Finite Group Databasebeta

← #13156323

Subgroups — #13156323

Order 1728.

522 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2C3×#63919C211yesmaximal, characteristic
3#11722056C211yesmaximal, characteristic
4#11722051C211yesmaximal, characteristic
5#11722039C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8#10590817—31nomaximal
9#10590489—31nomaximal
10#10590488—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#627—21no—
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815—21no—
20#88813—21no—
21C3×#3724—21no—
22C3×#888—31no—
23C3×#888—31no—
24#63919D311yescharacteristic
25#63844—31no—
26#63843—31no—
27Dic3×Dic6—31no—
28#63641—31no—
29#63637—31no—
30#63634—31no—
31C3×#818—31no—
32C3×#816—31no—
33C3×#810—31no—
34C3×#807—31no—
35#63576—31no—
36#63576—31no—
37#63576—61no—
38#63575D311yescharacteristic
39Dic3⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—61no—
42#63567D311yescharacteristic
43#63565—31no—
44#63565—31no—
45#63565—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×#634—21no—
52C3×#634Q812yescomplemented
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C3×#627—42no—
56C3⋊(C3⋊(C3⋊C8))—21no—
57C32×C3⋊C8—21no—
58#5091—91no—
59#4875—91no—
60#4793—91no—
61C2×#634C2×D311yescharacteristic
62C2×C3×Dic6—31no—
63C2×C3×Dic6—31no—
64C2×C3×Dic6—61no—
65C2×C3×Dic6—61no—
66C2×C3×Dic6—61no—
67C2×#627—21no—
68C2×C32×C8—31no—
69C32×C4⋊C4—31no—
70C32×C4⋊C4—31no—
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C4×C3⋊Dic3C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—61no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—61no—
79C2×C3⋊(C3⋊C8)—121no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—21no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—61no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97#3724—21no—
98#3724—61no—
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic32—61no—
102Dic32—61no—
103Dic32—61no—
104C22×C33C2×D411yescharacteristic
105C33×C4C2×Q811yescharacteristic
106C33×C4#4111yescharacteristic
107C3×C3⋊Dic3—22no—
108C3×C3⋊Dic3—21no—
109C3×C3⋊Dic3—41no—
110C3×C3⋊Dic3Q1612yes—
111C32×Dic3—41no—
112C32×Dic3—41no—
113C3×C4⋊Q8—91no—
114Q8×Dic3—91no—
115#888—31no—
116#888—31no—
117#887—91no—
118C4×Dic6—91no—
119C3×#107—91no—
120C3×#103—91no—
121#835—91no—
122#818—31no—
123#816—31no—
124Dic3⋊C8—91no—
125Dic3⋊C8—91no—
126Dic3⋊C8—181no—
127#810—31no—
128#810—91no—
129#810—91no—
130#810—181no—
131#807—31no—
132#807—91no—
133#807—91no—
134#807—91no—
135#807—181no—
136C4⋊(C3⋊C8)—91no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—41no—
141C2×C32×C4—61no—
142C2×C32×C4—61no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3#6711yescharacteristic
149C2×C3⋊Dic3#6711yescharacteristic
150#634—21no—
151#634Dic612yescomplemented
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—31no—
158C2×C3×Dic3—31no—
159C2×C3×Dic3—31no—
160C2×C3×Dic3—41no—
161C2×C3×Dic3—41no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3—121no—
170C3×Dic6—32no—
171C3×Dic6—32no—
172C3×Dic6—62no—
173C3×Dic6—61no—
174C3×Dic6—61no—
175C3×Dic6—121no—
176C3×Dic6—122no—
177C3×Dic6—122no—
178#627—42no—
179C32×C8—61no—
180C3⋊(C3⋊C8)—61no—
181C3⋊(C3⋊C8)—62no—
182C3⋊(C3⋊C8)—61no—
183C3⋊(C3⋊C8)—61no—
184C3⋊(C3⋊C8)—121no—
185C3⋊(C3⋊C8)—121no—
186C3⋊(C3⋊C8)—122no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—41no—
190#477—271noSylow
191C2×C33C22⋊Q811yescharacteristic
192C2×C33C2×Q1611yescharacteristic
193C2×C33AGL(1,ℤ/8)11yescharacteristic
194C2×C3×Q8—91no—
195C2×C3×Q8—181no—
196C2×Dic6—31no—
197C2×Dic6—31no—
198C2×Dic6—61no—
199C2×Dic6—61no—
200C2×Dic6—61no—
201C2×C3×C8—31no—
202C2×C3×C8—31no—
203C2×C3×C8—61no—
204C3×C4⋊C4—31no—
205C3×C4⋊C4—31no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—91no—
211C3×C4⋊C4—181no—
212C3×C42—91no—
213C4⋊Dic3—31no—
214C4⋊Dic3—31no—
215C4⋊Dic3—61no—
216C4⋊Dic3—181no—
217C4⋊Dic3D3211yescharacteristic
218C4⋊Dic3D3211yescharacteristic
219Dic3⋊C4—61no—
220Dic3⋊C4—61no—
221Dic3⋊C4—181no—
222Dic3⋊C4—181no—
223C4×Dic3—31no—
224C4×Dic3—31no—
225C4×Dic3—61no—
226C4×Dic3—181no—
227C4×Dic3—181no—
228C4×Dic3—181no—
229C2×C3⋊C8—91no—
230C2×C3⋊C8—91no—
231C2×C3⋊C8—91no—
232C2×C3⋊C8—181no—
233C2×C3⋊C8—181no—
234C2×C3⋊C8—181no—
235C2×C3⋊C8—361no—
236C2×C3⋊C8D3211yescharacteristic
237C22×C32—21no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—41no—
241C22×C32D3×D411yescharacteristic
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—121no—
253C32×C4—121no—
254C32×C4C2×Dic611yescharacteristic
255C32×C4#23511yescharacteristic
256C32×C4#23711yescharacteristic
257C32×C4D3×Q811yescharacteristic
258C32×C4D3×Q811yescharacteristic
259C32×C4#23911yescharacteristic
260C3⋊Dic3—22no—
261C3⋊Dic3—21no—
262C3⋊Dic3—41no—
263C3⋊Dic3#21612yes—
264C3×Dic3—32no—
265C3×Dic3—32no—
266C3×Dic3—41no—
267C3×Dic3—41no—
268C3×Dic3—41no—
269C3×Dic3—41no—
270C3×Dic3—62no—
271C3×Dic3—62no—
272C3×Dic3—61no—
273C3×Dic3—61no—
274C3×Dic3—62no—
275C3×Dic3—81no—
276C3×Dic3—81no—
277C3×Dic3—122no—
278C3×Dic3—121no—
279C3×Dic3—121no—
280C3×Dic3—121no—
281C3×Dic3—241no—
282C4⋊Q8—91no—
283C4×Q8—271no—
284#107—91no—
285#107—271no—
286C4⋊C8—271no—
287#103—91no—
288#103—271no—
289C33#47711yescharacteristic, complemented, Sylow
290C3×Q8—92no—
291C3×Q8—181no—
292C3×Q8—362no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—41no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—91no—
302C2×C3×C4—91no—
303C2×C3×C4—121no—
304C2×C3×C4—121no—
305C2×C3×C4—181no—
306C2×C3×C4—181no—
307C2×C3×C4C2×D3211yescharacteristic
308C2×C3×C4C2×D3211yescharacteristic
309C2×C3×C4C2×D3211yescharacteristic
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—31no—
313C2×Dic3—31no—
314C2×Dic3—31no—
315C2×Dic3—31no—
316C2×Dic3—61no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—121no—
323C2×Dic3—121no—
324Dic6—32no—
325Dic6—32no—
326Dic6—62no—
327Dic6—61no—
328Dic6—61no—
329Dic6—121no—
330Dic6—122no—
331Dic6—122no—
332C3×C8—61no—
333C3×C8—61no—
334C3×C8—121no—
335C3⋊C8—21no—
336C3⋊C8—182no—
337C3⋊C8—181no—
338C3⋊C8—181no—
339C3⋊C8—181no—
340C3⋊C8—361no—
341C3⋊C8—361no—
342C3⋊C8—362no—
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#87811yescharacteristic
356C2×C32#89611yescharacteristic
357C2×C32#91111yescharacteristic
358C2×C32#91411yescharacteristic
359C2×C32D3×Q1611yescharacteristic
360C2×C32D3×Q1611yescharacteristic
361C2×C32#93211yescharacteristic
362C2×C32C2×#21611yescharacteristic
363C2×C32#94611yescharacteristic
364C2×Q8—91no—
365C2×Q8—181no—
366C2×C8—31no—
367C2×C8—271no—
368C4⋊C4—31no—
369C4⋊C4—31no—
370C4⋊C4—91no—
371C4⋊C4—181no—
372C4⋊C4—541no—
373C42—91no—
374C42—541no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—41no—
379C22×C3#380211yescharacteristic
380C22×C3D3×#6711yescharacteristic
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—92no—
391C3×C4—121no—
392C3×C4—121no—
393C3×C4—121no—
394C3×C4—121no—
395C3×C4—181no—
396C3×C4—182no—
397C3×C4—241no—
398C3×C4—241no—
399C3×C4—361no—
400C3×C4D3×Dic611yescharacteristic
401C3×C4D3×Dic611yescharacteristic
402C3×C4#379611yescharacteristic
403C3×C4#379711yescharacteristic
404C3×C4#379911yescharacteristic
405C3×C4#380411yescharacteristic
406Dic3—32no—
407Dic3—32no—
408Dic3—41no—
409Dic3—41no—
410Dic3—62no—
411Dic3—62no—
412Dic3—62no—
413Dic3—61no—
414Dic3—61no—
415Dic3—122no—
416Dic3—121no—
417Dic3—121no—
418Dic3—121no—
419Dic3—241no—
420C32—21no—
421C32—21no—
422C32—21no—
423C32—41no—
424C32#479311yescharacteristic, complemented
425C32#487511yescharacteristic, complemented
426C32#509111yescharacteristic, complemented
427Q8—92no—
428Q8—181no—
429Q8—362no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—91no—
433C2×C4—91no—
434C2×C4—181no—
435C2×C4—181no—
436C2×C4D3311yescharacteristic
437C8—61no—
438C8—541no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3#6379811yescharacteristic
452C2×C3#6380211yescharacteristic
453C2×C3#6388711yescharacteristic
454C2×C3#6388811yescharacteristic
455C2×C3#6389911yescharacteristic
456C2×C3#6393211yescharacteristic
457C2×C3#6393411yescharacteristic
458C2×C3D3×#21611yescharacteristic
459C2×C3D3×#21611yescharacteristic
460C22#8895311yescharacteristic
461C4—92no—
462C4—121no—
463C4—121no—
464C4—181no—
465C4—182no—
466C4—361no—
467C4#8895711yescharacteristic
468C4#8896211yescharacteristic
469C3—21no—
470C3—21no—
471C3—21no—
472C3—41no—
473C3#1059048811yescharacteristic, complemented
474C3#1059048911yescharacteristic, complemented
475C3#1059081711yescharacteristic, complemented
476C2#1172284711yescharacteristic
477C2#1172287511yescharacteristic
478C2#1172297911yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#477
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
52. C3×#634Q8semidirect
151. #634Dic6semidirect
289. C33#477semidirect
424. C32#4793semidirect
425. C32#4875semidirect
426. C32#5091semidirect
473. C3#10590488semidirect
474. C3#10590489semidirect
475. C3#10590817semidirect

Maximal subgroups

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

subgroupindex
#117236682
C3×#639192
#117220562
#117220512
#117220392
C3×#635752
C3×#635672
#105908173
#105904893
#105904883