Finite Group Databasebeta

← #10593649⋊C3 #13146889

Subgroups — #10593649⋊C3 #13146889

Order 1728.

392 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 38 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
1D4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊((C3⋊S4)⋊C3)C211yesmaximal, characteristic, complemented
3(C3⋊(A4⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#10593649C311yesmaximal, characteristic, complemented
5C3×#5217—31nomaximal
6C22×C22⋊H(3,𝔽3)—21no—
7C2×(C3⋊S4)⋊C3—21no—
8C4×C22⋊H(3,𝔽3)C2211yescharacteristic
9#3753⋊C3—41nomaximal
10C22×C32×D4D311yescharacteristic, complemented
11C3×D4×A4—21no—
12C3×D4×A4C2×C311yescharacteristic, complemented
13C4⋊(C3⋊S4)C2×C311yescharacteristic, complemented
14C2×C3×#213—31no—
15C2×C3×#213—31no—
16C3×#930—31no—
17C3⋊(A4⋊C8)C2×C311yescharacteristic, complemented
18C3×#832—31no—
19C3×#832—31no—
20C3×#830—31no—
21C2×C22⋊H(3,𝔽3)—41no—
22C2×C22⋊H(3,𝔽3)D411yescharacteristic
23(C3⋊S4)⋊C3—41no—
24D4×H(3,𝔽3)—41no—
25C4⋊((C3⋊D3)⋊C3)—41no—
26(C3⋊(C3⋊C8))⋊C3—41no—
27#5415—91no—
28C3×#447—91no—
29#5217—31no—
30C24×C32—21no—
31C22×C3×A4—22no—
32C22×C3×A4—21no—
33C2×C3⋊S4—21no—
34C2×C32×D4—31no—
35C2×C32×D4—31no—
36C2×C32×D4—61no—
37C22×C32×C4C2×D311yescharacteristic
38C2×C3×#67—61no—
39C2×C3×D12—31no—
40C3×C4×A4—21no—
41C3×C4×A4C22×C311yescharacteristic
42#3753—41no—
43C3×#213—62no—
44C3×#213—61no—
45C3×#213—61no—
46C3×(C2×D3)⋊C4—61no—
47C3×C4⋊Dic3—31no—
48C2×C3×C3⋊C8—31no—
49C2×C3×C3⋊C8—31no—
50C22×H(3,𝔽3)—81no—
51C2×(C3⋊D3)⋊C3—81no—
52C22⋊H(3,𝔽3)D811yescharacteristic, complemented
53C4×H(3,𝔽3)—41no—
54C22×C3×D4—31no—
55C22×C3×D4C3×D311yescharacteristic
56D4×A4—31no—
57D4×A4—61no—
58C4⋊S4—91no—
59C2×C3×D8—91no—
60C2×C3×D8—91no—
61C3×#121—91no—
62C2×#213—31no—
63C2×#213—31no—
64#930—31no—
65A4⋊C8—91no—
66C3×#102—91no—
67C3×#102—91no—
68C3×C22⋊C8—91no—
69#832—31no—
70#832—31no—
71#830—31no—
72C23×C32—41no—
73C23×C32—61no—
74C23×C32—61no—
75C23×C32—121no—
76C23×C32#6711yescharacteristic
77C22×C3×D3—61no—
78C2×C3×A4—21no—
79C2×C3×A4—42no—
80C2×C3×A4—41no—
81C2×C3×A4C3×D411yescharacteristic
82C3⋊S4—41no—
83C32×D4—32no—
84C32×D4—31no—
85C32×D4—41no—
86C32×D4—61no—
87C32×D4—81no—
88C32×D4S411yescharacteristic, complemented
89C2×C32×C4—31no—
90C2×C32×C4—31no—
91C4⋊(C3⋊D3)—41no—
92C3×#67—121no—
93C3×#67—121no—
94C2×C3×Dic3—61no—
95C3×D12—61no—
96C3×D12—61no—
97C3⋊(C3⋊C8)—41no—
98C3×C3⋊C8—61no—
99C3×C3⋊C8—61no—
100#447—91noSylow
101C2×H(3,𝔽3)—41no—
102C2×H(3,𝔽3)—161no—
103(C3⋊D3)⋊C3—161no—
104C24×C3—21no—
105C24×C3—61no—
106C22×A4—61no—
107C22×A4—62no—
108C2×S4—181no—
109C2×C3×D4—31no—
110C2×C3×D4—31no—
111C2×C3×D4—31no—
112C2×C3×D4—31no—
113C2×C3×D4—61no—
114C2×C3×D4—61no—
115C2×C3×D4—61no—
116C2×C3×D4—61no—
117C2×C3×D4—62no—
118C2×C3×D4—91no—
119C2×C3×D4—181no—
120C22×C3×C4—31no—
121C22×C3×C4C2×C3×D311yescharacteristic
122C2×#67—61no—
123C2×D12—31no—
124C4×A4—31no—
125C4×A4—61no—
126C3×D8—182no—
127C3×D8—181no—
128C3×D8—181no—
129C2×C3×C8—91no—
130C2×C3×C8—91no—
131C3×C4⋊C4—91no—
132C3×C22⋊C4—181no—
133#213—62no—
134#213—61no—
135#213—61no—
136#213—361no—
137(C2×D3)⋊C4—61no—
138C4⋊Dic3—31no—
139C2×C3⋊C8—31no—
140C2×C3⋊C8—31no—
141C22×C32—21no—
142C22×C32—31no—
143C22×C32—31no—
144C22×C32—61no—
145C22×C32—81no—
146C22×C32—82no—
147C22×C32—121no—
148C22×C32—121no—
149C22×C32—121no—
150C22×C32—121no—
151C22×C32#21311yescharacteristic, complemented
152C2×C3⋊D3—81no—
153C2×C3×D3—121no—
154C2×C3×D3—121no—
155C2×C3×D3—121no—
156C3×A4—21no—
157C3×A4C3×D811yescharacteristic, complemented
158C32×C4—31no—
159C32×C4—41no—
160C32×C4—81no—
161C32×C4C2×S411yescharacteristic
162C3×Dic3—121no—
163C22×D4(C3⋊D3)⋊C311yescharacteristic, complemented
164C2×D8—91no—
165C2×D8—91no—
166#121—91no—
167#102—91no—
168#102—91no—
169C22⋊C8—91no—
170H(3,𝔽3)—41noSylow
171C23×C3—31no—
172C23×C3—41no—
173C23×C3—61no—
174C23×C3—61no—
175C23×C3—61no—
176C23×C3—61no—
177C23×C3—62no—
178C23×C3—62no—
179C23×C3—121no—
180C23×C3—121no—
181C23×C3—121no—
182C23×C3—122no—
183C23×C3—181no—
184C23×C3C3×#6711yescharacteristic
185C22×D3—61no—
186C2×A4—31no—
187C2×A4—61no—
188C2×A4—121no—
189C2×A4—122no—
190S4—361no—
191C3×D4—32no—
192C3×D4—32no—
193C3×D4—31no—
194C3×D4—31no—
195C3×D4—31no—
196C3×D4—61no—
197C3×D4—62no—
198C3×D4—61no—
199C3×D4—61no—
200C3×D4—62no—
201C3×D4—121no—
202C3×D4—181no—
203C3×D4—181no—
204C3×D4—241no—
205C3×D4—361no—
206C3×D4—361no—
207C3×D4C3×S411yescharacteristic
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—181no—
215#67—121no—
216#67—121no—
217C2×Dic3—61no—
218D12—61no—
219D12—61no—
220D12—361no—
221C3×C8—181no—
222C3×C8—181no—
223C3⋊C8—61no—
224C3⋊C8—61no—
225C3⋊C8—361no—
226C2×C32—31no—
227C2×C32—31no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—81no—
231C2×C32—121no—
232C2×C32—161no—
233C2×C32—162no—
234C2×C32GL(2,ℤ/4)11yescharacteristic
235C3⋊D3—161no—
236C3×D3—241no—
237C24—21no—
238C2×D4—31no—
239C2×D4—31no—
240C2×D4—61no—
241C2×D4—91no—
242C2×D4—181no—
243C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
244D8—182no—
245D8—181no—
246D8—181no—
247C2×C8—91no—
248C2×C8—91no—
249C4⋊C4—91no—
250C22⋊C4—181no—
251C22×C3—21no—
252C22×C3—31no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—62no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—121no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3—121no—
267C22×C3—121no—
268C22×C3—121no—
269C22×C3—121no—
270C22×C3—121no—
271C22×C3—122no—
272C22×C3—122no—
273C22×C3—122no—
274C22×C3—122no—
275C22×C3—241no—
276C22×C3—242no—
277C22×C3—361no—
278C22×C3—361no—
279C22×C3—361no—
280C22×C3C3×#21311yescharacteristic
281C2×D3—121no—
282C2×D3—121no—
283C2×D3—121no—
284C2×D3—721no—
285A4—31no—
286A4—61no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—61no—
291C3×C4—121no—
292C3×C4—241no—
293C3×C4—361no—
294C3×C4C2×C3×S411yescharacteristic
295Dic3—121no—
296C32—41no—
297C32—81no—
298C32#541511yescharacteristic, complemented
299C23—41no—
300C23—61no—
301C23—61no—
302C23—121no—
303C23—181no—
304C23#638⋊C311yescharacteristic
305D4—32no—
306D4—31no—
307D4—61no—
308D4—181no—
309D4—181no—
310D4—361no—
311D4—361no—
312D4(C3⋊S4)⋊C311yescharacteristic, complemented
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—181no—
316C8—181no—
317C8—181no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—41no—
324C2×C3—61no—
325C2×C3—61no—
326C2×C3—121no—
327C2×C3—122no—
328C2×C3—121no—
329C2×C3—121no—
330C2×C3—121no—
331C2×C3—241no—
332C2×C3—481no—
333C2×C3—482no—
334C2×C3—721no—
335C2×C3C3×GL(2,ℤ/4)11yescharacteristic
336D3—241no—
337D3—1441no—
338C22—21no—
339C22—31no—
340C22—31no—
341C22—61no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22—361no—
347C22—361no—
348C22—361no—
349C22#3753⋊C311yescharacteristic, complemented
350C4—31no—
351C4—361no—
352C4C2×(C3⋊S4)⋊C311yescharacteristic
353C3—31no—
354C3—121no—
355C3—241no—
356C3C3×#541511yescharacteristic
357C2—31no—
358C2—31no—
359C2—41no—
360C2—121no—
361C2—721no—
362C2#64269⋊C311yescharacteristic
subgroupG/H
centerC2#64269⋊C3
commutator, cosocleC3×C4×A4C22×C3
FrattiniC3×C4C2×C3×S4
FittingC22×C32×D4D3
socleC23×C3C3×#67
hypercenterD4(C3⋊S4)⋊C3
hypocenterC3×A4C3×D8
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
1. D4×C22⋊H(3,𝔽3)C2semidirect
2. C4⋊((C3⋊S4)⋊C3)C2semidirect
3. (C3⋊(A4⋊C8))⋊C3C2semidirect
4. #10593649C3semidirect
10. C22×C32×D4D3semidirect
12. C3×D4×A4C2×C3semidirect
13. C4⋊(C3⋊S4)C2×C3semidirect
17. C3⋊(A4⋊C8)C2×C3semidirect
52. C22⋊H(3,𝔽3)D8semidirect
88. C32×D4S4semidirect
151. C22×C32#213semidirect
157. C3×A4C3×D8semidirect
163. C22×D4(C3⋊D3)⋊C3semidirect
298. C32#5415semidirect
312. D4(C3⋊S4)⋊C3semidirect
349. C22#3753⋊C3semidirect

Maximal subgroups

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

subgroupindex
D4×C22⋊H(3,𝔽3)2
C4⋊((C3⋊S4)⋊C3)2
(C3⋊(A4⋊C8))⋊C32
#105936493
C3×#52173
#3753⋊C34