Finite Group Databasebeta

← #13166003

Subgroups — #13166003

Order 1728.

556 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic, complemented
2C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
3C3×A4×C3⋊C8C211yesmaximal, characteristic, complemented
4#10593636—31nomaximal
5#10593179—33nomaximal
6#10591164—31nomaximal
7C2×C3⋊(C3⋊S4)—21no—
8C2×C3×C3⋊S4—21no—
9C32×C4×A4C2211yescharacteristic
10#88791—41nomaximal
11C4⋊(C3⋊S4)—31no—
12C4⋊(C3⋊S4)—33no—
13C4⋊(C3⋊S4)—61no—
14C4⋊(C3⋊S4)—63no—
15C4⋊(C3⋊S4)D311yescharacteristic, complemented
16C3×C4⋊S4—33no—
17#64139—31no—
18C3×#930—31no—
19C22×C3×C3⋊C8D311yescharacteristic, complemented
20C3×C8×A4—31no—
21C2×#3714—31no—
22A4×C3⋊C8D313yescomplemented
23#63575—31no—
24#63564—31no—
25#63563—31no—
26C2×C32×A4D411yescharacteristic
27C3⋊(C3⋊S4)—41no—
28C3×C3⋊S4—41no—
29C4⋊(C3⋊(C3⋊D3))—41no—
30C3×C4⋊(C3⋊D3)—41no—
31C32×C3⋊C8—41no—
32#5402—93no—
33#5115—91no—
34#5038—91no—
35C2×C3⋊S4—21no—
36C2×C3⋊S4—62no—
37C2×C3⋊S4—66no—
38C2×C3⋊S4—63no—
39C2×C3⋊S4—61no—
40C2×C3×S4—63no—
41C22×C32×C4C2×D311yescharacteristic
42C2×#638—61no—
43C2×C4⋊(C3⋊D3)—31no—
44C2×C3×#67—61no—
45C2×C3×D12—31no—
46C3×C4×A4—21no—
47C3×C4×A4—23no—
48C3×C4×A4C2×D313yes—
49C3×C4×A4C2×D311yescharacteristic
50#3752—61no—
51C4⋊(C3⋊Dic3)—31no—
52#3745—121no—
53C3×(C2×D3)⋊C4—61no—
54C3×C4⋊Dic3—31no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57#3714—61no—
58#3714—61no—
59#3714—123no—
60C2×C3⋊(C3⋊D3)—81no—
61C2×C3×C3⋊D3—81no—
62C32×A4D811yescharacteristic, complemented
63C33×C4—41no—
64C4⋊S4—33no—
65C4⋊S4—93no—
66C4⋊S4—183no—
67C22×C3×C8—31no—
68C3×#121—91no—
69C2×#213—91no—
70#930—31no—
71#930—91no—
72#930—91no—
73#930—181no—
74C22×C3⋊C8C3⋊D311yescharacteristic, complemented
75C2×D24—91no—
76C8×A4—33no—
77#832—91no—
78#821—91no—
79#821—91no—
80#818—91no—
81#809—91no—
82#807—91no—
83C23×C32D1211yescharacteristic
84C22×C3⋊D3—61no—
85C22×C3×D3—61no—
86C2×C3×A4—21no—
87C2×C3×A4—23no—
88C2×C3×A4D1213yes—
89C2×C3×A4#6711yescharacteristic
90C3⋊S4—41no—
91C3⋊S4—126no—
92C3⋊S4—122no—
93C3⋊S4—123no—
94C3⋊S4—121no—
95C3×S4—123no—
96C2×C32×C4—31no—
97C2×C32×C4—31no—
98#638—121no—
99#638—121no—
100C2×C3⋊Dic3—61no—
101C4⋊(C3⋊D3)—41no—
102C4⋊(C3⋊D3)—61no—
103C4⋊(C3⋊D3)—61no—
104C4⋊(C3⋊D3)—121no—
105C4⋊(C3⋊D3)—123no—
106C4⋊(C3⋊D3)—241no—
107C4⋊(C3⋊D3)—243no—
108C3×#67—121no—
109C3×#67—121no—
110C2×C3×Dic3—61no—
111C3×D12—61no—
112C3×D12—61no—
113C3×D12—123no—
114C32×C8—121no—
115C3×C3⋊C8—31no—
116C3×C3⋊C8—43no—
117C3×C3⋊C8S411yescharacteristic, complemented
118#466—271noSylow
119C2×C33—41no—
120C3⋊(C3⋊D3)—161no—
121C3×C3⋊D3—161no—
122C2×S4—63no—
123C2×S4—186no—
124C2×S4—183no—
125C2×C3×D4—91no—
126C2×C3×D4—181no—
127C22×C3×C4—21no—
128C22×C3×C4D3211yescharacteristic
129C22×C3×C4C2×C3⋊D311yescharacteristic
130C2×#67—61no—
131C2×#67—182no—
132C2×#67—181no—
133C2×#67—181no—
134C2×D12—31no—
135C2×D12—91no—
136C2×D12—91no—
137C2×D12—181no—
138C4×A4—23no—
139C4×A4D3213yes—
140C2×C3×C8—91no—
141C2×C3×C8—91no—
142C3×C4⋊C4—91no—
143C3×C22⋊C4—181no—
144#213—181no—
145#213—181no—
146(C2×D3)⋊C4—61no—
147(C2×D3)⋊C4—181no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—182no—
150C4⋊Dic3—31no—
151C4⋊Dic3—91no—
152C4⋊Dic3—91no—
153C4⋊Dic3—181no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156D24—181no—
157D24—181no—
158D24—363no—
159C22×C32—31no—
160C22×C32—31no—
161C22×C32D2411yescharacteristic, complemented
162C2×C3⋊D3—81no—
163C2×C3⋊D3—121no—
164C2×C3⋊D3—121no—
165C2×C3⋊D3—121no—
166C2×C3⋊D3—243no—
167C2×C3⋊D3—241no—
168C2×C3⋊D3—246no—
169C2×C3⋊D3—242no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—243no—
174C3×A4—21no—
175C3×A4—23no—
176C3×A4D2413yescomplemented
177C3×A4#21311yescharacteristic, complemented
178C32×C4—31no—
179C32×C4—41no—
180C32×C4—43no—
181C32×C4—83no—
182C32×C4—81no—
183C32×C4C2×S411yescharacteristic
184C3⋊Dic3—121no—
185C3×Dic3—121no—
186C2×D8—271no—
187C22×C8—31no—
188#121—91no—
189#121—271no—
190#107—271no—
191#102—271no—
192#102—271no—
193C33—41noSylow
194C23×C3—21no—
195C23×C3—181no—
196C23×C3#62611yescharacteristic
197C23×C3C4⋊(C3⋊D3)11yescharacteristic
198C22×D3—61no—
199C22×D3—181no—
200C22×D3—181no—
201C22×D3—182no—
202C2×A4—23no—
203C2×A4#62613yes—
204S4—123no—
205S4—366no—
206S4—363no—
207C3×D4—181no—
208C3×D4—181no—
209C3×D4—361no—
210C3×D4—361no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—61no—
216C2×C3×C4—61no—
217C2×C3×C4—181no—
218#67—121no—
219#67—121no—
220#67—362no—
221#67—361no—
222#67—361no—
223#67—362no—
224#67—361no—
225#67—361no—
226C2×Dic3—61no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C2×Dic3—182no—
230D12—61no—
231D12—61no—
232D12—123no—
233D12—181no—
234D12—181no—
235D12—181no—
236D12—181no—
237D12—361no—
238D12—363no—
239D12—361no—
240D12—723no—
241C3×C8—31no—
242C3×C8—91no—
243C3×C8—123no—
244C3⋊C8—31no—
245C3⋊C8C3⋊S411yescharacteristic, complemented
246C2×C32—31no—
247C2×C32—31no—
248C2×C32—41no—
249C2×C32—43no—
250C2×C32—83no—
251C2×C32—81no—
252C2×C32C4⋊S411yescharacteristic
253C3⋊D3—161no—
254C3⋊D3—241no—
255C3⋊D3—483no—
256C3⋊D3—481no—
257C3⋊D3—486no—
258C3⋊D3—482no—
259C3×D3—241no—
260C3×D3—483no—
261C2×D4—91no—
262C2×D4—181no—
263C2×D4—271no—
264C2×D4—541no—
265C22×C4D3×C3⋊D311yescharacteristic
266D8—541no—
267D8—541no—
268C2×C8—91no—
269C2×C8—91no—
270C4⋊C4—91no—
271C4⋊C4—271no—
272C22⋊C4—181no—
273C22⋊C4—541no—
274C22×C3—21no—
275C22×C3—31no—
276C22×C3—31no—
277C22×C3—31no—
278C22×C3—31no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3#371411yescharacteristic, complemented
285C22×C3#374511yescharacteristic, complemented
286C2×D3—121no—
287C2×D3—121no—
288C2×D3—121no—
289C2×D3—243no—
290C2×D3—362no—
291C2×D3—361no—
292C2×D3—361no—
293C2×D3—361no—
294C2×D3—361no—
295C2×D3—362no—
296C2×D3—362no—
297C2×D3—361no—
298C2×D3—361no—
299C2×D3—723no—
300C2×D3—726no—
301A4—23no—
302A4#371413yescomplemented
303C3×C4—21no—
304C3×C4—31no—
305C3×C4—31no—
306C3×C4—43no—
307C3×C4—61no—
308C3×C4—83no—
309C3×C4—361no—
310C3×C4D3×S411yescharacteristic
311C3×C4C2×C3⋊S411yescharacteristic
312Dic3—121no—
313Dic3—361no—
314Dic3—361no—
315Dic3—362no—
316C32—41no—
317C32—43no—
318C32—83no—
319C32—81no—
320C32#540211yescharacteristic, complemented
321C23—181no—
322C23—541no—
323C23#629211yescharacteristic
324D4—181no—
325D4—181no—
326D4—361no—
327D4—361no—
328D4—541no—
329D4—541no—
330D4—1081no—
331D4—1081no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—181no—
335C2×C4—541no—
336C8—31no—
337C8—91no—
338C2×C3—21no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—43no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—83no—
347C2×C3—721no—
348C2×C3#6420711yescharacteristic
349C2×C3C4⋊(C3⋊S4)11yescharacteristic
350D3—241no—
351D3—483no—
352D3—722no—
353D3—721no—
354D3—721no—
355D3—1443no—
356D3—1446no—
357C22—31no—
358C22—31no—
359C22—361no—
360C22—361no—
361C22—361no—
362C22—1081no—
363C22—1081no—
364C22—1081no—
365C22#8879111yescharacteristic, complemented
366C4—31no—
367C4—361no—
368C4—1081no—
369C4D3×C3⋊S411yescharacteristic
370C3—21no—
371C3—43no—
372C3—83no—
373C3#1059317911yescharacteristic, complemented
374C3#1059363611yescharacteristic, complemented
375C2—31no—
376C2—31no—
377C2—721no—
378C2—2161no—
379C2#1172420211yescharacteristic
subgroupG/H
centerC2#11724202
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32D12
hypocenterC32×A4D8
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4⋊(C3⋊(C3⋊S4))C2semidirect
2. C3×C4⋊(C3⋊S4)C2semidirect
3. C3×A4×C3⋊C8C2semidirect
15. C4⋊(C3⋊S4)D3semidirect
19. C22×C3×C3⋊C8D3semidirect
22. A4×C3⋊C8D3semidirect
62. C32×A4D8semidirect
74. C22×C3⋊C8C3⋊D3semidirect
117. C3×C3⋊C8S4semidirect
161. C22×C32D24semidirect
176. C3×A4D24semidirect
177. C3×A4#213semidirect
245. C3⋊C8C3⋊S4semidirect
284. C22×C3#3714semidirect
285. C22×C3#3745semidirect
302. A4#3714semidirect
320. C32#5402semidirect
365. C22#88791semidirect
373. C3#10593179semidirect
374. C3#10593636semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊(C3⋊S4))2
C3×C4⋊(C3⋊S4)2
C3×A4×C3⋊C82
#105936363
#105931793
#105911643
#887914