Finite Group Databasebeta

← #10592838⋊C3 #13143220

Subgroups — #10592838⋊C3 #13143220

Order 1728.

607 conjugacy classes of subgroups.

The subgroup lattice has 451 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
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#63652⋊C3C214yesmaximal, complemented
5#10592838C311yesmaximal, characteristic, complemented
6C3×#5139—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3—21no—
9#3828⋊C3—21no—
10#3828⋊C3—21no—
11#3828⋊C3C412yescomplemented
12#3828⋊C3C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
14C2×C4×(C3⋊D3)⋊C3—21no—
15C2×#634⋊C3C411yescharacteristic, complemented
16M16×H(3,𝔽3)—21no—
17M16×H(3,𝔽3)C2212yescomplemented
18C2×C8×H(3,𝔽3)—21no—
19#3757⋊C3—21no—
20C4⋊((C3⋊Dic3)⋊C3)—21no—
21C4×(C3⋊Dic3)⋊C3C2212yescomplemented
22C2×#3828C2×C311yescharacteristic
23C2×C3×#235—31no—
24C2×C32×M16—21no—
25C2×C32×M16D311yescharacteristic, complemented
26C2×C32×M16C2×C311yescharacteristic, complemented
27#64135C2×C311yescharacteristic, complemented
28C3×#926—31no—
29#63652C2×C314yescomplemented
30C3×#825—34no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)C2×C411yescharacteristic
34C2×C4×H(3,𝔽3)C2×C411yescharacteristic
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#638⋊C3—22no—
40#638⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)C2×C412yes—
45C4×(C3⋊D3)⋊C3—22no—
46C4×(C3⋊D3)⋊C3—22no—
47#634⋊C3—21no—
48#634⋊C3C2×C412yes—
49C8×H(3,𝔽3)—22no—
50C3×#421—91no—
51#5139—31no—
52#5139—91no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×#638—21no—
57#3828—21no—
58#3828—21no—
59#3828C3×C412yescomplemented
60#3828C22×C312yes—
61C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
62C2×C4×C3⋊D3—21no—
63C2×#634C3×C411yescharacteristic, complemented
64C2×C3×#67—61no—
65C3×#235—32no—
66C3×#235—32no—
67C3×#235—61no—
68C3×#235—61no—
69C2×C3×D12—31no—
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—31no—
72C32×M16—22no—
73C32×M16—22no—
74C32×M16—21no—
75C32×M16—21no—
76C32×M16C2×D312yescomplemented
77C32×M16C22×C312yescomplemented
78C2×C32×C8—22no—
79C2×C32×C8—21no—
80C2×C32×C8—21no—
81#3757—21no—
82C4⋊(C3⋊Dic3)—21no—
83C4×C3⋊Dic3C22×C312yescomplemented
84C3×#217—61no—
85C3×C4⋊Dic3—61no—
86C3×C4×Dic3—32no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)C22⋊C411yescharacteristic
89C22×H(3,𝔽3)C22⋊C411yescharacteristic
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3—42no—
93C4×H(3,𝔽3)C22⋊C411yescharacteristic
94C4×H(3,𝔽3)C22⋊C411yescharacteristic
95C4×H(3,𝔽3)C22×C411yescharacteristic
96C4×H(3,𝔽3)C2×D411yescharacteristic
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—42no—
99C2×C3×#41—91no—
100C2×#235—31no—
101C2×#235—91no—
102C2×C3×M16—31no—
103C2×C3×M16—31no—
104C2×C3×M16—61no—
105C2×C3×M16C3×D311yescharacteristic
106C3×#117—91no—
107#926—31no—
108#926—91no—
109C3×#104—94no—
110#825—34no—
111#825—94no—
112C23×C32—21no—
113C23×C32D1211yescharacteristic
114C23×C32C3×D411yescharacteristic
115C22×C3⋊D3—21no—
116C22×C3×D3—61no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4C4×D311yescharacteristic
124C2×C32×C4C4×D311yescharacteristic
125C2×C32×C4D1211yescharacteristic
126C2×C32×C4#6711yescharacteristic
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4C2×C3×C411yescharacteristic
129C2×C32×C4C2×C3×C411yescharacteristic
130C2×C32×C4C3×D411yescharacteristic
131C2×C32×C4C3×D411yescharacteristic
132C2×C32×C4C3×D411yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C23×C311yescharacteristic
135#638—22no—
136#638—22no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—22no—
139C4⋊(C3⋊D3)—21no—
140C4⋊(C3⋊D3)C2×C3×C412yes—
141C4×C3⋊D3—22no—
142C4×C3⋊D3—22no—
143#634—21no—
144#634C2×C3×C412yes—
145C3×#67—62no—
146C3×#67—62no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—62no—
149C3×D12—32no—
150C3×D12—61no—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×Dic6—32no—
154C3×Dic6—61no—
155C32×C8—24no—
156C32×C8—22no—
157C32×C8—22no—
158#421—91noSylow
159C2×H(3,𝔽3)—21no—
160C2×H(3,𝔽3)—21no—
161C2×H(3,𝔽3)—21no—
162C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
163(C3⋊D3)⋊C3—42no—
164C3×#41—92no—
165C3×#41—92no—
166C3×#41—181no—
167C3×#41—181no—
168C2×C3×Q8—91no—
169C2×C3×D4—91no—
170C2×C3×D4—181no—
171C22×C3×C4—31no—
172C22×C3×C4—31no—
173C22×C3×C4—61no—
174C22×C3×C4—181no—
175C22×C3×C4C2×C3×D311yescharacteristic
176C2×#67—61no—
177C2×#67—181no—
178#235—32no—
179#235—32no—
180#235—61no—
181#235—61no—
182#235—92no—
183#235—92no—
184#235—181no—
185#235—181no—
186C2×D12—31no—
187C2×D12—91no—
188C2×C4×D3—61no—
189C2×C4×D3—181no—
190C2×Dic6—31no—
191C2×Dic6—91no—
192C3×M16—21no—
193C3×M16—32no—
194C3×M16—32no—
195C3×M16—61no—
196C3×M16—61no—
197C3×M16—62no—
198C3×M16—62no—
199C3×M16C2×C3×D312yes—
200C2×C3×C8—21no—
201C2×C3×C8—61no—
202C2×C3×C8—61no—
203C2×C3×C8—62no—
204C3×C4⋊C4—181no—
205C3×C22⋊C4—181no—
206C3×C42—92no—
207#217—61no—
208#217—181no—
209C4⋊Dic3—61no—
210C4⋊Dic3—181no—
211C4×Dic3—32no—
212C4×Dic3—92no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32(C2×D3)⋊C411yescharacteristic
220C22×C32(C2×D3)⋊C411yescharacteristic
221C22×C32C3×C22⋊C411yescharacteristic
222C22×C32C3×C22⋊C411yescharacteristic
223C22×C32C2×D1211yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C2×C3⋊D3—22no—
226C2×C3⋊D3—42no—
227C2×C3×D3—62no—
228C2×C3×D3—122no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4(C2×D3)⋊C411yescharacteristic
234C32×C4(C2×D3)⋊C411yescharacteristic
235C32×C4C3×C22⋊C411yescharacteristic
236C32×C4C3×C22⋊C411yescharacteristic
237C32×C4C2×C4×D311yescharacteristic
238C32×C4C2×#6711yescharacteristic
239C32×C4C22×C3×C411yescharacteristic
240C32×C4C2×C3×D411yescharacteristic
241C3⋊Dic3—22no—
242C3⋊Dic3—42no—
243C3×Dic3—62no—
244C3×Dic3—122no—
245C2×#41—91no—
246C2×M16(C3⋊D3)⋊C311yescharacteristic, complemented
247#117—91no—
248#104—94no—
249H(3,𝔽3)#42111yescharacteristic, complemented, Sylow
250C23×C3—31no—
251C23×C3—31no—
252C23×C3—61no—
253C23×C3—181no—
254C23×C3C3×D1211yescharacteristic
255C22×D3—61no—
256C22×D3—181no—
257C3×Q8—92no—
258C3×Q8—181no—
259C3×D4—92no—
260C3×D4—181no—
261C3×D4—182no—
262C3×D4—182no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—182no—
282C2×C3×C4—182no—
283C2×C3×C4—181no—
284C2×C3×C4—182no—
285C2×C3×C4C3×C4×D311yescharacteristic
286C2×C3×C4C3×C4×D311yescharacteristic
287C2×C3×C4C3×D1211yescharacteristic
288C2×C3×C4C3×#6711yescharacteristic
289C2×C3×C4C3×#6711yescharacteristic
290C2×C3×C4C22×C3×D311yescharacteristic
291#67—62no—
292#67—62no—
293#67—182no—
294#67—182no—
295C2×Dic3—61no—
296C2×Dic3—62no—
297C2×Dic3—181no—
298C2×Dic3—182no—
299D12—32no—
300D12—61no—
301D12—92no—
302D12—181no—
303C4×D3—62no—
304C4×D3—62no—
305C4×D3—182no—
306C4×D3—182no—
307Dic6—32no—
308Dic6—61no—
309Dic6—92no—
310Dic6—181no—
311C3×C8—22no—
312C3×C8—62no—
313C3×C8—62no—
314C3×C8—64no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—41no—
324C2×C32—41no—
325C2×C32C2×(C2×D3)⋊C411yescharacteristic
326C2×C32C2×C3×C22⋊C411yescharacteristic
327C3⋊D3—42no—
328C3×D3—122no—
329#41—92no—
330#41—92no—
331#41—181no—
332#41—181no—
333C2×Q8—91no—
334C2×D4—91no—
335C2×D4—181no—
336C22×C4—181no—
337C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
338M16—21no—
339M16C2×(C3⋊D3)⋊C312yescomplemented
340C2×C8—21no—
341C4⋊C4—181no—
342C22⋊C4—181no—
343C42—92no—
344C22×C3—31no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—31no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—41no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3—61no—
354C22×C3—121no—
355C22×C3—121no—
356C22×C3—122no—
357C22×C3—182no—
358C22×C3—362no—
359C22×C3C3×(C2×D3)⋊C411yescharacteristic
360C22×C3C3×(C2×D3)⋊C411yescharacteristic
361C22×C3C2×C3×D1211yescharacteristic
362C2×D3—62no—
363C2×D3—122no—
364C2×D3—182no—
365C2×D3—362no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—31no—
374C3×C4—61no—
375C3×C4—61no—
376C3×C4—61no—
377C3×C4—61no—
378C3×C4—182no—
379C3×C4—362no—
380C3×C4C3×(C2×D3)⋊C411yescharacteristic
381C3×C4C3×(C2×D3)⋊C411yescharacteristic
382C3×C4C2×C3×C4×D311yescharacteristic
383C3×C4C2×C3×#6711yescharacteristic
384Dic3—62no—
385Dic3—122no—
386Dic3—182no—
387Dic3—362no—
388C32—21no—
389C32#513911yescharacteristic, complemented
390C32C3×#42111yescharacteristic, complemented
391C23—181no—
392C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
393Q8—92no—
394Q8—181no—
395D4—92no—
396D4—182no—
397D4—182no—
398D4—181no—
399C2×C4—181no—
400C2×C4—182no—
401C2×C4—182no—
402C2×C4—182no—
403C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
404C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
405C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
406C2×C4#638⋊C311yescharacteristic
407C2×C4#638⋊C311yescharacteristic
408C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
409C8—22no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—61no—
416C2×C3—61no—
417C2×C3—61no—
418C2×C3—61no—
419C2×C3—61no—
420C2×C3—61no—
421C2×C3—62no—
422C2×C3—61no—
423C2×C3—121no—
424C2×C3—121no—
425C2×C3—362no—
426C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
427D3—122no—
428D3—362no—
429C22—41no—
430C22—182no—
431C22—362no—
432C22#3752⋊C311yescharacteristic
433C22#3752⋊C311yescharacteristic
434C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
435C4—182no—
436C4—362no—
437C4#3752⋊C311yescharacteristic
438C4#3752⋊C311yescharacteristic
439C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
440C4C2×#638⋊C311yescharacteristic
441C3—31no—
442C3—31no—
443C3—61no—
444C3C3×#513911yescharacteristic
445C2—21no—
446C2—21no—
447C2—21no—
448C2—362no—
449C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC4C2×#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC2×C3C2×C3×(C2×D3)⋊C4
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#421
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ 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. #64135⋊C3C2semidirect
4. #63652⋊C3C2semidirect
5. #10592838C3semidirect
11. #3828⋊C3C4semidirect
13. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
15. C2×#634⋊C3C4semidirect
17. M16×H(3,𝔽3)C22semidirect
21. C4×(C3⋊Dic3)⋊C3C22semidirect
25. C2×C32×M16D3semidirect
26. C2×C32×M16C2×C3semidirect
27. #64135C2×C3semidirect
29. #63652C2×C3semidirect
59. #3828C3×C4semidirect
61. C2×C4⋊(C3⋊D3)C3×C4semidirect
63. C2×#634C3×C4semidirect
76. C32×M16C2×D3semidirect
77. C32×M16C22×C3semidirect
83. C4×C3⋊Dic3C22×C3semidirect
246. C2×M16(C3⋊D3)⋊C3semidirect
249. H(3,𝔽3)#421semidirect
339. M16C2×(C3⋊D3)⋊C3semidirect
389. C32#5139semidirect
390. C32C3×#421semidirect

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
#64135⋊C32
#63652⋊C32
#105928383
C3×#51393