Finite Group Databasebeta

← #13156672

Subgroups — #13156672

Order 1728.

514 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723454C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic
4#11722057C211yesmaximal, characteristic
5#11722055C211yesmaximal, characteristic
6C3×#63573C211yesmaximal, characteristic
7C3×#63554C211yesmaximal, characteristic
8#10590892—31nomaximal
9#10590421—31nomaximal
10#10590305—31nomaximal
11C2×C3×#627—21no—
12C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21C3×Dic3⋊Dic3—21no—
22#64079—31no—
23C3×#888—31no—
24C3×#869—31no—
25#63918D311yescharacteristic
26#63869—31no—
27#63869—31no—
28#63869—61no—
29#63645—31no—
30#63634—31no—
31C3×#817—31no—
32C3×#815—31no—
33C3×#808—31no—
34C3×#803—31no—
35#63575—31no—
36#63575—31no—
37#63575—31no—
38#63575—31no—
39#63575—61no—
40#63575—61no—
41#63574—31no—
42#63573—31no—
43#63573—31no—
44#63573D311yescharacteristic
45#63554D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C3×#627—42no—
54C3×C3⋊(C3⋊C8)Q812yescomplemented
55C32×C3⋊C8Q812yescomplemented
56#5081—91no—
57#4856—91no—
58#4702—91no—
59C2×C3×Dic6—61no—
60C2×C3×Dic6—61no—
61C2×#627—21no—
62C2×C32×C8—31no—
63C32×C4⋊C4—31no—
64C32×C42—31no—
65C4⋊(C3⋊Dic3)—31no—
66C4⋊(C3⋊Dic3)—31no—
67C4⋊(C3⋊Dic3)—31no—
68C4⋊(C3⋊Dic3)—61no—
69C4⋊(C3⋊Dic3)—61no—
70C4⋊(C3⋊Dic3)—61no—
71C4⋊(C3⋊Dic3)—121no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74#3750—61no—
75#3750—61no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—31no—
79C3×C4⋊Dic3—31no—
80C3×C4⋊Dic3—31no—
81C3×C4⋊Dic3—31no—
82C3×C4⋊Dic3—61no—
83C3×C4⋊Dic3—61no—
84C3×C4⋊Dic3—61no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
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—62no—
98#3724—61no—
99#3724—61no—
100Dic3⋊Dic3—21no—
101Dic3⋊Dic3—62no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104C22×C33C2×D411yescharacteristic
105C33×C4C2×Q811yescharacteristic
106C33×C4C2×Q811yescharacteristic
107C3⋊(C3⋊Dic3)—41no—
108C3×C3⋊Dic3—41no—
109C3×C3⋊Dic3—41no—
110C32×Dic3—21no—
111C32×Dic3—21no—
112C32×Dic3—41no—
113C3×C4⋊Q8—91no—
114#890—91no—
115#888—31no—
116#870—91no—
117#870—91no—
118#870—181no—
119#869—31no—
120C3×#106—91no—
121C3×#97—91no—
122#818—91no—
123#818—91no—
124#818—91no—
125#818—181no—
126#817—31no—
127#817—91no—
128#815—31no—
129#808—31no—
130#808—91no—
131#808—91no—
132#807—91no—
133#807—91no—
134#807—91no—
135#807—181no—
136#803—31no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—31no—
141C2×C32×C4—31no—
142C2×C32×C4—41no—
143C2×C32×C4—61no—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—61no—
150C2×C3⋊Dic3—62no—
151C2×C3⋊Dic3—61no—
152C2×C3⋊Dic3—61no—
153C2×C3⋊Dic3—121no—
154C2×C3⋊Dic3—121no—
155C2×C3⋊Dic3—122no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—41no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—121no—
167C2×C3×Dic3D1211yescharacteristic
168C2×C3×Dic3D1211yescharacteristic
169C2×C3×Dic3#6711yescharacteristic
170C2×C3×Dic3#6711yescharacteristic
171C3×Dic6—122no—
172C3×Dic6—122no—
173#627—42no—
174C32×C8—32no—
175C3⋊(C3⋊C8)Dic612yescomplemented
176C3×C3⋊C8—22no—
177C3×C3⋊C8—32no—
178C3×C3⋊C8—32no—
179C3×C3⋊C8—62no—
180C3×C3⋊C8Dic612yescomplemented
181C3×C3⋊C8Dic612yescomplemented
182#501—271noSylow
183C2×C33C4⋊Q811yescharacteristic
184C2×C33AGL(1,ℤ/8)11yescharacteristic
185C2×C33#13711yescharacteristic
186C2×C3×Q8—181no—
187C2×Dic6—61no—
188C2×Dic6—61no—
189C2×C3×C8—31no—
190C2×C3×C8—31no—
191C2×C3×C8—61no—
192C2×C3×C8—91no—
193C3×C4⋊C4—31no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—91no—
197C3×C4⋊C4—91no—
198C3×C4⋊C4—181no—
199C3×C42—31no—
200C3×C42—31no—
201C3×C42—61no—
202C4⋊Dic3—31no—
203C4⋊Dic3—31no—
204C4⋊Dic3—31no—
205C4⋊Dic3—31no—
206C4⋊Dic3—61no—
207C4⋊Dic3—61no—
208C4⋊Dic3—61no—
209C4⋊Dic3—91no—
210C4⋊Dic3—91no—
211C4⋊Dic3—91no—
212C4⋊Dic3—181no—
213C4⋊Dic3—181no—
214C4⋊Dic3—181no—
215C4⋊Dic3—181no—
216C4⋊Dic3—361no—
217C4⋊Dic3D3211yescharacteristic
218Dic3⋊C4—61no—
219Dic3⋊C4—181no—
220Dic3⋊C4—181no—
221Dic3⋊C4—181no—
222Dic3⋊C4—181no—
223Dic3⋊C4—182no—
224Dic3⋊C4—182no—
225Dic3⋊C4—181no—
226C4×Dic3D3211yescharacteristic
227C2×C3⋊C8—31no—
228C2×C3⋊C8—31no—
229C2×C3⋊C8—61no—
230C2×C3⋊C8D3211yescharacteristic
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—41no—
235C22×C32C2×D1211yescharacteristic
236C22×C32D3×D411yescharacteristic
237C22×C32C2×#6711yescharacteristic
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—41no—
245C32×C4—41no—
246C32×C4—61no—
247C32×C4—61no—
248C32×C4—121no—
249C32×C4C2×Dic611yescharacteristic
250C32×C4C2×Dic611yescharacteristic
251C32×C4C2×Dic611yescharacteristic
252C32×C4D3×Q811yescharacteristic
253C32×C4D3×Q811yescharacteristic
254C32×C4D3×Q811yescharacteristic
255C3⋊Dic3—41no—
256C3⋊Dic3—41no—
257C3⋊Dic3—121no—
258C3⋊Dic3—122no—
259C3⋊Dic3—121no—
260C3⋊Dic3—121no—
261C3⋊Dic3—241no—
262C3⋊Dic3—241no—
263C3⋊Dic3—242no—
264C3×Dic3—21no—
265C3×Dic3—21no—
266C3×Dic3—21no—
267C3×Dic3—21no—
268C3×Dic3—41no—
269C3×Dic3—41no—
270C3×Dic3—41no—
271C3×Dic3—41no—
272C3×Dic3—81no—
273C3×Dic3—122no—
274C3×Dic3—121no—
275C3×Dic3—121no—
276C3×Dic3—121no—
277C3×Dic3—121no—
278C3×Dic3—241no—
279C4⋊Q8—91no—
280#125—271no—
281#107—271no—
282#107—271no—
283#106—91no—
284#106—271no—
285#97—91no—
286C33#50111yescharacteristic, complemented, Sylow
287C3×Q8—362no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—41no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—121no—
301C2×C3×C4—181no—
302C2×C3×C4—181no—
303C2×C3×C4C2×D3211yescharacteristic
304C2×C3×C4C2×D3211yescharacteristic
305C2×C3×C4C2×D3211yescharacteristic
306C2×Dic3—21no—
307C2×Dic3—62no—
308C2×Dic3—61no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—121no—
313C2×Dic3—182no—
314C2×Dic3—181no—
315C2×Dic3—181no—
316C2×Dic3—181no—
317C2×Dic3—361no—
318C2×Dic3—361no—
319C2×Dic3—362no—
320C2×Dic3#62611yescharacteristic
321C2×Dic3#62611yescharacteristic
322Dic6—122no—
323Dic6—122no—
324C3×C8—32no—
325C3×C8—32no—
326C3×C8—62no—
327C3×C8—92no—
328C3⋊C8—32no—
329C3⋊C8—32no—
330C3⋊C8—62no—
331C3⋊C8#62712yescomplemented
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32#86911yescharacteristic
345C2×C32#88811yescharacteristic
346C2×C32#90811yescharacteristic
347C2×C32#90911yescharacteristic
348C2×C32#91411yescharacteristic
349C2×C32#91511yescharacteristic
350C2×C32#93211yescharacteristic
351C2×C32#94211yescharacteristic
352C2×C32#94411yescharacteristic
353C2×Q8—181no—
354C2×C8—31no—
355C2×C8—91no—
356C4⋊C4—31no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—181no—
360C4⋊C4—271no—
361C4⋊C4—541no—
362C4⋊C4—541no—
363C42—31no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3D3×D1211yescharacteristic
369C22×C3C2×#62611yescharacteristic
370C22×C3D3×#6711yescharacteristic
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—21no—
374C3×C4—21no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—61no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—61no—
383C3×C4—121no—
384C3×C4—121no—
385C3×C4—121no—
386C3×C4—121no—
387C3×C4—241no—
388C3×C4—361no—
389C3×C4—361no—
390C3×C4D3×Dic611yescharacteristic
391C3×C4D3×Dic611yescharacteristic
392C3×C4D3×Dic611yescharacteristic
393C3×C4D3×Dic611yescharacteristic
394C3×C4#379711yescharacteristic
395C3×C4C2×#62711yescharacteristic
396Dic3—21no—
397Dic3—21no—
398Dic3—41no—
399Dic3—122no—
400Dic3—121no—
401Dic3—121no—
402Dic3—121no—
403Dic3—121no—
404Dic3—241no—
405Dic3—362no—
406Dic3—361no—
407Dic3—361no—
408Dic3—361no—
409Dic3—721no—
410Dic3—721no—
411Dic3—722no—
412C32—21no—
413C32—21no—
414C32—21no—
415C32—41no—
416C32#470211yescharacteristic, complemented
417C32#485611yescharacteristic, complemented
418C32#508111yescharacteristic, complemented
419Q8—362no—
420C2×C4—31no—
421C2×C4—31no—
422C2×C4—61no—
423C2×C4—181no—
424C2×C4—181no—
425C2×C4—541no—
426C2×C4D3311yescharacteristic
427C8—32no—
428C8—92no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3#6379411yescharacteristic
442C2×C3#6380011yescharacteristic
443C2×C3#6383011yescharacteristic
444C2×C3#6383311yescharacteristic
445C2×C3#6383911yescharacteristic
446C2×C3#6386611yescharacteristic
447C2×C3#6391811yescharacteristic
448C2×C3#6393411yescharacteristic
449C2×C3#6394611yescharacteristic
450C22D3×#62611yescharacteristic
451C4—61no—
452C4—61no—
453C4—121no—
454C4—361no—
455C4—361no—
456C4—1081no—
457C4D3×#62711yescharacteristic
458C4#8895711yescharacteristic
459C3—21no—
460C3—21no—
461C3—21no—
462C3—41no—
463C3#1059030511yescharacteristic, complemented
464C3#1059042111yescharacteristic, complemented
465C3#1059089211yescharacteristic, complemented
466C2#1172286811yescharacteristic
467C2#1172288411yescharacteristic
468C2#1172303311yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#501
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
54. C3×C3⋊(C3⋊C8)Q8semidirect
55. C32×C3⋊C8Q8semidirect
175. C3⋊(C3⋊C8)Dic6semidirect
180. C3×C3⋊C8Dic6semidirect
181. C3×C3⋊C8Dic6semidirect
286. C33#501semidirect
331. C3⋊C8#627semidirect
416. C32#4702semidirect
417. C32#4856semidirect
418. C32#5081semidirect
463. C3#10590305semidirect
464. C3#10590421semidirect
465. C3#10590892semidirect

Maximal subgroups

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

subgroupindex
#117234542
C3×#639182
#117220792
#117220572
#117220552
C3×#635732
C3×#635542
#105908923
#105904213
#105903053