Finite Group Databasebeta

← #10592705⋊C3 #13142699

Subgroups — #10592705⋊C3 #13142699

Order 1728.

629 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#63633⋊C3C214yesmaximal, complemented
5#10592705C311yesmaximal, characteristic, complemented
6C3×#5006—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3C2212yes—
9#3828⋊C3—21no—
10#3828⋊C3—21no—
11#3828⋊C3—21no—
12#3828⋊C3—21no—
13C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
14C2×C4×(C3⋊D3)⋊C3C412yescomplemented
15C2×#634⋊C3C411yescharacteristic, complemented
16C4⋊C4×H(3,𝔽3)—21no—
17C22⋊C4×H(3,𝔽3)C2212yes—
18C42×H(3,𝔽3)—21no—
19#3757⋊C3C2212yes—
20C4⋊((C3⋊Dic3)⋊C3)—21no—
21C4×(C3⋊Dic3)⋊C3—21no—
22C2×#3828C2×C311yescharacteristic
23C2×C3×#235—31no—
24C32×#117—21no—
25C32×#117D311yescharacteristic, complemented
26C32×#117C2×C311yescharacteristic, complemented
27#64135C2×C311yescharacteristic, complemented
28C3×#926—31no—
29#63633C2×C314yescomplemented
30C3×#806—34no—
31C23×H(3,𝔽3)C2311yescharacteristic
32C22×(C3⋊D3)⋊C3C2×C412yes—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)C2×C411yescharacteristic
35C2×C4×H(3,𝔽3)C2×C411yescharacteristic
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)D411yescharacteristic
40#638⋊C3—22no—
41#638⋊C3—22no—
42C2×(C3⋊Dic3)⋊C3—22no—
43C2×(C3⋊Dic3)⋊C3C2×C412yes—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)—21no—
46C4×(C3⋊D3)⋊C3—22no—
47C4×(C3⋊D3)⋊C3—22no—
48#634⋊C3—21no—
49#634⋊C3—21no—
50C3×#410—91no—
51#5006—31no—
52#5006—91no—
53C22×C32×C4—21no—
54C22×C32×C4C2×D311yescharacteristic
55C22×C32×C4C22×C311yescharacteristic
56C2×#638C22×C312yes—
57#3828—21no—
58#3828—21no—
59#3828—21no—
60#3828—21no—
61C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
62C2×C4×C3⋊D3C3×C412yescomplemented
63C2×#634C3×C411yescharacteristic, complemented
64C2×C3×#67—32no—
65C3×#235—61no—
66C3×#235—61no—
67C3×#235—61no—
68C3×#235—61no—
69C2×C3×D12—31no—
70C2×C3×C4×D3—32no—
71C2×C3×Dic6—31no—
72C32×C4⋊C4—22no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—21no—
75C32×C22⋊C4—22no—
76C32×C22⋊C4C2×D312yes—
77C32×C22⋊C4C22×C312yes—
78C32×C42—22no—
79C32×C42—21no—
80C32×C42—21no—
81#3757C22×C312yes—
82C4⋊(C3⋊Dic3)—21no—
83C4×C3⋊Dic3—21no—
84C3×#217—32no—
85C3×C4⋊Dic3—61no—
86C3×C4×Dic3—61no—
87C22×H(3,𝔽3)—41no—
88C22×H(3,𝔽3)C22×C411yescharacteristic
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3—42no—
93C4×H(3,𝔽3)—42no—
94C4×H(3,𝔽3)C22⋊C411yescharacteristic
95C4×H(3,𝔽3)C22⋊C411yescharacteristic
96C4×H(3,𝔽3)C22⋊C411yescharacteristic
97C4×H(3,𝔽3)C22⋊C411yescharacteristic
98(C3⋊Dic3)⋊C3—22no—
99(C3⋊Dic3)⋊C3—42no—
100C2×C3×#41—91no—
101C2×#235—31no—
102C2×#235—91no—
103C3×#117—31no—
104C3×#117—31no—
105C3×#117—61no—
106C3×#117—91no—
107C3×#117C3×D311yescharacteristic
108#926—31no—
109#926—91no—
110C3×#99—94no—
111#806—34no—
112#806—94no—
113C23×C32—21no—
114C23×C32C22×D311yescharacteristic
115C23×C32C23×C311yescharacteristic
116C22×C3⋊D3C2×C3×C412yes—
117C22×C3×D3—32no—
118C2×C32×C4—24no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4C4×D311yescharacteristic
128C2×C32×C4C4×D311yescharacteristic
129C2×C32×C4D1211yescharacteristic
130C2×C32×C4D1211yescharacteristic
131C2×C32×C4#6711yescharacteristic
132C2×C32×C4#6711yescharacteristic
133C2×C32×C4C2×C3×C411yescharacteristic
134C2×C32×C4C2×C3×C411yescharacteristic
135C2×C32×C4C3×D411yescharacteristic
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C3×D411yescharacteristic
138C2×C32×C4C3×D411yescharacteristic
139#638—22no—
140#638—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3C2×C3×C412yes—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)—21no—
145C4×C3⋊D3—22no—
146C4×C3⋊D3—22no—
147#634—21no—
148#634—21no—
149C3×#67—62no—
150C3×#67—62no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—62no—
153C3×D12—61no—
154C3×D12—61no—
155C3×C4×D3—62no—
156C3×C4×D3—62no—
157C3×Dic6—61no—
158C3×Dic6—61no—
159#410—91noSylow
160C2×H(3,𝔽3)—21no—
161C2×H(3,𝔽3)—21no—
162C2×H(3,𝔽3)—21no—
163C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
164(C3⋊D3)⋊C3—42no—
165C3×#41—181no—
166C3×#41—181no—
167C3×#41—181no—
168C3×#41—181no—
169C2×C3×Q8—91no—
170C2×C3×D4—91no—
171C2×C3×D4—92no—
172C22×C3×C4—31no—
173C22×C3×C4—31no—
174C22×C3×C4—61no—
175C22×C3×C4—92no—
176C22×C3×C4C2×C3×D311yescharacteristic
177C2×#67—32no—
178C2×#67—92no—
179#235—61no—
180#235—61no—
181#235—61no—
182#235—61no—
183#235—181no—
184#235—181no—
185#235—181no—
186#235—181no—
187C2×D12—31no—
188C2×D12—91no—
189C2×C4×D3—32no—
190C2×C4×D3—92no—
191C2×Dic6—31no—
192C2×Dic6—91no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—62no—
197C3×C4⋊C4—181no—
198C3×C22⋊C4—32no—
199C3×C22⋊C4—32no—
200C3×C22⋊C4—62no—
201C3×C22⋊C4—92no—
202C3×C22⋊C4C2×C3×D312yes—
203C3×C42—21no—
204C3×C42—61no—
205C3×C42—61no—
206C3×C42—62no—
207C3×C42—181no—
208#217—32no—
209#217—92no—
210C4⋊Dic3—61no—
211C4⋊Dic3—181no—
212C4×Dic3—61no—
213C4×Dic3—181no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—42no—
220C22×C32C2×C4×D311yescharacteristic
221C22×C32C2×D1211yescharacteristic
222C22×C32C2×#6711yescharacteristic
223C22×C32C22×C3×C411yescharacteristic
224C22×C32C2×C3×D411yescharacteristic
225C22×C32C2×C3×D411yescharacteristic
226C2×C3⋊D3—22no—
227C2×C3⋊D3—42no—
228C2×C3×D3—62no—
229C2×C3×D3—122no—
230C32×C4—21no—
231C32×C4—21no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—44no—
235C32×C4—42no—
236C32×C4—42no—
237C32×C4(C2×D3)⋊C411yescharacteristic
238C32×C4(C2×D3)⋊C411yescharacteristic
239C32×C4(C2×D3)⋊C411yescharacteristic
240C32×C4(C2×D3)⋊C411yescharacteristic
241C32×C4C3×C22⋊C411yescharacteristic
242C32×C4C3×C22⋊C411yescharacteristic
243C32×C4C3×C22⋊C411yescharacteristic
244C32×C4C3×C22⋊C411yescharacteristic
245C3⋊Dic3—22no—
246C3⋊Dic3—42no—
247C3×Dic3—62no—
248C3×Dic3—122no—
249C2×#41—91no—
250#117—91no—
251#117(C3⋊D3)⋊C311yescharacteristic, complemented
252#99—94no—
253H(3,𝔽3)#41011yescharacteristic, complemented, Sylow
254C23×C3—31no—
255C23×C3—31no—
256C23×C3—61no—
257C23×C3—92no—
258C23×C3C22×C3×D311yescharacteristic
259C22×D3—32no—
260C22×D3—92no—
261C3×Q8—181no—
262C3×Q8—181no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—182no—
266C3×D4—182no—
267C2×C3×C4—22no—
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—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—64no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—92no—
290C2×C3×C4—182no—
291C2×C3×C4—182no—
292C2×C3×C4—182no—
293C2×C3×C4C3×C4×D311yescharacteristic
294C2×C3×C4C3×C4×D311yescharacteristic
295C2×C3×C4C3×D1211yescharacteristic
296C2×C3×C4C3×D1211yescharacteristic
297C2×C3×C4C3×#6711yescharacteristic
298C2×C3×C4C3×#6711yescharacteristic
299#67—62no—
300#67—62no—
301#67—182no—
302#67—182no—
303C2×Dic3—32no—
304C2×Dic3—62no—
305C2×Dic3—92no—
306C2×Dic3—182no—
307D12—61no—
308D12—61no—
309D12—181no—
310D12—181no—
311C4×D3—62no—
312C4×D3—62no—
313C4×D3—182no—
314C4×D3—182no—
315Dic6—61no—
316Dic6—61no—
317Dic6—181no—
318Dic6—181no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32C2×(C2×D3)⋊C411yescharacteristic
330C2×C32C2×C3×C22⋊C411yescharacteristic
331C3⋊D3—42no—
332C3×D3—122no—
333#41—181no—
334#41—181no—
335#41—181no—
336#41—181no—
337C2×Q8—91no—
338C2×D4—92no—
339C2×D4—91no—
340C22×C4—92no—
341C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
342C4⋊C4—21no—
343C4⋊C4—181no—
344C22⋊C4—92no—
345C22⋊C4C2×(C3⋊D3)⋊C312yes—
346C42—21no—
347C42—181no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—31no—
351C22×C3—31no—
352C22×C3—31no—
353C22×C3—31no—
354C22×C3—41no—
355C22×C3—61no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—121no—
359C22×C3—121no—
360C22×C3—122no—
361C22×C3—182no—
362C22×C3—362no—
363C22×C3C2×C3×C4×D311yescharacteristic
364C22×C3C2×C3×D1211yescharacteristic
365C22×C3C2×C3×#6711yescharacteristic
366C2×D3—62no—
367C2×D3—122no—
368C2×D3—182no—
369C2×D3—362no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—42no—
379C3×C4—61no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—61no—
383C3×C4—122no—
384C3×C4—122no—
385C3×C4—124no—
386C3×C4—182no—
387C3×C4—362no—
388C3×C4C3×(C2×D3)⋊C411yescharacteristic
389C3×C4C3×(C2×D3)⋊C411yescharacteristic
390C3×C4C3×(C2×D3)⋊C411yescharacteristic
391C3×C4C3×(C2×D3)⋊C411yescharacteristic
392Dic3—62no—
393Dic3—122no—
394Dic3—182no—
395Dic3—362no—
396C32—21no—
397C32#500611yescharacteristic, complemented
398C32C3×#41011yescharacteristic, complemented
399C23—92no—
400C23C22×(C3⋊D3)⋊C311yescharacteristic
401Q8—181no—
402Q8—181no—
403D4—182no—
404D4—182no—
405D4—181no—
406D4—181no—
407C2×C4—22no—
408C2×C4—92no—
409C2×C4—182no—
410C2×C4—182no—
411C2×C4—182no—
412C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
413C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
414C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
415C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
416C2×C4#638⋊C311yescharacteristic
417C2×C4#638⋊C311yescharacteristic
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—61no—
426C2×C3—61no—
427C2×C3—61no—
428C2×C3—61no—
429C2×C3—62no—
430C2×C3—61no—
431C2×C3—121no—
432C2×C3—121no—
433C2×C3—362no—
434C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
435D3—122no—
436D3—362no—
437C22—41no—
438C22—182no—
439C22—362no—
440C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
441C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
442C22C2×#638⋊C311yescharacteristic
443C4—42no—
444C4—182no—
445C4—362no—
446C4#3752⋊C311yescharacteristic
447C4#3752⋊C311yescharacteristic
448C4#3752⋊C311yescharacteristic
449C4#3752⋊C311yescharacteristic
450C3—31no—
451C3—31no—
452C3—61no—
453C3C3×#500611yescharacteristic
454C2—21no—
455C2—21no—
456C2—21no—
457C2—362no—
458C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC4#3752⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC23×C3C22×C3×D3
FittingH(3,𝔽3)×#117C2
socleC2×C3C2×C3×(C2×D3)⋊C4
prime radicalC2×#3828⋊C3C2
cosocleC23×C32C23×C3
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#410
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#117C2semidirect
3. #64135⋊C3C2semidirect
4. #63633⋊C3C2semidirect
5. #10592705C3semidirect
13. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
14. C2×C4×(C3⋊D3)⋊C3C4semidirect
15. C2×#634⋊C3C4semidirect
25. C32×#117D3semidirect
26. C32×#117C2×C3semidirect
27. #64135C2×C3semidirect
29. #63633C2×C3semidirect
61. C2×C4⋊(C3⋊D3)C3×C4semidirect
62. C2×C4×C3⋊D3C3×C4semidirect
63. C2×#634C3×C4semidirect
251. #117(C3⋊D3)⋊C3semidirect
253. H(3,𝔽3)#410semidirect
397. C32#5006semidirect
398. C32C3×#410semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
H(3,𝔽3)×#1172
#64135⋊C32
#63633⋊C32
#105927053
C3×#50063