Finite Group Databasebeta

← #13156365

Subgroups — #13156365

Order 1728.

520 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723454C211yesmaximal, characteristic
2C3×#63844C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722054C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10590814—31nomaximal
9#10590544—31nomaximal
10#10590302—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88807—21no—
19#88806—21no—
20C3×#3724—21no—
21C3×Dic32—21no—
22#64079—31no—
23C3×#887—31no—
24C3×C4×Dic6—31no—
25#63869—31no—
26#63869—31no—
27#63869—61no—
28#63844D311yescharacteristic
29#63643—31no—
30#63635—31no—
31C3×#818—31no—
32C3×Dic3⋊C8—31no—
33C3×#807—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63575D311yescharacteristic
36#63573—31no—
37#63573—31no—
38#63573—61no—
39Dic3⋊(C3⋊C8)D311yescharacteristic
40#63567—31no—
41#63566—31no—
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63565—31no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634Q812yes—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C3×C3⋊(C3⋊C8)—21no—
56C32×C3⋊C8—21no—
57#5022—91no—
58#4877—91no—
59#4829—91no—
60C2×#634C2×D311yescharacteristic
61C2×C3×Dic6—31no—
62C2×C3×Dic6—31no—
63C2×C3×Dic6—61no—
64C2×C32×C8—31no—
65C32×C4⋊C4—31no—
66C32×C42—31no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—31no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—61no—
71C4⋊(C3⋊Dic3)—61no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)—121no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75#3750—61no—
76#3750—61no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3C2×D311yescharacteristic
83C3×C4⋊Dic3C2×D311yescharacteristic
84C3×Dic3⋊C4—61no—
85C3×Dic3⋊C4—61no—
86C3×C4×Dic3—21no—
87C3×C4×Dic3—61no—
88C3×C4×Dic3—61no—
89C3×C4×Dic3C2×D311yescharacteristic
90C3×C4×Dic3C2×D311yescharacteristic
91C2×C3×C3⋊C8—21no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—61no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97#3724—21no—
98#3724—62no—
99#3724—61no—
100#3724—61no—
101Dic3⋊Dic3—62no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104Dic32—21no—
105C22×C33C2×D411yescharacteristic
106C33×C4C2×Q811yescharacteristic
107C33×C4#4111yescharacteristic
108C3⋊(C3⋊Dic3)—41no—
109C3×C3⋊Dic3—22no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—21no—
112C32×Dic3—21no—
113C32×Dic3—41no—
114C3×C4×Q8—91no—
115#890—91no—
116#887—31no—
117#870—91no—
118#870—91no—
119#870—181no—
120C4×Dic6—31no—
121C3×#107—91no—
122C3×C4⋊C8—91no—
123#835—91no—
124#835—91no—
125#818—31no—
126#817—91no—
127#816—91no—
128#816—91no—
129#816—91no—
130#816—181no—
131Dic3⋊C8—31no—
132#810—91no—
133#808—91no—
134#808—91no—
135#808—181no—
136#807—31no—
137C4⋊(C3⋊C8)—31no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—31no—
142C2×C32×C4—31no—
143C2×C32×C4—41no—
144C2×C32×C4—61no—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3—61no—
151C2×C3⋊Dic3—62no—
152C2×C3⋊Dic3—61no—
153C2×C3⋊Dic3—61no—
154C2×C3⋊Dic3—121no—
155C2×C3⋊Dic3—121no—
156C2×C3⋊Dic3—122no—
157#634—21no—
158#634Dic612yes—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—41no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—121no—
170C2×C3×Dic3D1211yescharacteristic
171C2×C3×Dic3D1211yescharacteristic
172C2×C3×Dic3#6711yescharacteristic
173C2×C3×Dic3#6711yescharacteristic
174C3×Dic6—32no—
175C3×Dic6—32no—
176C3×Dic6—62no—
177C3×Dic6—61no—
178C3×Dic6—61no—
179C3×Dic6—121no—
180C32×C8—61no—
181C3⋊(C3⋊C8)—21no—
182C3×C3⋊C8—21no—
183C3×C3⋊C8—21no—
184C3×C3⋊C8—41no—
185C3×C3⋊C8—62no—
186C3×C3⋊C8—61no—
187C3×C3⋊C8—61no—
188#479—271noSylow
189C2×C33C22⋊Q811yescharacteristic
190C2×C33#13511yescharacteristic
191C2×C33#13711yescharacteristic
192C2×C3×Q8—91no—
193C2×Dic6—31no—
194C2×Dic6—31no—
195C2×Dic6—61no—
196C2×C3×C8—31no—
197C2×C3×C8—31no—
198C2×C3×C8—61no—
199C2×C3×C8—91no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—91no—
204C3×C4⋊C4—181no—
205C3×C42—31no—
206C3×C42—31no—
207C3×C42—61no—
208C3×C42—181no—
209C4⋊Dic3—31no—
210C4⋊Dic3—31no—
211C4⋊Dic3—61no—
212C4⋊Dic3—91no—
213C4⋊Dic3—91no—
214C4⋊Dic3—91no—
215C4⋊Dic3—181no—
216C4⋊Dic3—181no—
217C4⋊Dic3—181no—
218C4⋊Dic3—181no—
219C4⋊Dic3—361no—
220C4⋊Dic3D3211yescharacteristic
221Dic3⋊C4—61no—
222Dic3⋊C4—61no—
223Dic3⋊C4—181no—
224Dic3⋊C4—181no—
225Dic3⋊C4—181no—
226Dic3⋊C4—181no—
227Dic3⋊C4—182no—
228Dic3⋊C4—182no—
229Dic3⋊C4—181no—
230C4×Dic3—61no—
231C4×Dic3—61no—
232C4×Dic3D3211yescharacteristic
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—61no—
236C2×C3⋊C8D3211yescharacteristic
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—41no—
241C22×C32C2×D1211yescharacteristic
242C22×C32D3×D411yescharacteristic
243C22×C32C2×#6711yescharacteristic
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—21no—
250C32×C4—41no—
251C32×C4—41no—
252C32×C4—61no—
253C32×C4—61no—
254C32×C4—121no—
255C32×C4C2×Dic611yescharacteristic
256C32×C4#23511yescharacteristic
257C32×C4#23511yescharacteristic
258C32×C4#23711yescharacteristic
259C32×C4D3×Q811yescharacteristic
260C32×C4D3×Q811yescharacteristic
261C3⋊Dic3—22no—
262C3⋊Dic3—41no—
263C3⋊Dic3—121no—
264C3⋊Dic3—122no—
265C3⋊Dic3—121no—
266C3⋊Dic3—121no—
267C3⋊Dic3—241no—
268C3⋊Dic3—241no—
269C3⋊Dic3—242no—
270C3×Dic3—21no—
271C3×Dic3—21no—
272C3×Dic3—21no—
273C3×Dic3—21no—
274C3×Dic3—41no—
275C3×Dic3—41no—
276C3×Dic3—41no—
277C3×Dic3—41no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—81no—
281C3×Dic3—122no—
282C3×Dic3—122no—
283C3×Dic3—121no—
284C3×Dic3—121no—
285#125—271no—
286C4×Q8—91no—
287#107—91no—
288#106—271no—
289C4⋊C8—91no—
290#103—271no—
291#103—271no—
292C33#47911yescharacteristic, complemented, Sylow
293C3×Q8—92no—
294C3×Q8—181no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—31no—
299C2×C3×C4—31no—
300C2×C3×C4—31no—
301C2×C3×C4—31no—
302C2×C3×C4—41no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—121no—
308C2×C3×C4—181no—
309C2×C3×C4—181no—
310C2×C3×C4C2×D3211yescharacteristic
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3211yescharacteristic
313C2×Dic3—21no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—62no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—121no—
320C2×Dic3—182no—
321C2×Dic3—181no—
322C2×Dic3—181no—
323C2×Dic3—181no—
324C2×Dic3—361no—
325C2×Dic3—361no—
326C2×Dic3—362no—
327C2×Dic3#62611yescharacteristic
328C2×Dic3#62611yescharacteristic
329Dic6—32no—
330Dic6—32no—
331Dic6—62no—
332Dic6—61no—
333Dic6—61no—
334Dic6—121no—
335C3×C8—61no—
336C3×C8—61no—
337C3×C8—121no—
338C3×C8—181no—
339C3⋊C8—21no—
340C3⋊C8—62no—
341C3⋊C8—61no—
342C3⋊C8—61no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32#89611yescharacteristic
356C2×C32#89711yescharacteristic
357C2×C32#90411yescharacteristic
358C2×C32#90911yescharacteristic
359C2×C32#91211yescharacteristic
360C2×C32#91811yescharacteristic
361C2×C32#92411yescharacteristic
362C2×C32#94211yescharacteristic
363C2×C32#95011yescharacteristic
364C2×Q8—91no—
365C2×C8—31no—
366C2×C8—91no—
367C4⋊C4—31no—
368C4⋊C4—91no—
369C4⋊C4—181no—
370C4⋊C4—271no—
371C4⋊C4—541no—
372C4⋊C4—541no—
373C42—31no—
374C42—181no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—41no—
379C22×C3D3×D1211yescharacteristic
380C22×C3C2×#62611yescharacteristic
381C22×C3D3×#6711yescharacteristic
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—41no—
389C3×C4—41no—
390C3×C4—61no—
391C3×C4—61no—
392C3×C4—61no—
393C3×C4—61no—
394C3×C4—121no—
395C3×C4—121no—
396C3×C4—121no—
397C3×C4—121no—
398C3×C4—182no—
399C3×C4—241no—
400C3×C4—361no—
401C3×C4D3×Dic611yescharacteristic
402C3×C4D3×Dic611yescharacteristic
403C3×C4#379511yescharacteristic
404C3×C4#379711yescharacteristic
405C3×C4#379811yescharacteristic
406C3×C4#380411yescharacteristic
407Dic3—21no—
408Dic3—21no—
409Dic3—41no—
410Dic3—62no—
411Dic3—62no—
412Dic3—122no—
413Dic3—122no—
414Dic3—121no—
415Dic3—121no—
416Dic3—362no—
417Dic3—361no—
418Dic3—361no—
419Dic3—361no—
420Dic3—721no—
421Dic3—721no—
422Dic3—722no—
423C32—21no—
424C32—21no—
425C32—21no—
426C32—41no—
427C32#482911yescharacteristic, complemented
428C32#487711yescharacteristic, complemented
429C32#502211yescharacteristic, complemented
430Q8—92no—
431Q8—181no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—61no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—541no—
438C2×C4D3311yescharacteristic
439C8—61no—
440C8—181no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3#6380111yescharacteristic
454C2×C3#6380711yescharacteristic
455C2×C3#6382911yescharacteristic
456C2×C3#6383311yescharacteristic
457C2×C3#6385111yescharacteristic
458C2×C3#6385711yescharacteristic
459C2×C3#6389511yescharacteristic
460C2×C3#6393311yescharacteristic
461C2×C3#6394811yescharacteristic
462C22D3×#62611yescharacteristic
463C4—61no—
464C4—61no—
465C4—121no—
466C4—182no—
467C4—361no—
468C4—1081no—
469C4#8895711yescharacteristic
470C4#8896011yescharacteristic
471C3—21no—
472C3—21no—
473C3—21no—
474C3—41no—
475C3#1059030211yescharacteristic, complemented
476C3#1059054411yescharacteristic, complemented
477C3#1059081411yescharacteristic, complemented
478C2#1172285411yescharacteristic
479C2#1172288511yescharacteristic
480C2#1172297211yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#479
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
292. C33#479semidirect
427. C32#4829semidirect
428. C32#4877semidirect
429. C32#5022semidirect
475. C3#10590302semidirect
476. C3#10590544semidirect
477. C3#10590814semidirect

Maximal subgroups

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

subgroupindex
#117234542
C3×#638442
#117220762
#117220542
#117220402
C3×#635752
C3×Dic3⋊(C3⋊C8)2
#105908143
#105905443
#105903023