Finite Group Databasebeta

← #13156376

Subgroups — #13156376

Order 1728.

498 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2C3×#63872C211yesmaximal, characteristic
3#11722053C211yesmaximal, characteristic
4#11722051C211yesmaximal, characteristic
5#11722039C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic
7C3×#63567C211yesmaximal, characteristic
8#10590821—31nomaximal
9#10590492—31nomaximal
10#10590491—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—21no—
19#88813—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#890—31no—
23C3×#890—31no—
24#63872D311yescharacteristic
25#63844—31no—
26#63843—31no—
27Dic3×Dic6—31no—
28#63641—31no—
29#63637—31no—
30#63635—31no—
31C3×#818—31no—
32C3×#816—31no—
33C3×#810—31no—
34C3×#807—31no—
35#63575D311yescharacteristic
36#63574—31no—
37#63574—31no—
38#63574—61no—
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×#634Q812yes—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C32×C3⋊C8—21no—
57#5063—91no—
58#4877—91no—
59#4796—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×C4⋊C4—31no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—61no—
73C2×C3⋊(C3⋊C8)—61no—
74C2×C3⋊(C3⋊C8)—61no—
75C2×C3⋊(C3⋊C8)—121no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—61no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3C2×D311yescharacteristic
84C3×C4⋊Dic3C2×D311yescharacteristic
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×Dic3⋊C4—61no—
88C3×Dic3⋊C4—61no—
89C3×C4×Dic3—31no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3—61no—
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C2×D311yescharacteristic
95#3724—61no—
96Dic3⋊Dic3—21no—
97Dic3⋊Dic3—21no—
98Dic3⋊Dic3—61no—
99Dic3⋊Dic3—61no—
100Dic32—61no—
101Dic32—61no—
102Dic32—61no—
103C22×C33C2×D411yescharacteristic
104C33×C4C2×Q811yescharacteristic
105C33×C4#4111yescharacteristic
106C3×C3⋊Dic3—22no—
107C3×C3⋊Dic3—21no—
108C3×C3⋊Dic3—21no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—41no—
111C32×Dic3—41no—
112C3×#125—91no—
113Q8×Dic3—91no—
114#890—31no—
115#890—31no—
116#887—91no—
117C4×Dic6—91no—
118C3×#107—91no—
119C3×#103—91no—
120#835—91no—
121#818—31no—
122#816—31no—
123Dic3⋊C8—91no—
124Dic3⋊C8—91no—
125Dic3⋊C8—181no—
126#810—31no—
127#810—91no—
128#810—91no—
129#810—181no—
130#808—91no—
131#808—91no—
132#808—91no—
133#808—181no—
134#807—31no—
135C4⋊(C3⋊C8)—91no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—41no—
140C2×C32×C4—61no—
141C2×C32×C4—61no—
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D311yescharacteristic
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3#6711yescharacteristic
148C2×C3⋊Dic3#6711yescharacteristic
149#634—21no—
150#634Dic612yes—
151C2×C3×Dic3—21no—
152C2×C3×Dic3—21no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—31no—
158C2×C3×Dic3—31no—
159C2×C3×Dic3—41no—
160C2×C3×Dic3—41no—
161C2×C3×Dic3—61no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—121no—
168C2×C3×Dic3—121no—
169C3×Dic6—32no—
170C3×Dic6—32no—
171C3×Dic6—62no—
172C3×Dic6—61no—
173C3×Dic6—61no—
174C3×Dic6—121no—
175C32×C8—61no—
176C3⋊(C3⋊C8)—61no—
177C3⋊(C3⋊C8)—62no—
178C3⋊(C3⋊C8)—61no—
179C3⋊(C3⋊C8)—61no—
180C3⋊(C3⋊C8)—121no—
181C3⋊(C3⋊C8)—121no—
182C3⋊(C3⋊C8)—122no—
183C3×C3⋊C8—21no—
184C3×C3⋊C8—21no—
185C3×C3⋊C8—41no—
186#479—271noSylow
187C2×C33C22⋊Q811yescharacteristic
188C2×C33#13511yescharacteristic
189C2×C33#13711yescharacteristic
190C2×C3×Q8—91no—
191C2×Dic6—31no—
192C2×Dic6—31no—
193C2×Dic6—61no—
194C2×C3×C8—31no—
195C2×C3×C8—31no—
196C2×C3×C8—61no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—31no—
199C3×C4⋊C4—31no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—91no—
204C3×C4⋊C4—181no—
205C3×C4⋊C4—181no—
206C3×C42—91no—
207C4⋊Dic3—31no—
208C4⋊Dic3—31no—
209C4⋊Dic3—61no—
210C4⋊Dic3—61no—
211C4⋊Dic3—61no—
212C4⋊Dic3—181no—
213C4⋊Dic3D3211yescharacteristic
214C4⋊Dic3D3211yescharacteristic
215Dic3⋊C4—61no—
216Dic3⋊C4—61no—
217Dic3⋊C4—181no—
218Dic3⋊C4—181no—
219C4×Dic3—31no—
220C4×Dic3—31no—
221C4×Dic3—61no—
222C4×Dic3—181no—
223C4×Dic3—181no—
224C4×Dic3—181no—
225C2×C3⋊C8—91no—
226C2×C3⋊C8—91no—
227C2×C3⋊C8—91no—
228C2×C3⋊C8—181no—
229C2×C3⋊C8—181no—
230C2×C3⋊C8—181no—
231C2×C3⋊C8—361no—
232C2×C3⋊C8D3211yescharacteristic
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—41no—
237C22×C32D3×D411yescharacteristic
238C22×C32D3×D411yescharacteristic
239C22×C32C2×#6711yescharacteristic
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—41no—
247C32×C4—41no—
248C32×C4—121no—
249C32×C4—121no—
250C32×C4C2×Dic611yescharacteristic
251C32×C4#23511yescharacteristic
252C32×C4#23711yescharacteristic
253C32×C4D3×Q811yescharacteristic
254C32×C4D3×Q811yescharacteristic
255C32×C4#23911yescharacteristic
256C3⋊Dic3—21no—
257C3⋊Dic3—22no—
258C3⋊Dic3—21no—
259C3⋊Dic3—41no—
260C3×Dic3—41no—
261C3×Dic3—41no—
262C3×Dic3—41no—
263C3×Dic3—41no—
264C3×Dic3—62no—
265C3×Dic3—62no—
266C3×Dic3—61no—
267C3×Dic3—61no—
268C3×Dic3—61no—
269C3×Dic3—61no—
270C3×Dic3—62no—
271C3×Dic3—81no—
272C3×Dic3—81no—
273C3×Dic3—122no—
274C3×Dic3—121no—
275C3×Dic3—121no—
276C3×Dic3—121no—
277C3×Dic3—241no—
278#125—91no—
279C4×Q8—271no—
280#107—91no—
281#106—271no—
282C4⋊C8—271no—
283#103—91no—
284#103—271no—
285C33#47911yescharacteristic, complemented, Sylow
286C3×Q8—92no—
287C3×Q8—181no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—41no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—91no—
297C2×C3×C4—91no—
298C2×C3×C4—121no—
299C2×C3×C4—121no—
300C2×C3×C4—181no—
301C2×C3×C4—181no—
302C2×C3×C4C2×D3211yescharacteristic
303C2×C3×C4C2×D3211yescharacteristic
304C2×C3×C4C2×D3211yescharacteristic
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—31no—
310C2×Dic3—31no—
311C2×Dic3—61no—
312C2×Dic3—61no—
313C2×Dic3—61no—
314C2×Dic3—61no—
315C2×Dic3—61no—
316C2×Dic3—61no—
317C2×Dic3—121no—
318C2×Dic3—121no—
319Dic6—32no—
320Dic6—32no—
321Dic6—62no—
322Dic6—61no—
323Dic6—61no—
324Dic6—121no—
325C3×C8—61no—
326C3×C8—61no—
327C3×C8—121no—
328C3⋊C8—21no—
329C3⋊C8—182no—
330C3⋊C8—181no—
331C3⋊C8—181no—
332C3⋊C8—181no—
333C3⋊C8—361no—
334C3⋊C8—361no—
335C3⋊C8—362no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—41no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32#87811yescharacteristic
349C2×C32#89611yescharacteristic
350C2×C32#91211yescharacteristic
351C2×C32#91511yescharacteristic
352C2×C32#91811yescharacteristic
353C2×C32#91911yescharacteristic
354C2×C32#94611yescharacteristic
355C2×C32#95011yescharacteristic
356C2×C32#95111yescharacteristic
357C2×Q8—91no—
358C2×C8—31no—
359C2×C8—271no—
360C4⋊C4—31no—
361C4⋊C4—31no—
362C4⋊C4—91no—
363C4⋊C4—181no—
364C4⋊C4—181no—
365C4⋊C4—541no—
366C42—91no—
367C42—541no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—41no—
372C22×C3#380211yescharacteristic
373C22×C3D3×#6711yescharacteristic
374C22×C3D3×#6711yescharacteristic
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—41no—
382C3×C4—41no—
383C3×C4—121no—
384C3×C4—121no—
385C3×C4—121no—
386C3×C4—121no—
387C3×C4—181no—
388C3×C4—181no—
389C3×C4—182no—
390C3×C4—241no—
391C3×C4—241no—
392C3×C4—361no—
393C3×C4D3×Dic611yescharacteristic
394C3×C4D3×Dic611yescharacteristic
395C3×C4#379611yescharacteristic
396C3×C4#379711yescharacteristic
397C3×C4#379911yescharacteristic
398C3×C4#380411yescharacteristic
399Dic3—41no—
400Dic3—41no—
401Dic3—62no—
402Dic3—62no—
403Dic3—62no—
404Dic3—61no—
405Dic3—61no—
406Dic3—61no—
407Dic3—61no—
408Dic3—122no—
409Dic3—121no—
410Dic3—121no—
411Dic3—121no—
412Dic3—241no—
413C32—21no—
414C32—21no—
415C32—21no—
416C32—41no—
417C32#479611yescharacteristic, complemented
418C32#487711yescharacteristic, complemented
419C32#506311yescharacteristic, complemented
420Q8—92no—
421Q8—181no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—91no—
425C2×C4—91no—
426C2×C4—181no—
427C2×C4—181no—
428C2×C4D3311yescharacteristic
429C8—61no—
430C8—541no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—41no—
443C2×C3#6381011yescharacteristic
444C2×C3#6381111yescharacteristic
445C2×C3#6388711yescharacteristic
446C2×C3#6388811yescharacteristic
447C2×C3#6389911yescharacteristic
448C2×C3#6392911yescharacteristic
449C2×C3#6393311yescharacteristic
450C2×C3#6394311yescharacteristic
451C2×C3#6394911yescharacteristic
452C22#8895311yescharacteristic
453C4—121no—
454C4—121no—
455C4—181no—
456C4—181no—
457C4—182no—
458C4—361no—
459C4#8895711yescharacteristic
460C4#8896211yescharacteristic
461C3—21no—
462C3—21no—
463C3—21no—
464C3—41no—
465C3#1059049111yescharacteristic, complemented
466C3#1059049211yescharacteristic, complemented
467C3#1059082111yescharacteristic, complemented
468C2#1172286011yescharacteristic
469C2#1172289111yescharacteristic
470C2#1172297911yescharacteristic
subgroupG/H
centerC22#88953
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
285. C33#479semidirect
417. C32#4796semidirect
418. C32#4877semidirect
419. C32#5063semidirect
465. C3#10590491semidirect
466. C3#10590492semidirect
467. C3#10590821semidirect

Maximal subgroups

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

subgroupindex
#117236682
C3×#638722
#117220532
#117220512
#117220392
C3×#635752
C3×#635672
#105908213
#105904923
#105904913