Finite Group Databasebeta

← #13156731

Subgroups — #13156731

Order 1728.

1506 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6289C211yesmaximal, characteristic
2C2×#88804C212yesmaximal
3#11722082C211yesmaximal, characteristic, complemented
4C3×#63579C212yesmaximal, complemented
5C3×#63579C211yesmaximal, characteristic, complemented
6#10591371—32nomaximal
7#10591219—31nomaximal
8C23×C3⋊(C3⋊D3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C2×#6289—24no—
14C2×#6289C414yescomplemented
15#88804—24no—
16C22×C4×C3⋊D3—31no—
17C22×#624—32no—
18C22×#624—61no—
19C2×#3752—32no—
20C2×#3722—32no—
21C2×#3722—32no—
22C2×#3722—62no—
23C3×C3⋊#95—32no—
24C3×C3⋊#95—32no—
25C3×C3⋊#95—32no—
26#63579—32no—
27#63579—31no—
28#63579D312yescomplemented
29#63579D311yescharacteristic, complemented
30C23×C33C2311yescharacteristic
31C22×C3⋊(C3⋊D3)—21no—
32C22×C3⋊(C3⋊D3)—21no—
33C22×C3⋊(C3⋊D3)—22no—
34C22×C3⋊(C3⋊D3)C2×C412yes—
35C22×C3⋊(C3⋊D3)D414yes—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3—22no—
40C2×C3×C3⋊Dic3C2×C412yes—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3C2×C412yes—
46#6289—28no—
47#4664—91no—
48#4664—92no—
49C23×C3⋊D3—32no—
50C23×C3⋊D3—31no—
51C23×C3⋊D3—61no—
52C23×C3⋊D3—62no—
53C23×C3⋊D3—121no—
54C22×C32×C4—31no—
55C22×C32×C4—32no—
56C22×C3⋊Dic3C2×D312yes—
57C22×C3⋊Dic3C2×D311yescharacteristic
58C2×C4×C3⋊D3—34no—
59C2×C4×C3⋊D3—64no—
60C22×C3×Dic3—21no—
61C22×C3×Dic3—22no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3—62no—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C2×D312yes—
69C22×C3×Dic3C2×D312yes—
70C2×#624—38no—
71C2×#624—64no—
72C2×#624—64no—
73C2×#624—64no—
74C2×#624—68no—
75#3752—64no—
76#3722—64no—
77#3722—64no—
78#3722—68no—
79C22×C33C22×C411yescharacteristic
80C22×C33C2×D412yes—
81C22×C33#4111yescharacteristic
82C22×C33#4111yescharacteristic
83C22×C33#4112yes—
84C2×C3⋊(C3⋊D3)—22no—
85C2×C3⋊(C3⋊D3)—22no—
86C2×C3⋊(C3⋊D3)—24no—
87C2×C3⋊(C3⋊D3)—24no—
88C2×C3⋊(C3⋊D3)C22⋊C414yes—
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3—42no—
91C32×Dic3—22no—
92C32×Dic3—42no—
93C22×C4×D3—92no—
94C22×C4×D3—91no—
95C22×C4×D3—181no—
96C2×(C2×D3)⋊C4—92no—
97C2×(C2×D3)⋊C4—92no—
98C2×(C2×D3)⋊C4—92no—
99C2×(C2×D3)⋊C4—182no—
100C3×#95—92no—
101C3×#95—91no—
102C3⋊#95—32no—
103C3⋊#95—32no—
104C3⋊#95—32no—
105C3⋊#95—92no—
106C3⋊#95—91no—
107C23×C32—21no—
108C23×C32—22no—
109C23×C32—41no—
110C23×C32C22×D312yes—
111C23×C32C22×D311yescharacteristic
112C22×C3⋊D3—34no—
113C22×C3⋊D3—32no—
114C22×C3⋊D3—34no—
115C22×C3⋊D3—34no—
116C22×C3⋊D3—34no—
117C22×C3⋊D3—62no—
118C22×C3⋊D3—64no—
119C22×C3⋊D3—64no—
120C22×C3⋊D3—64no—
121C22×C3⋊D3—64no—
122C22×C3⋊D3—62no—
123C22×C3⋊D3—61no—
124C22×C3⋊D3—61no—
125C22×C3⋊D3—64no—
126C22×C3⋊D3—62no—
127C22×C3⋊D3—62no—
128C22×C3⋊D3—62no—
129C22×C3⋊D3—62no—
130C22×C3⋊D3—62no—
131C22×C3⋊D3—62no—
132C22×C3⋊D3—122no—
133C22×C3⋊D3—124no—
134C22×C3⋊D3—122no—
135C22×C3⋊D3—122no—
136C22×C3⋊D3—122no—
137C22×C3⋊D3—122no—
138C22×C3⋊D3—122no—
139C22×C3⋊D3—124no—
140C22×C3⋊D3—122no—
141C2×C32×C4—32no—
142C2×C32×C4—62no—
143C2×C32×C4—62no—
144C2×C32×C4—62no—
145C2×C32×C4—62no—
146C2×C3⋊Dic3—22no—
147C2×C3⋊Dic3—22no—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3C4×D312yes—
151C4×C3⋊D3—68no—
152C2×C3×Dic3—24no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—42no—
165C2×C3×Dic3—42no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—64no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—122no—
178C2×C3×Dic3—122no—
179C2×C3×Dic3C4×D314yes—
180#624—616no—
181#624—616no—
182#381—271noSylow
183C2×C33C2×C22⋊C411yescharacteristic
184C2×C33#11711yescharacteristic
185C2×C33#11711yescharacteristic
186C2×C33#12312yes—
187C2×C33#12312yes—
188C3⋊(C3⋊D3)—24no—
189C23×D3—92no—
190C23×D3—91no—
191C23×D3—181no—
192C23×D3—182no—
193C23×D3—361no—
194C22×C3×C4—32no—
195C22×C3×C4—32no—
196C22×C3×C4—32no—
197C22×C3×C4—61no—
198C22×C3×C4—62no—
199C22×C3×C4—91no—
200C22×C3×C4—92no—
201C22×Dic3—32no—
202C22×Dic3—32no—
203C22×Dic3—32no—
204C22×Dic3—61no—
205C22×Dic3—62no—
206C22×Dic3D3212yes—
207C22×Dic3D3211yescharacteristic
208C2×C4×D3—98no—
209C2×C4×D3—94no—
210C2×C4×D3—184no—
211C2×C4×D3—184no—
212C2×C4×D3—184no—
213C2×C4×D3—184no—
214C2×C4×D3—188no—
215(C2×D3)⋊C4—184no—
216(C2×D3)⋊C4—184no—
217(C2×D3)⋊C4—184no—
218(C2×D3)⋊C4—188no—
219C22×C32—22no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—22no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—41no—
232C22×C32—42no—
233C22×C32—41no—
234C22×C32—42no—
235C22×C32—41no—
236C22×C32C2×C4×D311yescharacteristic
237C22×C32C2×C4×D312yes—
238C22×C32#23512yes—
239C22×C32#23512yes—
240C22×C32#23512yes—
241C22×C32D3×D412yes—
242C22×C32D3×D412yes—
243C22×C32D3×D412yes—
244C22×C32#23712yes—
245C22×C32#23711yescharacteristic
246C22×C32#23912yes—
247C22×C32#23911yescharacteristic
248C2×C3⋊D3—38no—
249C2×C3⋊D3—34no—
250C2×C3⋊D3—64no—
251C2×C3⋊D3—68no—
252C2×C3⋊D3—62no—
253C2×C3⋊D3—64no—
254C2×C3⋊D3—62no—
255C2×C3⋊D3—64no—
256C2×C3⋊D3—64no—
257C2×C3⋊D3—68no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—64no—
260C2×C3⋊D3—64no—
261C2×C3⋊D3—64no—
262C2×C3⋊D3—64no—
263C2×C3⋊D3—64no—
264C2×C3⋊D3—64no—
265C2×C3⋊D3—64no—
266C2×C3⋊D3—124no—
267C2×C3⋊D3—124no—
268C2×C3⋊D3—124no—
269C2×C3⋊D3—124no—
270C2×C3⋊D3—124no—
271C2×C3⋊D3—124no—
272C2×C3⋊D3—124no—
273C2×C3⋊D3—124no—
274C2×C3⋊D3—124no—
275C2×C3⋊D3—128no—
276C2×C3⋊D3—128no—
277C32×C4—62no—
278C32×C4—122no—
279C3⋊Dic3—22no—
280C3⋊Dic3—42no—
281C3×Dic3—24no—
282C3×Dic3—24no—
283C3×Dic3—44no—
284C3×Dic3—42no—
285C3×Dic3—42no—
286C3×Dic3—64no—
287C3×Dic3—64no—
288C3×Dic3—124no—
289C3×Dic3—122no—
290C3×Dic3—122no—
291C23×C4—271no—
292C2×C22⋊C4—272no—
293#95—92no—
294#95—91no—
295#95—271no—
296C33#38111yescharacteristic, complemented, Sylow
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—41no—
300C23×C3C2×D3211yescharacteristic
301C23×C3C2×D3212yes—
302C22×D3—94no—
303C22×D3—94no—
304C22×D3—92no—
305C22×D3—94no—
306C22×D3—94no—
307C22×D3—182no—
308C22×D3—184no—
309C22×D3—184no—
310C22×D3—184no—
311C22×D3—184no—
312C22×D3—182no—
313C22×D3—184no—
314C22×D3—182no—
315C22×D3—182no—
316C22×D3—181no—
317C22×D3—181no—
318C22×D3—182no—
319C22×D3—182no—
320C22×D3—182no—
321C22×D3—182no—
322C22×D3—362no—
323C22×D3—364no—
324C22×D3—362no—
325C22×D3—362no—
326C22×D3—362no—
327C22×D3—362no—
328C22×D3—362no—
329C22×D3—364no—
330C22×D3—362no—
331C2×C3×C4—34no—
332C2×C3×C4—62no—
333C2×C3×C4—64no—
334C2×C3×C4—62no—
335C2×C3×C4—62no—
336C2×C3×C4—64no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—62no—
343C2×C3×C4—92no—
344C2×C3×C4—122no—
345C2×C3×C4—122no—
346C2×C3×C4—182no—
347C2×C3×C4—182no—
348C2×C3×C4—182no—
349C2×C3×C4—182no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—34no—
355C2×Dic3—62no—
356C2×Dic3—64no—
357C2×Dic3—62no—
358C2×Dic3—62no—
359C2×Dic3—64no—
360C2×Dic3—62no—
361C2×Dic3—62no—
362C2×Dic3—62no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—62no—
366C2×Dic3—122no—
367C2×Dic3—122no—
368C2×Dic3#62412yes—
369C4×D3—188no—
370C4×D3—1816no—
371C4×D3—1816no—
372C2×C32—22no—
373C2×C32—22no—
374C2×C32—21no—
375C2×C32—22no—
376C2×C32—22no—
377C2×C32—22no—
378C2×C32—22no—
379C2×C32—21no—
380C2×C32—22no—
381C2×C32—22no—
382C2×C32—21no—
383C2×C32—22no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—42no—
388C2×C32—42no—
389C2×C32#87211yescharacteristic
390C2×C32#87212yes—
391C2×C32D3×C22⋊C411yescharacteristic
392C2×C32D3×C22⋊C412yes—
393C2×C32#88312yes—
394C2×C32#88312yes—
395C2×C32#88312yes—
396C2×C32#89211yescharacteristic
397C2×C32#89212yes—
398C2×C32#89412yes—
399C2×C32#89412yes—
400C2×C32#89412yes—
401C3⋊D3—64no—
402C3⋊D3—68no—
403C3⋊D3—68no—
404C3⋊D3—128no—
405C3⋊D3—128no—
406C24—271no—
407C22×C4—31no—
408C22×C4—32no—
409C22×C4—91no—
410C22×C4—92no—
411C22×C4—274no—
412C22×C4—544no—
413C22⋊C4—544no—
414C22×C3—21no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—21no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—42no—
428C22×C3—41no—
429C22×C3—42no—
430C22×C3—41no—
431C22×C3#379612yes—
432C22×C3#379811yescharacteristic
433C22×C3#379811yescharacteristic
434C22×C3#379912yes—
435C22×C3#379912yes—
436C22×C3C4×D3212yes—
437C22×C3#380212yes—
438C22×C3#380412yes—
439C22×C3#380412yes—
440C22×C3C2×#62411yescharacteristic
441C22×C3#381112yes—
442C22×C3#381112yes—
443C2×D3—98no—
444C2×D3—94no—
445C2×D3—184no—
446C2×D3—188no—
447C2×D3—184no—
448C2×D3—184no—
449C2×D3—188no—
450C2×D3—188no—
451C2×D3—184no—
452C2×D3—184no—
453C2×D3—182no—
454C2×D3—184no—
455C2×D3—184no—
456C2×D3—184no—
457C2×D3—184no—
458C2×D3—182no—
459C2×D3—184no—
460C2×D3—184no—
461C2×D3—364no—
462C2×D3—364no—
463C2×D3—364no—
464C2×D3—364no—
465C2×D3—364no—
466C2×D3—364no—
467C2×D3—364no—
468C2×D3—364no—
469C2×D3—364no—
470C2×D3—368no—
471C2×D3—368no—
472C3×C4—64no—
473C3×C4—64no—
474C3×C4—124no—
475C3×C4—122no—
476C3×C4—122no—
477C3×C4—182no—
478C3×C4—362no—
479Dic3—22no—
480Dic3—42no—
481Dic3—64no—
482Dic3—64no—
483Dic3—124no—
484Dic3—122no—
485Dic3—122no—
486C32—22no—
487C32—21no—
488C32—41no—
489C32#466411yescharacteristic, complemented
490C32#466412yescomplemented
491C23—272no—
492C23—274no—
493C23—541no—
494C23—542no—
495C23—541no—
496C23D3311yescharacteristic
497C2×C4—32no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—62no—
502C2×C4—92no—
503C2×C4—182no—
504C2×C4—182no—
505C2×C4—182no—
506C2×C4—182no—
507C2×C4—548no—
508C2×C3—21no—
509C2×C3—22no—
510C2×C3—21no—
511C2×C3—22no—
512C2×C3—21no—
513C2×C3—22no—
514C2×C3—22no—
515C2×C3—22no—
516C2×C3—22no—
517C2×C3—22no—
518C2×C3—22no—
519C2×C3—22no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3—42no—
524C2×C3—42no—
525C2×C3#6383612yes—
526C2×C3#6383612yes—
527C2×C3#6384911yescharacteristic
528C2×C3#6385312yes—
529C2×C3#6385312yes—
530C2×C3#6385412yes—
531C2×C3#6385412yes—
532C2×C3#6387411yescharacteristic
533C2×C3#6389712yes—
534C2×C3#6392112yes—
535C2×C3#6397411yescharacteristic
536C2×C3#6397512yes—
537D3—188no—
538D3—188no—
539D3—184no—
540D3—368no—
541D3—368no—
542C22—274no—
543C22—542no—
544C22—542no—
545C22—544no—
546C22—544no—
547C22D3×#62411yescharacteristic
548C22#8895412yes—
549C22#8895911yescharacteristic
550C22#8896212yes—
551C22#8896511yescharacteristic
552C4—62no—
553C4—122no—
554C4—182no—
555C4—362no—
556C3—21no—
557C3—22no—
558C3—41no—
559C3#1059121911yescharacteristic, complemented
560C3#1059137112yescomplemented
561C2—544no—
562C2#1172290611yescharacteristic
563C2#1172291811yescharacteristic
564C2#1172292011yescharacteristic
565C2#1172299612yes—
566C2#1172300012yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC23×C3⋊(C3⋊D3)C22
hypocenterC33#381
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. #11722082C2semidirect
4. C3×#63579C2semidirect
5. C3×#63579C2semidirect
14. C2×#6289C4semidirect
28. #63579D3semidirect
29. #63579D3semidirect
296. C33#381semidirect
489. C32#4664semidirect
490. C32#4664semidirect
559. C3#10591219semidirect
560. C3#10591371semidirect

Maximal subgroups

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

subgroupindex
C22×#62892
C2×#888042
#117220822
C3×#635792
#105913713
#105912193