Finite Group Databasebeta

← #13175498

Subgroups — #13175498

Order 1728.

1932 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
1C22×#6293C211yesmaximal, characteristic, complemented
2C2×#89028C211yesmaximal, characteristic
3#11723927C211yesmaximal, characteristic, complemented
4C3×#64086C211yesmaximal, characteristic, complemented
5C32×#886C211yesmaximal, characteristic, complemented
6#11723612C211yesmaximal, characteristic, complemented
7#11723603C211yesmaximal, characteristic, complemented
8#11723535C212yesmaximal, complemented
9#11723495C212yesmaximal, complemented
10#11723491C212yesmaximal, complemented
11#11723419C212yesmaximal, complemented
12#10596119—31nomaximal
13#10594081—34nomaximal
14C2×#6312C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×#67C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C2×#6293—24no—
20C2×#6293C2211yescharacteristic
21C2×#6293C2212yes—
22C2×#6293C2212yes—
23C2×#6291C2211yescharacteristic
24C2×#6289C2211yescharacteristic
25#89028—24no—
26C2×D3×C3⋊Dic3C2211yescharacteristic
27C33×C22⋊C4C2211yescharacteristic
28#88859C2212yes—
29C4⋊(C3⋊(C3⋊Dic3))C2212yes—
30C3×#3757C2211yescharacteristic
31C3×#3750C2212yes—
32C3×C4×C3⋊Dic3C2212yes—
33C32×(C2×D3)⋊C4C2212yes—
34C32×C4⋊Dic3C2212yes—
35#88807C2212yes—
36#88807C2212yes—
37#88804C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×#3830—31no—
40C22×#634—31no—
41C22×#627—34no—
42C2×#3804—34no—
43C32×#123—31no—
44#64144—31no—
45#64138—31no—
46#64105—32no—
47#64097—32no—
48#64095—31no—
49#64095—34no—
50#64095—64no—
51#64094—32no—
52#64087—32no—
53#64086D311yescharacteristic, complemented
54C3×#886D314yescomplemented
55C3×#877—34no—
56#63966—34no—
57#63962—34no—
58#63918—38no—
59#63895—38no—
60#63893—38no—
61#63852—38no—
62C23×C33C2311yescharacteristic
63C22×C3⋊(C3⋊D3)C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C2×C33×C4C2312yes—
66#6312—22no—
67C2×C3⋊(C3⋊Dic3)—22no—
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C2×C3×C3⋊Dic3D414yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C32×#67—22no—
75C2×C32×Dic3—22no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2312yes—
78#6293—22no—
79#6293—24no—
80#6293—24no—
81#6291—22no—
82#6289—22no—
83D3×C3⋊Dic3—22no—
84#5648—94no—
85#5526—91no—
86C2×C32×D4—31no—
87C22×C32×C4—31no—
88C2×#638—31no—
89C2×#638—34no—
90C2×#638—31no—
91C2×#638—64no—
92C22×C3⋊Dic3—31no—
93C22×C3⋊Dic3—34no—
94C22×C3⋊Dic3—31no—
95C22×C3⋊Dic3—64no—
96C22×C3⋊Dic3C2×D311yescharacteristic
97#3830—64no—
98C2×C4×C3⋊D3—31no—
99C2×#634—32no—
100C2×#634—32no—
101C2×#634—31no—
102C2×#634—64no—
103C2×C3×#67C2×D314yes—
104C22×C3×Dic3—34no—
105C22×C3×Dic3C2×D314yes—
106C2×#627—34no—
107C2×#627—38no—
108C2×#627—38no—
109C2×#627—616no—
110C2×#626—34no—
111C2×#624—34no—
112#3804—616no—
113C2×D3×Dic3—34no—
114C32×C4⋊C4—32no—
115C32×C22⋊C4—24no—
116C32×C22⋊C4—32no—
117C32×C22⋊C4C2×D314yes—
118C32×C22⋊C4C2×D311yescharacteristic
119#3757—32no—
120#3757C2×D311yescharacteristic
121#3752—32no—
122#3752—38no—
123#3752—32no—
124#3752—68no—
125C4⋊(C3⋊Dic3)—32no—
126C4⋊(C3⋊Dic3)—38no—
127C4⋊(C3⋊Dic3)—68no—
128#3750—32no—
129#3750—32no—
130#3750C2×D312yes—
131C4×C3⋊Dic3C2×D312yes—
132C3×#217—34no—
133C3×(C2×D3)⋊C4C2×D318yes—
134C3×C4⋊Dic3C2×D318yes—
135C3×Dic3⋊C4—38no—
136C3×C4×Dic3—38no—
137Dic3⋊Dic3—38no—
138Dic3⋊Dic3—38no—
139#3722—38no—
140(C2×D3)⋊Dic3—38no—
141C22×C33—22no—
142C22×C33C2×D412yes—
143C22×C33C2411yescharacteristic
144C2×C3⋊(C3⋊D3)—21no—
145C2×C3⋊(C3⋊D3)—22no—
146C2×C32×D3—21no—
147C2×C32×D3—22no—
148C33×C4—22no—
149C3⋊(C3⋊Dic3)—21no—
150C3⋊(C3⋊Dic3)—22no—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3C2×D412yes—
153C3×C3⋊Dic3C2×D412yes—
154C32×Dic3—21no—
155C32×Dic3—22no—
156C2×#237—94no—
157C2×#235—91no—
158C22×Dic6—94no—
159C22×Dic6—91no—
160C3×#123—34no—
161C3×#117—91no—
162#935—94no—
163#929—94no—
164#897—92no—
165#897—92no—
166#896—98no—
167#888—98no—
168#886—91no—
169#886—91no—
170#886—94no—
171#886—91no—
172#886—184no—
173#886C3⋊D311yescharacteristic, complemented
174#885—98no—
175#878—98no—
176#877—34no—
177#875—92no—
178#869—92no—
179C23×C32—24no—
180C23×C32—31no—
181C23×C32C22×D314yes—
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3—31no—
184C22×C3⋊D3—34no—
185C22×C3⋊D3—64no—
186C22×C3×D3C22×D314yes—
187C32×D4—62no—
188C2×C32×C4—28no—
189C2×C32×C4—31no—
190C2×C32×C4—32no—
191C2×C32×C4—62no—
192C2×C32×C4C22×D318yes—
193C2×C32×C4C22×D312yes—
194#638—62no—
195#638—62no—
196#638—68no—
197#638—128no—
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—38no—
201C2×C3⋊Dic3—34no—
202C2×C3⋊Dic3—64no—
203C2×C3⋊Dic3—68no—
204C2×C3⋊Dic3—62no—
205C2×C3⋊Dic3—62no—
206C2×C3⋊Dic3—68no—
207C2×C3⋊Dic3—128no—
208C2×C3⋊Dic3D1214yes—
209C2×C3⋊Dic3C22×D312yes—
210C2×C3⋊Dic3C22×D311yescharacteristic
211C2×C3⋊Dic3C22×D311yescharacteristic
212C4×C3⋊D3—62no—
213#634—64no—
214#634—64no—
215#634—62no—
216C3×#67—28no—
217C2×C3×Dic3—28no—
218C2×C3×Dic3—316no—
219C2×C3×Dic3—34no—
220C2×C3×Dic3—34no—
221C2×C3×Dic3—38no—
222C2×C3×Dic3C22×D318yes—
223C2×C3×Dic3C22×D314yes—
224#627—68no—
225#627—616no—
226#627—616no—
227#626—68no—
228#624—68no—
229D3×Dic3—68no—
230#536—271noSylow
231C2×C33—22no—
232C2×C33C22×D411yescharacteristic
233C2×C332−1+412yes—
234C3⋊(C3⋊D3)—41no—
235C32×D3—41no—
236C2×C3×D4—34no—
237C22×C3×C4—34no—
238C22×C3×C4—91no—
239C2×#67—94no—
240C2×#67—94no—
241C2×#67—91no—
242C2×#67—184no—
243C2×#67C2×C3⋊D311yescharacteristic
244C22×Dic3—34no—
245C22×Dic3—94no—
246C22×Dic3—94no—
247C22×Dic3—91no—
248C22×Dic3—184no—
249C22×Dic3C2×C3⋊D311yescharacteristic
250#237—1816no—
251#235—184no—
252C2×D12—91no—
253C2×C4×D3—94no—
254C2×C4×D3—91no—
255C2×C4×D3—91no—
256C2×Dic6—98no—
257C2×Dic6—98no—
258C2×Dic6—94no—
259C2×Dic6—91no—
260C2×Dic6—92no—
261C2×Dic6—92no—
262C2×Dic6—184no—
263C2×Dic6—1816no—
264C3×C4⋊C4—38no—
265C3×C4⋊C4—92no—
266C3×C22⋊C4—24no—
267C3×C22⋊C4—38no—
268C3×C22⋊C4—91no—
269C3×C22⋊C4D3214yes—
270C3×C22⋊C4C2×C3⋊D311yescharacteristic
271C3×C42—92no—
272#217—34no—
273#217—98no—
274(C2×D3)⋊C4—98no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—98no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—188no—
280(C2×D3)⋊C4C2×C3⋊D312yes—
281C4⋊Dic3—98no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284C4⋊Dic3—92no—
285C4⋊Dic3—188no—
286C4⋊Dic3C2×C3⋊D312yes—
287Dic3⋊C4—38no—
288Dic3⋊C4—98no—
289Dic3⋊C4—98no—
290C4×Dic3—38no—
291C22×C32—22no—
292C22×C32—24no—
293C22×C32—28no—
294C22×C32—28no—
295C22×C32—48no—
296C22×C32—61no—
297C22×C32—62no—
298C22×C32C2×D1212yes—
299C22×C32D3×D418yes—
300C22×C32C23×D314yes—
301C22×C32C23×D311yescharacteristic
302C2×C3⋊D3—61no—
303C2×C3⋊D3—62no—
304C2×C3⋊D3—64no—
305C2×C3⋊D3—68no—
306C2×C3⋊D3—124no—
307C2×C3⋊D3—128no—
308C2×C3×D3—24no—
309C2×C3×D3—28no—
310C32×C4—22no—
311C32×C4—28no—
312C32×C4—48no—
313C32×C4—62no—
314C32×C4—61no—
315C3⋊Dic3—22no—
316C3⋊Dic3—62no—
317C3⋊Dic3—61no—
318C3⋊Dic3—64no—
319C3⋊Dic3—68no—
320C3⋊Dic3—128no—
321C3⋊Dic3—124no—
322C3⋊Dic3C2×D1212yes—
323C3⋊Dic3C2×D1212yes—
324C3×Dic3—24no—
325C3×Dic3—28no—
326C3×Dic3—38no—
327C3×Dic3—38no—
328C3×Dic3—68no—
329C2×#41—271no—
330C22×Q8—271no—
331C4⋊Q8—272no—
332#124—272no—
333#123—31no—
334#123—271no—
335#123—271no—
336#123—271no—
337C22⋊Q8—272no—
338C22⋊Q8—272no—
339#117—91no—
340C33#53611yescharacteristic, complemented, Sylow
341C23×C3—24no—
342C23×C3—34no—
343C23×C3C2×D3214yes—
344C23×C3C22×C3⋊D311yescharacteristic
345C22×D3—94no—
346C22×D3—91no—
347C22×D3—184no—
348C22×D3C22×C3⋊D311yescharacteristic
349C3×D4—68no—
350C2×C3×C4—28no—
351C2×C3×C4—34no—
352C2×C3×C4—38no—
353C2×C3×C4—68no—
354C2×C3×C4—91no—
355C2×C3×C4—91no—
356C2×C3×C4—94no—
357C2×C3×C4—92no—
358C2×C3×C4C2×D3218yes—
359C2×C3×C4C22×C3⋊D312yes—
360#67—22no—
361#67—188no—
362#67—188no—
363#67—182no—
364#67—368no—
365C2×Dic3—22no—
366C2×Dic3—34no—
367C2×Dic3—34no—
368C2×Dic3—316no—
369C2×Dic3—38no—
370C2×Dic3—94no—
371C2×Dic3—98no—
372C2×Dic3—91no—
373C2×Dic3—92no—
374C2×Dic3—184no—
375C2×Dic3—188no—
376C2×Dic3—188no—
377C2×Dic3—188no—
378C2×Dic3—182no—
379C2×Dic3—368no—
380C2×Dic3C22×C3⋊D311yescharacteristic
381C2×Dic3C22×C3⋊D312yes—
382D12—182no—
383C4×D3—182no—
384C4×D3—188no—
385C4×D3—182no—
386Dic6—184no—
387Dic6—184no—
388Dic6—182no—
389Dic6—1816no—
390Dic6—1816no—
391Dic6—188no—
392C2×C32—22no—
393C2×C32—24no—
394C2×C32—28no—
395C2×C32—28no—
396C2×C32—48no—
397C2×C32—121no—
398C2×C32C22×D1211yescharacteristic
399C2×C32C2×D3×D414yes—
400C2×C32#101018yes—
401C2×C32#101012yes—
402C3⋊D3—121no—
403C3⋊D3—124no—
404C3⋊D3—244no—
405C3×D3—44no—
406#41—544no—
407C2×Q8—272no—
408C2×Q8—272no—
409C2×Q8—271no—
410C2×Q8—544no—
411C2×D4—31no—
412C2×D4—271no—
413C2×D4—271no—
414C22×C4—31no—
415C22×C4—91no—
416C22×C4—271no—
417C22×C4—271no—
418C22×C4—271no—
419C4⋊C4—32no—
420C4⋊C4—92no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C22⋊C4—32no—
425C22⋊C4—91no—
426C22⋊C4—272no—
427C22⋊C4—272no—
428C22⋊C4—272no—
429C22⋊C4D3×C3⋊D311yescharacteristic
430C42—92no—
431C22×C3—28no—
432C22×C3—22no—
433C22×C3—24no—
434C22×C3—28no—
435C22×C3—48no—
436C22×C3—64no—
437C22×C3—68no—
438C22×C3D3×D1218yes—
439C22×C3D4×C3⋊D312yes—
440C22×C3C22×D3214yes—
441C22×C3C23×C3⋊D311yescharacteristic
442C2×D3—21no—
443C2×D3—22no—
444C2×D3—184no—
445C2×D3—188no—
446C2×D3—181no—
447C2×D3—182no—
448C2×D3—364no—
449C2×D3—368no—
450C3×C4—28no—
451C3×C4—22no—
452C3×C4—48no—
453C3×C4—68no—
454C3×C4—64no—
455C3×C4—92no—
456C3×C4—92no—
457C3×C4—182no—
458Dic3—22no—
459Dic3—21no—
460Dic3—38no—
461Dic3—38no—
462Dic3—68no—
463Dic3—188no—
464Dic3—184no—
465Dic3—182no—
466Dic3—181no—
467Dic3—368no—
468Dic3—364no—
469C32—24no—
470C32#552611yescharacteristic, complemented
471C32#564814yescomplemented
472C23—31no—
473C23—271no—
474C23C2×D3×C3⋊D311yescharacteristic
475Q8—544no—
476Q8—544no—
477Q8—542no—
478D4—62no—
479D4—542no—
480D4—542no—
481C2×C4—32no—
482C2×C4—31no—
483C2×C4—62no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—94no—
487C2×C4—92no—
488C2×C4—272no—
489C2×C4—271no—
490C2×C4—542no—
491C2×C4—542no—
492C2×C4—542no—
493C2×C4C2×D3×C3⋊D312yes—
494C2×C3—22no—
495C2×C3—28no—
496C2×C3—24no—
497C2×C3—28no—
498C2×C3—48no—
499C2×C3—124no—
500C2×C3C2×D3×D1214yes—
501C2×C3#6430918yes—
502C2×C3C2×D4×C3⋊D311yescharacteristic
503C2×C3#6436712yes—
504D3—41no—
505D3—361no—
506D3—364no—
507D3—724no—
508C22—22no—
509C22—61no—
510C22—62no—
511C22—541no—
512C22—542no—
513C22C3⋊D3×D1212yes—
514C22C22×D3×C3⋊D311yescharacteristic
515C4—22no—
516C4—62no—
517C4—61no—
518C4—92no—
519C4—92no—
520C4—182no—
521C4—541no—
522C4—542no—
523C3—24no—
524C3#1059408114yescomplemented
525C3#1059611911yescharacteristic, complemented
526C2—22no—
527C2—121no—
528C2—1081no—
529C2C2×C3⋊D3×D1211yescharacteristic
530C2#1172458212yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×#89028C2
hypercenterC22⋊C4D3×C3⋊D3
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6293C2semidirect
3. #11723927C2semidirect
4. C3×#64086C2semidirect
5. C32×#886C2semidirect
6. #11723612C2semidirect
7. #11723603C2semidirect
8. #11723535C2semidirect
9. #11723495C2semidirect
10. #11723491C2semidirect
11. #11723419C2semidirect
53. #64086D3semidirect
54. C3×#886D3semidirect
173. #886C3⋊D3semidirect
340. C33#536semidirect
470. C32#5526semidirect
471. C32#5648semidirect
524. C3#10594081semidirect
525. C3#10596119semidirect

Maximal subgroups

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

subgroupindex
C22×#62932
C2×#890282
#117239272
C3×#640862
C32×#8862
#117236122
#117236032
#117235352
#117234952
#117234912
#117234192
#105961193
#105940813