Finite Group Databasebeta

← #13159162

Subgroups — #13159162

Order 1728.

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

normal subgroups only

This group has 980 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
1C3×#64104C211yesmaximal, characteristic, complemented
2#11723526C211yesmaximal, characteristic, complemented
3C2×#88791C211yesmaximal, characteristic, complemented
4C2×#88787C211yesmaximal, characteristic, complemented
5#11722179C211yesmaximal, characteristic, complemented
6C32×#808C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592564—31nomaximal
9#10590586—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×#6292—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3752—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88804—21no—
21#88791—22no—
22#88787—22no—
23C2×#3745—31no—
24C2×#3743—31no—
25#64104—31no—
26#64104D311yescharacteristic, complemented
27C3×#895—34no—
28#63909—34no—
29C2×#3714—34no—
30C2×#3711—34no—
31C32×#106—31no—
32#63636—31no—
33#63636—31no—
34#63636—34no—
35#63636—64no—
36C3×#808D314yescomplemented
37#63562—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3D411yescharacteristic
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))—21no—
45C2×C3×C3⋊Dic3D411yescharacteristic
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—21no—
48C3×C4⋊(C3⋊D3)—42no—
49C3×C4×C3⋊D3—21no—
50C3×C4×C3⋊D3—21no—
51C2×C32×Dic3—21no—
52#6292—42no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C32×C3⋊C8D412yescomplemented
55#5039—91no—
56#4865—94no—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×C4⋊(C3⋊D3)—61no—
62C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×D12—64no—
66C2×C3×C4×D3—34no—
67C2×#626—64no—
68C2×C32×C8—31no—
69C32×C4⋊C4—24no—
70C32×C4⋊C4—31no—
71C32×C4⋊C4C2×D314yes—
72C32×C4⋊C4C2×D311yescharacteristic
73#3752—21no—
74#3752—61no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—34no—
77C2×C3⋊(C3⋊C8)—64no—
78#3745—62no—
79#3743—62no—
80C3×(C2×D3)⋊C4—64no—
81C3×C4⋊Dic3C2×D314yes—
82C2×C3×C3⋊C8C2×D314yes—
83#3722—64no—
84#3714—68no—
85#3711—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C3⋊(C3⋊D3)—41no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C3×C3⋊D3—41no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C33×C4—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3×C3⋊Dic3—21no—
100C32×Dic3—41no—
101C3×#121—91no—
102C2×#213—91no—
103#930—91no—
104C2×#208—91no—
105C2×D24—94no—
106C2×ΓL(1,𝔽25)—94no—
107#895—34no—
108#895—94no—
109C3×#106—34no—
110#832—91no—
111#809—94no—
112#809—94no—
113#809—91no—
114#809—184no—
115#808C3⋊D311yescharacteristic, complemented
116C22×C3⋊D3—21no—
117C22×C3⋊D3—61no—
118C22×C3⋊D3—64no—
119C22×C3⋊D3—124no—
120C22×C3⋊D3#6711yescharacteristic
121C22×C3×D3—34no—
122C22×C3×D3—64no—
123C2×C32×C4—21no—
124C2×C32×C4—24no—
125C2×C32×C4—24no—
126C2×C32×C4—44no—
127C2×C32×C4—61no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130C2×C3⋊Dic3#6711yescharacteristic
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—42no—
134C4⋊(C3⋊D3)—68no—
135C4⋊(C3⋊D3)—61no—
136C4⋊(C3⋊D3)—61no—
137C4⋊(C3⋊D3)—64no—
138C4⋊(C3⋊D3)—64no—
139C4⋊(C3⋊D3)—124no—
140C4⋊(C3⋊D3)—122no—
141C4×C3⋊D3—21no—
142C4×C3⋊D3—21no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—34no—
145C3×D12—64no—
146C3×D12—64no—
147C3×D12—128no—
148C3×C4×D3—64no—
149C3×C4×D3—64no—
150#626—128no—
151C32×C8—32no—
152C3⋊(C3⋊C8)—61no—
153C3⋊(C3⋊C8)—64no—
154C3⋊(C3⋊C8)—124no—
155C3×C3⋊C8D1218yescomplemented
156#469—271noSylow
157C2×C33#12111yescharacteristic
158C2×C33AGL(1,ℤ/8)11yescharacteristic
159C2×C33AGL(1,ℤ/8)11yescharacteristic
160C3⋊(C3⋊D3)—81no—
161C3×C3⋊D3—41no—
162C3×C3⋊D3—81no—
163C2×C3×D4—91no—
164C2×C3×D4—181no—
165C22×C3×C4—91no—
166C2×#67—181no—
167C2×D12—34no—
168C2×D12—64no—
169C2×D12—94no—
170C2×D12—91no—
171C2×D12—184no—
172C2×D12—184no—
173C2×C4×D3—34no—
174C2×C3×C8—34no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—34no—
177C3×C4⋊C4D3214yes—
178C3×C4⋊C4C2×C3⋊D311yescharacteristic
179C3×C22⋊C4—181no—
180#213—182no—
181(C2×D3)⋊C4—64no—
182(C2×D3)⋊C4—181no—
183(C2×D3)⋊C4—184no—
184C4⋊Dic3C2×C3⋊D311yescharacteristic
185#208—182no—
186C2×C3⋊C8—94no—
187C2×C3⋊C8—91no—
188C2×C3⋊C8—184no—
189C2×C3⋊C8C2×C3⋊D311yescharacteristic
190D24—188no—
191ΓL(1,𝔽25)—188no—
192C22×C32—24no—
193C22×C32D3×D414yes—
194C22×C32C2×#6711yescharacteristic
195C2×C3⋊D3—21no—
196C2×C3⋊D3—21no—
197C2×C3⋊D3—21no—
198C2×C3⋊D3—41no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—128no—
202C2×C3⋊D3—124no—
203C2×C3⋊D3—121no—
204C2×C3⋊D3—121no—
205C2×C3⋊D3—121no—
206C2×C3⋊D3—124no—
207C2×C3⋊D3—124no—
208C2×C3⋊D3—244no—
209C2×C3⋊D3—244no—
210C2×C3×D3—64no—
211C2×C3×D3—64no—
212C2×C3×D3—64no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C2×C3×D3—124no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—41no—
219C32×C4—44no—
220C32×C4—84no—
221C32×C4—121no—
222C32×C4C2×D1214yes—
223C32×C4D3×D411yescharacteristic
224C32×C4#23711yescharacteristic
225C32×C4#23914yes—
226C3⋊Dic3—21no—
227C3×Dic3—44no—
228C3×Dic3—64no—
229C2×D8—271no—
230C2×M16—271no—
231#121—91no—
232#121—271no—
233#106—31no—
234#102—271no—
235#102—271no—
236C33#46911yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C23×C3—181no—
239C22×D3—34no—
240C22×D3—64no—
241C22×D3—184no—
242C22×D3—181no—
243C22×D3—364no—
244C3×D4—181no—
245C3×D4—181no—
246C3×D4—362no—
247C2×C3×C4—24no—
248C2×C3×C4—21no—
249C2×C3×C4—24no—
250C2×C3×C4—44no—
251C2×C3×C4—64no—
252C2×C3×C4—91no—
253C2×C3×C4—181no—
254C2×C3×C4—181no—
255C2×C3×C4C2×D3214yes—
256C2×C3×C4C22×C3⋊D311yescharacteristic
257#67—362no—
258C2×Dic3—21no—
259C2×Dic3—34no—
260D12—64no—
261D12—64no—
262D12—128no—
263D12—188no—
264D12—184no—
265D12—184no—
266D12—181no—
267D12—181no—
268D12—364no—
269D12—368no—
270C4×D3—64no—
271C4×D3—64no—
272C3×C8—38no—
273C3⋊C8—184no—
274C3⋊C8—181no—
275C3⋊C8—364no—
276C3⋊C8C4⋊(C3⋊D3)12yescomplemented
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32#89514yes—
281C2×C32#91414yes—
282C2×C32#91414yes—
283C2×C32#93211yescharacteristic
284C2×C32#93911yescharacteristic
285C2×C32#94911yescharacteristic
286C3⋊D3—41no—
287C3⋊D3—81no—
288C3⋊D3—248no—
289C3⋊D3—244no—
290C3⋊D3—241no—
291C3×D3—124no—
292C3×D3—244no—
293C2×D4—91no—
294C2×D4—181no—
295C2×D4—271no—
296C2×D4—541no—
297C22×C4—91no—
298D8—542no—
299M16—542no—
300C2×C8—31no—
301C2×C8—271no—
302C4⋊C4—31no—
303C4⋊C4D3×C3⋊D311yescharacteristic
304C22⋊C4—181no—
305C22⋊C4—541no—
306C22×C3—24no—
307C22×C3—181no—
308C22×C3—181no—
309C22×C3—181no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3—361no—
313C22×C3D3×#6714yes—
314C22×C3D4×C3⋊D311yescharacteristic
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—64no—
318C2×D3—124no—
319C2×D3—124no—
320C2×D3—124no—
321C2×D3—368no—
322C2×D3—361no—
323C2×D3—361no—
324C2×D3—361no—
325C2×D3—364no—
326C2×D3—364no—
327C2×D3—364no—
328C2×D3—724no—
329C2×D3—724no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—41no—
333C3×C4—44no—
334C3×C4—84no—
335C3×C4—124no—
336C3×C4—181no—
337C3×C4#379614yes—
338C3×C4D3×D1214yes—
339C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
340C3×C4#383211yescharacteristic
341Dic3—41no—
342Dic3—64no—
343C32—24no—
344C32#486514yescomplemented
345C32#503911yescharacteristic, complemented
346C23—91no—
347C23—181no—
348C23—541no—
349D4—181no—
350D4—181no—
351D4—362no—
352D4—541no—
353D4—541no—
354D4—1082no—
355C2×C4—21no—
356C2×C4—61no—
357C2×C4—91no—
358C2×C4—181no—
359C2×C4—181no—
360C2×C4C2×D3×C3⋊D311yescharacteristic
361C8—32no—
362C8—541no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—361no—
367C2×C3—721no—
368C2×C3#6388614yes—
369C2×C3#6393414yes—
370C2×C3#6393914yes—
371C2×C3#6410411yescharacteristic
372C2×C3#6412311yescharacteristic
373C2×C3#6412311yescharacteristic
374D3—124no—
375D3—244no—
376D3—728no—
377D3—721no—
378D3—724no—
379C22—181no—
380C22—181no—
381C22—181no—
382C22—361no—
383C22—361no—
384C22—361no—
385C22—1081no—
386C22—1081no—
387C22—1081no—
388C22C3⋊D3×#6711yescharacteristic
389C4—41no—
390C4—121no—
391C4—181no—
392C4#8901711yescharacteristic
393C4D3×C4⋊(C3⋊D3)11yescharacteristic
394C3—24no—
395C3#1059058614yescomplemented
396C3#1059256411yescharacteristic, complemented
397C2—361no—
398C2—721no—
399C2—2161no—
400C2#1172348211yescharacteristic
401C2#1172356211yescharacteristic
402C2#1172357011yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#469
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64104C2semidirect
2. #11723526C2semidirect
3. C2×#88791C2semidirect
4. C2×#88787C2semidirect
5. #11722179C2semidirect
6. C32×#808C2semidirect
7. #11722033C2semidirect
26. #64104D3semidirect
36. C3×#808D3semidirect
54. C32×C3⋊C8D4semidirect
115. #808C3⋊D3semidirect
155. C3×C3⋊C8D12semidirect
236. C33#469semidirect
276. C3⋊C8C4⋊(C3⋊D3)semidirect
344. C32#4865semidirect
345. C32#5039semidirect
395. C3#10590586semidirect
396. C3#10592564semidirect

Maximal subgroups

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

subgroupindex
C3×#641042
#117235262
C2×#887912
C2×#887872
#117221792
C32×#8082
#117220332
#105925643
#105905863