Finite Group Databasebeta

← #13155699

Subgroups — #13155699

Order 1728.

578 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723668C211yesmaximal, characteristic
2C3×#63843C211yesmaximal, characteristic
3C2×C3×#3719C211yesmaximal, characteristic
4#11722039C211yesmaximal, characteristic
5#11722037C211yesmaximal, characteristic
6#11722020C211yesmaximal, characteristic
7C3×#63573C211yesmaximal, characteristic
8#10591576—31nomaximal
9#10590819—31nomaximal
10#10590480—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815—21no—
19#88813—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#3719C414yescomplemented
23C3×Q8×Dic3—31no—
24C2×C3×#216—31no—
25C2×C3×Dic12—31no—
26C3×#887—31no—
27#63844—31no—
28#63843—31no—
29#63843D311yescharacteristic
30Dic3×Dic6—31no—
31C2×#3719D311yescharacteristic
32#63662—31no—
33#63642—31no—
34#63637—31no—
35C3×#817—31no—
36C3×#808—31no—
37#63573D311yescharacteristic
38#63565—31no—
39#63565—31no—
40#63565—31no—
41#63565—31no—
42#63565—61no—
43#63565—61no—
44#63554—31no—
45#63554—31no—
46#63554—61no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×#634—21no—
53C3×#634C2×C412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C32×Dic6—21no—
57C32×Dic6C2×C412yes—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8C2×C412yes—
60#5184—91no—
61#5030—91no—
62#4853—91no—
63C2×C32×Q8—31no—
64C2×#634C2×D311yescharacteristic
65C2×C3×Dic6—21no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C32×C8—31no—
72C32×C4⋊C4—31no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4×C3⋊Dic3C2×D311yescharacteristic
75C2×C3⋊(C3⋊C8)—31no—
76C2×C3⋊(C3⋊C8)—31no—
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—61no—
79C2×C3⋊(C3⋊C8)—61no—
80C2×C3⋊(C3⋊C8)—61no—
81C2×C3⋊(C3⋊C8)—121no—
82C3×#216—34no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4—61no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—61no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—61no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C3×Dic12—34no—
100#3724—61no—
101Dic3⋊Dic3—21no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104Dic32—21no—
105Dic32—61no—
106Dic32—61no—
107Dic32—61no—
108#3719Dic314yescomplemented
109C22×C33C2×D411yescharacteristic
110C33×C4C22×C411yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—22no—
117C32×Dic3—41no—
118C2×C3×Q16—91no—
119C3×C4×Q8—91no—
120Q8×Dic3—31no—
121Q8×Dic3—91no—
122C2×#216—31no—
123C2×Dic12—31no—
124#887—31no—
125#887—91no—
126C4×Dic6—91no—
127C3×#106—91no—
128#835—91no—
129#835—91no—
130#835—91no—
131#835—181no—
132#817—31no—
133#815—91no—
134#815—91no—
135#815—181no—
136#810—91no—
137#810—91no—
138#810—91no—
139#810—181no—
140#808—31no—
141#803—91no—
142C32×Q8—32no—
143C32×Q8—61no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4—61no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3#6711yescharacteristic
156C2×C3⋊Dic3#6711yescharacteristic
157#634—21no—
158#634C2×Dic312yes—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—31no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—41no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—121no—
176C2×C3×Dic3—121no—
177C3×Dic6—21no—
178C3×Dic6—21no—
179C3×Dic6—22no—
180C3×Dic6—32no—
181C3×Dic6—32no—
182C3×Dic6—41no—
183C3×Dic6—62no—
184C3×Dic6—61no—
185C3×Dic6—61no—
186C3×Dic6—121no—
187C3×Dic6C4×D312yes—
188C3×Dic6C2×Dic312yes—
189C32×C8—32no—
190C3⋊(C3⋊C8)—61no—
191C3⋊(C3⋊C8)—62no—
192C3⋊(C3⋊C8)—61no—
193C3⋊(C3⋊C8)—61no—
194C3⋊(C3⋊C8)—121no—
195C3⋊(C3⋊C8)—121no—
196C3⋊(C3⋊C8)—122no—
197C3×C3⋊C8—22no—
198C3×C3⋊C8C4×D312yes—
199C3×C3⋊C8C2×Dic312yes—
200#441—271noSylow
201C2×C33C4×D411yescharacteristic
202C2×C33#13711yescharacteristic
203C2×C33#13711yescharacteristic
204C2×C3×Q8—31no—
205C2×C3×Q8—31no—
206C2×C3×Q8—61no—
207C2×C3×Q8—91no—
208C2×Dic6—31no—
209C2×Dic6—31no—
210C2×Dic6—61no—
211C2×Dic6D3211yescharacteristic
212C3×Q16—94no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—181no—
221C3×C42—91no—
222C3×C42—181no—
223#216—34no—
224C4⋊Dic3—31no—
225C4⋊Dic3—31no—
226C4⋊Dic3—61no—
227C4⋊Dic3—61no—
228C4⋊Dic3—181no—
229C4⋊Dic3D3211yescharacteristic
230Dic3⋊C4—61no—
231Dic3⋊C4—181no—
232Dic3⋊C4—181no—
233C4×Dic3—31no—
234C4×Dic3—31no—
235C4×Dic3—61no—
236C4×Dic3—61no—
237C4×Dic3—61no—
238C4×Dic3—181no—
239C4×Dic3—181no—
240C4×Dic3—181no—
241C2×C3⋊C8—91no—
242C2×C3⋊C8—91no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—181no—
245C2×C3⋊C8—181no—
246C2×C3⋊C8—181no—
247C2×C3⋊C8—361no—
248C2×C3⋊C8D3211yescharacteristic
249Dic12—34no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32D3×D411yescharacteristic
255C22×C32D3×D411yescharacteristic
256C22×C32C2×#6711yescharacteristic
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—41no—
264C32×C4—41no—
265C32×C4—62no—
266C32×C4—121no—
267C32×C4C2×C4×D311yescharacteristic
268C32×C4C2×C4×D311yescharacteristic
269C32×C4#23511yescharacteristic
270C32×C4#23711yescharacteristic
271C32×C4#23911yescharacteristic
272C32×C4C22×Dic311yescharacteristic
273C3⋊Dic3—22no—
274C3⋊Dic3—21no—
275C3⋊Dic3—21no—
276C3⋊Dic3—41no—
277C3×Dic3—22no—
278C3×Dic3—22no—
279C3×Dic3—42no—
280C3×Dic3—41no—
281C3×Dic3—41no—
282C3×Dic3—62no—
283C3×Dic3—61no—
284C3×Dic3—61no—
285C3×Dic3—61no—
286C3×Dic3—61no—
287C3×Dic3—62no—
288C3×Dic3—81no—
289C3×Dic3—122no—
290C3×Dic3—121no—
291C3×Dic3—121no—
292C3×Dic3—121no—
293C3×Dic3—121no—
294C3×Dic3—241no—
295C2×Q16—91no—
296C4×Q8—91no—
297C4×Q8—271no—
298#106—91no—
299#103—271no—
300#103—271no—
301#97—271no—
302C33#44111yescharacteristic, complemented, Sylow
303C3×Q8—32no—
304C3×Q8—32no—
305C3×Q8—62no—
306C3×Q8—61no—
307C3×Q8—61no—
308C3×Q8—92no—
309C3×Q8—121no—
310C3×Q8—181no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—91no—
320C2×C3×C4—91no—
321C2×C3×C4—121no—
322C2×C3×C4—121no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4C2×D3211yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—31no—
331C2×Dic3—31no—
332C2×Dic3—31no—
333C2×Dic3—31no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—121no—
341C2×Dic3—121no—
342Dic6—21no—
343Dic6—32no—
344Dic6—32no—
345Dic6—62no—
346Dic6—61no—
347Dic6—61no—
348Dic6—121no—
349Dic6D3×Dic312yes—
350C3×C8—32no—
351C3×C8—32no—
352C3×C8—62no—
353C3⋊C8—182no—
354C3⋊C8—181no—
355C3⋊C8—181no—
356C3⋊C8—181no—
357C3⋊C8—361no—
358C3⋊C8—361no—
359C3⋊C8—362no—
360C3⋊C8D3×Dic312yes—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32#89311yescharacteristic
374C2×C32#91511yescharacteristic
375C2×C32#91511yescharacteristic
376C2×C32#91811yescharacteristic
377C2×C32#91811yescharacteristic
378C2×C32C4×#6711yescharacteristic
379C2×C32D4×Dic311yescharacteristic
380C2×C32#94211yescharacteristic
381C2×C32#95111yescharacteristic
382C2×Q8—31no—
383C2×Q8—91no—
384Q16—94no—
385C2×C8—31no—
386C2×C8—271no—
387C4⋊C4—31no—
388C4⋊C4—91no—
389C4⋊C4—181no—
390C4⋊C4—541no—
391C42—91no—
392C42—181no—
393C42—541no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—41no—
398C22×C3#380211yescharacteristic
399C22×C3D3×#6711yescharacteristic
400C22×C3D3×#6711yescharacteristic
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—62no—
410C3×C4—62no—
411C3×C4—122no—
412C3×C4—121no—
413C3×C4—121no—
414C3×C4—181no—
415C3×C4—181no—
416C3×C4—182no—
417C3×C4—241no—
418C3×C4—361no—
419C3×C4#379611yescharacteristic
420C3×C4#379911yescharacteristic
421C3×C4C4×D3211yescharacteristic
422C3×C4C2×D3×Dic311yescharacteristic
423C3×C4C2×D3×Dic311yescharacteristic
424C3×C4#380411yescharacteristic
425Dic3—22no—
426Dic3—41no—
427Dic3—62no—
428Dic3—61no—
429Dic3—61no—
430Dic3—61no—
431Dic3—61no—
432Dic3—62no—
433Dic3—122no—
434Dic3—121no—
435Dic3—121no—
436Dic3—121no—
437Dic3—121no—
438Dic3—241no—
439C32—21no—
440C32—21no—
441C32—21no—
442C32—41no—
443C32#485311yescharacteristic, complemented
444C32#503011yescharacteristic, complemented
445C32#518411yescharacteristic, complemented
446Q8—32no—
447Q8—61no—
448Q8—92no—
449Q8—181no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—91no—
453C2×C4—91no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4D3311yescharacteristic
457C8—32no—
458C8—541no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3#6381111yescharacteristic
472C2×C3#6381111yescharacteristic
473C2×C3#6389811yescharacteristic
474C2×C3#6390411yescharacteristic
475C2×C3#6392911yescharacteristic
476C2×C3#6394311yescharacteristic
477C2×C3#6394611yescharacteristic
478C2×C3#6394811yescharacteristic
479C2×C3Dic3×#6711yescharacteristic
480C22#8895311yescharacteristic
481C4—62no—
482C4—121no—
483C4—181no—
484C4—181no—
485C4—182no—
486C4—361no—
487C4D32×Dic311yescharacteristic
488C4#8896211yescharacteristic
489C3—21no—
490C3—21no—
491C3—21no—
492C3—41no—
493C3#1059048011yescharacteristic, complemented
494C3#1059081911yescharacteristic, complemented
495C3#1059157611yescharacteristic, complemented
496C2#1172289111yescharacteristic
497C2#1172289211yescharacteristic
498C2#1172292711yescharacteristic
subgroupG/H
centerC22#88953
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
22. C3×#3719C4semidirect
108. #3719Dic3semidirect
302. C33#441semidirect
443. C32#4853semidirect
444. C32#5030semidirect
445. C32#5184semidirect
493. C3#10590480semidirect
494. C3#10590819semidirect
495. C3#10591576semidirect

Maximal subgroups

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

subgroupindex
#117236682
C3×#638432
C2×C3×#37192
#117220392
#117220372
#117220202
C3×#635732
#105915763
#105908193
#105904803