Finite Group Databasebeta

← #13156980

Subgroups — #13156980

Order 1728.

846 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3805C211yesmaximal, characteristic
2#11723715C211yesmaximal, characteristic, complemented
3#11723595C211yesmaximal, characteristic, complemented
4#11722062C212yesmaximal, complemented
5C3×#63580C212yesmaximal, complemented
6#10591528—32nomaximal
7#10590406—31nomaximal
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C32×#67C2212yes—
10C2×C3×#627C411yescharacteristic, complemented
11C2×C3×#625C411yescharacteristic, complemented
12C3×#3805—21no—
13C3×#3805—21no—
14C3×#3805—21no—
15C3×#3805—21no—
16C2×C3×D3×Dic3C412yescomplemented
17#88864C2211yescharacteristic
18C3×#3757C2212yes—
19C32×#217C2211yescharacteristic
20#88815—21no—
21#88813—21no—
22#88807—21no—
23Dic3×C3⋊Dic3—21no—
24C2×C3×#237—32no—
25C2×#3805D311yescharacteristic
26#64086—31no—
27#63957—31no—
28#63955—32no—
29#63955—32no—
30#63955—61no—
31#63661—32no—
32C3×#834—32no—
33C3×#806—32no—
34#63580—32no—
35#63580—32no—
36#63580—62no—
37#63580D312yescomplemented
38C23×C33C2311yescharacteristic
39C22×C32×D3C2×C412yes—
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C3×C3⋊Dic3D411yescharacteristic
47C2×C3×C3⋊Dic3D411yescharacteristic
48C32×#67—22no—
49C32×#67—22no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3C2×C412yes—
52C3×#627—21no—
53C3×#627—21no—
54C3×#625—21no—
55C3×#625—21no—
56C3×D3×Dic3—22no—
57C3×D3×Dic3—22no—
58#5240—91no—
59#4740—92no—
60C2×C32×D4—32no—
61C22×C3⋊Dic3C2×D311yescharacteristic
62C2×C3×#67—22no—
63C2×C3×#67—32no—
64C2×C3×#67C2×D312yes—
65C2×C3×#67C2×D312yes—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—61no—
69C3×#237—62no—
70C3×#237—62no—
71C3×#237—62no—
72C3×#237—62no—
73C2×C3×C4×D3—32no—
74C2×C3×Dic6—32no—
75C2×#627Dic311yescharacteristic, complemented
76C2×#625Dic311yescharacteristic, complemented
77#3805—21no—
78#3805—21no—
79#3805—21no—
80#3805—21no—
81C2×D3×Dic3Dic312yescomplemented
82C32×C22⋊C4—31no—
83#3757—31no—
84#3757—32no—
85#3757—61no—
86#3757—62no—
87#3757—121no—
88#3757C2×D312yes—
89#3750—61no—
90C4×C3⋊Dic3—61no—
91C3×#217—21no—
92C3×#217—32no—
93C3×#217—32no—
94C3×#217—62no—
95C3×#217C2×D312yes—
96#3724—61no—
97Dic3⋊Dic3—62no—
98Dic3⋊Dic3—62no—
99Dic3⋊Dic3—62no—
100Dic32—62no—
101Dic32—62no—
102Dic32—62no—
103Dic32—61no—
104C22×C33—41no—
105C22×C33C22×C411yescharacteristic
106C22×C33C2×D411yescharacteristic
107C22×C33C2×D411yescharacteristic
108C2×C32×D3—22no—
109C2×C32×D3—42no—
110C3⋊(C3⋊Dic3)—41no—
111C3×C3⋊Dic3—42no—
112C3×C3⋊Dic3C22⋊C411yescharacteristic
113C3×C3⋊Dic3C22⋊C411yescharacteristic
114C3×C3⋊Dic3C22⋊C411yescharacteristic
115C3×C3⋊Dic3C22⋊C411yescharacteristic
116C32×Dic3—22no—
117C32×Dic3—41no—
118C2×C3×#41—91no—
119C2×#237—32no—
120#926—91no—
121#926—91no—
122#877—92no—
123#877—92no—
124#877—181no—
125C3×#99—92no—
126#834—32no—
127#834—92no—
128#834—92no—
129#834—182no—
130#806—32no—
131#806—92no—
132C23×C32—21no—
133C23×C32—22no—
134C23×C32—32no—
135C23×C32—41no—
136C23×C32C22×D312yes—
137C23×C32C22×D311yescharacteristic
138C22×C3×D3—22no—
139C22×C3×D3C4×D312yes—
140C22×C3×D3C2×Dic312yes—
141C32×D4—62no—
142C32×D4—62no—
143C2×C32×C4—32no—
144C2×C32×C4—61no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—62no—
148C2×C3⋊Dic3—62no—
149C2×C3⋊Dic3—122no—
150C2×C3⋊Dic3—122no—
151C2×C3⋊Dic3C2×Dic311yescharacteristic
152C2×C3⋊Dic3C2×Dic311yescharacteristic
153C2×C3⋊Dic3#6711yescharacteristic
154C2×C3⋊Dic3#6711yescharacteristic
155C2×C3⋊Dic3#6711yescharacteristic
156C2×C3⋊Dic3#6711yescharacteristic
157C3×#67—22no—
158C3×#67—22no—
159C3×#67—22no—
160C3×#67—22no—
161C3×#67—42no—
162C3×#67—42no—
163C3×#67—62no—
164C3×#67—62no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—122no—
185C2×C3×Dic3C4×D312yes—
186C2×C3×Dic3C2×Dic312yes—
187C3×C4×D3—62no—
188C3×C4×D3—62no—
189C3×Dic6—62no—
190C3×Dic6—62no—
191#627—21no—
192#627—21no—
193#625—21no—
194#625—21no—
195D3×Dic3—22no—
196D3×Dic3—22no—
197#410—271noSylow
198C2×C33—21no—
199C2×C33—21no—
200C2×C33—21no—
201C2×C33C2×C22⋊C411yescharacteristic
202C32×D3—42no—
203C3×#41—181no—
204C3×#41—181no—
205C3×#41—181no—
206C3×#41—181no—
207C2×C3×Q8—91no—
208C2×C3×D4—32no—
209C2×C3×D4—32no—
210C2×C3×D4—62no—
211C2×C3×D4—91no—
212C22×C3×C4—92no—
213C22×C3×C4—91no—
214C2×#67—32no—
215C2×#67D3212yes—
216C22×Dic3—32no—
217C22×Dic3—32no—
218C22×Dic3—61no—
219#237—62no—
220#237—62no—
221#237—62no—
222#237—62no—
223C2×C4×D3—32no—
224C2×Dic6—32no—
225C3×C22⋊C4—32no—
226C3×C22⋊C4—61no—
227C3×C22⋊C4—92no—
228#217—32no—
229#217—32no—
230#217—62no—
231#217—92no—
232#217—91no—
233#217—181no—
234#217—182no—
235#217—361no—
236#217D3211yescharacteristic
237C4⋊Dic3—181no—
238C4⋊Dic3—181no—
239Dic3⋊C4—182no—
240Dic3⋊C4—182no—
241Dic3⋊C4—182no—
242C4×Dic3—181no—
243C4×Dic3—182no—
244C4×Dic3—182no—
245C4×Dic3—181no—
246C4×Dic3—182no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32—41no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—42no—
259C22×C32—62no—
260C22×C32—82no—
261C22×C32—82no—
262C22×C32—122no—
263C22×C32C2×C4×D312yes—
264C22×C32D3×D412yes—
265C22×C32D3×D412yes—
266C22×C32C22×Dic311yescharacteristic
267C22×C32C2×#6711yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C2×C3×D3—22no—
270C2×C3×D3—22no—
271C2×C3×D3—42no—
272C2×C3×D3—42no—
273C2×C3×D3—42no—
274C2×C3×D3—82no—
275C32×C4—62no—
276C32×C4—121no—
277C3⋊Dic3—42no—
278C3⋊Dic3—121no—
279C3⋊Dic3—122no—
280C3⋊Dic3—122no—
281C3⋊Dic3—242no—
282C3⋊Dic3—242no—
283C3⋊Dic3#21711yescharacteristic
284C3⋊Dic3#21711yescharacteristic
285C3⋊Dic3#21711yescharacteristic
286C3⋊Dic3#21711yescharacteristic
287C3×Dic3—22no—
288C3×Dic3—22no—
289C3×Dic3—32no—
290C3×Dic3—32no—
291C3×Dic3—32no—
292C3×Dic3—32no—
293C3×Dic3—42no—
294C3×Dic3—42no—
295C3×Dic3—42no—
296C3×Dic3—61no—
297C3×Dic3—61no—
298C3×Dic3—61no—
299C3×Dic3—61no—
300C3×Dic3—122no—
301C3×Dic3—122no—
302C3×Dic3—242no—
303C2×#41—91no—
304#117—271no—
305#117—271no—
306#99—92no—
307#99—272no—
308C33#41011yescharacteristic, complemented, Sylow
309C23×C3—21no—
310C23×C3—22no—
311C23×C3—32no—
312C23×C3—32no—
313C23×C3—41no—
314C23×C3—62no—
315C23×C3C2×D3211yescharacteristic
316C23×C3C2×D3212yes—
317C22×D3D3×Dic312yes—
318C3×Q8—181no—
319C3×Q8—181no—
320C3×D4—62no—
321C3×D4—62no—
322C3×D4—62no—
323C3×D4—62no—
324C3×D4—122no—
325C3×D4—122no—
326C3×D4—181no—
327C3×D4—181no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—91no—
336C2×C3×C4—91no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—182no—
340C2×C3×C4—182no—
341C2×C3×C4—182no—
342#67—22no—
343#67—22no—
344#67—62no—
345#67—62no—
346C2×Dic3—21no—
347C2×Dic3—32no—
348C2×Dic3—32no—
349C2×Dic3—32no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—32no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—62no—
362C2×Dic3—62no—
363C2×Dic3—122no—
364C2×Dic3—182no—
365C2×Dic3—182no—
366C2×Dic3—181no—
367C2×Dic3—362no—
368C2×Dic3—362no—
369C2×Dic3D3×Dic312yes—
370C4×D3—62no—
371C4×D3—62no—
372Dic6—62no—
373Dic6—62no—
374C2×C32—21no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—22no—
379C2×C32—22no—
380C2×C32—22no—
381C2×C32—21no—
382C2×C32—22no—
383C2×C32—41no—
384C2×C32—42no—
385C2×C32—42no—
386C2×C32—42no—
387C2×C32—42no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—81no—
391C2×C32—81no—
392C2×C32—122no—
393C2×C32D3×C22⋊C412yes—
394C2×C32C2×#21711yescharacteristic
395C3×D3—42no—
396C3×D3—42no—
397C3×D3—82no—
398#41—181no—
399#41—181no—
400#41—181no—
401#41—181no—
402C2×Q8—91no—
403C2×D4—32no—
404C2×D4—91no—
405C22×C4—92no—
406C22×C4—91no—
407C4⋊C4—541no—
408C4⋊C4—541no—
409C22⋊C4—31no—
410C22⋊C4—92no—
411C22⋊C4—271no—
412C42—541no—
413C42—541no—
414C22×C3—21no—
415C22×C3—22no—
416C22×C3—21no—
417C22×C3—22no—
418C22×C3—21no—
419C22×C3—22no—
420C22×C3—41no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3—122no—
431C22×C3—122no—
432C22×C3—122no—
433C22×C3—242no—
434C22×C3C4×D3211yescharacteristic
435C22×C3#380211yescharacteristic
436C22×C3C2×D3×Dic312yes—
437C22×C3D3×#6712yes—
438C22×C3D3×#6712yes—
439C22×C3#381111yescharacteristic
440C2×D3—22no—
441C2×D3—42no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4—91no—
445C3×C4—91no—
446C3×C4—91no—
447C3×C4—91no—
448C3×C4—122no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4—362no—
452Dic3—22no—
453Dic3—32no—
454Dic3—32no—
455Dic3—32no—
456Dic3—32no—
457Dic3—41no—
458Dic3—61no—
459Dic3—61no—
460Dic3—61no—
461Dic3—61no—
462Dic3—122no—
463Dic3—122no—
464Dic3—242no—
465Dic3—361no—
466Dic3—362no—
467Dic3—362no—
468Dic3—722no—
469Dic3—722no—
470C32—22no—
471C32—21no—
472C32—41no—
473C32#474012yescomplemented
474C32#524011yescharacteristic, complemented
475C23—32no—
476C23D3311yescharacteristic
477Q8—181no—
478Q8—181no—
479D4—62no—
480D4—62no—
481D4—181no—
482D4—181no—
483C2×C4—32no—
484C2×C4—61no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—91no—
491C2×C4—182no—
492C2×C4—182no—
493C2×C4—182no—
494C2×C4—541no—
495C2×C3—22no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—22no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—22no—
504C2×C3—42no—
505C2×C3—42no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—42no—
509C2×C3—42no—
510C2×C3—41no—
511C2×C3—81no—
512C2×C3—81no—
513C2×C3—122no—
514C2×C3—122no—
515C2×C3—242no—
516C2×C3#6392111yescharacteristic
517C2×C3D3×#21712yes—
518D3—42no—
519C22—41no—
520C22—62no—
521C22—122no—
522C22D32×Dic311yescharacteristic
523C22#8895211yescharacteristic
524C22#8895311yescharacteristic
525C4—62no—
526C4—91no—
527C4—91no—
528C4—91no—
529C4—91no—
530C4—121no—
531C4—362no—
532C4—1081no—
533C3—21no—
534C3—22no—
535C3—41no—
536C3#1059040611yescharacteristic, complemented
537C3#1059152812yescomplemented
538C2—21no—
539C2—21no—
540C2—21no—
541C2—122no—
542C2#1172290211yescharacteristic
subgroupG/H
centerC2#11722902
commutatorC22×C33C22×C4
Frattini, hypercenterC23D33
Fitting, cosocleC23×C33C23
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3805C2
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11723715C2semidirect
3. #11723595C2semidirect
4. #11722062C2semidirect
5. C3×#63580C2semidirect
10. C2×C3×#627C4semidirect
11. C2×C3×#625C4semidirect
16. C2×C3×D3×Dic3C4semidirect
37. #63580D3semidirect
75. C2×#627Dic3semidirect
76. C2×#625Dic3semidirect
81. C2×D3×Dic3Dic3semidirect
308. C33#410semidirect
473. C32#4740semidirect
474. C32#5240semidirect
536. C3#10590406semidirect
537. C3#10591528semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38052
#117237152
#117235952
#117220622
C3×#635802
#105915283
#105904063