Finite Group Databasebeta

← #13173412

Subgroups — #13173412

Order 1728.

1000 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 89 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×D4×C3⋊D3C211yesmaximal, characteristic
2C2×#88994C212yesmaximal, complemented
3#11723111C212yesmaximal, complemented
4#11723099C212yesmaximal, complemented
5#10593817—31nomaximal
6C23×C3×C3⋊D3C411yescharacteristic, complemented
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3⋊#157—22no—
9C2×C33×D4C411yescharacteristic, complemented
10C2×C3×#638C411yescharacteristic, complemented
11C2×C3×#638C2211yescharacteristic
12C3×D4×C3⋊D3—22no—
13C3×D4×C3⋊D3—22no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
16#88994—22no—
17#88994C2214yescomplemented
18C2×D4×C3⋊D3D311yescharacteristic
19C2×C3×D3×D4—62no—
20C2×C22⋊#157—32no—
21#63786—32no—
22#63774—32no—
23C23×C33C2×C411yescharacteristic
24C23×C33C2×C411yescharacteristic, complemented
25C22×C3×C3⋊D3—21no—
26C22×C3×C3⋊D3—21no—
27C22×C3×C3⋊D3—42no—
28C22×C3×C3⋊D3—42no—
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C22×C3×C3⋊D3C2×C411yescharacteristic, complemented
31C22×C3×C3⋊D3D412yes—
32C22×C3×C3⋊D3D412yes—
33C22×C3×C3⋊D3C2311yescharacteristic
34C2×C3⋊#157—24no—
35C2×C3⋊#157—44no—
36C33×D4—22no—
37C2×C33×C4C2×C411yescharacteristic, complemented
38C3×#638—22no—
39C3×#638—22no—
40C2×C3×C3⋊Dic3C2×C411yescharacteristic, complemented
41C3×C4⋊(C3⋊D3)—22no—
42C3×C4×C3⋊D3—21no—
43C3×C4×C3⋊D3—21no—
44C2×#834—91nomaximal
45C23×C3⋊D3Dic311yescharacteristic, complemented
46C23×C3⋊D3C2×D311yescharacteristic
47C23×C3×D3—62no—
48C23×C3×D3—62no—
49C22×#157—62no—
50C2×C32×D4—22no—
51C2×C32×D4—42no—
52C2×C32×D4Dic311yescharacteristic, complemented
53C2×#638Dic311yescharacteristic, complemented
54C2×#638C2×D311yescharacteristic
55D4×C3⋊D3—22no—
56D4×C3⋊D3—22no—
57C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C4×C3⋊D3Dic311yescharacteristic, complemented
59C2×C3×#67—62no—
60C2×C3×#67—62no—
61C3×D3×D4—68no—
62C3×D3×D4—68no—
63C2×C3×D12—62no—
64C2×C3×C4×D3—62no—
65C22⋊#157—34no—
66C22⋊#157—62no—
67C22×C33—21no—
68C22×C33—41no—
69C22×C33—41no—
70C22×C33C22⋊C412yes—
71C22×C33C22×C411yescharacteristic
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3—24no—
74C2×C3×C3⋊D3—21no—
75C2×C3×C3⋊D3—42no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3—42no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3C22⋊C411yescharacteristic
81C2×C3×C3⋊D3C22⋊C411yescharacteristic
82C2×C3×C3⋊D3C2×D412yes—
83C3⋊#157—44no—
84C33×C4—21no—
85C3×C3⋊Dic3—21no—
86C22×C3×D4—91no—
87C2×D3×D4—62no—
88C2×#217—92no—
89#834—92no—
90#834—92no—
91C23×C32—22no—
92C23×C32—22no—
93C23×C32—42no—
94C23×C32—42no—
95C23×C32C2×Dic311yescharacteristic
96C23×C32C2×Dic311yescharacteristic, complemented
97C22×C3⋊D3—21no—
98C22×C3⋊D3—21no—
99C22×C3⋊D3—42no—
100C22×C3⋊D3—42no—
101C22×C3⋊D3C2×Dic311yescharacteristic
102C22×C3⋊D3C2×Dic311yescharacteristic, complemented
103C22×C3⋊D3#6712yes—
104C22×C3⋊D3#6712yes—
105C22×C3⋊D3C22×D311yescharacteristic
106C22×C3×D3—62no—
107C22×C3×D3—62no—
108C22×C3×D3—64no—
109C22×C3×D3—64no—
110C22×C3×D3—62no—
111C22×C3×D3—64no—
112C22×C3×D3—64no—
113C22×C3×D3—128no—
114C22×C3×D3—128no—
115C2×#157—64no—
116C2×#157—124no—
117C32×D4—22no—
118C32×D4—28no—
119C32×D4—48no—
120C2×C32×C4—22no—
121C2×C32×C4—42no—
122C2×C32×C4C2×Dic311yescharacteristic, complemented
123#638—22no—
124#638—22no—
125C2×C3⋊Dic3C2×Dic311yescharacteristic, complemented
126C4⋊(C3⋊D3)—22no—
127C4×C3⋊D3—21no—
128C4×C3⋊D3—21no—
129C3×#67—68no—
130C3×#67—68no—
131C2×C3×Dic3—62no—
132C3×D12—68no—
133C3×C4×D3—64no—
134C3×C4×D3—64no—
135C2×#99—271noSylow
136C2×C33—21no—
137C2×C33—22no—
138C2×C33—41no—
139C2×C33C2×C22⋊C411yescharacteristic
140C3×C3⋊D3—21no—
141C3×C3⋊D3—22no—
142C3×C3⋊D3—41no—
143C3×C3⋊D3#9912yescomplemented
144C24×C3—91no—
145C24×C3—91no—
146C23×D3—62no—
147C23×D3—62no—
148C2×C3×D4—22no—
149C2×C3×D4—42no—
150C2×C3×D4—91no—
151C2×C3×D4—91no—
152C2×C3×D4—91no—
153C2×C3×D4—182no—
154C2×C3×D4—182no—
155C2×C3×D4#15711yescharacteristic, complemented
156C22×C3×C4—91no—
157C2×#67—62no—
158C2×#67—62no—
159C22×Dic3—182no—
160D3×D4—68no—
161D3×D4—68no—
162C2×D12—62no—
163C2×C4×D3—62no—
164#217—94no—
165#217—182no—
166C22×C32—21no—
167C22×C32—24no—
168C22×C32—22no—
169C22×C32—24no—
170C22×C32—44no—
171C22×C32—42no—
172C22×C32—44no—
173C22×C32—41no—
174C22×C32—41no—
175C22×C32—44no—
176C22×C32—44no—
177C22×C32—84no—
178C22×C32—84no—
179C22×C32#21712yes—
180C22×C32C22×Dic311yescharacteristic
181C2×C3⋊D3—22no—
182C2×C3⋊D3—21no—
183C2×C3⋊D3—24no—
184C2×C3⋊D3—42no—
185C2×C3⋊D3—41no—
186C2×C3⋊D3—42no—
187C2×C3⋊D3—42no—
188C2×C3⋊D3—41no—
189C2×C3⋊D3#21711yescharacteristic
190C2×C3⋊D3#21711yescharacteristic
191C2×C3⋊D3C2×#6712yes—
192C2×C3×D3—62no—
193C2×C3×D3—62no—
194C2×C3×D3—64no—
195C2×C3×D3—68no—
196C2×C3×D3—64no—
197C2×C3×D3—128no—
198C2×C3×D3—128no—
199C2×C3×D3—128no—
200C2×C3×D3—128no—
201C2×C3×D3—124no—
202C2×C3×D3—124no—
203#157—124no—
204C32×C4—21no—
205C32×C4—24no—
206C32×C4—44no—
207C3⋊Dic3—21no—
208C3×Dic3—64no—
209C22×D4—91no—
210C2×C22⋊C4—272no—
211#99—272no—
212#99—272no—
213C33C2×#9911yescharacteristic, complemented, Sylow
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—91no—
219C23×C3—92no—
220C23×C3—91no—
221C23×C3—92no—
222C23×C3—91no—
223C23×C3—181no—
224C23×C3—181no—
225C23×C3—362no—
226C23×C3—362no—
227C23×C3C2×#15711yescharacteristic
228C23×C3C2×#15711yescharacteristic, complemented
229C22×D3—62no—
230C22×D3—62no—
231C22×D3—64no—
232C22×D3—64no—
233C22×D3—62no—
234C22×D3—64no—
235C22×D3—64no—
236C22×D3—128no—
237C22×D3—128no—
238C3×D4—28no—
239C3×D4—22no—
240C3×D4—48no—
241C3×D4—182no—
242C3×D4—182no—
243C3×D4—182no—
244C2×C3×C4—22no—
245C2×C3×C4—42no—
246C2×C3×C4—91no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4C2×#15711yescharacteristic, complemented
250#67—68no—
251#67—68no—
252C2×Dic3—62no—
253C2×Dic3—184no—
254C2×Dic3—364no—
255D12—68no—
256C4×D3—64no—
257C4×D3—64no—
258C2×C32—21no—
259C2×C32—24no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—44no—
263C2×C32—42no—
264C2×C32—44no—
265C2×C32—41no—
266C2×C32—44no—
267C2×C32—84no—
268C2×C32—84no—
269C2×C32C2×#21711yescharacteristic
270C3⋊D3—21no—
271C3⋊D3—22no—
272C3⋊D3—41no—
273C3⋊D3#83412yescomplemented
274C3×D3—64no—
275C3×D3—64no—
276C3×D3—124no—
277C3×D3—124no—
278C24—91no—
279C24—91no—
280C2×D4—91no—
281C2×D4—91no—
282C2×D4—91no—
283C2×D4—182no—
284C2×D4—182no—
285C2×D4C3⋊#15711yescharacteristic, complemented
286C22×C4—91no—
287C22×C4—542no—
288C22⋊C4—274no—
289C22⋊C4—542no—
290C22×C3—24no—
291C22×C3—21no—
292C22×C3—22no—
293C22×C3—24no—
294C22×C3—44no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—44no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—44no—
301C22×C3—84no—
302C22×C3—84no—
303C22×C3—91no—
304C22×C3—92no—
305C22×C3—91no—
306C22×C3—184no—
307C22×C3—182no—
308C22×C3—181no—
309C22×C3—361no—
310C22×C3—362no—
311C22×C3—361no—
312C22×C3—362no—
313C22×C3—362no—
314C22×C3C22⋊#15712yes—
315C22×C3C22×#15711yescharacteristic
316C2×D3—62no—
317C2×D3—62no—
318C2×D3—64no—
319C2×D3—68no—
320C2×D3—64no—
321C2×D3—128no—
322C2×D3—128no—
323C2×D3—128no—
324C2×D3—128no—
325C2×D3—124no—
326C2×D3—124no—
327C3×C4—24no—
328C3×C4—21no—
329C3×C4—44no—
330C3×C4—181no—
331Dic3—64no—
332Dic3—364no—
333C32—22no—
334C32—42no—
335C32C2×#83411yescharacteristic, complemented
336C23—91no—
337C23—92no—
338C23—91no—
339C23—92no—
340C23—91no—
341C23—181no—
342C23—181no—
343C23—362no—
344C23—362no—
345C23C2×C3⋊#15711yescharacteristic
346C23C2×C3⋊#15711yescharacteristic, complemented
347D4—22no—
348D4—182no—
349D4—182no—
350D4—182no—
351C2×C4—91no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4—544no—
355C2×C4—1084no—
356C2×C4C2×C3⋊#15711yescharacteristic, complemented
357C2×C3—24no—
358C2×C3—21no—
359C2×C3—22no—
360C2×C3—22no—
361C2×C3—44no—
362C2×C3—44no—
363C2×C3—44no—
364C2×C3—41no—
365C2×C3—42no—
366C2×C3—84no—
367C2×C3—84no—
368C2×C3—92no—
369C2×C3—182no—
370C2×C3—181no—
371C2×C3—361no—
372C2×C3C2×C22⋊#15711yescharacteristic
373D3—64no—
374D3—64no—
375D3—124no—
376D3—124no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—91no—
381C22—91no—
382C22—92no—
383C22—184no—
384C22—182no—
385C22—181no—
386C22—361no—
387C22—362no—
388C22—361no—
389C22—362no—
390C22—362no—
391C22#8899412yes—
392C22C22×C3⋊#15711yescharacteristic
393C4—21no—
394C4—181no—
395C4—1084no—
396C3—22no—
397C3—42no—
398C3#1059381711yescharacteristic, complemented
399C2—21no—
400C2—22no—
401C2—41no—
402C2—92no—
403C2—182no—
404C2—181no—
405C2—361no—
406C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC2×C33×D4C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×D4C3⋊#157
hypocenterC33C2×#99
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#88994C2semidirect
3. #11723111C2semidirect
4. #11723099C2semidirect
6. C23×C3×C3⋊D3C4semidirect
9. C2×C33×D4C4semidirect
10. C2×C3×#638C4semidirect
15. C2×C3×C4×C3⋊D3C4semidirect
17. #88994C22semidirect
24. C23×C33C2×C4semidirect
30. C22×C3×C3⋊D3C2×C4semidirect
37. C2×C33×C4C2×C4semidirect
40. C2×C3×C3⋊Dic3C2×C4semidirect
45. C23×C3⋊D3Dic3semidirect
52. C2×C32×D4Dic3semidirect
53. C2×#638Dic3semidirect
58. C2×C4×C3⋊D3Dic3semidirect
96. C23×C32C2×Dic3semidirect
102. C22×C3⋊D3C2×Dic3semidirect
122. C2×C32×C4C2×Dic3semidirect
125. C2×C3⋊Dic3C2×Dic3semidirect
143. C3×C3⋊D3#99semidirect
155. C2×C3×D4#157semidirect
213. C33C2×#99semidirect
228. C23×C3C2×#157semidirect
249. C2×C3×C4C2×#157semidirect
273. C3⋊D3#834semidirect
285. C2×D4C3⋊#157semidirect
335. C32C2×#834semidirect
346. C23C2×C3⋊#157semidirect
356. C2×C4C2×C3⋊#157semidirect
398. C3#10593817semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#889942
#117231112
#117230992
#105938173
C2×#8349