Finite Group Databasebeta

← #13155848

Subgroups — #13155848

Order 1728.

1000 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 1000 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#11723431C211yesmaximal, characteristic, complemented
3C2×#88794C212yesmaximal, complemented
4#11722072C211yesmaximal, characteristic, complemented
5C3×#63562C212yesmaximal, complemented
6#10591459—32nomaximal
7#10590597—31nomaximal
8C22×C3×D32—21no—
9C2×#6308C2211yescharacteristic, complemented
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12C2212yes—
12C2×C3×#625—21no—
13C3×#3802—24no—
14C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
15C32×C4⋊Dic3C2211yescharacteristic, complemented
16Dic3⋊(C3⋊Dic3)—21no—
17#88803—21no—
18#88794—24no—
19C2×C3×D3×D4—32no—
20C2×#3802D311yescharacteristic
21C2×#3754—32no—
22#64106—31no—
23#63857—32no—
24#63857—61no—
25C2×#3716—32no—
26C2×#3716—32no—
27C2×#3716—62no—
28C3×#832—32no—
29C3×#809—32no—
30#63562D312yescomplemented
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—
40#6308—22no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C3×C4×C3⋊D3—22no—
43C2×C32×Dic3—21no—
44C32×D12—22no—
45C32×D12D414yes—
46C3×#625—24no—
47C3×C3⋊(C3⋊C8)—22no—
48#5218—91no—
49#4776—92no—
50C23×C3×D3—62no—
51C22×D32—21no—
52C2×C32×D4—32no—
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×#634—31no—
55C2×#634—32no—
56C2×#634—62no—
57C2×#634—61no—
58C2×#634—121no—
59C2×C3×#67—62no—
60C3×D3×D4—64no—
61C3×D3×D4—64no—
62C2×C3×D12—22no—
63C2×C3×D12C2×D312yes—
64C2×C3×D12C2×D312yes—
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—61no—
67C2×#625—21no—
68#3802—24no—
69C32×C4⋊C4—31no—
70#3754—64no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—61no—
73C2×C3⋊(C3⋊C8)C2×D312yescomplemented
74C3×C4⋊Dic3—21no—
75C3×C4⋊Dic3C2×D312yescomplemented
76C2×C3×C3⋊C8—32no—
77C2×C3×C3⋊C8—32no—
78C2×C3×C3⋊C8—62no—
79Dic3⋊Dic3—62no—
80Dic3⋊Dic3—62no—
81(C2×D3)⋊Dic3—62no—
82(C2×D3)⋊Dic3—62no—
83#3716—68no—
84#3716—64no—
85#3716—64no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3SD1612yescomplemented
90C2×C32×D3—24no—
91C2×C32×D3—42no—
92C2×C32×D3—42no—
93C3×D32—48no—
94C33×C4C2×D412yes—
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—21no—
97C32×Dic3—41no—
98C22×C3×D4—91no—
99C2×D3×D4—32no—
100C2×#214—92no—
101C2×#214—92no—
102C2×#214—182no—
103#924—91no—
104C2×#204—92no—
105#897—92no—
106#897—181no—
107C3×#102—92no—
108#832—32no—
109#830—91no—
110(C2×D3)⋊C8—92no—
111#809—32no—
112C23×C32—62no—
113C22×C3⋊D3#6711yescharacteristic
114C22×C3×D3—22no—
115C22×C3×D3—22no—
116C22×C3×D3—32no—
117C22×C3×D3—42no—
118C22×C3×D3—61no—
119C22×C3×D3—68no—
120C22×C3×D3—124no—
121C22×C3×D3—124no—
122C2×D32—24no—
123C2×D32—42no—
124C2×D32—42no—
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—
140#634—62no—
141#634—64no—
142#634—62no—
143#634—62no—
144#634—122no—
145#634—122no—
146#634—124no—
147C3×#67—68no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—61no—
152C3×D12—22no—
153C3×D12—22no—
154C3×D12—24no—
155C3×D12—42no—
156C3×D12D1214yes—
157C3×D12#6714yes—
158C3×C4×D3—62no—
159C3×C4×D3—62no—
160C3×C4×D3—122no—
161#625—24no—
162C3⋊(C3⋊C8)—22no—
163C3×C3⋊C8—64no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—62no—
166#450—271noSylow
167C2×C33#12011yescharacteristic
168C2×C33C2×SD1611yescharacteristic
169C2×C33AGL(1,ℤ/8)11yescharacteristic
170C3×C3⋊D3—22no—
171C32×D3—44no—
172C24×C3—181no—
173C23×D3—62no—
174C2×C3×D4—32no—
175C2×C3×D4—32no—
176C2×C3×D4—62no—
177C2×C3×D4—181no—
178C2×C3×D4—184no—
179C22×C3×C4—91no—
180C2×#67—62no—
181D3×D4—64no—
182D3×D4—64no—
183C2×D12D3212yes—
184C2×C4×D3—32no—
185C2×C4×D3—61no—
186C2×Dic6—92no—
187C2×Dic6—91no—
188C2×Dic6—182no—
189C2×Dic6—181no—
190C2×Dic6—361no—
191C2×C3×C8—92no—
192C3×C4⋊C4—32no—
193C3×C4⋊C4—61no—
194#217—181no—
195#214—184no—
196#214—184no—
197#214—188no—
198(C2×D3)⋊C4—182no—
199(C2×D3)⋊C4—182no—
200C4⋊Dic3—182no—
201C4⋊Dic3—182no—
202C4⋊Dic3D3211yescharacteristic, complemented
203Dic3⋊C4—181no—
204C2×C3⋊C8—32no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—62no—
207#204—184no—
208C22×C32—22no—
209C22×C32—21no—
210C22×C32—41no—
211C22×C32—64no—
212C22×C32—122no—
213C22×C32—122no—
214C22×C32D3×D412yes—
215C22×C32C2×#6711yescharacteristic
216C2×C3⋊D3—21no—
217C2×C3⋊D3—21no—
218C2×C3⋊D3#21412yescomplemented
219C2×C3×D3—24no—
220C2×C3×D3—24no—
221C2×C3×D3—34no—
222C2×C3×D3—44no—
223C2×C3×D3—42no—
224C2×C3×D3—42no—
225C2×C3×D3—42no—
226C2×C3×D3—42no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C2×C3×D3—62no—
230C2×C3×D3—82no—
231C2×C3×D3—82no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C2×C3×D3—1216no—
235D32—48no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—22no—
239C32×C4—42no—
240C32×C4—121no—
241C32×C4C2×D1212yes—
242C32×C4D3×D412yes—
243C32×C4C2×#6712yes—
244C3⋊Dic3—21no—
245C3⋊Dic3—124no—
246C3⋊Dic3—122no—
247C3⋊Dic3—124no—
248C3⋊Dic3—122no—
249C3⋊Dic3—121no—
250C3×Dic3—42no—
251C3×Dic3—42no—
252C3×Dic3—62no—
253C3×Dic3—121no—
254C22×D4—91no—
255C2×SD16—272no—
256C22⋊Q8—271no—
257#102—92no—
258C22⋊C8—271no—
259C33#45011yescharacteristic, complemented, Sylow
260C23×C3—62no—
261C23×C3—62no—
262C23×C3—91no—
263C23×C3—122no—
264C23×C3—184no—
265C23×C3—362no—
266C23×C3—362no—
267C22×D3—22no—
268C22×D3—32no—
269C22×D3—61no—
270C22×D3—68no—
271C22×D3—124no—
272C22×D3—124no—
273C3×D4—34no—
274C3×D4—34no—
275C3×D4—64no—
276C3×D4—62no—
277C3×D4—62no—
278C3×D4—122no—
279C3×D4—184no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—41no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—91no—
286C2×C3×C4—182no—
287C2×C3×C4C2×D3211yescharacteristic
288C2×C3×C4C2×D3212yes—
289#67—68no—
290C2×Dic3—21no—
291C2×Dic3—32no—
292C2×Dic3—61no—
293C2×Dic3—182no—
294C2×Dic3—182no—
295C2×Dic3—181no—
296C2×Dic3—362no—
297C2×Dic3—362no—
298D12—22no—
299D12#62614yes—
300C4×D3—62no—
301C4×D3—62no—
302C4×D3—122no—
303Dic6—184no—
304Dic6—182no—
305Dic6—182no—
306Dic6—182no—
307Dic6—362no—
308Dic6—362no—
309Dic6—364no—
310C3×C8—182no—
311C3⋊C8—64no—
312C3⋊C8—62no—
313C3⋊C8—62no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—22no—
317C2×C32—22no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C2×C32—124no—
324C2×C32#88212yes—
325C2×C32#91112yes—
326C2×C32D3×SD1612yes—
327C2×C32#93211yescharacteristic
328C2×C32C2×#21411yescharacteristic
329C2×C32#95311yescharacteristic
330C3⋊D3—22no—
331C3×D3—44no—
332C3×D3—44no—
333C3×D3—64no—
334C3×D3—84no—
335C3×D3—122no—
336C24—181no—
337C2×Q8—271no—
338C2×D4—32no—
339C2×D4—181no—
340C2×D4—184no—
341C22×C4—91no—
342SD16—544no—
343C2×C8—92no—
344C4⋊C4—31no—
345C4⋊C4—541no—
346C22⋊C4—541no—
347C22×C3—21no—
348C22×C3—22no—
349C22×C3—41no—
350C22×C3—64no—
351C22×C3—64no—
352C22×C3—92no—
353C22×C3—124no—
354C22×C3—122no—
355C22×C3—122no—
356C22×C3—122no—
357C22×C3—122no—
358C22×C3—181no—
359C22×C3—181no—
360C22×C3—242no—
361C22×C3—242no—
362C22×C3—368no—
363C22×C3D3×#6712yes—
364C22×C3#381111yescharacteristic
365C2×D3—24no—
366C2×D3—34no—
367C2×D3—42no—
368C2×D3—42no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—62no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—1216no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—42no—
379C3×C4—122no—
380C3×C4—122no—
381C3×C4—181no—
382C3×C4D3×D1212yes—
383C3×C4C2×#62612yes—
384C3×C4D3×#6712yes—
385Dic3—41no—
386Dic3—62no—
387Dic3—121no—
388Dic3—364no—
389Dic3—362no—
390Dic3—364no—
391Dic3—361no—
392Dic3—362no—
393C32—21no—
394C32—22no—
395C32—41no—
396C32#477612yescomplemented
397C32#521811yescharacteristic, complemented
398C23—62no—
399C23—91no—
400C23—184no—
401C23—362no—
402C23—362no—
403Q8—542no—
404D4—34no—
405D4—62no—
406D4—184no—
407C2×C4—61no—
408C2×C4—91no—
409C2×C4—182no—
410C2×C4—541no—
411C2×C4D3311yescharacteristic
412C8—182no—
413C2×C3—21no—
414C2×C3—22no—
415C2×C3—21no—
416C2×C3—22no—
417C2×C3—21no—
418C2×C3—22no—
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—124no—
423C2×C3—124no—
424C2×C3—182no—
425C2×C3—244no—
426C2×C3#6392311yescharacteristic
427C2×C3#6392711yescharacteristic
428C2×C3D3×#21412yes—
429C2×C3#6393212yes—
430C2×C3#6394011yescharacteristic
431C2×C3#6397712yes—
432D3—44no—
433D3—64no—
434D3—122no—
435C22—64no—
436C22—92no—
437C22—122no—
438C22—122no—
439C22—181no—
440C22—181no—
441C22—368no—
442C22#8895211yescharacteristic
443C4—121no—
444C4—181no—
445C4—1081no—
446C4D3×#62612yes—
447C3—21no—
448C3—22no—
449C3—41no—
450C3#1059059711yescharacteristic, complemented
451C3#1059145912yescomplemented
452C2—124no—
453C2—182no—
454C2#1172283711yescharacteristic
455C2#1172287211yescharacteristic
456C2#1172294811yescharacteristic
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. #11723431C2semidirect
3. C2×#88794C2semidirect
4. #11722072C2semidirect
5. C3×#63562C2semidirect
9. C2×#6308C22semidirect
14. C2×C3×C3⋊(C3⋊C8)C22semidirect
15. C32×C4⋊Dic3C22semidirect
30. #63562D3semidirect
73. C2×C3⋊(C3⋊C8)C2×D3semidirect
75. C3×C4⋊Dic3C2×D3semidirect
89. C2×C3×C3⋊D3SD16semidirect
202. C4⋊Dic3D32semidirect
218. C2×C3⋊D3#214semidirect
259. C33#450semidirect
396. C32#4776semidirect
397. C32#5218semidirect
450. C3#10590597semidirect
451. C3#10591459semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38022
#117234312
C2×#887942
#117220722
C3×#635622
#105914593
#105905973