Finite Group Databasebeta

← #13173063

Subgroups — #13173063

Order 1728.

1402 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1402 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C22×C3⋊(C3⋊S4)C211yesmaximal, characteristic
2C2×C32×C4×A4C211yesmaximal, characteristic, complemented
3C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4#10593647—312nomaximal
5#10592912—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×C3⋊(C3⋊S4)—21no—
8C2×C3⋊(C3⋊S4)—21no—
9C2×C3⋊(C3⋊S4)C412yescomplemented
10C32×C4×A4—21no—
11C3⋊(C3⋊SL(2,ℤ/4))—21no—
12#88859—41nomaximal
13C22×C3⋊S4—312no—
14C23×C32×C4D311yescharacteristic, complemented
15C22×#638—31no—
16C2×C3×C4×A4D3112yescomplemented
17C2×C3⋊SL(2,ℤ/4)—312no—
18C2×#3757—31no—
19C2×#3752—31no—
20C2×#3752—31no—
21#63658—31no—
22#63658—31no—
23C22×C3⋊(C3⋊D3)—41no—
24C2×C32×A4C2×C411yescharacteristic
25C2×C32×A4D411yescharacteristic
26C2×C32×A4D411yescharacteristic
27C3⋊(C3⋊S4)—22no—
28C2×C33×C4—41no—
29C2×C3⋊(C3⋊Dic3)—41no—
30(C2×S4)⋊C4—99no—
31#5213—94no—
32C24×C32C2×D311yescharacteristic
33C23×C3⋊D3—31no—
34C22×C3×A4C2×D3112yes—
35C2×C3⋊S4—324no—
36C2×C3⋊S4—612no—
37C2×C3⋊S4—612no—
38C22×C32×C4—21no—
39C22×C32×C4—31no—
40C22×C32×C4—31no—
41C22×C32×C4—61no—
42C2×#638—32no—
43C2×#638—32no—
44C2×#638—61no—
45C2×#638—61no—
46C2×#638—61no—
47C2×#638—61no—
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—31no—
51C3×C4×A4—212no—
52C3⋊SL(2,ℤ/4)—612no—
53#3757—61no—
54#3757—61no—
55#3752—62no—
56#3752—61no—
57#3752—61no—
58#3752—1212no—
59C22×C33—41no—
60C2×C3⋊(C3⋊D3)—42no—
61C2×C3⋊(C3⋊D3)—81no—
62C2×C3⋊(C3⋊D3)—81no—
63C32×A4C22⋊C411yescharacteristic, complemented
64C33×C4—81no—
65C3⋊(C3⋊Dic3)—81no—
66C22×S4—99no—
67C23×C3×C4C3⋊D314yescomplemented
68C22×#67—94no—
69C2×C4×A4C3⋊D319yescomplemented
70C2×SL(2,ℤ/4)—99no—
71C2×#217—94no—
72C2×(C2×D3)⋊C4—94no—
73C2×(C2×D3)⋊C4—94no—
74C3⋊#95—94no—
75C3⋊#95—94no—
76C23×C32—31no—
77C23×C32—31no—
78C23×C32—31no—
79C23×C32—31no—
80C23×C32C4×D311yescharacteristic
81C23×C32D1211yescharacteristic
82C23×C32#6711yescharacteristic
83C22×C3⋊D3—32no—
84C22×C3⋊D3—61no—
85C22×C3⋊D3—61no—
86C22×C3⋊D3—61no—
87C22×C3⋊D3—61no—
88C22×C3⋊D3—61no—
89C22×C3⋊D3—61no—
90C22×C3⋊D3—1212no—
91C2×C3×A4C4×D3112yes—
92C2×C3×A4D12112yes—
93C2×C3×A4#67112yes—
94C3⋊S4—624no—
95C2×C32×C4—32no—
96C2×C32×C4—31no—
97C2×C32×C4—412no—
98C2×C32×C4—61no—
99C2×C32×C4—61no—
100C2×C32×C4—61no—
101C2×C32×C4S411yescharacteristic, complemented
102#638—62no—
103#638—62no—
104#638—62no—
105#638—62no—
106C2×C3⋊Dic3—32no—
107C2×C3⋊Dic3—61no—
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—1212no—
116#386—271noSylow
117C2×C33—41no—
118C2×C33—41no—
119C2×C33—41no—
120C3⋊(C3⋊D3)—82no—
121C24×C3C2×C3⋊D314yes—
122C23×D3—94no—
123C22×A4C2×C3⋊D319yes—
124C2×S4—918no—
125C2×S4—189no—
126C2×S4—189no—
127C22×C3×C4—24no—
128C22×C3×C4—34no—
129C22×C3×C4—34no—
130C22×C3×C4—64no—
131C2×#67—98no—
132C2×#67—98no—
133C2×#67—184no—
134C2×#67—184no—
135C2×#67—184no—
136C2×#67—184no—
137C22×Dic3—94no—
138C22×Dic3—94no—
139C22×Dic3—94no—
140C4×A4—29no—
141SL(2,ℤ/4)—189no—
142#217—184no—
143#217—184no—
144(C2×D3)⋊C4—188no—
145(C2×D3)⋊C4—184no—
146(C2×D3)⋊C4—184no—
147(C2×D3)⋊C4—369no—
148C22×C32—31no—
149C22×C32—31no—
150C22×C32—31no—
151C22×C32—31no—
152C22×C32—31no—
153C22×C32—31no—
154C22×C32—31no—
155C22×C32—31no—
156C22×C32—31no—
157C22×C32—412no—
158C22×C32—61no—
159C22×C32(C2×D3)⋊C411yescharacteristic, complemented
160C22×C32C2×S411yescharacteristic
161C2×C3⋊D3—62no—
162C2×C3⋊D3—62no—
163C2×C3⋊D3—62no—
164C2×C3⋊D3—1224no—
165C2×C3⋊D3—121no—
166C2×C3⋊D3—121no—
167C2×C3⋊D3—121no—
168C2×C3⋊D3—121no—
169C2×C3⋊D3—2412no—
170C2×C3⋊D3—2412no—
171C3×A4(C2×D3)⋊C4112yescomplemented
172C32×C4—21no—
173C32×C4—61no—
174C32×C4—812no—
175C3⋊Dic3—62no—
176C3⋊Dic3—121no—
177C3⋊Dic3—121no—
178C3⋊Dic3—2412no—
179C22×D4—271no—
180C23×C4C3⋊(C3⋊D3)11yescharacteristic, complemented
181C2×C22⋊C4—271no—
182C2×C22⋊C4—271no—
183C2×C22⋊C4—271no—
184#95—271no—
185#95—271no—
186C33—41noSylow
187C23×C3—34no—
188C23×C3—34no—
189C23×C3—34no—
190C23×C3—34no—
191C23×C3C4×C3⋊D314yes—
192C23×C3C4⋊(C3⋊D3)14yes—
193C23×C3#63814yes—
194C22×D3—98no—
195C22×D3—184no—
196C22×D3—184no—
197C22×D3—184no—
198C22×D3—184no—
199C22×D3—184no—
200C22×D3—184no—
201C22×D3—369no—
202C2×A4C4×C3⋊D319yes—
203C2×A4C4⋊(C3⋊D3)19yes—
204C2×A4#63819yes—
205S4—1818no—
206C2×C3×C4—38no—
207C2×C3×C4—34no—
208C2×C3×C4—49no—
209C2×C3×C4—64no—
210C2×C3×C4—64no—
211C2×C3×C4—64no—
212C2×C3×C4C3⋊S414yescomplemented
213#67—188no—
214#67—188no—
215#67—188no—
216#67—188no—
217C2×Dic3—98no—
218C2×Dic3—184no—
219C2×Dic3—184no—
220C2×Dic3—184no—
221C2×Dic3—184no—
222C2×Dic3—184no—
223C2×Dic3—184no—
224C2×Dic3—184no—
225C2×Dic3—184no—
226C2×Dic3—369no—
227C2×C32—31no—
228C2×C32—31no—
229C2×C32—31no—
230C2×C32—31no—
231C2×C32—412no—
232C2×C32—412no—
233C2×C32—412no—
234C2×C32C4×S411yescharacteristic
235C2×C32C4⋊S411yescharacteristic
236C2×C32GL(2,ℤ/4)11yescharacteristic
237C3⋊D3—122no—
238C3⋊D3—2424no—
239C24—271no—
240C24C2×C3⋊(C3⋊D3)11yescharacteristic
241C2×D4—272no—
242C2×D4—272no—
243C2×D4—541no—
244C2×D4—541no—
245C2×D4—541no—
246C2×D4—541no—
247C22×C4—21no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—61no—
251C22×C4—271no—
252C22×C4—271no—
253C22×C4—271no—
254C22⋊C4—542no—
255C22⋊C4—541no—
256C22⋊C4—541no—
257C22⋊C4—541no—
258C22⋊C4—541no—
259C22×C3—34no—
260C22×C3—34no—
261C22×C3—34no—
262C22×C3—34no—
263C22×C3—34no—
264C22×C3—34no—
265C22×C3—34no—
266C22×C3—34no—
267C22×C3—34no—
268C22×C3—49no—
269C22×C3—64no—
270C22×C3#375214yescomplemented
271C22×C3C2×C3⋊S414yes—
272C2×D3—188no—
273C2×D3—188no—
274C2×D3—188no—
275C2×D3—3618no—
276C2×D3—364no—
277C2×D3—364no—
278C2×D3—364no—
279C2×D3—364no—
280C2×D3—729no—
281C2×D3—729no—
282A4#375219yescomplemented
283C3×C4—24no—
284C3×C4—64no—
285C3×C4—89no—
286Dic3—188no—
287Dic3—364no—
288Dic3—364no—
289Dic3—729no—
290C32—412no—
291C32(C2×S4)⋊C411yescharacteristic, complemented
292C23—31no—
293C23—31no—
294C23—31no—
295C23—31no—
296C23—272no—
297C23—541no—
298C23—541no—
299C23—541no—
300C23—541no—
301C23—541no—
302C23—541no—
303C23C4×C3⋊(C3⋊D3)11yescharacteristic
304C23C4⋊(C3⋊(C3⋊D3))11yescharacteristic
305C23#631211yescharacteristic
306D4—542no—
307D4—542no—
308D4—542no—
309D4—542no—
310C2×C4—32no—
311C2×C4—31no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—272no—
316C2×C4—541no—
317C2×C4—541no—
318C2×C4—541no—
319C2×C4—541no—
320C2×C4—541no—
321C2×C4—541no—
322C2×C4—541no—
323C2×C4—541no—
324C2×C4C3⋊(C3⋊S4)11yescharacteristic, complemented
325C2×C3—34no—
326C2×C3—34no—
327C2×C3—34no—
328C2×C3—34no—
329C2×C3—49no—
330C2×C3—49no—
331C2×C3—49no—
332C2×C3C4×C3⋊S414yes—
333C2×C3C4⋊(C3⋊S4)14yes—
334C2×C3#6426914yes—
335D3—368no—
336D3—7218no—
337C22—31no—
338C22—31no—
339C22—31no—
340C22—31no—
341C22—31no—
342C22—31no—
343C22—31no—
344C22—31no—
345C22—31no—
346C22—61no—
347C22—542no—
348C22—542no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22—1081no—
353C22—1081no—
354C22#8885911yescharacteristic, complemented
355C22C2×C3⋊(C3⋊S4)11yescharacteristic
356C4—21no—
357C4—61no—
358C4—542no—
359C4—1081no—
360C4—1081no—
361C3—49no—
362C3#1059364714yescomplemented
363C2—31no—
364C2—31no—
365C2—31no—
366C2—31no—
367C2—1082no—
368C2C4×C3⋊(C3⋊S4)11yescharacteristic
369C2C4⋊(C3⋊(C3⋊S4))11yescharacteristic
370C2#1172447311yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutatorC2×C32×A4C2×C4
FittingC23×C32×C4D3
socleC24×C32C2×D3
prime radicalC22×C3⋊(C3⋊S4)C2
cosocleC22×C32×A4C22
hypercenterC2×C4C3⋊(C3⋊S4)
hypocenterC32×A4C22⋊C4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4×A4C2semidirect
3. C2×C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
9. C2×C3⋊(C3⋊S4)C4semidirect
14. C23×C32×C4D3semidirect
16. C2×C3×C4×A4D3semidirect
63. C32×A4C22⋊C4semidirect
67. C23×C3×C4C3⋊D3semidirect
69. C2×C4×A4C3⋊D3semidirect
101. C2×C32×C4S4semidirect
159. C22×C32(C2×D3)⋊C4semidirect
171. C3×A4(C2×D3)⋊C4semidirect
180. C23×C4C3⋊(C3⋊D3)semidirect
212. C2×C3×C4C3⋊S4semidirect
270. C22×C3#3752semidirect
282. A4#3752semidirect
291. C32(C2×S4)⋊C4semidirect
324. C2×C4C3⋊(C3⋊S4)semidirect
354. C22#88859semidirect
362. C3#10593647semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊S4)2
C2×C32×C4×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
#105936473
#105929123
#888594