Finite Group Databasebeta

← #10592832⋊C3 #13143216

Subgroups — #10592832⋊C3 #13143216

Order 1728.

635 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C2×M16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#63643⋊C3C212yesmaximal, complemented
6#10592832C311yesmaximal, characteristic, complemented
7C3×#5133—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—21no—
10C22×(C3⋊Dic3)⋊C3—21no—
11#3828⋊C3—22no—
12#3828⋊C3C414yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×#634⋊C3C2211yescharacteristic
16M16×H(3,𝔽3)—22no—
17C2×C8×H(3,𝔽3)C2212yes—
18C4⋊((C3⋊Dic3)⋊C3)—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2212yes—
20C2×#3828C2×C311yescharacteristic
21C2×C3×#235—31no—
22C2×C32×M16—21no—
23C2×C32×M16D311yescharacteristic, complemented
24C2×C32×M16C2×C311yescharacteristic, complemented
25C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
26C2×C3×C4⋊Dic3—31no—
27#63648C2×C312yescomplemented
28#63643C2×C312yescomplemented
29C3×#821—32no—
30C3×#816—32no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)C2×C412yes—
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)D412yes—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37#638⋊C3—24no—
38C2×(C3⋊Dic3)⋊C3—21no—
39C2×(C3⋊Dic3)⋊C3—22no—
40C2×(C3⋊Dic3)⋊C3—42no—
41C4⋊((C3⋊D3)⋊C3)—21no—
42C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
43C4×(C3⋊D3)⋊C3—24no—
44#634⋊C3—21no—
45#634⋊C3C2×C412yescomplemented
46C8×H(3,𝔽3)—22no—
47C3×#417—91no—
48#5133—31no—
49#5133—91no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C2×#638—21no—
54C22×C3⋊Dic3—21no—
55#3828—22no—
56#3828C3×C414yescomplemented
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×#634C22×C311yescharacteristic
60C2×C3×#67—61no—
61C22×C3×Dic3—61no—
62C3×#235—34no—
63C3×#235—62no—
64C2×C3×D12—31no—
65C2×C3×C4×D3—61no—
66C2×C3×Dic6—31no—
67C32×M16—24no—
68C32×M16—22no—
69C32×M16—22no—
70C2×C32×C8—22no—
71C2×C32×C8C2×D312yes—
72C2×C32×C8C22×C312yes—
73C4⋊(C3⋊Dic3)—21no—
74C4⋊(C3⋊Dic3)C22×C312yes—
75C3×C4⋊Dic3—32no—
76C3×C4⋊Dic3—61no—
77C22×H(3,𝔽3)—21no—
78C22×H(3,𝔽3)—21no—
79C22×H(3,𝔽3)C22⋊C412yes—
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C2×(C3⋊D3)⋊C3—22no—
82C2×(C3⋊D3)⋊C3—41no—
83C2×(C3⋊D3)⋊C3—41no—
84C4×H(3,𝔽3)C22⋊C412yes—
85C4×H(3,𝔽3)C22×C411yescharacteristic
86C4×H(3,𝔽3)C2×D411yescharacteristic
87(C3⋊Dic3)⋊C3—22no—
88(C3⋊Dic3)⋊C3—42no—
89C2×C3×#41—91no—
90C2×#235—31no—
91C2×#235—91no—
92C2×C3×M16—31no—
93C2×C3×M16—31no—
94C2×C3×M16—61no—
95C2×C3×M16C3×D311yescharacteristic
96C2×C3×C4⋊C4—91no—
97C2×C4⋊Dic3—31no—
98C2×C4⋊Dic3—91no—
99C3×#103—92no—
100C3×#102—92no—
101#821—32no—
102#821—92no—
103#816—32no—
104#816—92no—
105C23×C32—21no—
106C23×C32D1211yescharacteristic
107C23×C32C3×D411yescharacteristic
108C22×C3⋊D3—21no—
109C22×C3×D3—61no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4C4×D312yes—
115C2×C32×C4D1211yescharacteristic
116C2×C32×C4#6712yes—
117C2×C32×C4C2×C3×C412yes—
118C2×C32×C4C3×D411yescharacteristic
119C2×C32×C4C3×D412yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C23×C311yescharacteristic
122#638—24no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—42no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)C2×C3×C412yescomplemented
128C4×C3⋊D3—24no—
129#634—21no—
130#634C2×C3×C412yescomplemented
131C3×#67—64no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—122no—
135C3×D12—32no—
136C3×D12—61no—
137C3×C4×D3—64no—
138C3×Dic6—32no—
139C3×Dic6—61no—
140C32×C8—24no—
141C32×C8—22no—
142C32×C8—22no—
143#417—91noSylow
144C2×H(3,𝔽3)—22no—
145C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
146C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
147C2×H(3,𝔽3)#13711yescharacteristic
148(C3⋊D3)⋊C3—42no—
149C3×#41—94no—
150C3×#41—182no—
151C2×C3×Q8—91no—
152C2×C3×D4—91no—
153C2×C3×D4—181no—
154C22×C3×C4—31no—
155C22×C3×C4—31no—
156C22×C3×C4—61no—
157C22×C3×C4—181no—
158C22×C3×C4—181no—
159C22×C3×C4C2×C3×D311yescharacteristic
160C2×#67—61no—
161C2×#67—181no—
162C22×Dic3—61no—
163C22×Dic3—181no—
164#235—34no—
165#235—62no—
166#235—94no—
167#235—182no—
168C2×D12—31no—
169C2×D12—91no—
170C2×C4×D3—61no—
171C2×C4×D3—181no—
172C2×Dic6—31no—
173C2×Dic6—91no—
174C3×M16—22no—
175C3×M16—62no—
176C3×M16—62no—
177C3×M16—64no—
178C2×C3×C8—32no—
179C2×C3×C8—32no—
180C2×C3×C8—62no—
181C2×C3×C8C2×C3×D312yes—
182C3×C4⋊C4—92no—
183C3×C4⋊C4—181no—
184C4⋊Dic3—32no—
185C4⋊Dic3—61no—
186C4⋊Dic3—92no—
187C4⋊Dic3—181no—
188C22×C32—21no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—22no—
193C22×C32—21no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32(C2×D3)⋊C412yes—
197C22×C32C3×C22⋊C412yes—
198C22×C32C2×D1211yescharacteristic
199C22×C32C2×C3×D411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—62no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C32×C4—22no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4(C2×D3)⋊C412yes—
210C32×C4C3×C22⋊C412yes—
211C32×C4C2×C4×D311yescharacteristic
212C32×C4C2×#6711yescharacteristic
213C32×C4C22×C3×C411yescharacteristic
214C32×C4C2×C3×D411yescharacteristic
215C3⋊Dic3—22no—
216C3⋊Dic3—42no—
217C3×Dic3—62no—
218C3×Dic3—122no—
219C2×#41—91no—
220C2×M16(C3⋊D3)⋊C311yescharacteristic, complemented
221C2×C4⋊C4—91no—
222#103—92no—
223#102—92no—
224H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
225C23×C3—31no—
226C23×C3—31no—
227C23×C3—61no—
228C23×C3—181no—
229C23×C3C3×D1211yescharacteristic
230C22×D3—61no—
231C22×D3—181no—
232C3×Q8—92no—
233C3×Q8—181no—
234C3×D4—92no—
235C3×D4—181no—
236C3×D4—184no—
237C2×C3×C4—32no—
238C2×C3×C4—32no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—31no—
244C2×C3×C4—31no—
245C2×C3×C4—62no—
246C2×C3×C4—61no—
247C2×C3×C4—62no—
248C2×C3×C4—61no—
249C2×C3×C4—184no—
250C2×C3×C4—181no—
251C2×C3×C4—182no—
252C2×C3×C4—362no—
253C2×C3×C4C3×C4×D312yes—
254C2×C3×C4C3×D1211yescharacteristic
255C2×C3×C4C3×#6712yes—
256C2×C3×C4C22×C3×D311yescharacteristic
257#67—64no—
258#67—184no—
259C2×Dic3—61no—
260C2×Dic3—62no—
261C2×Dic3—122no—
262C2×Dic3—181no—
263C2×Dic3—182no—
264C2×Dic3—362no—
265D12—32no—
266D12—61no—
267D12—92no—
268D12—181no—
269C4×D3—64no—
270C4×D3—184no—
271Dic6—32no—
272Dic6—61no—
273Dic6—92no—
274Dic6—181no—
275C3×C8—22no—
276C3×C8—62no—
277C3×C8—62no—
278C3×C8—64no—
279C2×C32—22no—
280C2×C32—22no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—42no—
285C2×C32#90811yescharacteristic
286C2×C32#90911yescharacteristic
287C2×C32C2×(C2×D3)⋊C411yescharacteristic
288C2×C32C2×C3×C22⋊C411yescharacteristic
289C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
290C2×C32C3×#13711yescharacteristic
291C3⋊D3—42no—
292C3×D3—122no—
293#41—94no—
294#41—182no—
295C2×Q8—91no—
296C2×D4—91no—
297C2×D4—181no—
298C22×C4—181no—
299C22×C4—181no—
300C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
301M16—22no—
302C2×C8C2×(C3⋊D3)⋊C312yes—
303C4⋊C4—92no—
304C4⋊C4—181no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—32no—
310C22×C3—32no—
311C22×C3—61no—
312C22×C3—61no—
313C22×C3—61no—
314C22×C3—61no—
315C22×C3—61no—
316C22×C3—62no—
317C22×C3—121no—
318C22×C3—121no—
319C22×C3—182no—
320C22×C3—361no—
321C22×C3—361no—
322C22×C3C3×(C2×D3)⋊C412yes—
323C22×C3C2×C3×D1211yescharacteristic
324C2×D3—62no—
325C2×D3—121no—
326C2×D3—121no—
327C2×D3—182no—
328C2×D3—361no—
329C2×D3—361no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—31no—
334C3×C4—32no—
335C3×C4—32no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—62no—
339C3×C4—182no—
340C3×C4—362no—
341C3×C4C3×(C2×D3)⋊C412yes—
342C3×C4C2×C3×C4×D311yescharacteristic
343C3×C4C2×C3×#6711yescharacteristic
344Dic3—62no—
345Dic3—122no—
346Dic3—182no—
347Dic3—362no—
348C32—21no—
349C32#513311yescharacteristic, complemented
350C32C3×#41711yescharacteristic, complemented
351C23—181no—
352C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
353Q8—92no—
354Q8—181no—
355D4—92no—
356D4—184no—
357D4—181no—
358C2×C4—181no—
359C2×C4—184no—
360C2×C4—182no—
361C2×C4—362no—
362C2×C4C4×(C3⋊D3)⋊C312yes—
363C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
364C2×C4#638⋊C312yes—
365C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
366C8—22no—
367C2×C3—22no—
368C2×C3—31no—
369C2×C3—31no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—62no—
375C2×C3—62no—
376C2×C3—61no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—122no—
380C2×C3—362no—
381C2×C3C3×#90811yescharacteristic
382C2×C3C3×#90911yescharacteristic
383C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
384D3—122no—
385D3—362no—
386C22—21no—
387C22—21no—
388C22—182no—
389C22—361no—
390C22—361no—
391C22#3752⋊C312yes—
392C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
393C4—182no—
394C4—362no—
395C4#3752⋊C312yes—
396C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
397C4C2×#638⋊C311yescharacteristic
398C3—31no—
399C3—31no—
400C3—61no—
401C3C3×#513311yescharacteristic
402C2—22no—
403C2—362no—
404C2#64117⋊C311yescharacteristic
405C2#64118⋊C311yescharacteristic
406C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#417
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×M16×H(3,𝔽3)C2semidirect
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. #63648⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #10592832C3semidirect
12. #3828⋊C3C4semidirect
23. C2×C32×M16D3semidirect
24. C2×C32×M16C2×C3semidirect
25. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
27. #63648C2×C3semidirect
28. #63643C2×C3semidirect
42. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
45. #634⋊C3C2×C4semidirect
56. #3828C3×C4semidirect
127. C4⋊(C3⋊D3)C2×C3×C4semidirect
130. #634C2×C3×C4semidirect
220. C2×M16(C3⋊D3)⋊C3semidirect
224. H(3,𝔽3)#417semidirect
349. C32#5133semidirect
350. C32C3×#417semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×M16×H(3,𝔽3)2
C2×C4⋊((C3⋊Dic3)⋊C3)2
#63648⋊C32
#63643⋊C32
#105928323
C3×#51333