Finite Group Databasebeta

← #10592470⋊C3 #13141788

Subgroups — #10592470⋊C3 #13141788

Order 1728.

617 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3830⋊C3C211yesmaximal, characteristic
2C2×#3743⋊C3C211yesmaximal, characteristic, complemented
3C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#102C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#63643⋊C3C211yesmaximal, characteristic, complemented
7#63637⋊C3C211yesmaximal, characteristic, complemented
8#10592470C311yesmaximal, characteristic, complemented
9C3×#4771—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×#638⋊C3—21no—
12C22×(C3⋊Dic3)⋊C3—21no—
13#3830⋊C3—22no—
14#3830⋊C3C414yescomplemented
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2211yescharacteristic
18C2×C8×H(3,𝔽3)C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
21#3750⋊C3—21no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
23#3743⋊C3—22no—
24C2×#3830C2×C311yescharacteristic
25C2×C3×#237—31no—
26C2×#3743C2×C311yescharacteristic, complemented
27C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
28C2×C3×ΓL(1,𝔽25)—31no—
29C3×D3×C4⋊C4—31no—
30C32×#102—21no—
31C32×#102D311yescharacteristic, complemented
32C32×#102C2×C311yescharacteristic, complemented
33#63659C2×C311yescharacteristic, complemented
34#63643C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36C3×#832—31no—
37C3×#816—31no—
38C3×#810—31no—
39C23×H(3,𝔽3)—21no—
40C22×(C3⋊D3)⋊C3D411yescharacteristic
41D4×H(3,𝔽3)—21no—
42D4×H(3,𝔽3)C2×C412yescomplemented
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45#638⋊C3—24no—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—24no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3D411yescharacteristic
50C4×(C3⋊D3)⋊C3—42no—
51C4×(C3⋊D3)⋊C3C2×C412yes—
52C4×(C3⋊D3)⋊C3D412yes—
53#634⋊C3—21no—
54#634⋊C3C2×C412yescomplemented
55C8×H(3,𝔽3)—21no—
56(C3⋊(C3⋊C8))⋊C3—21no—
57C3×#417—91no—
58#4771—31no—
59#4771—91no—
60C2×C32×D4—21no—
61C2×C32×D4C2×D311yescharacteristic
62C2×C32×D4C22×C311yescharacteristic
63C2×#638—21no—
64C22×C3⋊Dic3—21no—
65#3830—22no—
66#3830C3×C414yescomplemented
67C2×C4×C3⋊D3—21no—
68C2×C4×C3⋊D3C22×C311yescharacteristic
69C2×#634C22×C311yescharacteristic
70C2×C3×#67—61no—
71C22×C3×Dic3—61no—
72C3×#237—34no—
73C3×#237—62no—
74C2×C3×C4×D3—31no—
75C2×C3×C4×D3—61no—
76C2×C3×Dic6—31no—
77C2×C32×C8—21no—
78C2×C32×C8C2×D311yescharacteristic
79C2×C32×C8C22×C311yescharacteristic
80C32×C4⋊C4—21no—
81C32×C4⋊C4C2×D311yescharacteristic
82C32×C4⋊C4C22×C311yescharacteristic
83C4⋊(C3⋊Dic3)C22×C311yescharacteristic
84#3750—21no—
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
86#3743—22no—
87C3×C4⋊Dic3—31no—
88C3×Dic3⋊C4—61no—
89C2×C3×C3⋊C8—31no—
90C3×ΓL(1,𝔽25)—62no—
91C22×H(3,𝔽3)—22no—
92C22×H(3,𝔽3)—41no—
93C22×H(3,𝔽3)—41no—
94C22×H(3,𝔽3)C2×D411yescharacteristic
95C2×(C3⋊D3)⋊C3—21no—
96C2×(C3⋊D3)⋊C3—21no—
97C2×(C3⋊D3)⋊C3C22⋊C412yes—
98C4×H(3,𝔽3)—41no—
99C4×H(3,𝔽3)C22×C411yescharacteristic
100C4×H(3,𝔽3)C2×D411yescharacteristic
101(C3⋊Dic3)⋊C3—22no—
102(C3⋊Dic3)⋊C3—41no—
103(C3⋊Dic3)⋊C3C22⋊C412yes—
104C2×C3×#41—91no—
105C2×#237—31no—
106C2×#237—91no—
107C2×C3×M16—91no—
108C2×C3×C4⋊C4—91no—
109C2×ΓL(1,𝔽25)—31no—
110C2×ΓL(1,𝔽25)—91no—
111D3×C4⋊C4—31no—
112D3×C4⋊C4—91no—
113C3×#103—91no—
114C3×#103—91no—
115C3×#102—31no—
116C3×#102—31no—
117C3×#102—61no—
118C3×#102—91no—
119C3×#102C3×D311yescharacteristic
120#832—31no—
121#832—91no—
122#816—31no—
123#816—91no—
124#810—31no—
125#810—91no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—41no—
129C22×C3⋊D3C3×D411yescharacteristic
130C22×C3×D3—31no—
131C32×D4—21no—
132C32×D4—21no—
133C32×D4—22no—
134C32×D4—41no—
135C32×D4C4×D312yescomplemented
136C32×D4C2×C3×C412yescomplemented
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C23×C311yescharacteristic
143#638—24no—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—24no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3C3×D411yescharacteristic
148C4×C3⋊D3—42no—
149C4×C3⋊D3C2×C3×C412yes—
150C4×C3⋊D3C3×D412yes—
151#634—21no—
152#634C2×C3×C412yescomplemented
153C3×#67—64no—
154C2×C3×Dic3—31no—
155C2×C3×Dic3—64no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3—61no—
158C3×C4×D3—32no—
159C3×C4×D3—32no—
160C3×C4×D3—122no—
161C3×Dic6—32no—
162C3×Dic6—61no—
163C32×C8—21no—
164C32×C8—21no—
165C32×C8—41no—
166C3⋊(C3⋊C8)—21no—
167C3×C3⋊C8—61no—
168#417—91noSylow
169C2×H(3,𝔽3)—42no—
170C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
171C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
172C2×H(3,𝔽3)#13711yescharacteristic
173(C3⋊D3)⋊C3—22no—
174C3×#41—94no—
175C3×#41—182no—
176C2×C3×Q8—91no—
177C2×C3×D4—31no—
178C2×C3×D4—31no—
179C2×C3×D4—61no—
180C2×C3×D4—181no—
181C2×C3×D4C2×C3×D311yescharacteristic
182C22×C3×C4—91no—
183C22×C3×C4—181no—
184C22×C3×C4—181no—
185C2×#67—61no—
186C2×#67—181no—
187C22×Dic3—61no—
188C22×Dic3—181no—
189#237—34no—
190#237—62no—
191#237—94no—
192#237—182no—
193C2×C4×D3—31no—
194C2×C4×D3—61no—
195C2×C4×D3—91no—
196C2×C4×D3—181no—
197C2×Dic6—31no—
198C2×Dic6—91no—
199C3×M16—182no—
200C2×C3×C8—31no—
201C2×C3×C8—31no—
202C2×C3×C8—61no—
203C2×C3×C8—91no—
204C2×C3×C8C2×C3×D311yescharacteristic
205C3×C4⋊C4—31no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—91no—
209C3×C4⋊C4—181no—
210C3×C4⋊C4C2×C3×D311yescharacteristic
211C4⋊Dic3—31no—
212C4⋊Dic3—91no—
213Dic3⋊C4—61no—
214Dic3⋊C4—181no—
215C2×C3⋊C8—31no—
216C2×C3⋊C8—91no—
217ΓL(1,𝔽25)—62no—
218ΓL(1,𝔽25)—182no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—42no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—81no—
228C22×C32—81no—
229C22×C32D3×D411yescharacteristic
230C22×C32C2×C3×D411yescharacteristic
231C2×C3⋊D3—21no—
232C2×C3⋊D3—21no—
233C2×C3⋊D3C3×C22⋊C412yes—
234C2×C3×D3—32no—
235C2×C3×D3—61no—
236C2×C3×D3—61no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—41no—
240C32×C4—41no—
241C32×C4—81no—
242C32×C4C2×C4×D311yescharacteristic
243C32×C4D3×D411yescharacteristic
244C32×C4C22×C3×C411yescharacteristic
245C32×C4C2×C3×D411yescharacteristic
246C3⋊Dic3—22no—
247C3⋊Dic3—41no—
248C3⋊Dic3C3×C22⋊C412yes—
249C3×Dic3—32no—
250C3×Dic3—62no—
251C3×Dic3—121no—
252C2×#41—91no—
253C2×M16—91no—
254C2×C4⋊C4—91no—
255#103—91no—
256#103—91no—
257#102—91no—
258#102(C3⋊D3)⋊C311yescharacteristic, complemented
259H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
260C23×C3—21no—
261C23×C3—61no—
262C23×C3—61no—
263C23×C3—91no—
264C23×C3—121no—
265C22×D3—31no—
266C22×D3—91no—
267C3×Q8—92no—
268C3×Q8—181no—
269C3×D4—21no—
270C3×D4—32no—
271C3×D4—32no—
272C3×D4—61no—
273C3×D4—61no—
274C3×D4—62no—
275C3×D4—121no—
276C3×D4—184no—
277C3×D4C3×C4×D312yes—
278C2×C3×C4—21no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—92no—
285C2×C3×C4—92no—
286C2×C3×C4—91no—
287C2×C3×C4—121no—
288C2×C3×C4—184no—
289C2×C3×C4—181no—
290C2×C3×C4—181no—
291C2×C3×C4—362no—
292C2×C3×C4C22×C3×D311yescharacteristic
293#67—64no—
294#67—184no—
295C2×Dic3—31no—
296C2×Dic3—64no—
297C2×Dic3—61no—
298C2×Dic3—61no—
299C2×Dic3—91no—
300C2×Dic3—184no—
301C2×Dic3—181no—
302C2×Dic3—181no—
303C4×D3—32no—
304C4×D3—32no—
305C4×D3—92no—
306C4×D3—92no—
307C4×D3—122no—
308C4×D3—362no—
309Dic6—32no—
310Dic6—61no—
311Dic6—92no—
312Dic6—181no—
313C3×C8—21no—
314C3×C8—61no—
315C3×C8—61no—
316C3×C8—121no—
317C3×C8—181no—
318C3⋊C8—61no—
319C3⋊C8—181no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—42no—
324C2×C32—42no—
325C2×C32—82no—
326C2×C32D3×C22⋊C411yescharacteristic
327C2×C32#91111yescharacteristic
328C2×C32#91511yescharacteristic
329C2×C32C2×C3×C22⋊C411yescharacteristic
330C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
331C2×C32C3×#13711yescharacteristic
332C3⋊D3—22no—
333C3×D3—62no—
334#41—94no—
335#41—182no—
336C2×Q8—91no—
337C2×D4—181no—
338C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
339C22×C4—91no—
340C22×C4—181no—
341C22×C4—181no—
342M16—182no—
343C2×C8—91no—
344C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
345C4⋊C4—91no—
346C4⋊C4—181no—
347C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
348C22×C3—22no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—62no—
354C22×C3—62no—
355C22×C3—61no—
356C22×C3—92no—
357C22×C3—122no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—181no—
363C22×C3—181no—
364C22×C3—241no—
365C22×C3—241no—
366C22×C3C3×D3×D411yescharacteristic
367C2×D3—32no—
368C2×D3—61no—
369C2×D3—61no—
370C2×D3—92no—
371C2×D3—181no—
372C2×D3—181no—
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—41no—
378C3×C4—61no—
379C3×C4—61no—
380C3×C4—92no—
381C3×C4—121no—
382C3×C4—121no—
383C3×C4—182no—
384C3×C4—241no—
385C3×C4—361no—
386C3×C4C2×C3×C4×D311yescharacteristic
387C3×C4C3×D3×D411yescharacteristic
388Dic3—32no—
389Dic3—62no—
390Dic3—92no—
391Dic3—121no—
392Dic3—182no—
393Dic3—361no—
394C32—21no—
395C32#477111yescharacteristic, complemented
396C32C3×#41711yescharacteristic, complemented
397C23—21no—
398C23—91no—
399Q8—92no—
400Q8—181no—
401D4—21no—
402D4—184no—
403D4C4×(C3⋊D3)⋊C312yescomplemented
404C2×C4—21no—
405C2×C4—91no—
406C2×C4—92no—
407C2×C4—92no—
408C2×C4—184no—
409C2×C4—181no—
410C2×C4—181no—
411C2×C4—362no—
412C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
413C8—21no—
414C8—181no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—42no—
422C2×C3—61no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—122no—
426C2×C3—122no—
427C2×C3—182no—
428C2×C3—242no—
429C2×C3C3×D3×C22⋊C411yescharacteristic
430C2×C3C3×#91111yescharacteristic
431C2×C3C3×#91511yescharacteristic
432D3—62no—
433D3—182no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—92no—
438C22—181no—
439C22—181no—
440C22D4×(C3⋊D3)⋊C311yescharacteristic
441C4—41no—
442C4—92no—
443C4—182no—
444C4—361no—
445C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
446C4D4×(C3⋊D3)⋊C311yescharacteristic
447C3—31no—
448C3—31no—
449C3—61no—
450C3C3×#477111yescharacteristic
451C2—42no—
452C2—182no—
453C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
454C2#64120⋊C311yescharacteristic
455C2#64124⋊C311yescharacteristic
subgroupG/H
centerC22D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×#3830⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#102(C3⋊D3)⋊C3
hypocenterC32C3×#417
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#3743⋊C3C2semidirect
3. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
4. H(3,𝔽3)×#102C2semidirect
5. #63659⋊C3C2semidirect
6. #63643⋊C3C2semidirect
7. #63637⋊C3C2semidirect
8. #10592470C3semidirect
14. #3830⋊C3C4semidirect
26. C2×#3743C2×C3semidirect
27. C4⋊C4×C3⋊D3C2×C3semidirect
31. C32×#102D3semidirect
32. C32×#102C2×C3semidirect
33. #63659C2×C3semidirect
34. #63643C2×C3semidirect
35. #63637C2×C3semidirect
42. D4×H(3,𝔽3)C2×C4semidirect
54. #634⋊C3C2×C4semidirect
66. #3830C3×C4semidirect
135. C32×D4C4×D3semidirect
136. C32×D4C2×C3×C4semidirect
152. #634C2×C3×C4semidirect
258. #102(C3⋊D3)⋊C3semidirect
259. H(3,𝔽3)#417semidirect
395. C32#4771semidirect
396. C32C3×#417semidirect
403. D4C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
C2×#3743⋊C32
C4⋊C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1022
#63659⋊C32
#63643⋊C32
#63637⋊C32
#105924703
C3×#47713