D3—182no—
303C4×D3—182no—
304C4×D3—182no—
305C4×D3—182no—
306C4×D3—182no—
307C4×D3—182no—
308C3×C8—182no—
309C3×C8—182no—
310C3⋊C8—62no—
311C3⋊C8—62no—
312C3⋊C8—122no—
313C2×C32—22no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—42no—
321C2×C32—81no—
322C2×C32C2×(C2×D3)⋊C411yescharacteristic
323C3⋊D3—24no—
324C3⋊D3—21no—
325C3⋊D3D3×M1612yes—
326C3×D3—62no—
327C3×D3—62no—
328C3×D3—62no—
329C3×D3—64no—
330C3×D3—64no—
331C3×D3—121no—
332C3×D3—124no—
333C24—271no—
334C22×C4—91no—
335C22×C4—181no—
336C22×C4—272no—
337C22×C4—272no—
338C22×C4—271no—
339C22×C4—541no—
340C22×C4—541no—
341C22×C4—541no—
342M16—181no—
343M16—181no—
344M16—181no—
345M16—541no—
346C2×C8—92no—
347C2×C8—272no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—91no—
353C22×C3—91no—
354C22×C3—182no—
355C22×C3—182no—
356C22×C3—184no—
357C22×C3D32⋊C411yescharacteristic
358C2×D3—61no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—61no—
362C2×D3—62no—
363C2×D3—64no—
364C2×D3—64no—
365C2×D3—64no—
366C2×D3—62no—
367C2×D3—94no—
368C2×D3—122no—
369C2×D3—122no—
370C2×D3—124no—
371C2×D3—184no—
372C2×D3—182no—
373C2×D3—184no—
374C2×D3—188no—
375C2×D3—184no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—91no—
385C3×C4—91no—
386C3×C4—182no—
387C3×C4D32⋊C411yescharacteristic
388C3×C4C2×GO+(4,𝔽2)11yescharacteristic
389Dic3—61no—
390Dic3—61no—
391Dic3—61no—
392Dic3—61no—
393Dic3—62no—
394Dic3—62no—
395Dic3—122no—
396C32—21no—
397C32—21no—
398C32—41no—
399C32—41no—
400C32#512611yescharacteristic, complemented
401C23—91no—
402C23—181no—
403C23—272no—
404C23—541no—
405C23—544no—
406C2×C4—91no—
407C2×C4—91no—
408C2×C4—91no—
409C2×C4—91no—
410C2×C4—91no—
411C2×C4—181no—
412C2×C4—182no—
413C2×C4—182no—
414C2×C4—272no—
415C2×C4—542no—
416C2×C4—542no—
417C2×C4—541no—
418C2×C4#157⋊D311yescharacteristic
419C8—182no—
420C8—182no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—41no—
426C2×C3—42no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—81no—
430C2×C3—92no—
431C2×C3—184no—
432C2×C3—181no—
433C2×C3C2×D32⋊C411yescharacteristic
434D3—62no—
435D3—62no—
436D3—62no—
437D3—64no—
438D3—64no—
439D3—121no—
440D3—124no—
441C22—91no—
442C22—91no—
443C22—182no—
444C22—184no—
445C22—182no—
446C22—274no—
447C22—542no—
448C22—544no—
449C22#8893911yescharacteristic
450C4—91no—
451C4—91no—
452C4—182no—
453C4#8893911yescharacteristic
454C4C2×#157⋊D311yescharacteristic
455C3—21no—
456C3—21no—
457C3—41no—
458C3—41no—
459C3#1059330711yescharacteristic, complemented
460C2—21no—
461C2—92no—
462C2—181no—
463C2—184no—
464C2C2×#8893911yescharacteristic
subgroupG/H
center, FrattiniC4C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC22×#1113C4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4#157⋊D3
hypocenterC33#407
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64283C2semidirect
3. #11723656C2semidirect
4. #11722780C2semidirect
17. C3×#3787C22semidirect
20. #88808C22semidirect
23. #64283D3semidirect
72. #3787C2×D3semidirect
89. #1113M16semidirect
252. C33#407semidirect
400. C32#5126semidirect
459. C3#10593307semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C3×#642832
#117236562
#117227802
#105933073
#51269