Finite Group Databasebeta

← #13156570

Subgroups — #13156570

Order 1728.

550 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723443C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722042C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic
8#10590898—31nomaximal
9#10590562—31nomaximal
10#10590295—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18Dic3⋊(C3⋊Dic3)—21no—
19Dic3⋊(C3⋊Dic3)—21no—
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22#64078—31no—
23C3×#890—31no—
24C3×#870—31no—
25#63869D311yescharacteristic
26#63866—31no—
27#63866—31no—
28#63866—61no—
29#63643—31no—
30#63637—31no—
31C3×#816—31no—
32C3×#815—31no—
33C3×#810—31no—
34C3×#803—31no—
35#63567—31no—
36#63567—31no—
37#63567—31no—
38#63567—31no—
39#63567—31no—
40#63567—61no—
41#63567—61no—
42#63567D311yescharacteristic
43#63566—31no—
44#63565—31no—
45#63554D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48#6308—21no—
49#6308—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5069—91no—
60#4797—91no—
61#4703—91no—
62C2×#634—31no—
63C2×#634—31no—
64C2×#634—31no—
65C2×#634—61no—
66C2×#634—61no—
67C2×#634—61no—
68C2×#634—121no—
69C2×#634C2×D311yescharacteristic
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—61no—
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75C32×C42—31no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—31no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×Dic3⋊C4—61no—
88C3×Dic3⋊C4—61no—
89C3×Dic3⋊C4—61no—
90C3×C4×Dic3—21no—
91C3×C4×Dic3C2×D311yescharacteristic
92C3×C4×Dic3C2×D311yescharacteristic
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—21no—
100Dic3⋊Dic3—21no—
101Dic3⋊Dic3—62no—
102Dic3⋊Dic3—62no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—61no—
107C22×C33C2×D411yescharacteristic
108C33×C4#4111yescharacteristic
109C33×C4#4111yescharacteristic
110C3⋊(C3⋊Dic3)—41no—
111C3×C3⋊Dic3—41no—
112C3×C3⋊Dic3—41no—
113C32×Dic3—21no—
114C32×Dic3—21no—
115C32×Dic3—41no—
116C3×#125—91no—
117#944—91no—
118#890—31no—
119#870—31no—
120#869—91no—
121#869—91no—
122#869—181no—
123C3×#103—91no—
124C3×#97—91no—
125#835—91no—
126#816—31no—
127#816—91no—
128#816—91no—
129#816—91no—
130#816—91no—
131#816—181no—
132#815—31no—
133#810—31no—
134#810—91no—
135#810—91no—
136#810—91no—
137#810—91no—
138#810—181no—
139#803—31no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—31no—
144C2×C32×C4—31no—
145C2×C32×C4—41no—
146C2×C32×C4—61no—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—21no—
152C2×C3⋊Dic3—61no—
153C2×C3⋊Dic3—62no—
154C2×C3⋊Dic3—61no—
155C2×C3⋊Dic3—61no—
156C2×C3⋊Dic3—121no—
157C2×C3⋊Dic3—121no—
158C2×C3⋊Dic3—122no—
159#634—21no—
160#634—21no—
161#634—61no—
162#634—61no—
163#634—62no—
164#634—62no—
165#634—61no—
166#634—61no—
167#634—61no—
168#634—61no—
169#634—121no—
170#634—121no—
171#634—121no—
172#634—121no—
173#634—122no—
174#634—122no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—121no—
186C2×C3×Dic3D1211yescharacteristic
187C2×C3×Dic3D1211yescharacteristic
188C2×C3×Dic3#6711yescharacteristic
189C2×C3×Dic3#6711yescharacteristic
190C3×Dic6—61no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—61no—
194C3×Dic6—121no—
195C3×Dic6—121no—
196C32×C8—61no—
197C3⋊(C3⋊C8)—21no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—41no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—121no—
204#491—271noSylow
205C2×C33#12411yescharacteristic
206C2×C33#13711yescharacteristic
207C2×C33#13711yescharacteristic
208C2×C3×Q8—91no—
209C2×Dic6—31no—
210C2×Dic6—31no—
211C2×Dic6—61no—
212C2×Dic6—91no—
213C2×Dic6—91no—
214C2×Dic6—91no—
215C2×Dic6—181no—
216C2×Dic6—181no—
217C2×Dic6—181no—
218C2×Dic6—361no—
219C2×C3×C8—31no—
220C2×C3×C8—31no—
221C2×C3×C8—61no—
222C2×C3×C8—91no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4—181no—
228C3×C4⋊C4—181no—
229C3×C42—31no—
230C3×C42—31no—
231C3×C42—61no—
232C4⋊Dic3—31no—
233C4⋊Dic3—31no—
234C4⋊Dic3—61no—
235C4⋊Dic3—61no—
236C4⋊Dic3—181no—
237C4⋊Dic3—181no—
238C4⋊Dic3—181no—
239C4⋊Dic3—181no—
240C4⋊Dic3—182no—
241C4⋊Dic3—182no—
242C4⋊Dic3D3211yescharacteristic
243Dic3⋊C4—61no—
244Dic3⋊C4—61no—
245Dic3⋊C4—61no—
246Dic3⋊C4—181no—
247Dic3⋊C4—181no—
248C4×Dic3D3211yescharacteristic
249C2×C3⋊C8—31no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—61no—
252C2×C3⋊C8D3211yescharacteristic
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32C2×D1211yescharacteristic
258C22×C32D3×D411yescharacteristic
259C22×C32C2×#6711yescharacteristic
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—41no—
267C32×C4—41no—
268C32×C4—61no—
269C32×C4—61no—
270C32×C4—121no—
271C32×C4#23511yescharacteristic
272C32×C4#23511yescharacteristic
273C32×C4#23511yescharacteristic
274C32×C4#23711yescharacteristic
275C32×C4#23911yescharacteristic
276C32×C4#23911yescharacteristic
277C3⋊Dic3—41no—
278C3⋊Dic3—41no—
279C3⋊Dic3—122no—
280C3⋊Dic3—122no—
281C3⋊Dic3—122no—
282C3⋊Dic3—124no—
283C3⋊Dic3—121no—
284C3⋊Dic3—121no—
285C3⋊Dic3—121no—
286C3×Dic3—22no—
287C3×Dic3—22no—
288C3×Dic3—21no—
289C3×Dic3—21no—
290C3×Dic3—21no—
291C3×Dic3—21no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—81no—
295C3×Dic3—122no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—241no—
301C4⋊Q8—271no—
302#125—91no—
303#103—91no—
304#103—271no—
305#103—271no—
306#103—271no—
307#97—91no—
308C33#49111yescharacteristic, complemented, Sylow
309C3×Q8—181no—
310C3×Q8—181no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—41no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—121no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—21no—
330C2×Dic3—62no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—121no—
336C2×Dic3—182no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—361no—
341C2×Dic3—361no—
342C2×Dic3—362no—
343C2×Dic3#62611yescharacteristic
344C2×Dic3#62611yescharacteristic
345Dic6—61no—
346Dic6—61no—
347Dic6—61no—
348Dic6—61no—
349Dic6—121no—
350Dic6—121no—
351Dic6—182no—
352Dic6—182no—
353Dic6—181no—
354Dic6—181no—
355Dic6—181no—
356Dic6—181no—
357Dic6—181no—
358Dic6—181no—
359Dic6—361no—
360Dic6—361no—
361Dic6—361no—
362Dic6—361no—
363Dic6—362no—
364Dic6—362no—
365C3×C8—61no—
366C3×C8—61no—
367C3×C8—121no—
368C3×C8—181no—
369C3⋊C8—21no—
370C3⋊C8—61no—
371C3⋊C8—61no—
372C3⋊C8—121no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32#87511yescharacteristic
386C2×C32#88511yescharacteristic
387C2×C32#90911yescharacteristic
388C2×C32#90911yescharacteristic
389C2×C32#91511yescharacteristic
390C2×C32#91811yescharacteristic
391C2×C32#94711yescharacteristic
392C2×C32#95111yescharacteristic
393C2×C32#95111yescharacteristic
394C2×Q8—91no—
395C2×Q8—271no—
396C2×C8—31no—
397C2×C8—91no—
398C4⋊C4—31no—
399C4⋊C4—91no—
400C4⋊C4—181no—
401C4⋊C4—181no—
402C4⋊C4—541no—
403C4⋊C4—541no—
404C42—31no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3D3×D1211yescharacteristic
410C22×C3C2×#62611yescharacteristic
411C22×C3D3×#6711yescharacteristic
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4—62no—
421C3×C4—62no—
422C3×C4—61no—
423C3×C4—61no—
424C3×C4—61no—
425C3×C4—61no—
426C3×C4—121no—
427C3×C4—121no—
428C3×C4—241no—
429C3×C4—361no—
430C3×C4—361no—
431C3×C4#379611yescharacteristic
432C3×C4#379811yescharacteristic
433C3×C4#379911yescharacteristic
434C3×C4#379911yescharacteristic
435C3×C4#379911yescharacteristic
436C3×C4#380411yescharacteristic
437Dic3—21no—
438Dic3—21no—
439Dic3—41no—
440Dic3—122no—
441Dic3—121no—
442Dic3—121no—
443Dic3—121no—
444Dic3—121no—
445Dic3—241no—
446Dic3—362no—
447Dic3—362no—
448Dic3—362no—
449Dic3—364no—
450Dic3—361no—
451Dic3—361no—
452Dic3—361no—
453C32—21no—
454C32—21no—
455C32—21no—
456C32—41no—
457C32#470311yescharacteristic, complemented
458C32#479711yescharacteristic, complemented
459C32#506911yescharacteristic, complemented
460Q8—181no—
461Q8—181no—
462Q8—541no—
463Q8—541no—
464C2×C4—31no—
465C2×C4—31no—
466C2×C4—61no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—541no—
470C2×C4D3311yescharacteristic
471C8—61no—
472C8—181no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3#6380011yescharacteristic
486C2×C3#6380111yescharacteristic
487C2×C3#6383111yescharacteristic
488C2×C3#6383111yescharacteristic
489C2×C3#6385211yescharacteristic
490C2×C3#6386411yescharacteristic
491C2×C3#6390711yescharacteristic
492C2×C3#6392911yescharacteristic
493C2×C3#6394311yescharacteristic
494C22D3×#62611yescharacteristic
495C4—61no—
496C4—61no—
497C4—121no—
498C4—361no—
499C4—361no—
500C4—1081no—
501C4#8896211yescharacteristic
502C4#8896511yescharacteristic
503C3—21no—
504C3—21no—
505C3—21no—
506C3—41no—
507C3#1059029511yescharacteristic, complemented
508C3#1059056211yescharacteristic, complemented
509C3#1059089811yescharacteristic, complemented
510C2#1172288611yescharacteristic
511C2#1172288711yescharacteristic
512C2#1172300311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#491
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
308. C33#491semidirect
457. C32#4703semidirect
458. C32#4797semidirect
459. C32#5069semidirect
507. C3#10590295semidirect
508. C3#10590562semidirect
509. C3#10590898semidirect

Maximal subgroups

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

subgroupindex
#117234432
C3×#638692
#117220762
#117220422
#117220412
C3×#635672
C3×#635542
#105908983
#105905623
#105902953