Finite Group Databasebeta

← #13181396

Subgroups — #13181396

Order 1728.

1666 conjugacy classes of subgroups; only the 215 normal subgroups are shown.

normal subgroups only

This group has 1666 conjugacy classes of subgroups — too many to draw. Only its 215 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×C32×C4×A4C211yesmaximal, characteristic, complemented
2#11724473C212yesmaximal, complemented
3C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic, complemented
4C4×C3⋊(C3⋊S4)C212yesmaximal, complemented
5#11724463C211yesmaximal, characteristic, complemented
6#10596575—312nomaximal
7#10596262—31nomaximal
8C22×C32×A4C2211yescharacteristic
9C2×C3⋊(C3⋊S4)C2212yes—
10#89076—41nomaximal
11C32×C4×A4C2211yescharacteristic
12C32×C4×A4C2211yescharacteristic
13C3⋊(C3⋊SL(2,ℤ/4))C2212yes—
14C23×C32×C4D311yescharacteristic, complemented
15C2×#3828—31no—
16C2×C3×C4×A4D3112yescomplemented
17#64269—324no—
18C4⋊(C3⋊S4)—312no—
19C4×C3⋊S4—324no—
20#64259—312no—
21#64162—32no—
22#64139—31no—
23#64139—31no—
24#64138—32no—
25C4×#638—32no—
26C4×#638—32no—
27#64135—31no—
28#64133—31no—
29#64133—31no—
30C2×C32×A4—21no—
31C2×C32×A4C2311yescharacteristic
32C3⋊(C3⋊S4)—22no—
33C2×C33×C4—41no—
34#6312—42no—
35C4⋊(C3⋊(C3⋊D3))—41no—
36C4×C3⋊(C3⋊D3)—42no—
37#6308—41no—
38#5912—99no—
39#5791—94no—
40C24×C32C2×D311yescharacteristic
41C22×C3×A4C2×D3112yes—
42C2×C3⋊S4—324no—
43C22×C32×C4—31no—
44C22×C32×C4—31no—
45C22×C32×C4—31no—
46C22×C32×C4—31no—
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C2×#638—32no—
50C2×#638—32no—
51C2×#638—32no—
52#3828—61no—
53#3828—61no—
54#3828—61no—
55#3828—61no—
56#3828—1212no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4×C3⋊D3—32no—
59C2×#634—31no—
60C3×C4×A4C2×D3112yes—
61C3×C4×A4C2×D3112yes—
62C3⋊SL(2,ℤ/4)—324no—
63#3757—32no—
64#3757—32no—
65#3757—32no—
66#3752—32no—
67#3752—32no—
68C4⋊(C3⋊Dic3)—31no—
69C4⋊(C3⋊Dic3)—31no—
70#3750—32no—
71#3750—32no—
72C4×C3⋊Dic3—32no—
73C22×C33—41no—
74C2×C3⋊(C3⋊D3)—42no—
75C32×A4#4111yescharacteristic, complemented
76C33×C4—41no—
77C33×C4—41no—
78C3⋊(C3⋊Dic3)—42no—
79C23×C3×C4C3⋊D314yescomplemented
80C2×#235—94no—
81C2×C4×A4C3⋊D319yescomplemented
82GL(2,ℤ/4)—918no—
83C4⋊S4—99no—
84C4×S4—918no—
85A4⋊Q8—99no—
86#953—98no—
87#930—94no—
88#930—94no—
89#929—98no—
90C4×#67—98no—
91C4×#67—98no—
92#926—94no—
93#924—94no—
94#924—94no—
95C23×C32—21no—
96C23×C32—31no—
97C23×C32—31no—
98C23×C32—61no—
99C23×C32C22×D311yescharacteristic
100C22×C3⋊D3—32no—
101C2×C3×A4—212no—
102C2×C3×A4C22×D3112yes—
103C3⋊S4—624no—
104C2×C32×C4—31no—
105C2×C32×C4—31no—
106C2×C32×C4—31no—
107C2×C32×C4—31no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—31no—
111C2×C32×C4—412no—
112C2×C32×C4—61no—
113C2×C32×C4S411yescharacteristic, complemented
114#638—62no—
115#638—62no—
116#638—64no—
117#638—62no—
118#638—62no—
119#638—1224no—
120C2×C3⋊Dic3—32no—
121C2×C3⋊Dic3—32no—
122C2×C3⋊Dic3—32no—
123C4⋊(C3⋊D3)—61no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—1212no—
126C4×C3⋊D3—62no—
127C4×C3⋊D3—62no—
128C4×C3⋊D3—1224no—
129#634—61no—
130#634—61no—
131#634—1212no—
132#525—271noSylow
133C2×C33—41no—
134C2×C33—81no—
135C3⋊(C3⋊D3)—82no—
136C24×C3C2×C3⋊D314yes—
137C22×A4C2×C3⋊D319yes—
138C2×S4—918no—
139C22×C3×C4—34no—
140C22×C3×C4—34no—
141C22×C3×C4—34no—
142C22×C3×C4—34no—
143C22×C3×C4C2×C3⋊D314yes—
144C22×C3×C4C2×C3⋊D314yes—
145C2×#67—98no—
146C2×#67—98no—
147C2×#67—98no—
148#235—184no—
149#235—184no—
150#235—184no—
151#235—184no—
152#235—369no—
153C2×D12—94no—
154C2×C4×D3—98no—
155C2×Dic6—94no—
156C4×A4C2×C3⋊D319yes—
157C4×A4C2×C3⋊D319yes—
158SL(2,ℤ/4)—918no—
159#217—98no—
160#217—98no—
161#217—98no—
162(C2×D3)⋊C4—98no—
163(C2×D3)⋊C4—98no—
164C4⋊Dic3—94no—
165C4⋊Dic3—94no—
166Dic3⋊C4—98no—
167Dic3⋊C4—98no—
168C4×Dic3—98no—
169C22×C32—31no—
170C22×C32—31no—
171C22×C32—32no—
172C22×C32—31no—
173C22×C32—412no—
174C22×C32—61no—
175C22×C32—61no—
176C22×C32—61no—
177C22×C32#23511yescharacteristic, complemented
178C22×C32C2×S411yescharacteristic
179C2×C3⋊D3—62no—
180C2×C3⋊D3—62no—
181C2×C3⋊D3—62no—
182C2×C3⋊D3—1224no—
183C3×A4#235112yescomplemented
184C32×C4—31no—
185C32×C4—31no—
186C32×C4—412no—
187C32×C4—412no—
188C32×C4C2×S411yescharacteristic
189C32×C4C2×S411yescharacteristic
190C3⋊Dic3—62no—
191C3⋊Dic3—62no—
192C3⋊Dic3—62no—
193C3⋊Dic3—1224no—
194C2×#41—271no—
195C23×C4C3⋊(C3⋊D3)11yescharacteristic, complemented
196#123—272no—
197C22⋊Q8—271no—
198C22⋊Q8—271no—
199#121—271no—
200#121—271no—
201#120—272no—
202C4×D4—272no—
203C4×D4—272no—
204#117—271no—
205C33—41noSylow
206C23×C3—24no—
207C23×C3—34no—
208C23×C3—34no—
209C23×C3—64no—
210C23×C3C22×C3⋊D314yes—
211C22×D3—98no—
212C2×A4—29no—
213C2×A4C22×C3⋊D319yes—
214S4—1818no—
215C2×C3×C4—34no—
216C2×C3×C4—34no—
217C2×C3×C4—34no—
218C2×C3×C4—34no—
219C2×C3×C4—34no—
220C2×C3×C4—34no—
221C2×C3×C4—34no—
222C2×C3×C4—49no—
223C2×C3×C4—64no—
224C2×C3×C4C3⋊S414yescomplemented
225#67—188no—
226#67—188no—
227#67—1816no—
228#67—188no—
229#67—188no—
230#67—3618no—
231C2×Dic3—98no—
232C2×Dic3—98no—
233C2×Dic3—98no—
234D12—184no—
235D12—184no—
236D12—369no—
237C4×D3—188no—
238C4×D3—188no—
239C4×D3—3618no—
240Dic6—184no—
241Dic6—184no—
242Dic6—369no—
243C2×C32—21no—
244C2×C32—31no—
245C2×C32—31no—
246C2×C32—412no—
247C2×C32—61no—
248C2×C32—812no—
249C2×C32C22×S411yescharacteristic
250C3⋊D3—122no—
251C3⋊D3—2424no—
252C24C2×C3⋊(C3⋊D3)11yescharacteristic
253#41—541no—
254#41—541no—
255#41—541no—
256#41—541no—
257C2×Q8—271no—
258C2×D4—271no—
259C2×D4—272no—
260C2×D4—272no—
261C2×D4—272no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—272no—
267C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
268C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
269C4⋊C4—271no—
270C4⋊C4—271no—
271C4⋊C4—272no—
272C4⋊C4—272no—
273C22⋊C4—272no—
274C22⋊C4—272no—
275C22⋊C4—272no—
276C22⋊C4—272no—
277C22⋊C4—272no—
278C42—272no—
279C22×C3—34no—
280C22×C3—34no—
281C22×C3—38no—
282C22×C3—34no—
283C22×C3—49no—
284C22×C3—64no—
285C22×C3—64no—
286C22×C3—64no—
287C22×C3#382814yescomplemented
288C22×C3C2×C3⋊S414yes—
289C2×D3—188no—
290C2×D3—188no—
291C2×D3—188no—
292C2×D3—3618no—
293A4#382819yescomplemented
294C3×C4—34no—
295C3×C4—34no—
296C3×C4—49no—
297C3×C4—49no—
298C3×C4C2×C3⋊S414yes—
299C3×C4C2×C3⋊S414yes—
300Dic3—188no—
301Dic3—188no—
302Dic3—188no—
303Dic3—3618no—
304C32—412no—
305C32#591211yescharacteristic, complemented
306C23—21no—
307C23—31no—
308C23—31no—
309C23—61no—
310C23—272no—
311C23C22×C3⋊(C3⋊D3)11yescharacteristic
312Q8—541no—
313Q8—541no—
314D4—541no—
315D4—542no—
316D4—542no—
317D4—544no—
318D4—542no—
319D4—542no—
320D4—541no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4—272no—
330C2×C4—272no—
331C2×C4—272no—
332C2×C4—542no—
333C2×C4—542no—
334C2×C4C3⋊(C3⋊S4)11yescharacteristic, complemented
335C2×C3—24no—
336C2×C3—34no—
337C2×C3—34no—
338C2×C3—49no—
339C2×C3—64no—
340C2×C3—89no—
341C2×C3C22×C3⋊S414yes—
342D3—368no—
343D3—7218no—
344C22—31no—
345C22—31no—
346C22—32no—
347C22—31no—
348C22—61no—
349C22—61no—
350C22—61no—
351C22—542no—
352C22—542no—
353C22—542no—
354C22#8907611yescharacteristic, complemented
355C22C2×C3⋊(C3⋊S4)11yescharacteristic
356C4—31no—
357C4—31no—
358C4—542no—
359C4—542no—
360C4—542no—
361C4C2×C3⋊(C3⋊S4)11yescharacteristic
362C4C2×C3⋊(C3⋊S4)11yescharacteristic
363C3—49no—
364C3#1059657514yescomplemented
365C2—21no—
366C2—31no—
367C2—31no—
368C2—61no—
369C2—1082no—
370C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
centerC4C2×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×A4C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC23×C32×C4D3
socleC23×C32C22×D3
hypercenterC2×C4C3⋊(C3⋊S4)
hypocenterC32×A4#41
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×A4C2semidirect
2. #11724473C2semidirect
3. C4⋊(C3⋊(C3⋊S4))C2semidirect
4. C4×C3⋊(C3⋊S4)C2semidirect
5. #11724463C2semidirect
14. C23×C32×C4D3semidirect
16. C2×C3×C4×A4D3semidirect
75. C32×A4#41semidirect
79. C23×C3×C4C3⋊D3semidirect
81. C2×C4×A4C3⋊D3semidirect
113. C2×C32×C4S4semidirect
177. C22×C32#235semidirect
183. C3×A4#235semidirect
195. C23×C4C3⋊(C3⋊D3)semidirect
224. C2×C3×C4C3⋊S4semidirect
287. C22×C3#3828semidirect
293. A4#3828semidirect
305. C32#5912semidirect
334. C2×C4C3⋊(C3⋊S4)semidirect
354. C22#89076semidirect
364. C3#10596575semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
#117244732
C4⋊(C3⋊(C3⋊S4))2
C4×C3⋊(C3⋊S4)2
#117244632
#105965753
#105962623
#890764