Finite Group Databasebeta

← #13159963

Subgroups — #13159963

Order 1728.

968 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 168 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×#3744C211yesmaximal, characteristic, complemented
2#11723517C211yesmaximal, characteristic
3Dic3×#634C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592547—31nomaximal
9#10590799—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×#6292—21no—
14C2×#6292—21no—
15C2×C33×C8C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C3×#3744—22no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21Dic3×C3⋊Dic3—21no—
22#88797—22no—
23C2×#3744—31no—
24C2×#3744D311yescharacteristic, complemented
25#64083—31no—
26C4×#634—31no—
27C2×C3×#204—34no—
28#63911—34no—
29Dic3×Dic6—34no—
30C2×#3717—34no—
31C32×C4⋊C8—31no—
32#63648—31no—
33#63648—31no—
34#63648—34no—
35#63648—64no—
36C3×Dic3⋊C8D314yescomplemented
37#63563—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C2×C3×C3⋊Dic3—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×#634—21no—
48C3×#634D412yescomplemented
49C2×C32×Dic3D411yescharacteristic
50C2×C32×Dic3D411yescharacteristic
51#6292—42no—
52#6292—42no—
53C33×C8—21no—
54C32×C3⋊C8—21no—
55#5163—91no—
56#4848—94no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—64no—
60C2×C4⋊(C3⋊D3)—61no—
61C2×C4⋊(C3⋊D3)—61no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×#634C2×D311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×Dic6—34no—
66C2×#626—64no—
67C2×#626—64no—
68C2×C32×C8—24no—
69C2×C32×C8—31no—
70C2×C32×C8C2×D314yes—
71C2×C32×C8C2×D311yescharacteristic
72C32×C42—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—34no—
75C4⋊(C3⋊Dic3)—64no—
76#3750—61no—
77C4×C3⋊Dic3—61no—
78#3744—22no—
79#3744—62no—
80C3×C4×Dic3C2×D314yes—
81C2×C3×C3⋊C8C2×D314yes—
82C3×#204—68no—
83Dic3⋊Dic3—64no—
84Dic32—64no—
85#3717—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
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×C4C2×D411yescharacteristic
94C33×C4#4111yescharacteristic
95C3⋊(C3⋊Dic3)—41no—
96C3×C3⋊Dic3—22no—
97C32×Dic3—21no—
98C32×Dic3SD1612yescomplemented
99C2×C3×SD16—91no—
100Q8×Dic3—91no—
101C2×#215—91no—
102#940—91no—
103C2×#204—34no—
104C2×#204—94no—
105#874—94no—
106C4×Dic6—94no—
107C3×C4⋊C8—34no—
108#832—91no—
109#821—94no—
110#821—94no—
111#821—91no—
112#821—184no—
113Dic3⋊C8C3⋊D311yescharacteristic, complemented
114C22×C3⋊D3—21no—
115C22×C3⋊D3—61no—
116C22×C3⋊D3—64no—
117C22×C3⋊D3—124no—
118C22×C3×D3—64no—
119C2×C32×C4—24no—
120C2×C32×C4—31no—
121C2×C32×C4—31no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—124no—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)—68no—
131C4⋊(C3⋊D3)—61no—
132C4⋊(C3⋊D3)—64no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—124no—
136C4⋊(C3⋊D3)—122no—
137C4⋊(C3⋊D3)—122no—
138#634—21no—
139#634#6712yescomplemented
140C2×C3×Dic3—64no—
141C2×C3×Dic3D1214yes—
142C2×C3×Dic3D1214yes—
143C3×D12—64no—
144C3×D12—64no—
145C3×Dic6—38no—
146C3×Dic6—64no—
147#626—128no—
148#626—128no—
149C32×C8—21no—
150C32×C8—24no—
151C32×C8—44no—
152C32×C8—61no—
153C3×C3⋊C8—24no—
154#460—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33C2×SD1611yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3⋊(C3⋊D3)—81no—
159C3×C3⋊D3—81no—
160C2×C3×Q8—91no—
161C2×C3×D4—91no—
162C2×#67—181no—
163C2×#67—181no—
164C2×D12—34no—
165C2×D12—94no—
166C2×D12—91no—
167C2×D12—184no—
168C2×D12—184no—
169C2×D12—184no—
170C2×Dic6—34no—
171C3×SD16—182no—
172C2×C3×C8—24no—
173C2×C3×C8—34no—
174C2×C3×C8D3214yes—
175C2×C3×C8C2×C3⋊D311yescharacteristic
176C3×C42—34no—
177#215—182no—
178C4⋊Dic3—94no—
179C4⋊Dic3—91no—
180C4⋊Dic3—184no—
181C4⋊Dic3—181no—
182Dic3⋊C4—184no—
183C4×Dic3—184no—
184C4×Dic3—181no—
185C4×Dic3C2×C3⋊D311yescharacteristic
186C2×C3⋊C8C2×C3⋊D311yescharacteristic
187#204—68no—
188#204—188no—
189C22×C32—24no—
190C22×C32C2×D1214yes—
191C22×C32D3×D411yescharacteristic
192C2×C3⋊D3—41no—
193C2×C3⋊D3—41no—
194C2×C3⋊D3—41no—
195C2×C3⋊D3—128no—
196C2×C3⋊D3—124no—
197C2×C3⋊D3—121no—
198C2×C3⋊D3—121no—
199C2×C3⋊D3—121no—
200C2×C3⋊D3—124no—
201C2×C3⋊D3—124no—
202C2×C3⋊D3—244no—
203C2×C3⋊D3—244no—
204C2×C3×D3—124no—
205C2×C3×D3—124no—
206C2×C3×D3—124no—
207C32×C4—24no—
208C32×C4—24no—
209C32×C4—32no—
210C32×C4—61no—
211C32×C4D3×D414yes—
212C32×C4#23711yescharacteristic
213C32×C4#23914yes—
214C32×C4C2×#6711yescharacteristic
215C3⋊Dic3—22no—
216C3⋊Dic3—121no—
217C3⋊Dic3—124no—
218C3⋊Dic3—244no—
219C3×Dic3—24no—
220C3×Dic3—68no—
221C3×Dic3#20418yescomplemented
222C2×SD16—91no—
223C2×SD16—271no—
224#127—271no—
225C4×Q8—271no—
226C4⋊C8—31no—
227#102—271no—
228#102—271no—
229C33#46011yescharacteristic, complemented, Sylow
230C23×C3—181no—
231C22×D3—64no—
232C22×D3—184no—
233C22×D3—181no—
234C22×D3—364no—
235C3×Q8—92no—
236C3×Q8—181no—
237C3×D4—181no—
238C3×D4—181no—
239C2×C3×C4—24no—
240C2×C3×C4—34no—
241C2×C3×C4—34no—
242C2×C3×C4—181no—
243C2×C3×C4C2×D3214yes—
244C2×C3×C4C22×C3⋊D311yescharacteristic
245#67—362no—
246#67—362no—
247C2×Dic3—64no—
248C2×Dic3—184no—
249C2×Dic3—181no—
250C2×Dic3—364no—
251C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
252C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
253D12—64no—
254D12—64no—
255D12—188no—
256D12—184no—
257D12—184no—
258D12—181no—
259D12—181no—
260D12—364no—
261D12—368no—
262D12—368no—
263Dic6—38no—
264Dic6—64no—
265C3×C8—24no—
266C3×C8—21no—
267C3×C8—44no—
268C3×C8—64no—
269C3⋊C8—21no—
270C2×C32—24no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32#89514yes—
274C2×C32C2×#20414yes—
275C2×C32#90814yes—
276C2×C32D3×SD1611yescharacteristic
277C2×C32#91411yescharacteristic
278C2×C32#93911yescharacteristic
279C3⋊D3—81no—
280C3⋊D3—248no—
281C3⋊D3—241no—
282C3⋊D3—244no—
283C3×D3—244no—
284C2×Q8—91no—
285C2×D4—91no—
286C2×D4—271no—
287C2×D4—541no—
288C2×D4—541no—
289SD16—182no—
290SD16—542no—
291C2×C8—31no—
292C2×C8D3×C3⋊D311yescharacteristic
293C4⋊C4—271no—
294C4⋊C4—541no—
295C42—31no—
296C42—541no—
297C22×C3—24no—
298C22×C3—361no—
299C22×C3—361no—
300C22×C3—361no—
301C22×C3D3×D1214yes—
302C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
303C2×D3—124no—
304C2×D3—124no—
305C2×D3—124no—
306C2×D3—368no—
307C2×D3—364no—
308C2×D3—364no—
309C2×D3—361no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—364no—
313C2×D3—724no—
314C2×D3—724no—
315C3×C4—24no—
316C3×C4—24no—
317C3×C4—38no—
318C3×C4—64no—
319C3×C4—182no—
320C3×C4#379614yes—
321C3×C4D3×#6714yes—
322C3×C4D4×C3⋊D311yescharacteristic
323C3×C4#383211yescharacteristic
324Dic3—21no—
325Dic3—68no—
326Dic3—364no—
327Dic3—361no—
328Dic3—724no—
329Dic3#374412yescomplemented
330C32—24no—
331C32#484814yescomplemented
332C32#516311yescharacteristic, complemented
333C23—181no—
334C23—541no—
335Q8—92no—
336Q8—181no—
337D4—181no—
338D4—181no—
339D4—541no—
340D4—541no—
341D4—1082no—
342D4—1082no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—181no—
346C2×C4—541no—
347C2×C4C2×D3×C3⋊D311yescharacteristic
348C8—21no—
349C8—61no—
350C2×C3—24no—
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—721no—
354C2×C3D3×#20414yes—
355C2×C3#6379414yes—
356C2×C3#6388614yes—
357C2×C3#6410411yescharacteristic
358C2×C3C2×#374411yescharacteristic
359C2×C3#6411711yescharacteristic
360D3—244no—
361D3—728no—
362D3—721no—
363D3—724no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22D3×C4⋊(C3⋊D3)11yescharacteristic
371C4—32no—
372C4—61no—
373C4—182no—
374C4—1081no—
375C4#8901711yescharacteristic
376C4C3⋊D3×#6711yescharacteristic
377C3—24no—
378C3#1059079914yescomplemented
379C3#1059254711yescharacteristic, complemented
380C2—721no—
381C2—2161no—
382C2D3×#374411yescharacteristic
383C2#1172331511yescharacteristic
384C2#1172348211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723517C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#460
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3744C2semidirect
3. Dic3×#634C2semidirect
4. C2×#88797C2semidirect
5. #11722191C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. #11722034C2semidirect
24. C2×#3744D3semidirect
36. C3×Dic3⋊C8D3semidirect
48. C3×#634D4semidirect
98. C32×Dic3SD16semidirect
113. Dic3⋊C8C3⋊D3semidirect
139. #634#67semidirect
221. C3×Dic3#204semidirect
229. C33#460semidirect
329. Dic3#3744semidirect
331. C32#4848semidirect
332. C32#5163semidirect
378. C3#10590799semidirect
379. C3#10592547semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
#117235172
Dic3×#6342
C2×#887972
#117221912
C32×Dic3⋊C82
#117220342
#105925473
#105907993