Finite Group Databasebeta

← #13156234

Subgroups — #13156234

Order 1728.

606 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#11723349C211yesmaximal, characteristic, complemented
2C3×#63823C212yesmaximal, complemented
3C2×C3×#3713C211yesmaximal, characteristic, complemented
4#11722081C211yesmaximal, characteristic, complemented
5#11722025C212yesmaximal, complemented
6#10591678—31nomaximal
7#10591505—32nomaximal
8C32×#235C2211yescharacteristic
9C2×C32×D12C2212yes—
10#88855C2211yescharacteristic
11C3×#3748C2212yes—
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13#88786—21no—
14D3×C3⋊(C3⋊C8)—21no—
15C3×#3715—22no—
16C3×#3713—22no—
17C3×#3713—21no—
18C3×#3713—21no—
19#64157—31no—
20C3×#949—32no—
21C3×#932—32no—
22C2×C3×#213—32no—
23#63823D312yescomplemented
24C2×#3713D311yescharacteristic, complemented
25#63814—32no—
26#63814—61no—
27#63660—32no—
28#63578—31no—
29#63577—32no—
30#63558—32no—
31#63558—32no—
32#63558—62no—
33C22×C32×D3—22no—
34C2×C33×C4C2311yescharacteristic
35C32×#67—21no—
36C32×D12—22no—
37C32×D12—22no—
38C32×D12D411yescharacteristic
39C32×C4×D3—21no—
40C32×Dic6D411yescharacteristic
41C3⋊(C3⋊(C3⋊C8))—21no—
42C3×C3⋊(C3⋊C8)—22no—
43C3×C3⋊(C3⋊C8)D411yescharacteristic
44C3×C3⋊(C3⋊C8)D411yescharacteristic
45#5199—92no—
46#5160—91no—
47C32×#41—31no—
48C2×C32×D4—32no—
49C3×#235—21no—
50C3×#235C2×D312yes—
51C2×C3×D12—22no—
52C2×C3×D12C2×D312yes—
53C2×C3×D12C2×D312yes—
54#3748—31no—
55#3748—32no—
56#3748—61no—
57#3748—61no—
58#3748—62no—
59#3748—121no—
60#3748C2×D312yes—
61C2×C3⋊(C3⋊C8)—61no—
62C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
63C3×#215—62no—
64C3×#214—62no—
65C3×#213—62no—
66C3×#213—62no—
67C3×#213—62no—
68C3×#213—62no—
69C3×#208—32no—
70C3×#208—32no—
71C3×#208—62no—
72C2×C3×C3⋊C8—32no—
73C2×C3×C3⋊C8—61no—
74#3715—22no—
75#3713—22no—
76#3713—21no—
77#3713—21no—
78#3711—62no—
79#3711—62no—
80D3×C3⋊C8—62no—
81D3×C3⋊C8—62no—
82C22×C33#4111yescharacteristic
83C2×C32×D3—21no—
84C2×C32×D3—42no—
85C2×C32×D3—82no—
86C33×C4C2×D411yescharacteristic
87C33×C4C2×D411yescharacteristic
88C32×Dic3—21no—
89C3×AGL(1,ℤ/8)—92no—
90C2×C3×D8—91no—
91#949—32no—
92#948—92no—
93#948—181no—
94#932—32no—
95C2×#213—32no—
96#901—91no—
97#833—92no—
98#833—92no—
99#833—182no—
100#823—92no—
101#822—92no—
102#819—91no—
103C23×C32—62no—
104C22×C3×D3—24no—
105C22×C3×D3—22no—
106C22×C3×D3—22no—
107C32×Q8—31no—
108C32×D4—31no—
109C32×D4—61no—
110C32×D4—62no—
111C32×D4—62no—
112C2×C32×C4—21no—
113C2×C32×C4—22no—
114C2×C32×C4—41no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D312yes—
117C2×C32×C4C22×D311yescharacteristic
118C3×#67—22no—
119C3×#67—22no—
120C3×D12—22no—
121C3×D12—22no—
122C3×D12—22no—
123C3×D12—22no—
124C3×D12—21no—
125C3×D12—42no—
126C3×D12—42no—
127C3×D12#6712yes—
128C3×C4×D3—22no—
129C3×C4×D3—22no—
130C3×Dic6—21no—
131C3×Dic6#6712yes—
132C3⋊(C3⋊C8)—22no—
133C3⋊(C3⋊C8)—61no—
134C3⋊(C3⋊C8)—62no—
135C3⋊(C3⋊C8)—62no—
136C3⋊(C3⋊C8)—122no—
137C3⋊(C3⋊C8)—122no—
138C3⋊(C3⋊C8)#6711yescharacteristic
139C3⋊(C3⋊C8)#6711yescharacteristic
140C3×C3⋊C8—32no—
141C3×C3⋊C8—32no—
142C3×C3⋊C8—61no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—62no—
146C3×C3⋊C8—122no—
147#472—271noSylow
148C2×C33—21no—
149C2×C33#12111yescharacteristic
150C32×D3—41no—
151C32×D3—82no—
152C3×#41—32no—
153C3×#41—61no—
154C2×C3×D4—32no—
155C2×C3×D4—32no—
156C2×C3×D4—62no—
157#235D3211yescharacteristic
158C2×D12D3212yes—
159C3×SD16—182no—
160C3×D8—181no—
161C3×D8—181no—
162C3×D8—182no—
163C3×M16—92no—
164C2×C3×C8—91no—
165#215—62no—
166#214—62no—
167#213—62no—
168#213—62no—
169#213—62no—
170#213—62no—
171#208—32no—
172#208—32no—
173#208—62no—
174#208—92no—
175#208—91no—
176#208—181no—
177#208—182no—
178#208—182no—
179#208—182no—
180#208—361no—
181C2×C3⋊C8—32no—
182C2×C3⋊C8—61no—
183C2×C3⋊C8—182no—
184C2×C3⋊C8—182no—
185ΓL(1,𝔽25)—181no—
186D3×C8—181no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—41no—
190C22×C32—61no—
191C22×C32—122no—
192C22×C32—242no—
193C22×C32#23711yescharacteristic
194C22×C32#23712yes—
195C2×C3×D3—22no—
196C2×C3×D3—22no—
197C2×C3×D3—44no—
198C2×C3×D3—42no—
199C2×C3×D3—42no—
200C2×C3×D3—82no—
201C2×C3×D3—82no—
202C2×C3×D3—84no—
203C32×C4—21no—
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—21no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4—61no—
210C32×C4D3×D411yescharacteristic
211C32×C4D3×D412yes—
212C32×C4C2×#6712yes—
213C32×C4C2×#6711yescharacteristic
214C3×Dic3—22no—
215C3×Dic3—22no—
216AGL(1,ℤ/8)—92no—
217C2×D8—91no—
218#131—271no—
219#108—271no—
220#100—272no—
221C33#47211yescharacteristic, complemented, Sylow
222C23×C3—64no—
223C23×C3—62no—
224C23×C3—62no—
225C22×D3—22no—
226C3×Q8—32no—
227C3×Q8—61no—
228C3×D4—32no—
229C3×D4—61no—
230C3×D4—62no—
231C3×D4—62no—
232C3×D4—62no—
233C3×D4—62no—
234C3×D4—62no—
235C3×D4—62no—
236C3×D4—122no—
237C3×D4—122no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—41no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4C2×D3211yescharacteristic
244C2×C3×C4C2×D3212yes—
245#67—21no—
246D12—22no—
247D12—22no—
248D12#62511yescharacteristic
249C4×D3—21no—
250Dic6#62511yescharacteristic
251C3×C8—91no—
252C3×C8—91no—
253C3×C8—182no—
254C3⋊C8—32no—
255C3⋊C8—32no—
256C3⋊C8—61no—
257C3⋊C8—61no—
258C3⋊C8—62no—
259C3⋊C8—62no—
260C3⋊C8—122no—
261C3⋊C8—182no—
262C3⋊C8—182no—
263C3⋊C8—181no—
264C3⋊C8—362no—
265C3⋊C8—362no—
266C2×C32—21no—
267C2×C32—22no—
268C2×C32—22no—
269C2×C32—21no—
270C2×C32—41no—
271C2×C32—42no—
272C2×C32—41no—
273C2×C32—81no—
274C2×C32—121no—
275C2×C32—242no—
276C2×C32#93811yescharacteristic
277C2×C32#93912yes—
278C3×D3—42no—
279C3×D3—42no—
280C3×D3—84no—
281C3×D3—82no—
282C3×D3—82no—
283#41—31no—
284C2×D4—32no—
285SD16—182no—
286D8—181no—
287D8—181no—
288D8—182no—
289M16—92no—
290M16—271no—
291M16—541no—
292C2×C8—91no—
293C2×C8—541no—
294C22×C3—21no—
295C22×C3—22no—
296C22×C3—41no—
297C22×C3—62no—
298C22×C3—62no—
299C22×C3—124no—
300C22×C3—122no—
301C22×C3—122no—
302C22×C3—242no—
303C22×C3—242no—
304C22×C3—244no—
305C22×C3#380512yes—
306C22×C3#380511yescharacteristic
307C2×D3—21no—
308C2×D3—42no—
309C2×D3—82no—
310C3×C4—22no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—22no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—62no—
317C3×C4—62no—
318C3×C4C2×#62511yescharacteristic
319C3×C4D3×#6712yes—
320C3×C4D3×#6712yes—
321C3×C4#381111yescharacteristic
322Dic3—21no—
323C32—22no—
324C32—21no—
325C32—41no—
326C32#516011yescharacteristic, complemented
327C32#519912yescomplemented
328C23—62no—
329Q8—31no—
330D4—31no—
331D4—61no—
332D4—62no—
333D4—62no—
334C2×C4—61no—
335C2×C4D3311yescharacteristic
336C8—91no—
337C8—91no—
338C8—182no—
339C8—541no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—42no—
345C2×C3—41no—
346C2×C3—41no—
347C2×C3—81no—
348C2×C3—122no—
349C2×C3—122no—
350C2×C3—244no—
351C2×C3—242no—
352C2×C3—242no—
353C2×C3#6397012yes—
354C2×C3#6398011yescharacteristic
355D3—41no—
356D3—82no—
357C22—61no—
358C22—122no—
359C22—242no—
360C22#8896311yescharacteristic
361C4—61no—
362C4D3×#62511yescharacteristic
363C4#8895211yescharacteristic
364C3—21no—
365C3—22no—
366C3—41no—
367C3#1059150512yescomplemented
368C3#1059167811yescharacteristic, complemented
369C2—21no—
370C2—121no—
371C2—242no—
372C2#1172295311yescharacteristic
subgroupG/H
centerC2#11722953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC2×C33#121
hypocenterC33#472
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723349C2semidirect
2. C3×#63823C2semidirect
3. C2×C3×#3713C2semidirect
4. #11722081C2semidirect
5. #11722025C2semidirect
23. #63823D3semidirect
24. C2×#3713D3semidirect
221. C33#472semidirect
326. C32#5160semidirect
327. C32#5199semidirect
367. C3#10591505semidirect
368. C3#10591678semidirect

Maximal subgroups

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

subgroupindex
#117233492
C3×#638232
C2×C3×#37132
#117220812
#117220252
#105916783
#105915053