Finite Group Databasebeta

← #13159160

Subgroups — #13159160

Order 1728.

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

normal subgroups only

This group has 924 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
1C3×#64104C211yesmaximal, characteristic, complemented
2#11723522C211yesmaximal, characteristic, complemented
3C2×#88789C211yesmaximal, characteristic, complemented
4C2×#88787C211yesmaximal, characteristic, complemented
5#11722178C211yesmaximal, characteristic, complemented
6C32×#809C211yesmaximal, characteristic, complemented
7#11722034C211yesmaximal, characteristic, complemented
8#10592481—31nomaximal
9#10590504—34nomaximal
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C33×C4⋊C4C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×#3752—21no—
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88789—22no—
22#88787—22no—
23#64147—31no—
24C2×#3753—31no—
25C2×#3743—31no—
26#64104D311yescharacteristic, complemented
27C3×#895—34no—
28#63908—34no—
29C2×#3713—34no—
30C2×#3711—34no—
31C32×#102—31no—
32#63648—31no—
33#63635—31no—
34#63635—34no—
35#63635—64no—
36C3×#809D314yescomplemented
37#63563—34no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3D411yescharacteristic
40C22×C32×D3—21no—
41C2×C33×C4—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C3×C4⋊(C3⋊D3)—42no—
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C32×D12—21no—
51C32×D12—21no—
52#6290—42no—
53C3⋊(C3⋊(C3⋊C8))D412yescomplemented
54C32×C3⋊C8—21no—
55#5039—91no—
56#4782—94no—
57C2×C32×D4—31no—
58C2×#638—61no—
59C2×C4⋊(C3⋊D3)—21no—
60C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×C3×D12—34no—
63C2×C3×D12—64no—
64C2×C3×D12C2×D314yes—
65C2×C3×C4×D3—34no—
66C2×#625—64no—
67C2×C32×C8—31no—
68C32×C4⋊C4—24no—
69C32×C4⋊C4C2×D314yes—
70C32×C4⋊C4C2×D311yescharacteristic
71#3757—61no—
72#3753—62no—
73#3752—21no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—34no—
76C4⋊(C3⋊Dic3)—64no—
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—34no—
79C2×C3⋊(C3⋊C8)—64no—
80#3743—62no—
81C3×(C2×D3)⋊C4—64no—
82C2×C3×C3⋊C8C2×D314yes—
83(C2×D3)⋊Dic3—64no—
84#3713—68no—
85#3711—68no—
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C32×D3—41no—
94C2×C32×D3—41no—
95C2×C32×D3—41no—
96C33×C4—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C3×C3⋊Dic3—21no—
101C3×#121—91no—
102#938—94no—
103C2×#213—94no—
104C2×#213—91no—
105#930—91no—
106C2×#208—91no—
107C2×ΓL(1,𝔽25)—94no—
108#895—34no—
109C3×#102—34no—
110#832—91no—
111#821—94no—
112#809C3⋊D311yescharacteristic, complemented
113#808—94no—
114#808—91no—
115#808—184no—
116C23×C32—61no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3#6711yescharacteristic
119C22×C3×D3—24no—
120C22×C3×D3—34no—
121C22×C3×D3—64no—
122C32×D4—61no—
123C32×D4—61no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4—24no—
127C2×C32×C4—44no—
128C2×C32×C4C22×D314yes—
129C2×C32×C4C22×D311yescharacteristic
130#638—122no—
131C2×C3⋊Dic3—61no—
132C2×C3⋊Dic3—64no—
133C2×C3⋊Dic3—124no—
134C2×C3⋊Dic3#6711yescharacteristic
135C4⋊(C3⋊D3)—21no—
136C4⋊(C3⋊D3)—21no—
137C4⋊(C3⋊D3)—42no—
138C4×C3⋊D3—21no—
139C4×C3⋊D3—21no—
140C2×C3×Dic3—34no—
141C3×D12—24no—
142C3×D12—24no—
143C3×D12—64no—
144C3×D12—64no—
145C3×D12—128no—
146C3×C4×D3—64no—
147C3×C4×D3—64no—
148#625—128no—
149C32×C8—61no—
150C3⋊(C3⋊C8)—32no—
151C3⋊(C3⋊C8)—38no—
152C3⋊(C3⋊C8)—68no—
153C3×C3⋊C8—24no—
154#469—271noSylow
155C2×C33#12111yescharacteristic
156C2×C33AGL(1,ℤ/8)11yescharacteristic
157C2×C33AGL(1,ℤ/8)11yescharacteristic
158C3×C3⋊D3—41no—
159C3×C3⋊D3—81no—
160C32×D3—81no—
161C2×C3×D4—34no—
162C2×C3×D4—91no—
163C2×C3×D4—181no—
164C22×C3×C4—91no—
165C2×#67—181no—
166C2×#67—184no—
167C2×D12—34no—
168C2×D12—64no—
169C2×D12C2×C3⋊D311yescharacteristic
170C2×C4×D3—34no—
171C2×C3×C8—34no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4D3214yes—
174C3×C4⋊C4C2×C3⋊D311yescharacteristic
175C3×C22⋊C4—181no—
176#217—184no—
177#213—182no—
178#213—188no—
179(C2×D3)⋊C4—64no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—94no—
182C4⋊Dic3—91no—
183C4⋊Dic3—184no—
184#208—182no—
185C2×C3⋊C8—94no—
186C2×C3⋊C8—91no—
187C2×C3⋊C8—184no—
188C2×C3⋊C8C2×C3⋊D311yescharacteristic
189ΓL(1,𝔽25)—188no—
190C22×C32—24no—
191C22×C32—121no—
192C22×C32—121no—
193C22×C32—121no—
194C22×C32D3×D414yes—
195C22×C32C2×#6711yescharacteristic
196C2×C3⋊D3—21no—
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3⋊D3—41no—
200C2×C3⋊D3—41no—
201C2×C3⋊D3—41no—
202C2×C3×D3—44no—
203C2×C3×D3—44no—
204C2×C3×D3—44no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207C2×C3×D3—64no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C2×C3×D3—124no—
211C32×C4—24no—
212C32×C4—24no—
213C32×C4—41no—
214C32×C4—44no—
215C32×C4—84no—
216C32×C4#23514yes—
217C32×C4D3×D414yes—
218C32×C4D3×D411yescharacteristic
219C32×C4#23711yescharacteristic
220C3⋊Dic3—21no—
221C3⋊Dic3—121no—
222C3⋊Dic3—124no—
223C3⋊Dic3—244no—
224C3×Dic3—64no—
225C2×D8—271no—
226C2×M16—271no—
227#121—91no—
228#121—271no—
229#106—271no—
230#102—31no—
231#102—271no—
232C33#46911yescharacteristic, complemented, Sylow
233C23×C3—64no—
234C23×C3—91no—
235C23×C3—181no—
236C22×D3—21no—
237C22×D3—34no—
238C22×D3—64no—
239C3×D4—64no—
240C3×D4—64no—
241C3×D4—181no—
242C3×D4—181no—
243C3×D4—362no—
244C2×C3×C4—24no—
245C2×C3×C4—21no—
246C2×C3×C4—24no—
247C2×C3×C4—44no—
248C2×C3×C4—91no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—362no—
254#67—368no—
255C2×Dic3—34no—
256C2×Dic3—184no—
257C2×Dic3—181no—
258C2×Dic3—364no—
259D12—21no—
260D12—21no—
261D12—64no—
262D12—64no—
263D12—128no—
264C4×D3—64no—
265C4×D3—64no—
266C3×C8—64no—
267C3⋊C8—21no—
268C3⋊C8—98no—
269C3⋊C8—92no—
270C3⋊C8—188no—
271C2×C32—24no—
272C2×C32—24no—
273C2×C32—24no—
274C2×C32—241no—
275C2×C32#88414yes—
276C2×C32#91114yes—
277C2×C32#91414yes—
278C2×C32#93211yescharacteristic
279C2×C32#93911yescharacteristic
280C2×C32#94911yescharacteristic
281C3⋊D3—41no—
282C3⋊D3—81no—
283C3×D3—84no—
284C3×D3—124no—
285C3×D3—244no—
286C2×D4—31no—
287C2×D4—91no—
288C2×D4—181no—
289C2×D4—541no—
290C22×C4—91no—
291D8—542no—
292M16—542no—
293C2×C8—31no—
294C2×C8—271no—
295C4⋊C4—271no—
296C4⋊C4D3×C3⋊D311yescharacteristic
297C22⋊C4—181no—
298C22⋊C4—541no—
299C22×C3—24no—
300C22×C3—124no—
301C22×C3—124no—
302C22×C3—124no—
303C22×C3—181no—
304C22×C3—181no—
305C22×C3—181no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3D3×#6714yes—
310C22×C3D4×C3⋊D311yescharacteristic
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—64no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—124no—
318C2×D3—124no—
319C2×D3—124no—
320C3×C4—24no—
321C3×C4—24no—
322C3×C4—41no—
323C3×C4—44no—
324C3×C4—84no—
325C3×C4—181no—
326C3×C4#379514yes—
327C3×C4#380214yes—
328C3×C4#382811yescharacteristic
329C3×C4D4×C3⋊D311yescharacteristic
330Dic3—64no—
331Dic3—361no—
332Dic3—364no—
333Dic3—724no—
334C32—24no—
335C32#478214yescomplemented
336C32#503911yescharacteristic, complemented
337C23—61no—
338C23—91no—
339C23—181no—
340D4—61no—
341D4—61no—
342D4—181no—
343D4—181no—
344D4—362no—
345D4—1082no—
346C2×C4—21no—
347C2×C4—91no—
348C2×C4—181no—
349C2×C4—181no—
350C2×C4—541no—
351C2×C4C2×D3×C3⋊D311yescharacteristic
352C8—61no—
353C8—272no—
354C2×C3—24no—
355C2×C3—24no—
356C2×C3—24no—
357C2×C3—244no—
358C2×C3—361no—
359C2×C3—721no—
360C2×C3#6388514yes—
361C2×C3#6393214yes—
362C2×C3#6393914yes—
363C2×C3#6409311yescharacteristic
364C2×C3#6412011yescharacteristic
365C2×C3#6412311yescharacteristic
366D3—81no—
367D3—124no—
368D3—244no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22—181no—
373C22—181no—
374C22—181no—
375C22—361no—
376C22—361no—
377C22—361no—
378C22C3⋊D3×#6711yescharacteristic
379C4—41no—
380C4—181no—
381C4—1081no—
382C4#8902111yescharacteristic
383C4#8902611yescharacteristic
384C3—24no—
385C3#1059050414yescomplemented
386C3#1059248111yescharacteristic, complemented
387C2—241no—
388C2—361no—
389C2—721no—
390C2#1172351911yescharacteristic
391C2#1172356011yescharacteristic
392C2#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. #11723522C2semidirect
3. C2×#88789C2semidirect
4. C2×#88787C2semidirect
5. #11722178C2semidirect
6. C32×#809C2semidirect
7. #11722034C2semidirect
26. #64104D3semidirect
36. C3×#809D3semidirect
53. C3⋊(C3⋊(C3⋊C8))D4semidirect
112. #809C3⋊D3semidirect
232. C33#469semidirect
335. C32#4782semidirect
336. C32#5039semidirect
385. C3#10590504semidirect
386. C3#10592481semidirect

Maximal subgroups

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

subgroupindex
C3×#641042
#117235222
C2×#887892
C2×#887872
#117221782
C32×#8092
#117220342
#105924813
#105905043