Finite Group Databasebeta

← #13180738

Subgroups — #13180738

Order 1728.

1134 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 89 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
1A4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2#11724463C211yesmaximal, characteristic, complemented
3C32×C4×S4C211yesmaximal, characteristic, complemented
4#11724189C212yesmaximal, complemented
5C3⋊D3×SL(2,ℤ/4)C212yesmaximal, complemented
6#10596424—34nomaximal
7#10596021—31nomaximal
8C2×A4×C3⋊D3C2212yes—
9C2×C32×S4C2211yescharacteristic
10C32×C4×A4C2211yescharacteristic
11#89021—41nomaximal
12C3⋊(C3⋊SL(2,ℤ/4))C2212yes—
13C32×SL(2,ℤ/4)C2211yescharacteristic
14C2×#3828—31no—
15C22×C4⋊(C3⋊D3)D311yescharacteristic, complemented
16A4×D12—34no—
17#64259—34no—
18#64259—64no—
19C3×C4×S4D314yescomplemented
20#64209—38no—
21D3×SL(2,ℤ/4)—38no—
22C32×C4×D4—31no—
23#64146—32no—
24#64139—31no—
25#64133—31no—
26#64103—32no—
27#64093—32no—
28C22⋊C4×C3⋊D3—32no—
29#64084—31no—
30C4×C4⋊(C3⋊D3)—31no—
31C2×C32×A4C2311yescharacteristic
32A4×C3⋊D3—22no—
33C32×S4—21no—
34#6308—41no—
35C3×C4⋊(C3⋊D3)—41no—
36C32×C4×D3—41no—
37#6290—42no—
38Dic3×C3⋊D3—42no—
39#5914—91no—
40#5550—94no—
41C23×C3⋊D3C2×D312yes—
42C2×D3×A4—38no—
43C2×C3×S4C2×D314yes—
44C2×C32×D4—31no—
45C22×C32×C4—31no—
46C22×C32×C4C2×D311yescharacteristic
47C2×#638—32no—
48C2×#638—32no—
49#3828—62no—
50#3828—61no—
51#3828—61no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—32no—
55C2×C4×C3⋊D3—32no—
56C2×C4×C3⋊D3—32no—
57C2×#634—31no—
58C3×C4×A4—24no—
59C3×C4×A4C2×D314yes—
60#3795—124no—
61C3⋊SL(2,ℤ/4)—38no—
62C3⋊SL(2,ℤ/4)—68no—
63C3×SL(2,ℤ/4)C2×D314yes—
64C32×C4⋊C4—31no—
65C32×C22⋊C4—31no—
66C32×C22⋊C4—31no—
67C32×C42—31no—
68#3757—32no—
69#3752—32no—
70#3752—32no—
71#3752—32no—
72#3752—32no—
73C4⋊(C3⋊Dic3)—31no—
74#3750—32no—
75C2×C3×C3⋊D3—42no—
76C2×C32×D3—41no—
77C32×A4#4111yescharacteristic, complemented
78C33×C4—41no—
79C3⋊(C3⋊Dic3)—42no—
80C32×Dic3—41no—
81C2×#235—94no—
82C22×D12—34no—
83D4×A4—91no—
84GL(2,ℤ/4)—92no—
85C2×SL(2,ℤ/4)—92no—
86C4×S4C3⋊D311yescharacteristic, complemented
87A4⋊Q8—91no—
88A4⋊Q8—184no—
89C3×C4×D4—34no—
90#937—98no—
91#930—94no—
92#924—94no—
93#894—98no—
94#884—98no—
95D3×C22⋊C4—98no—
96#875—94no—
97C4×D12—94no—
98C23×C32—31no—
99C23×C32C22×D311yescharacteristic
100C22×C3⋊D3—22no—
101C22×C3⋊D3—32no—
102C22×C3⋊D3—32no—
103C22×C3⋊D3—62no—
104C2×C3×A4—24no—
105C2×C3×A4C22×D314yes—
106D3×A4—68no—
107C3×S4—24no—
108C32×D4—61no—
109C32×D4—61no—
110C2×C32×C4—31no—
111C2×C32×C4—31no—
112C2×C32×C4—31no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—61no—
116C2×C32×C4—61no—
117#638—62no—
118#638—62no—
119#638—122no—
120C2×C3⋊Dic3—32no—
121C2×C3⋊Dic3—32no—
122C4⋊(C3⋊D3)—32no—
123C4⋊(C3⋊D3)—31no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)S411yescharacteristic, complemented
126C4×C3⋊D3—62no—
127C4×C3⋊D3—62no—
128C4×C3⋊D3—122no—
129#634—61no—
130#634—61no—
131#634—124no—
132#634—244no—
133C3×D12—124no—
134C3×C4×D3—44no—
135#625—128no—
136D3×Dic3—128no—
137#546—271noSylow
138C2×C33—41no—
139C3×C3⋊D3—82no—
140C32×D3—81no—
141C23×D3—38no—
142C22×A4—92no—
143C2×S4C2×C3⋊D311yescharacteristic
144C2×C3×D4—34no—
145C22×C3×C4—34no—
146C22×C3×C4D3214yes—
147C2×#67—98no—
148C2×#67—98no—
149#237—361no—
150#235—184no—
151#235—188no—
152#235—184no—
153C2×D12—94no—
154C2×D12—94no—
155C2×D12—98no—
156C2×C4×D3—98no—
157C2×C4×D3—98no—
158C2×Dic6—94no—
159C4×A4—24no—
160C4×A4C2×C3⋊D311yescharacteristic
161SL(2,ℤ/4)—92no—
162SL(2,ℤ/4)—188no—
163SL(2,ℤ/4)C2×C3⋊D311yescharacteristic
164C3×C4⋊C4—34no—
165C3×C22⋊C4—34no—
166C3×C22⋊C4—34no—
167C3×C42—34no—
168#217—98no—
169(C2×D3)⋊C4—98no—
170(C2×D3)⋊C4—98no—
171(C2×D3)⋊C4—98no—
172(C2×D3)⋊C4—98no—
173C4⋊Dic3—94no—
174Dic3⋊C4—98no—
175C22×C32—31no—
176C22×C32—31no—
177C22×C32—61no—
178C22×C32—61no—
179C22×C32—61no—
180C22×C32#23711yescharacteristic, complemented
181C2×C3⋊D3—32no—
182C2×C3⋊D3—62no—
183C2×C3⋊D3—62no—
184C2×C3⋊D3—62no—
185C2×C3⋊D3—62no—
186C2×C3⋊D3C2×S412yes—
187C2×C3×D3—44no—
188C2×C3×D3—128no—
189C3×A4—24no—
190C3×A4#23514yescomplemented
191C32×C4—31no—
192C32×C4—44no—
193C32×C4—61no—
194C32×C4—61no—
195C32×C4—61no—
196C32×C4—84no—
197C32×C4C2×S411yescharacteristic
198C3⋊Dic3—62no—
199C3⋊Dic3—62no—
200C3⋊Dic3—128no—
201C3⋊Dic3—248no—
202C3×Dic3—44no—
203C2×#41—271no—
204C22×D4—91no—
205#124—271no—
206#123—272no—
207C22⋊Q8—271no—
208#121—272no—
209#121—271no—
210#120—272no—
211C4×D4—31no—
212C4×D4—271no—
213C2×C22⋊C4—272no—
214C33—41noSylow
215C23×C3—34no—
216C23×C3C2×D3214yes—
217C22×D3—68no—
218C22×D3—98no—
219C22×D3—98no—
220C22×D3—188no—
221C2×A4—24no—
222C2×A4—182no—
223C2×A4C22×C3⋊D311yescharacteristic
224S4—21no—
225C3×D4—64no—
226C3×D4—64no—
227C3×D4—361no—
228C2×C3×C4—34no—
229C2×C3×C4—34no—
230C2×C3×C4—34no—
231C2×C3×C4—34no—
232C2×C3×C4—34no—
233C2×C3×C4—64no—
234C2×C3×C4—64no—
235#67—188no—
236#67—188no—
237#67—362no—
238#67—368no—
239C2×Dic3—98no—
240C2×Dic3—98no—
241C2×Dic3—362no—
242D12—34no—
243D12—98no—
244D12—94no—
245D12—184no—
246C4×D3—41no—
247C4×D3—188no—
248C4×D3—188no—
249C4×D3—368no—
250Dic6—184no—
251Dic6—184no—
252Dic6—361no—
253Dic6—724no—
254C2×C32—31no—
255C2×C32—31no—
256C2×C32—44no—
257C2×C32—84no—
258C2×C32—121no—
259C2×C32C22×S411yescharacteristic
260C3⋊D3—22no—
261C3⋊D3—62no—
262C3×D3—84no—
263C3×D3—248no—
264C24—92no—
265#41—541no—
266#41—542no—
267#41—541no—
268C2×Q8—271no—
269C2×D4—31no—
270C2×D4—271no—
271C2×D4—272no—
272C2×D4—271no—
273C2×D4—272no—
274C2×D4—272no—
275C22×C4—31no—
276C22×C4—272no—
277C22×C4—272no—
278C22×C4D3×C3⋊D311yescharacteristic
279C4⋊C4—31no—
280C4⋊C4—271no—
281C4⋊C4—272no—
282C22⋊C4—31no—
283C22⋊C4—31no—
284C22⋊C4—272no—
285C22⋊C4—272no—
286C22⋊C4—272no—
287C22⋊C4—272no—
288C22⋊C4—272no—
289C42—31no—
290C22×C3—34no—
291C22×C3—34no—
292C22×C3—64no—
293C22×C3—64no—
294C22×C3—64no—
295C22×C3—362no—
296C22×C3#379514yescomplemented
297C2×D3—38no—
298C2×D3—41no—
299C2×D3—98no—
300C2×D3—188no—
301C2×D3—188no—
302C2×D3—188no—
303C2×D3—188no—
304A4—24no—
305A4#382811yescharacteristic, complemented
306C3×C4—34no—
307C3×C4—41no—
308C3×C4—64no—
309C3×C4—64no—
310C3×C4—64no—
311C3×C4—84no—
312C3×C4D3×S414yes—
313Dic3—41no—
314Dic3—188no—
315Dic3—188no—
316Dic3—362no—
317Dic3—728no—
318C32—44no—
319C32—84no—
320C32#591411yescharacteristic, complemented
321C23—31no—
322C23—182no—
323C23—272no—
324C23—272no—
325C23—542no—
326C23C2×D3×C3⋊D311yescharacteristic
327Q8—541no—
328Q8—541no—
329D4—61no—
330D4—61no—
331D4—91no—
332D4—272no—
333D4—271no—
334D4—541no—
335D4—542no—
336D4—542no—
337D4—1082no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—272no—
346C2×C4—272no—
347C2×C4—542no—
348C2×C4—542no—
349C2×C4—1082no—
350C2×C3—34no—
351C2×C3—34no—
352C2×C3—41no—
353C2×C3—84no—
354C2×C3—124no—
355C2×C3—722no—
356C2×C3C2×D3×S414yes—
357D3—68no—
358D3—81no—
359D3—188no—
360C22—31no—
361C22—31no—
362C22—61no—
363C22—61no—
364C22—61no—
365C22—92no—
366C22—272no—
367C22—542no—
368C22—542no—
369C22—542no—
370C22—542no—
371C22#8902111yescharacteristic, complemented
372C4—31no—
373C4—61no—
374C4—61no—
375C4—61no—
376C4—542no—
377C4—542no—
378C4C3⋊D3×S411yescharacteristic
379C3—41no—
380C3—84no—
381C3#1059642414yescomplemented
382C2—31no—
383C2—31no—
384C2—121no—
385C2—182no—
386C2—542no—
387C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊D3×S4
commutator, cosocleC2×C32×A4C23
FittingC22×C32×C4C2×D3
socleC23×C32C22×D3
hypercenterC4C3⋊D3×S4
hypocenterC32×A4#41
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. A4×C4⋊(C3⋊D3)C2semidirect
2. #11724463C2semidirect
3. C32×C4×S4C2semidirect
4. #11724189C2semidirect
5. C3⋊D3×SL(2,ℤ/4)C2semidirect
15. C22×C4⋊(C3⋊D3)D3semidirect
19. C3×C4×S4D3semidirect
77. C32×A4#41semidirect
86. C4×S4C3⋊D3semidirect
125. C4⋊(C3⋊D3)S4semidirect
180. C22×C32#237semidirect
190. C3×A4#235semidirect
296. C22×C3#3795semidirect
305. A4#3828semidirect
320. C32#5914semidirect
371. C22#89021semidirect
381. C3#10596424semidirect

Maximal subgroups

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

subgroupindex
A4×C4⋊(C3⋊D3)2
#117244632
C32×C4×S42
#117241892
C3⋊D3×SL(2,ℤ/4)2
#105964243
#105960213
#890214