Finite Group Databasebeta

← #13161043

Subgroups — #13161043

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
1C2×C3×#3753C211yesmaximal, characteristic, complemented
2C32×#938C211yesmaximal, characteristic, complemented
3#11723515C211yesmaximal, characteristic, complemented
4C2×#88786C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722054C211yesmaximal, characteristic, complemented
7#11722033C211yesmaximal, characteristic, complemented
8#10592738—31nomaximal
9#10591451—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×#67—21no—
13C2×C32×C4×D3C2211yescharacteristic
14C2×#6290—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3753—22no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×#217—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21#88803—21no—
22#88786—22no—
23C32×#121—31no—
24C2×#3753D311yescharacteristic, complemented
25#64139—31no—
26C2×#3748—31no—
27C3×#938D314yescomplemented
28C2×C3×#213—34no—
29#63908—34no—
30C2×#3711—34no—
31#63659—31no—
32#63659—34no—
33#63659—64no—
34#63636—31no—
35#63635—31no—
36#63573—34no—
37#63562—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C22×C32×D3D411yescharacteristic
41C33×D4—21no—
42C33×D4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)—21no—
47C32×#67—42no—
48C2×C32×Dic3—21no—
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3—21no—
51C32×C4×D3—21no—
52#6290—42no—
53C3⋊(C3⋊(C3⋊C8))—21no—
54C3×C3⋊(C3⋊C8)D412yescomplemented
55#5159—91no—
56#5039—94no—
57C2×C32×D4—24no—
58C2×C32×D4—61no—
59C2×C32×D4C2×D314yes—
60C2×C32×D4C2×D311yescharacteristic
61C22×C32×C4—31no—
62C2×#638—61no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C3×#67—24no—
65C2×C3×D12—34no—
66C2×C3×C4×D3C2×D314yes—
67C2×#625—64no—
68C32×C4⋊C4—31no—
69C32×C22⋊C4—61no—
70#3753—22no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—64no—
75#3748—62no—
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—34no—
78C2×C3⋊(C3⋊C8)—64no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×#217—24no—
81C3×#213—68no—
82C3×C4⋊Dic3C2×D314yes—
83C2×C3×C3⋊C8—34no—
84(C2×D3)⋊Dic3—64no—
85#3711—68no—
86C22×C33—41no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33C2×D411yescharacteristic
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C32×D3—21no—
94C2×C32×D3—21no—
95C2×C32×D3—21no—
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C3⋊(C3⋊Dic3)—41no—
99C32×Dic3—21no—
100C32×Dic3—41no—
101C2×C3×D8—91no—
102C3×#121—34no—
103#938—91no—
104#938C3⋊D311yescharacteristic, complemented
105C2×#213—34no—
106#930—94no—
107C2×#208—94no—
108C2×ΓL(1,𝔽25)—91no—
109#832—94no—
110#832—91no—
111#832—91no—
112#832—184no—
113#817—91no—
114#809—94no—
115#808—94no—
116C23×C32—21no—
117C23×C32—24no—
118C23×C32—31no—
119C23×C32—44no—
120C22×C3⋊D3—21no—
121C22×C3×D3—64no—
122C22×C3×D3#6714yes—
123C32×D4—28no—
124C32×D4—24no—
125C32×D4—21no—
126C32×D4—21no—
127C32×D4—24no—
128C32×D4—44no—
129C32×D4—122no—
130C2×C32×C4—24no—
131C2×C32×C4—31no—
132C2×C32×C4—61no—
133C2×C32×C4—61no—
134C2×C32×C4—61no—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C22×D311yescharacteristic
137#638—122no—
138C2×C3⋊Dic3—61no—
139C2×C3⋊Dic3—64no—
140C2×C3⋊Dic3—124no—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)—21no—
143C3×#67—48no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3#6714yes—
146C3×D12—64no—
147C3×D12—64no—
148C3×C4×D3—24no—
149C3×C4×D3—24no—
150#625—128no—
151C3⋊(C3⋊C8)—61no—
152C3⋊(C3⋊C8)—64no—
153C3⋊(C3⋊C8)—124no—
154C3⋊(C3⋊C8)#6712yescomplemented
155C3×C3⋊C8—38no—
156#469—271noSylow
157C2×C33—81no—
158C2×C33#12111yescharacteristic
159C2×C33AGL(1,ℤ/8)11yescharacteristic
160C2×C33AGL(1,ℤ/8)11yescharacteristic
161C3×C3⋊D3—81no—
162C32×D3—41no—
163C2×C3×D4—24no—
164C2×C3×D4—64no—
165C2×C3×D4—91no—
166C2×C3×D4D3214yes—
167C2×C3×D4C2×C3⋊D311yescharacteristic
168C22×C3×C4—34no—
169C2×#67—21no—
170C2×#67—184no—
171C2×#67—181no—
172C2×D12—34no—
173C2×C4×D3C2×C3⋊D311yescharacteristic
174C3×D8—182no—
175C2×C3×C8—91no—
176C3×C4⋊C4—34no—
177C3×C22⋊C4—64no—
178#217—21no—
179#217—181no—
180#213—68no—
181(C2×D3)⋊C4—184no—
182C4⋊Dic3—94no—
183C4⋊Dic3—91no—
184C4⋊Dic3—184no—
185C4⋊Dic3C2×C3⋊D311yescharacteristic
186#208—188no—
187C2×C3⋊C8—34no—
188C2×C3⋊C8—94no—
189C2×C3⋊C8—91no—
190C2×C3⋊C8—184no—
191ΓL(1,𝔽25)—182no—
192C22×C32—24no—
193C22×C32—48no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—44no—
198C22×C32—44no—
199C22×C32—44no—
200C22×C32—61no—
201C22×C32—61no—
202C22×C32—61no—
203C22×C32—84no—
204C22×C32—84no—
205C22×C32D3×D411yescharacteristic
206C22×C32C2×#6714yes—
207C2×C3⋊D3—41no—
208C2×C3⋊D3—41no—
209C2×C3⋊D3—41no—
210C2×C3×D3—24no—
211C2×C3×D3—24no—
212C2×C3×D3—24no—
213C2×C3×D3—124no—
214C2×C3×D3—124no—
215C2×C3×D3—124no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—61no—
219C32×C4—121no—
220C32×C4D3×D414yes—
221C32×C4#23714yes—
222C32×C4#23711yescharacteristic
223C32×C4C2×#6711yescharacteristic
224C3⋊Dic3—121no—
225C3⋊Dic3—124no—
226C3⋊Dic3—244no—
227C3×Dic3—24no—
228C3×Dic3—44no—
229C2×D8—91no—
230C2×M16—271no—
231#121—31no—
232#121—271no—
233#106—271no—
234#102—271no—
235#102—271no—
236C33#46911yescharacteristic, complemented, Sylow
237C23×C3—24no—
238C23×C3—21no—
239C23×C3—34no—
240C23×C3—44no—
241C23×C3—181no—
242C22×D3—64no—
243C22×D3#63811yescharacteristic
244C3×D4—28no—
245C3×D4—24no—
246C3×D4—24no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—44no—
250C3×D4—128no—
251C3×D4—181no—
252C3×D4—181no—
253C2×C3×C4—24no—
254C2×C3×C4—34no—
255C2×C3×C4—64no—
256C2×C3×C4—64no—
257C2×C3×C4—64no—
258C2×C3×C4C2×D3214yes—
259C2×C3×C4C22×C3⋊D311yescharacteristic
260#67—42no—
261#67—362no—
262#67—368no—
263C2×Dic3—21no—
264C2×Dic3—184no—
265C2×Dic3—181no—
266C2×Dic3—364no—
267C2×Dic3#63811yescharacteristic
268D12—64no—
269D12—64no—
270C4×D3—21no—
271C4×D3—21no—
272C3×C8—92no—
273C3⋊C8—38no—
274C3⋊C8—184no—
275C3⋊C8—181no—
276C3⋊C8—364no—
277C2×C32—24no—
278C2×C32—24no—
279C2×C32—24no—
280C2×C32—88no—
281C2×C32—81no—
282C2×C32—84no—
283C2×C32—121no—
284C2×C32#91111yescharacteristic
285C2×C32#91111yescharacteristic
286C2×C32#93214yes—
287C2×C32#93811yescharacteristic
288C2×C32#93914yes—
289C2×C32#94914yes—
290C3⋊D3—81no—
291C3×D3—44no—
292C3×D3—244no—
293C2×D4—61no—
294C2×D4—91no—
295C2×D4—541no—
296C2×D4D3×C3⋊D311yescharacteristic
297C22×C4—31no—
298D8—182no—
299M16—542no—
300C2×C8—91no—
301C2×C8—271no—
302C4⋊C4—31no—
303C4⋊C4—271no—
304C22⋊C4—61no—
305C22⋊C4—541no—
306C22×C3—24no—
307C22×C3—48no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—64no—
315C22×C3—64no—
316C22×C3—64no—
317C22×C3—84no—
318C22×C3—84no—
319C22×C3—361no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3D3×#6714yes—
323C22×C3C2×#63811yescharacteristic
324C2×D3—21no—
325C2×D3—21no—
326C2×D3—21no—
327C2×D3—124no—
328C2×D3—124no—
329C2×D3—124no—
330C3×C4—24no—
331C3×C4—24no—
332C3×C4—64no—
333C3×C4—124no—
334C3×C4#380514yes—
335C3×C4D3×#6714yes—
336C3×C4D4×C3⋊D311yescharacteristic
337C3×C4#383011yescharacteristic
338Dic3—21no—
339Dic3—41no—
340Dic3—361no—
341Dic3—364no—
342Dic3—724no—
343C32—24no—
344C32#503914yescomplemented
345C32#515911yescharacteristic, complemented
346C23—21no—
347C23—31no—
348C23—181no—
349D4—21no—
350D4—21no—
351D4—122no—
352D4—181no—
353D4—181no—
354D4—1082no—
355C2×C4—31no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—541no—
360C2×C4C2×D3×C3⋊D311yescharacteristic
361C8—92no—
362C8—541no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—88no—
367C2×C3—84no—
368C2×C3—81no—
369C2×C3—124no—
370C2×C3—721no—
371C2×C3#6392514yes—
372C2×C3#6393214yes—
373C2×C3#6397014yes—
374C2×C3#6414111yescharacteristic
375C2×C3#6414811yescharacteristic
376C2×C3#6415811yescharacteristic
377D3—41no—
378D3—244no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—361no—
386C22—361no—
387C22—361no—
388C22D3×#63811yescharacteristic
389C4—61no—
390C4—121no—
391C4—1081no—
392C4#8902911yescharacteristic
393C4C3⋊D3×#6711yescharacteristic
394C3—24no—
395C3#1059145114yescomplemented
396C3#1059273811yescharacteristic, complemented
397C2—81no—
398C2—121no—
399C2—721no—
400C2#1172354411yescharacteristic
401C2#1172355511yescharacteristic
402C2#1172361511yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#469
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3753C2semidirect
2. C32×#938C2semidirect
3. #11723515C2semidirect
4. C2×#88786C2semidirect
5. #11722202C2semidirect
6. #11722054C2semidirect
7. #11722033C2semidirect
24. C2×#3753D3semidirect
27. C3×#938D3semidirect
54. C3×C3⋊(C3⋊C8)D4semidirect
104. #938C3⋊D3semidirect
154. C3⋊(C3⋊C8)#67semidirect
236. C33#469semidirect
344. C32#5039semidirect
345. C32#5159semidirect
395. C3#10591451semidirect
396. C3#10592738semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×#9382
#117235152
C2×#887862
#117222022
#117220542
#117220332
#105927383
#105914513