Finite Group Databasebeta

← #13156368

Subgroups — #13156368

Order 1728.

552 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723399C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic
4#11722042C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10590809—31nomaximal
9#10590490—31nomaximal
10#10590292—31nomaximal
11C2×#6308C2211yescharacteristic
12C3×C4⋊(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#88806—21no—
19Dic3×C3⋊Dic3—21no—
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C4×#634—31no—
23C3×#890—31no—
24C3×#870—31no—
25#63869D311yescharacteristic
26#63844—31no—
27#63844—31no—
28#63844—61no—
29#63643—31no—
30#63637—31no—
31C3×#817—31no—
32C3×Dic3⋊C8—31no—
33C3×#808—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63576—31no—
36#63575—31no—
37#63575—31no—
38#63573D311yescharacteristic
39Dic3⋊(C3⋊C8)D311yescharacteristic
40#63567—31no—
41#63567—31no—
42#63567—31no—
43#63567—31no—
44#63567—61no—
45#63567—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48#6308—21no—
49#6308Q812yes—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C3×C3⋊(C3⋊C8)—21no—
56C32×C3⋊C8—21no—
57#5063—91no—
58#4857—91no—
59#4829—91no—
60C2×#634—31no—
61C2×#634—31no—
62C2×#634—31no—
63C2×#634—61no—
64C2×#634—61no—
65C2×#634—61no—
66C2×#634—121no—
67C2×C32×C8—31no—
68C32×C4⋊C4—31no—
69C32×C42—31no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72#3750—61no—
73C4×C3⋊Dic3—61no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×C4⋊Dic3—21no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
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×Dic3⋊C4—61no—
86C3×Dic3⋊C4—61no—
87C3×Dic3⋊C4—61no—
88C3×C4×Dic3—21no—
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—21no—
102Dic32—62no—
103Dic32—61no—
104Dic32—61no—
105C22×C33C2×D411yescharacteristic
106C33×C4C2×Q811yescharacteristic
107C33×C4#4111yescharacteristic
108C3⋊(C3⋊Dic3)—22no—
109C3×C3⋊Dic3—41no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—21no—
112C32×Dic3—21no—
113C32×Dic3—41no—
114C3×#125—91no—
115#890—31no—
116#887—91no—
117#870—31no—
118C4×Dic6—91no—
119C4×Dic6—91no—
120C4×Dic6—181no—
121C3×#106—91no—
122C3×C4⋊C8—91no—
123#818—91no—
124#817—31no—
125#816—91no—
126#816—91no—
127#816—91no—
128#816—181no—
129Dic3⋊C8—31no—
130#810—91no—
131#810—91no—
132#810—91no—
133#810—181no—
134#808—31no—
135#807—91no—
136#807—91no—
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—32no—
158#634—32no—
159#634—32no—
160#634—62no—
161#634—62no—
162#634—62no—
163#634—61no—
164#634—62no—
165#634—61no—
166#634—61no—
167#634—122no—
168#634—121no—
169#634—121no—
170#634—122no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—121no—
182C2×C3×Dic3D1211yescharacteristic
183C2×C3×Dic3D1211yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C32×C8—61no—
187C3⋊(C3⋊C8)—21no—
188C3×C3⋊C8—21no—
189C3×C3⋊C8—21no—
190C3×C3⋊C8—41no—
191C3×C3⋊C8—61no—
192C3×C3⋊C8—61no—
193C3×C3⋊C8—121no—
194#479—271noSylow
195C2×C33C22⋊Q811yescharacteristic
196C2×C33#13511yescharacteristic
197C2×C33#13711yescharacteristic
198C2×Dic6—91no—
199C2×Dic6—91no—
200C2×Dic6—91no—
201C2×Dic6—181no—
202C2×Dic6—181no—
203C2×Dic6—181no—
204C2×Dic6—361no—
205C2×C3×C8—31no—
206C2×C3×C8—31no—
207C2×C3×C8—61no—
208C2×C3×C8—91no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—91no—
214C3×C4⋊C4—181no—
215C3×C4⋊C4—181no—
216C3×C42—31no—
217C3×C42—31no—
218C3×C42—61no—
219C4⋊Dic3—31no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—31no—
223C4⋊Dic3—61no—
224C4⋊Dic3—61no—
225C4⋊Dic3—61no—
226C4⋊Dic3D3211yescharacteristic
227Dic3⋊C4—61no—
228Dic3⋊C4—61no—
229Dic3⋊C4—61no—
230Dic3⋊C4—181no—
231Dic3⋊C4—181no—
232Dic3⋊C4—182no—
233Dic3⋊C4—181no—
234C4×Dic3—181no—
235C4×Dic3—181no—
236C4×Dic3—182no—
237C4×Dic3—181no—
238C4×Dic3D3211yescharacteristic
239C2×C3⋊C8—31no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—61no—
242C2×C3⋊C8D3211yescharacteristic
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—41no—
247C22×C32C2×D1211yescharacteristic
248C22×C32D3×D411yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—41no—
257C32×C4—41no—
258C32×C4—61no—
259C32×C4—61no—
260C32×C4—121no—
261C32×C4#23511yescharacteristic
262C32×C4#23711yescharacteristic
263C32×C4D3×Q811yescharacteristic
264C32×C4D3×Q811yescharacteristic
265C32×C4D3×Q811yescharacteristic
266C32×C4#23911yescharacteristic
267C3⋊Dic3—41no—
268C3⋊Dic3—41no—
269C3⋊Dic3—62no—
270C3⋊Dic3—64no—
271C3⋊Dic3—62no—
272C3⋊Dic3—62no—
273C3⋊Dic3—122no—
274C3⋊Dic3—122no—
275C3⋊Dic3—124no—
276C3×Dic3—22no—
277C3×Dic3—21no—
278C3×Dic3—21no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—41no—
282C3×Dic3—41no—
283C3×Dic3—41no—
284C3×Dic3—81no—
285C3×Dic3—122no—
286C3×Dic3—121no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289C3×Dic3—121no—
290C3×Dic3—241no—
291#125—91no—
292C4×Q8—271no—
293#107—271no—
294#106—91no—
295C4⋊C8—91no—
296#103—271no—
297#103—271no—
298C33#47911yescharacteristic, complemented, Sylow
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—31no—
303C2×C3×C4—31no—
304C2×C3×C4—31no—
305C2×C3×C4—31no—
306C2×C3×C4—41no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—121no—
312C2×C3×C4—181no—
313C2×C3×C4—181no—
314C2×C3×C4C2×D3211yescharacteristic
315C2×C3×C4C2×D3211yescharacteristic
316C2×C3×C4C2×D3211yescharacteristic
317C2×Dic3—21no—
318C2×Dic3—62no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—121no—
324C2×Dic3—182no—
325C2×Dic3—181no—
326C2×Dic3—181no—
327C2×Dic3—181no—
328C2×Dic3—361no—
329C2×Dic3—361no—
330C2×Dic3—362no—
331C2×Dic3#62611yescharacteristic
332C2×Dic3#62611yescharacteristic
333Dic6—92no—
334Dic6—92no—
335Dic6—92no—
336Dic6—182no—
337Dic6—182no—
338Dic6—182no—
339Dic6—182no—
340Dic6—181no—
341Dic6—181no—
342Dic6—181no—
343Dic6—362no—
344Dic6—361no—
345Dic6—361no—
346Dic6—362no—
347C3×C8—61no—
348C3×C8—61no—
349C3×C8—121no—
350C3×C8—181no—
351C3⋊C8—21no—
352C3⋊C8—61no—
353C3⋊C8—61no—
354C3⋊C8—121no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32—41no—
367C2×C32#89611yescharacteristic
368C2×C32#89711yescharacteristic
369C2×C32#90411yescharacteristic
370C2×C32#90911yescharacteristic
371C2×C32#91511yescharacteristic
372C2×C32#91611yescharacteristic
373C2×C32#94611yescharacteristic
374C2×C32#95011yescharacteristic
375C2×C32#95111yescharacteristic
376C2×Q8—271no—
377C2×C8—31no—
378C2×C8—91no—
379C4⋊C4—31no—
380C4⋊C4—91no—
381C4⋊C4—91no—
382C4⋊C4—181no—
383C4⋊C4—181no—
384C4⋊C4—541no—
385C42—31no—
386C42—541no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—41no—
391C22×C3D3×D1211yescharacteristic
392C22×C3C2×#62611yescharacteristic
393C22×C3D3×#6711yescharacteristic
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—41no—
401C3×C4—41no—
402C3×C4—62no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—121no—
408C3×C4—121no—
409C3×C4—121no—
410C3×C4—241no—
411C3×C4—361no—
412C3×C4—361no—
413C3×C4#379611yescharacteristic
414C3×C4#379711yescharacteristic
415C3×C4#379711yescharacteristic
416C3×C4#379711yescharacteristic
417C3×C4#379911yescharacteristic
418C3×C4#380411yescharacteristic
419Dic3—21no—
420Dic3—21no—
421Dic3—41no—
422Dic3—122no—
423Dic3—121no—
424Dic3—121no—
425Dic3—121no—
426Dic3—121no—
427Dic3—184no—
428Dic3—182no—
429Dic3—182no—
430Dic3—182no—
431Dic3—241no—
432Dic3—362no—
433Dic3—362no—
434Dic3—364no—
435C32—21no—
436C32—21no—
437C32—21no—
438C32—41no—
439C32#482911yescharacteristic, complemented
440C32#485711yescharacteristic, complemented
441C32#506311yescharacteristic, complemented
442Q8—272no—
443Q8—541no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—61no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—541no—
450C2×C4D3311yescharacteristic
451C8—61no—
452C8—181no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3#6380011yescharacteristic
466C2×C3#6380611yescharacteristic
467C2×C3#6382911yescharacteristic
468C2×C3#6383111yescharacteristic
469C2×C3#6387111yescharacteristic
470C2×C3#6388811yescharacteristic
471C2×C3#6389511yescharacteristic
472C2×C3#6392911yescharacteristic
473C2×C3#6394711yescharacteristic
474C22D3×#62611yescharacteristic
475C4—61no—
476C4—61no—
477C4—121no—
478C4—361no—
479C4—361no—
480C4—542no—
481C4#8895811yescharacteristic
482C4#8896211yescharacteristic
483C3—21no—
484C3—21no—
485C3—21no—
486C3—41no—
487C3#1059029211yescharacteristic, complemented
488C3#1059049011yescharacteristic, complemented
489C3#1059080911yescharacteristic, complemented
490C2#1172285511yescharacteristic
491C2#1172288611yescharacteristic
492C2#1172297311yescharacteristic
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
298. C33#479semidirect
439. C32#4829semidirect
440. C32#4857semidirect
441. C32#5063semidirect
487. C3#10590292semidirect
488. C3#10590490semidirect
489. C3#10590809semidirect

Maximal subgroups

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

subgroupindex
#117233992
C3×#638692
#117220802
#117220422
#117220412
C3×#635732
C3×Dic3⋊(C3⋊C8)2
#105908093
#105904903
#105902923