Finite Group Databasebeta

← #13161153

Subgroups — #13161153

Order 1728.

1016 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 146 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×#88862C211yesmaximal, characteristic, complemented
2C3×#64156C211yesmaximal, characteristic, complemented
3#11723494C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic, complemented
6#11722051C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592518—31nomaximal
9#10591568—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6289—21no—
14#88862—22no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752—21no—
17C3×#3752—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88804—21no—
22#88791—22no—
23#64156D311yescharacteristic, complemented
24C2×#3755—31no—
25C2×#3755—34no—
26C2×#3755—64no—
27C2×#3745—31no—
28#64102—31no—
29C3×#947—34no—
30#63894—34no—
31C2×#3714—34no—
32C32×#103—31no—
33#63641—31no—
34#63636—31no—
35C3×#835D314yescomplemented
36Dic3⋊(C3⋊C8)—34no—
37#63562—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C33×Q8—21no—
41C33×Q8—21no—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C4⋊(C3⋊(C3⋊D3))—21no—
45C4⋊(C3⋊(C3⋊D3))D412yes—
46C2×C3×C3⋊Dic3D411yescharacteristic
47C2×C3×C3⋊Dic3D411yescharacteristic
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C2×C32×Dic3—21no—
51#6289—42no—
52C3⋊(C3⋊(C3⋊C8))—21no—
53C32×C3⋊C8—21no—
54#5057—91no—
55#4819—94no—
56C2×C32×Q8—24no—
57C2×C32×Q8C2×D314yes—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C4⋊(C3⋊D3)—31no—
60C2×C4⋊(C3⋊D3)—34no—
61C2×C4⋊(C3⋊D3)—64no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C4×C3⋊D3—61no—
64C2×C3×D12—34no—
65C2×#624—64no—
66C2×C32×C8—31no—
67C32×C4⋊C4—31no—
68#3755—62no—
69#3755—68no—
70#3755—616no—
71#3752—21no—
72#3752—21no—
73#3752—61no—
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3745—62no—
79C3×(C2×D3)⋊C4—64no—
80C3×(C2×D3)⋊C4—64no—
81C3×C4⋊Dic3C2×D314yes—
82C3×C4×Dic3—34no—
83C2×C3×C3⋊C8C2×D314yes—
84#3722—64no—
85#3714—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—22no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C33×C4—41no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—21no—
98C32×Dic3—41no—
99C3×#124—91no—
100#947—34no—
101C2×#215—94no—
102C2×#215—91no—
103C2×#215—184no—
104C2×#213—91no—
105C2×D24—94no—
106#893—94no—
107C4×D12—91no—
108C3×#103—34no—
109#835C3⋊D311yescharacteristic, complemented
110#832—91no—
111Dic3⋊C8—94no—
112#809—94no—
113C4⋊(C3⋊C8)—91no—
114C22×C3⋊D3—21no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—124no—
118C22×C3×D3—64no—
119C32×Q8—28no—
120C32×Q8—24no—
121C32×Q8—21no—
122C32×Q8—21no—
123C32×Q8—24no—
124C32×Q8—44no—
125C2×C32×C4—21no—
126C2×C32×C4—24no—
127C2×C32×C4—24no—
128C2×C32×C4—44no—
129C2×C32×C4—61no—
130C2×C32×C4C22×D314yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C3⋊Dic3#6711yescharacteristic
133C2×C3⋊Dic3#6711yescharacteristic
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—32no—
137C4⋊(C3⋊D3)—38no—
138C4⋊(C3⋊D3)—68no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—124no—
142C4×C3⋊D3—122no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—34no—
146C3×D12—64no—
147C3×D12—64no—
148#624—128no—
149C32×C8—61no—
150C3⋊(C3⋊C8)—68no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—61no—
153C3×C3⋊C8—24no—
154#463—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33#13511yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3⋊(C3⋊D3)—42no—
159C3×C3⋊D3—81no—
160C2×C3×Q8—24no—
161C2×C3×Q8D3214yes—
162C2×C3×Q8C2×C3⋊D311yescharacteristic
163C2×C3×D4—91no—
164C2×D12—34no—
165C2×D12—94no—
166C2×D12—91no—
167C2×D12—184no—
168C2×C4×D3—181no—
169C2×C4×D3—184no—
170C2×C3×C8—34no—
171C3×C4⋊C4—34no—
172C3×C22⋊C4—181no—
173C3×C22⋊C4—181no—
174C3×C42—91no—
175#215—188no—
176#215—182no—
177#215—1816no—
178#213—182no—
179(C2×D3)⋊C4—64no—
180(C2×D3)⋊C4—64no—
181(C2×D3)⋊C4—181no—
182(C2×D3)⋊C4—184no—
183C4⋊Dic3C2×C3⋊D311yescharacteristic
184C4×Dic3—34no—
185C2×C3⋊C8—94no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—184no—
188C2×C3⋊C8C2×C3⋊D311yescharacteristic
189D24—188no—
190C22×C32—24no—
191C22×C32D3×D414yes—
192C22×C32C2×#6711yescharacteristic
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—62no—
197C2×C3⋊D3—68no—
198C2×C3⋊D3—128no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—121no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—244no—
204C2×C3⋊D3—244no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C2×C3×D3—124no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—48no—
211C32×C4—44no—
212C32×C4—41no—
213C32×C4—121no—
214C32×C4#23514yes—
215C32×C4D3×D414yes—
216C32×C4D3×D411yescharacteristic
217C32×C4#23711yescharacteristic
218C3⋊Dic3—21no—
219C3⋊Dic3—21no—
220C3×Dic3—44no—
221C3×Dic3—64no—
222C3×Dic3—64no—
223C2×SD16—271no—
224C2×D8—271no—
225#124—91no—
226C4×D4—271no—
227C4⋊C8—271no—
228#103—31no—
229#102—271no—
230C33#46311yescharacteristic, complemented, Sylow
231C23×C3—181no—
232C22×D3—64no—
233C22×D3—184no—
234C22×D3—181no—
235C22×D3—364no—
236C3×Q8—28no—
237C3×Q8—24no—
238C3×Q8—24no—
239C3×Q8—21no—
240C3×Q8—21no—
241C3×Q8—44no—
242C3×D4—181no—
243C3×D4—181no—
244C2×C3×C4—24no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4—44no—
248C2×C3×C4—64no—
249C2×C3×C4—91no—
250C2×C3×C4—91no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253C2×Dic3—21no—
254C2×Dic3—34no—
255C2×Dic3—34no—
256D12—64no—
257D12—64no—
258D12—98no—
259D12—92no—
260D12—188no—
261D12—184no—
262D12—181no—
263D12—364no—
264C4×D3—362no—
265C4×D3—368no—
266C3×C8—64no—
267C3⋊C8—21no—
268C3⋊C8—188no—
269C3⋊C8—184no—
270C3⋊C8—181no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32#88414yes—
275C2×C32#91414yes—
276C2×C32#91914yes—
277C2×C32#93811yescharacteristic
278C2×C32#94911yescharacteristic
279C2×C32#95011yescharacteristic
280C3⋊D3—81no—
281C3⋊D3—122no—
282C3⋊D3—128no—
283C3⋊D3—248no—
284C3×D3—244no—
285C2×Q8D3×C3⋊D311yescharacteristic
286C2×D4—91no—
287C2×D4—271no—
288C22×C4—541no—
289SD16—542no—
290D8—542no—
291C2×C8—31no—
292C2×C8—271no—
293C4⋊C4—31no—
294C22⋊C4—181no—
295C22⋊C4—181no—
296C22⋊C4—541no—
297C42—91no—
298C22×C3—24no—
299C22×C3—361no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3D3×#6714yes—
303C22×C3D4×C3⋊D311yescharacteristic
304C2×D3—124no—
305C2×D3—124no—
306C2×D3—124no—
307C2×D3—182no—
308C2×D3—188no—
309C2×D3—368no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—364no—
313C2×D3—364no—
314C2×D3—724no—
315C2×D3—724no—
316C3×C4—24no—
317C3×C4—24no—
318C3×C4—48no—
319C3×C4—41no—
320C3×C4—44no—
321C3×C4—124no—
322C3×C4—181no—
323C3×C4—181no—
324C3×C4#380414yes—
325C3×C4#381114yes—
326C3×C4#382811yescharacteristic
327C3×C4D4×C3⋊D311yescharacteristic
328Dic3—41no—
329Dic3—64no—
330Dic3—64no—
331C32—24no—
332C32#481914yescomplemented
333C32#505711yescharacteristic, complemented
334C23—181no—
335C23—541no—
336Q8—21no—
337Q8—21no—
338D4—181no—
339D4—181no—
340D4—272no—
341D4—541no—
342C2×C4—21no—
343C2×C4—61no—
344C2×C4—91no—
345C2×C4—91no—
346C2×C4—1082no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—61no—
349C8—541no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—721no—
354C2×C3#6393914yes—
355C2×C3#6394914yes—
356C2×C3#6395814yes—
357C2×C3#6409311yescharacteristic
358C2×C3#6412311yescharacteristic
359C2×C3#6412811yescharacteristic
360D3—244no—
361D3—362no—
362D3—368no—
363D3—728no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22—542no—
368C22—1081no—
369C22—1081no—
370C22C3⋊D3×#6711yescharacteristic
371C4—41no—
372C4—121no—
373C4—181no—
374C4—181no—
375C4#8903111yescharacteristic
376C4#8903911yescharacteristic
377C3—24no—
378C3#1059156814yescomplemented
379C3#1059251811yescharacteristic, complemented
380C2—721no—
381C2—1082no—
382C2#1172357011yescharacteristic
383C2#1172358511yescharacteristic
384C2#1172362311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#88791C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88862C2semidirect
2. C3×#64156C2semidirect
3. #11723494C2semidirect
5. C32×#835C2semidirect
6. #11722051C2semidirect
7. #11722033C2semidirect
23. #64156D3semidirect
35. C3×#835D3semidirect
109. #835C3⋊D3semidirect
230. C33#463semidirect
332. C32#4819semidirect
333. C32#5057semidirect
378. C3#10591568semidirect
379. C3#10592518semidirect

Maximal subgroups

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

subgroupindex
C2×#888622
C3×#641562
#117234942
C2×#887912
C32×#8352
#117220512
#117220332
#105925183
#105915683