Finite Group Databasebeta

← D3×#63781 #13166518

Subgroups — D3×#63781 #13166518

Order 1728.

1240 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C22×D3×#157C212yesmaximal
2C2×C4×D3×C3⋊D3C211yesmaximal, characteristic
3#11723106C211yesmaximal, characteristic, complemented
4C3×#63781C211yesmaximal, characteristic, complemented
5#11722722C212yesmaximal, complemented
6C2×#63781—31nomaximal
7C22×D3×C3⋊D3C2211yescharacteristic
8C22×C3⋊#157C2212yes—
9C22×C3×#157C2212yes—
10C2×D3×#157—24no—
11C2×D3×#157—24no—
12C2×D3×#157C418yescomplemented
13C2×C4×C3⋊(C3⋊D3)C411yescharacteristic, complemented
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×C4×D3C411yescharacteristic, complemented
16C2×#6289C411yescharacteristic, complemented
17C2×Dic3×C3⋊D3C2211yescharacteristic
18C2×D3×C3⋊Dic3C411yescharacteristic, complemented
19C4×D3×C3⋊D3—24no—
20C23×#157—32no—
21C22×C4×C3⋊D3—31no—
22C2×C4×D32—62no—
23#63781—31no—
24#63781—32no—
25#63781D311yescharacteristic, direct product factor
26C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
27C22×C3×C3⋊D3C2311yescharacteristic
28C22×C32×D3C2×C411yescharacteristic
29C2×D3×C3⋊D3C2×C414yes—
30C2×D3×C3⋊D3D412yes—
31C2×D3×C3⋊D3D414yes—
32C2×D3×C3⋊D3Q812yes—
33C2×C3⋊#157—22no—
34C2×C3⋊#157—22no—
35C2×C3⋊#157C2×C414yescomplemented
36C2×C3×#157—22no—
37C2×C3×#157—22no—
38C2×C3×#157C2×C414yescomplemented
39D3×#157—216no—
40C2×C33×C4C2×C411yescharacteristic, complemented
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic, complemented
42C4×C3⋊(C3⋊D3)—22no—
43C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
44C3×C4×C3⋊D3—22no—
45C2×C32×Dic3C2×C411yescharacteristic, complemented
46C32×C4×D3—22no—
47#6289—22no—
48Dic3×C3⋊D3—22no—
49D3×C3⋊Dic3—22no—
50D3×#95—91nomaximal
51C23×C3⋊D3—31no—
52C22×D32—62no—
53C22×#157—32no—
54C22×#157—38no—
55C22×#157—64no—
56C22×#157—64no—
57C22×#157C2×D312yes—
58C22×C32×C4—31no—
59C22×C3⋊Dic3—31no—
60C2×C4×C3⋊D3—31no—
61C2×C4×C3⋊D3—31no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—64no—
64C2×C4×C3⋊D3—62no—
65C2×C4×C3⋊D3—122no—
66C2×C4×C3⋊D3C2×D311yescharacteristic
67C2×C3×C4×D3—22no—
68C2×C3×C4×D3—62no—
69C2×#624—62no—
70C2×D3×Dic3—62no—
71C2×D3×Dic3—62no—
72C4×D32—616no—
73C22×C33C22×C411yescharacteristic
74C2×C3⋊(C3⋊D3)C4212yes—
75C2×C3⋊(C3⋊D3)C22⋊C412yes—
76C2×C3⋊(C3⋊D3)C4⋊C412yes—
77C2×C3×C3⋊D3C22×C412yes—
78C2×C3×C3⋊D3C2×D411yescharacteristic
79C2×C3×C3⋊D3C2×D412yes—
80C2×C3×C3⋊D3C2×Q811yescharacteristic
81C2×C32×D3C4212yes—
82C2×C32×D3C22⋊C412yes—
83C2×C32×D3C4⋊C412yes—
84D3×C3⋊D3C22⋊C418yes—
85D3×C3⋊D3C4⋊C418yes—
86C3⋊#157—24no—
87C3×#157—24no—
88C33×C4—21no—
89C3⋊(C3⋊Dic3)—21no—
90C3×C3⋊Dic3—21no—
91C32×Dic3—21no—
92C22×C4×D3—92no—
93C22×C4×D3—91no—
94C22×C4×D3—182no—
95C3×#95—91no—
96C3⋊#95—91no—
97C3⋊#95—92no—
98C23×C32—31no—
99C22×C3⋊D3—31no—
100C22×C3⋊D3—34no—
101C22×C3⋊D3—32no—
102C22×C3⋊D3—34no—
103C22×C3⋊D3—32no—
104C22×C3⋊D3—62no—
105C22×C3⋊D3—122no—
106C22×C3⋊D3C22×D311yescharacteristic
107C22×C3×D3—22no—
108C22×C3×D3—62no—
109C2×D32—68no—
110C2×D32—64no—
111C2×D32—68no—
112C2×D32—64no—
113C2×#157—22no—
114C2×#157—22no—
115C2×#157—34no—
116C2×#157—616no—
117C2×#157—62no—
118C2×#157—62no—
119C2×#157C4×D314yescomplemented
120C2×C32×C4—22no—
121C2×C32×C4—31no—
122C2×C32×C4—42no—
123C2×C32×C4—62no—
124C2×C32×C4C4×D311yescharacteristic, complemented
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—62no—
127C2×C3⋊Dic3—62no—
128C2×C3⋊Dic3—122no—
129C2×C3⋊Dic3C4×D311yescharacteristic, complemented
130C4×C3⋊D3—22no—
131C4×C3⋊D3—62no—
132C4×C3⋊D3—62no—
133C4×C3⋊D3—62no—
134C4×C3⋊D3—68no—
135C4×C3⋊D3—128no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—62no—
138C3×C4×D3—28no—
139C3×C4×D3—68no—
140#624—68no—
141D3×Dic3—68no—
142D3×Dic3—68no—
143C2×#95—271noSylow
144C2×C33C2×C4211yescharacteristic
145C2×C33C2×C22⋊C411yescharacteristic
146C2×C33C2×C4⋊C411yescharacteristic
147C3⋊(C3⋊D3)#9514yescomplemented
148C3×C3⋊D3C2×C22⋊C412yes—
149C3×C3⋊D3C2×C4⋊C412yes—
150C32×D3#9514yescomplemented
151C23×D3—91no—
152C23×D3—182no—
153C22×C3×C4—62no—
154C22×C3×C4—92no—
155C22×C3×C4—91no—
156C22×Dic3—92no—
157C22×Dic3—91no—
158C22×Dic3—182no—
159C2×C4×D3—62no—
160C2×C4×D3—98no—
161C2×C4×D3—91no—
162C2×C4×D3—91no—
163C2×C4×D3—91no—
164C2×C4×D3—182no—
165C2×C4×D3—182no—
166C2×C4×D3—182no—
167C2×C4×D3—184no—
168C2×C4×D3—1816no—
169C2×C4×D3—184no—
170C2×C4×D3—184no—
171C2×C4×D3—362no—
172C2×C4×D3#15711yescharacteristic, complemented
173C22×C32—22no—
174C22×C32—32no—
175C22×C32—32no—
176C22×C32—32no—
177C22×C32—42no—
178C22×C32C2×C4×D311yescharacteristic
179C2×C3⋊D3—32no—
180C2×C3⋊D3—32no—
181C2×C3⋊D3—32no—
182C2×C3⋊D3—38no—
183C2×C3⋊D3—38no—
184C2×C3⋊D3—64no—
185C2×C3⋊D3—64no—
186C2×C3⋊D3—64no—
187C2×C3⋊D3—124no—
188C2×C3⋊D3—124no—
189C2×C3⋊D3—124no—
190C2×C3⋊D3C2×C4×D312yes—
191C2×C3⋊D3D3×D411yescharacteristic
192C2×C3⋊D3D3×D412yes—
193C2×C3⋊D3D3×Q811yescharacteristic
194C2×C3×D3—24no—
195C2×C3×D3—24no—
196C2×C3×D3—24no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C2×C3×D3—62no—
200C2×C3×D3—62no—
201D32—616no—
202D32—616no—
203#157—24no—
204#157—64no—
205C32×C4—21no—
206C32×C4—24no—
207C32×C4—44no—
208C32×C4—61no—
209C3⋊Dic3—21no—
210C3⋊Dic3—61no—
211C3⋊Dic3—64no—
212C3⋊Dic3—124no—
213C3×Dic3—24no—
214C3×Dic3—64no—
215C23×C4—271no—
216C23×C4—272no—
217#95—91no—
218#95—272no—
219#95—271no—
220C33C2×#9511yescharacteristic, complemented, Sylow
221C23×C3—62no—
222C23×C3—91no—
223C22×D3—62no—
224C22×D3—91no—
225C22×D3—94no—
226C22×D3—92no—
227C22×D3—92no—
228C22×D3—94no—
229C22×D3—188no—
230C22×D3—188no—
231C22×D3—184no—
232C22×D3—184no—
233C22×D3—182no—
234C22×D3—362no—
235C22×D3C2×#15711yescharacteristic
236C2×C3×C4—22no—
237C2×C3×C4—42no—
238C2×C3×C4—62no—
239C2×C3×C4—68no—
240C2×C3×C4—94no—
241C2×C3×C4—91no—
242C2×C3×C4—182no—
243C2×C3×C4—182no—
244C2×C3×C4—182no—
245C2×C3×C4C2×#15711yescharacteristic, complemented
246C2×Dic3—62no—
247C2×Dic3—94no—
248C2×Dic3—91no—
249C2×Dic3—182no—
250C2×Dic3—182no—
251C2×Dic3—188no—
252C2×Dic3—182no—
253C2×Dic3—182no—
254C2×Dic3—362no—
255C2×Dic3C2×#15711yescharacteristic, complemented
256C4×D3—22no—
257C4×D3—68no—
258C4×D3—182no—
259C4×D3—188no—
260C4×D3—182no—
261C4×D3—188no—
262C4×D3—188no—
263C4×D3—1816no—
264C4×D3—182no—
265C4×D3—368no—
266C2×C32—22no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—34no—
270C2×C32—42no—
271C2×C32—42no—
272C2×C32—42no—
273C2×C32C42×D311yescharacteristic
274C2×C32D3×C22⋊C411yescharacteristic
275C2×C32D3×C4⋊C411yescharacteristic
276C3⋊D3—34no—
277C3⋊D3—68no—
278C3⋊D3—128no—
279C3⋊D3D3×C22⋊C412yes—
280C3⋊D3D3×C4⋊C412yes—
281C3×D3—28no—
282C3×D3—64no—
283C3×D3—64no—
284C24—271no—
285C22×C4—31no—
286C22×C4—92no—
287C22×C4—91no—
288C22×C4—271no—
289C22×C4—271no—
290C22×C4—271no—
291C22×C4—278no—
292C22×C4—272no—
293C22×C4—271no—
294C22×C4—544no—
295C22×C4—544no—
296C22×C4—544no—
297C22×C3—22no—
298C22×C3—42no—
299C22×C3—64no—
300C22×C3—64no—
301C22×C3—64no—
302C22×C3—92no—
303C22×C3—91no—
304C22×C3—91no—
305C22×C3—92no—
306C22×C3C22×#15711yescharacteristic
307C2×D3—64no—
308C2×D3—64no—
309C2×D3—62no—
310C2×D3—62no—
311C2×D3—92no—
312C2×D3—92no—
313C2×D3—92no—
314C2×D3—98no—
315C2×D3—98no—
316C2×D3—1816no—
317C2×D3—1816no—
318C2×D3—184no—
319C2×D3—184no—
320C2×D3—184no—
321C2×D3—364no—
322C2×D3—364no—
323C2×D3—364no—
324C2×D3C4×#15712yes—
325C2×D3C4⋊#15712yes—
326C2×D3C22⋊#15712yes—
327C3×C4—24no—
328C3×C4—21no—
329C3×C4—44no—
330C3×C4—64no—
331C3×C4—184no—
332C3×C4—181no—
333Dic3—21no—
334Dic3—64no—
335Dic3—184no—
336Dic3—184no—
337Dic3—181no—
338Dic3—364no—
339C32—22no—
340C32—42no—
341C32D3×#9511yescharacteristic, complemented
342C23—31no—
343C23—91no—
344C23—274no—
345C23—272no—
346C23—274no—
347C23—272no—
348C23—271no—
349C2×C4—31no—
350C2×C4—62no—
351C2×C4—94no—
352C2×C4—91no—
353C2×C4—182no—
354C2×C4—182no—
355C2×C4—182no—
356C2×C4—274no—
357C2×C4—271no—
358C2×C4—542no—
359C2×C4—542no—
360C2×C4—542no—
361C2×C4—542no—
362C2×C4—5416no—
363C2×C4—542no—
364C2×C4—542no—
365C2×C4D3×#15711yescharacteristic, complemented
366C2×C3—22no—
367C2×C3—22no—
368C2×C3—22no—
369C2×C3—42no—
370C2×C3—42no—
371C2×C3—42no—
372C2×C3—68no—
373C2×C3—92no—
374C2×C3—92no—
375C2×C3C2×C4×#15711yescharacteristic
376C2×C3C2×C4⋊#15711yescharacteristic
377C2×C3C2×C22⋊#15711yescharacteristic
378D3—64no—
379D3—64no—
380D3—94no—
381D3—188no—
382D3—368no—
383D3#6378114yesdirect product factor
384C22—32no—
385C22—32no—
386C22—32no—
387C22—92no—
388C22—91no—
389C22—91no—
390C22—92no—
391C22—278no—
392C22—278no—
393C22—272no—
394C22—272no—
395C22—272no—
396C22C2×D3×#15711yescharacteristic
397C4—21no—
398C4—61no—
399C4—184no—
400C4—181no—
401C4—544no—
402C4—541no—
403C3—22no—
404C3—42no—
405C3C2×#6378111yescharacteristic, complemented
406C2—34no—
407C2—92no—
408C2—92no—
409C2—274no—
410C2C4×D3×#15711yescharacteristic
411C2D3×C4⋊#15711yescharacteristic
412C2D3×C22⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×#157
commutatorC2×C33C2×C42
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×D3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#95
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723106C2semidirect
4. C3×#63781C2semidirect
5. #11722722C2semidirect
12. C2×D3×#157C4semidirect
13. C2×C4×C3⋊(C3⋊D3)C4semidirect
15. C2×C32×C4×D3C4semidirect
16. C2×#6289C4semidirect
18. C2×D3×C3⋊Dic3C4semidirect
25. #63781D3direct (N × H)
35. C2×C3⋊#157C2×C4semidirect
38. C2×C3×#157C2×C4semidirect
40. C2×C33×C4C2×C4semidirect
41. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
43. C2×C3×C3⋊Dic3C2×C4semidirect
45. C2×C32×Dic3C2×C4semidirect
119. C2×#157C4×D3semidirect
124. C2×C32×C4C4×D3semidirect
129. C2×C3⋊Dic3C4×D3semidirect
147. C3⋊(C3⋊D3)#95semidirect
150. C32×D3#95semidirect
172. C2×C4×D3#157semidirect
220. C33C2×#95semidirect
245. C2×C3×C4C2×#157semidirect
255. C2×Dic3C2×#157semidirect
341. C32D3×#95semidirect
365. C2×C4D3×#157semidirect
383. D3#63781direct (N × H)
405. C3C2×#63781semidirect

Maximal subgroups

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

subgroupindex
C22×D3×#1572
C2×C4×D3×C3⋊D32
#117231062
C3×#637812
#117227222
C2×#637813
D3×#959