Finite Group Databasebeta

← #13161440

Subgroups — #13161440

Order 1728.

1548 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88864C211yesmaximal, characteristic
2C2×C3×#3757C211yesmaximal, characteristic
3C2×C32×#217C211yesmaximal, characteristic
4C2×#88807C211yesmaximal, characteristic, complemented
5#11722061C211yesmaximal, characteristic, complemented
6#11722061C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592651—31nomaximal
9#10591717—34nomaximal
10C24×C33C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17#88864—22no—
18C3×#3757—22no—
19C32×#217—22no—
20#88807—22no—
21C2×C32×C22⋊C4—31no—
22C2×#3757—31no—
23C2×#3757—34no—
24C2×#3757—64no—
25C2×#3757D311yescharacteristic
26C2×#3750—31no—
27C2×C3×#217—34no—
28C2×C3×#217D314yes—
29C2×Dic3⋊Dic3—34no—
30#63658—31no—
31#63658—31no—
32#63658—31no—
33#63579—34no—
34#63579—34no—
35#63579—34no—
36C23×C33—21no—
37C23×C33—21no—
38C23×C33—21no—
39C23×C33—21no—
40C23×C33—21no—
41C23×C33—21no—
42C23×C33D412yes—
43C23×C33C2311yescharacteristic
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)—21no—
46C2×C3⋊(C3⋊Dic3)—21no—
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D412yes—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62#4953—91no—
63#4952—94no—
64C24×C32—24no—
65C24×C32C2×D311yescharacteristic
66C24×C32C2×D314yes—
67C22×C32×C4—31no—
68C22×C32×C4—31no—
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—31no—
71C22×C3⋊Dic3—34no—
72C22×C3⋊Dic3—34no—
73C22×C3⋊Dic3—64no—
74C22×C3⋊Dic3—64no—
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3×Dic3—34no—
78C22×C3×Dic3—34no—
79C22×C3×Dic3C2×D314yes—
80C22×C3×Dic3C2×D314yes—
81C32×C22⋊C4—62no—
82#3757—22no—
83#3757—616no—
84#3757—62no—
85#3757—68no—
86#3750—62no—
87C3×#217—28no—
88C3×#217—68no—
89Dic3⋊Dic3—68no—
90C22×C33—22no—
91C22×C33—22no—
92C22×C33—22no—
93C22×C33—41no—
94C22×C33—41no—
95C22×C33—41no—
96C22×C33—41no—
97C22×C33C2×D411yescharacteristic
98C22×C33C2×D411yescharacteristic
99C22×C33#4111yescharacteristic
100C22×C33#4111yescharacteristic
101C22×C33#4111yescharacteristic
102C22×C33#4111yescharacteristic
103C22×C33#4111yescharacteristic
104C3⋊(C3⋊Dic3)—41no—
105C3⋊(C3⋊Dic3)—41no—
106C3×C3⋊Dic3—22no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—41no—
109C32×Dic3—41no—
110C2×C3×C22⋊C4—34no—
111C2×C3×C22⋊C4—91no—
112C2×#217—34no—
113C2×#217—94no—
114C2×#217—91no—
115C2×#217—184no—
116C2×#217C3⋊D311yescharacteristic
117C2×C4⋊Dic3—91no—
118C2×Dic3⋊C4—94no—
119C3⋊#95—94no—
120C3⋊#95—94no—
121C3⋊#95—94no—
122C3⋊#95—91no—
123C3⋊#95—91no—
124C3⋊#95—91no—
125C23×C32—28no—
126C23×C32—28no—
127C23×C32—21no—
128C23×C32—24no—
129C23×C32—24no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—28no—
136C23×C32—24no—
137C23×C32—24no—
138C23×C32—24no—
139C23×C32—24no—
140C23×C32—24no—
141C23×C32—44no—
142C23×C32—44no—
143C23×C32—44no—
144C23×C32—44no—
145C23×C32D1212yes—
146C23×C32#6718yes—
147C23×C32C22×D314yes—
148C23×C32C22×D311yescharacteristic
149C2×C32×C4—61no—
150C2×C32×C4—61no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—68no—
161C2×C3⋊Dic3—68no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—64no—
169C2×C3⋊Dic3—64no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—64no—
172C2×C3⋊Dic3—64no—
173C2×C3⋊Dic3—64no—
174C2×C3⋊Dic3—124no—
175C2×C3⋊Dic3—124no—
176C2×C3⋊Dic3—124no—
177C2×C3⋊Dic3—124no—
178C2×C3⋊Dic3#6712yes—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—38no—
186C2×C3×Dic3—64no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—64no—
191#397—271noSylow
192C2×C33—42no—
193C2×C33#12111yescharacteristic
194C2×C33#12311yescharacteristic
195C2×C33#12311yescharacteristic
196C2×C33#12311yescharacteristic
197C2×C33#12411yescharacteristic
198C2×C33#12611yescharacteristic
199C2×C33#12611yescharacteristic
200C24×C3—24no—
201C24×C3D3214yes—
202C24×C3C2×C3⋊D311yescharacteristic
203C22×C3×C4—34no—
204C22×C3×C4—34no—
205C22×C3×C4—91no—
206C22×C3×C4—91no—
207C22×Dic3—34no—
208C22×Dic3—34no—
209C22×Dic3—94no—
210C22×Dic3—94no—
211C22×Dic3—91no—
212C22×Dic3—91no—
213C22×Dic3—184no—
214C22×Dic3—184no—
215C22×Dic3C2×C3⋊D311yescharacteristic
216C22×Dic3C2×C3⋊D311yescharacteristic
217C3×C22⋊C4—68no—
218C3×C22⋊C4—182no—
219#217—22no—
220#217—68no—
221#217—1816no—
222#217—188no—
223#217—182no—
224C4⋊Dic3—182no—
225Dic3⋊C4—188no—
226C22×C32—216no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—28no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—28no—
239C22×C32—28no—
240C22×C32—48no—
241C22×C32—48no—
242C22×C32—48no—
243C22×C32—48no—
244C22×C32—48no—
245C22×C32—48no—
246C22×C32—44no—
247C22×C32—44no—
248C22×C32—41no—
249C22×C32—41no—
250C22×C32—44no—
251C22×C32—44no—
252C22×C32—41no—
253C22×C32—41no—
254C22×C32C2×D1211yescharacteristic
255C22×C32#23514yes—
256C22×C32#23514yes—
257C22×C32#23511yescharacteristic
258C22×C32D3×D414yes—
259C22×C32#23714yes—
260C22×C32#23714yes—
261C22×C32#23714yes—
262C22×C32#23711yescharacteristic
263C22×C32#23711yescharacteristic
264C22×C32#23711yescharacteristic
265C22×C32#23711yescharacteristic
266C22×C32C2×#6714yes—
267C22×C32C2×#6711yescharacteristic
268C32×C4—121no—
269C32×C4—121no—
270C3⋊Dic3—22no—
271C3⋊Dic3—41no—
272C3⋊Dic3—128no—
273C3⋊Dic3—128no—
274C3⋊Dic3—121no—
275C3⋊Dic3—124no—
276C3⋊Dic3—121no—
277C3⋊Dic3—124no—
278C3×Dic3—44no—
279C3×Dic3—44no—
280C3×Dic3—68no—
281C3×Dic3—124no—
282C2×C4⋊C4—271no—
283C2×C22⋊C4—31no—
284C2×C22⋊C4—91no—
285C2×C22⋊C4—271no—
286#95—271no—
287#95—271no—
288#95—271no—
289C33#39711yescharacteristic, complemented, Sylow
290C23×C3—28no—
291C23×C3—28no—
292C23×C3—24no—
293C23×C3—24no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—24no—
297C23×C3—24no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—28no—
305C23×C3—24no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—44no—
309C23×C3—44no—
310C23×C3#62618yes—
311C23×C3#63812yes—
312C23×C3C2×D3214yes—
313C23×C3C22×C3⋊D311yescharacteristic
314C2×C3×C4—64no—
315C2×C3×C4—64no—
316C2×C3×C4—64no—
317C2×C3×C4—64no—
318C2×C3×C4—64no—
319C2×C3×C4—64no—
320C2×C3×C4—92no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—38no—
333C2×Dic3—64no—
334C2×Dic3—64no—
335C2×Dic3—64no—
336C2×Dic3—64no—
337C2×Dic3—64no—
338C2×Dic3—188no—
339C2×Dic3—188no—
340C2×Dic3—184no—
341C2×Dic3—184no—
342C2×Dic3—184no—
343C2×Dic3—184no—
344C2×Dic3—184no—
345C2×Dic3—184no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—364no—
353C2×Dic3—364no—
354C2×Dic3—364no—
355C2×Dic3—364no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—416no—
364C2×C32—48no—
365C2×C32—42no—
366C2×C32#87914yes—
367C2×C32#87914yes—
368C2×C32#87911yescharacteristic
369C2×C32#87911yescharacteristic
370C2×C32#88314yes—
371C2×C32#88514yes—
372C2×C32#88611yescharacteristic
373C2×C32#88611yescharacteristic
374C2×C32#92914yes—
375C2×C32#93011yescharacteristic
376C2×C32#93514yes—
377C2×C32#93511yescharacteristic
378C2×C32#93611yescharacteristic
379C2×C32#93914yes—
380C24D3×C3⋊D311yescharacteristic
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—271no—
386C22×C4—271no—
387C4⋊C4—542no—
388C22⋊C4—62no—
389C22⋊C4—182no—
390C22⋊C4—542no—
391C22×C3—216no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—28no—
396C22×C3—28no—
397C22×C3—28no—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—48no—
406C22×C3—48no—
407C22×C3—48no—
408C22×C3—48no—
409C22×C3—48no—
410C22×C3—48no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3#380414yes—
420C22×C3#380414yes—
421C22×C3#380414yes—
422C22×C3#380514yes—
423C22×C3#380514yes—
424C22×C3C2×#62614yes—
425C22×C3D3×#6714yes—
426C22×C3#382811yescharacteristic
427C22×C3#382811yescharacteristic
428C22×C3D4×C3⋊D311yescharacteristic
429C22×C3#383011yescharacteristic
430C22×C3#383011yescharacteristic
431C22×C3#383011yescharacteristic
432C22×C3C2×#63811yescharacteristic
433C3×C4—124no—
434C3×C4—124no—
435C3×C4—182no—
436C3×C4—361no—
437Dic3—41no—
438Dic3—41no—
439Dic3—68no—
440Dic3—124no—
441Dic3—368no—
442Dic3—368no—
443Dic3—361no—
444Dic3—361no—
445Dic3—364no—
446Dic3—364no—
447C32—24no—
448C32#495214yescomplemented
449C32#495311yescharacteristic, complemented
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23#629112yes—
457C23C2×D3×C3⋊D311yescharacteristic
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—92no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—541no—
471C2×C4—541no—
472C2×C4—541no—
473C2×C4—541no—
474C2×C4—541no—
475C2×C4—541no—
476C2×C3—24no—
477C2×C3—24no—
478C2×C3—24no—
479C2×C3—24no—
480C2×C3—24no—
481C2×C3—24no—
482C2×C3—24no—
483C2×C3—416no—
484C2×C3—42no—
485C2×C3—48no—
486C2×C3#6395614yes—
487C2×C3#6395614yes—
488C2×C3#6395914yes—
489C2×C3#6396214yes—
490C2×C3#6396614yes—
491C2×C3#6396914yes—
492C2×C3#6397814yes—
493C2×C3#6408811yescharacteristic
494C2×C3#6408811yescharacteristic
495C2×C3#6409211yescharacteristic
496C2×C3#6409411yescharacteristic
497C2×C3#6413811yescharacteristic
498C2×C3#6414411yescharacteristic
499C2×C3#6414811yescharacteristic
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22#8902811yescharacteristic
508C22#8902911yescharacteristic
509C22#8902911yescharacteristic
510C22#8903111yescharacteristic
511C22#8903111yescharacteristic
512C22C2×#629111yescharacteristic
513C22C3⋊D3×#6711yescharacteristic
514C4—121no—
515C4—121no—
516C4—182no—
517C4—361no—
518C4—1081no—
519C4—1081no—
520C3—24no—
521C3#1059171714yescomplemented
522C3#1059265111yescharacteristic, complemented
523C2—42no—
524C2#1172359111yescharacteristic
525C2#1172359611yescharacteristic
526C2#1172359711yescharacteristic
527C2#1172360311yescharacteristic
528C2#1172361211yescharacteristic
529C2#1172362111yescharacteristic
530C2#1172363511yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
Fitting, prime radicalC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#397
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#88807C2semidirect
5. #11722061C2semidirect
6. #11722061C2semidirect
7. #11722061C2semidirect
289. C33#397semidirect
448. C32#4952semidirect
449. C32#4953semidirect
521. C3#10591717semidirect
522. C3#10592651semidirect

Maximal subgroups

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

subgroupindex
C2×#888642
C2×C3×#37572
C2×C32×#2172
C2×#888072
#117220612
#105926513
#105917173