Finite Group Databasebeta

← #13167583

Subgroups — #13167583

Order 1728.

2060 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 2060 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C23×#1113C211yesmaximal, characteristic
2C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
3C2×#88939C214yesmaximal, complemented
4#11723723C211yesmaximal, characteristic, complemented
5#10593512—31nomaximal
6C23×C3×C3⋊D3—21no—
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×#1113—21no—
9C22×#1113—22no—
10C22×#1113—28no—
11C22×#1113C414yescomplemented
12C22×#1113C2212yes—
13C22×C3×#157C2212yescomplemented
14C2×#6294C2212yescomplemented
15C3×C22⋊#157—22no—
16#88939—28no—
17C3×#3757—21no—
18#88814—21no—
19C23×D32—31no—
20C23×D32—61no—
21C2×C22⋊#157D311yescharacteristic, complemented
22C2×D32⋊C4—34no—
23#63974—31no—
24D3×#217—61no—
25C23×C33D411yescharacteristic
26C22×C3×C3⋊D3—28no—
27C22×C3×C3⋊D3—24no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3—24no—
30C22×C3×C3⋊D3C2×C412yes—
31C22×C3×C3⋊D3D411yescharacteristic
32C22×C3×C3⋊D3D412yes—
33C22×C3×C3⋊D3C2311yescharacteristic
34C2×#1113—28no—
35C2×#1113—216no—
36C2×#1113—24no—
37C2×#1113—24no—
38C2×#1113—22no—
39C2×#1113—216no—
40C2×#1113C2×C414yes—
41C2×#1113D418yes—
42C2×C3×#157—24no—
43C2×C3×#157—22no—
44C2×C3×C3⋊Dic3—22no—
45#6294—24no—
46#4955—91nomaximal
47C23×C3⋊D3—21no—
48C23×C3⋊D3C2×D311yescharacteristic
49C23×C3×D3—61no—
50C23×C3×D3—61no—
51C23×C3×D3—61no—
52C23×C3×D3—61no—
53C23×C3×D3—121no—
54C22×D32—32no—
55C22×D32—34no—
56C22×D32—68no—
57C22×D32—64no—
58C22×D32—62no—
59C22×D32—616no—
60C22×D32—62no—
61C22×D32—64no—
62C22×D32—61no—
63C22×D32—62no—
64C22×#157C2×D312yescomplemented
65C2×#624—32no—
66C2×D3×Dic3—62no—
67C22⋊#157—22no—
68D32⋊C4—68no—
69#3757—21no—
70C3×#217—61no—
71C3×#217—61no—
72C3×#217—121no—
73#3722—61no—
74(C2×D3)⋊Dic3—62no—
75C22×C33—22no—
76C22×C33C22⋊C412yescomplemented
77C22×C33C2×D411yescharacteristic
78C2×C3×C3⋊D3—216no—
79C2×C3×C3⋊D3—24no—
80C2×C3×C3⋊D3—22no—
81C2×C3×C3⋊D3—28no—
82C2×C3×C3⋊D3—24no—
83C2×C3×C3⋊D3—24no—
84C2×C3×C3⋊D3C22⋊C412yescomplemented
85C2×C3×C3⋊D3C22×C411yescharacteristic
86C2×C3×C3⋊D3C2×D414yes—
87C2×C3×C3⋊D3C2×D411yescharacteristic
88#1113—216no—
89#1113—24no—
90#1113—28no—
91#1113C22⋊C418yescomplemented
92C3×#157—42no—
93C3×C3⋊Dic3—42no—
94C24×D3—91no—
95C24×D3—181no—
96C2×C3×C22⋊C4—91no—
97C2×#217—91no—
98C2×(C2×D3)⋊C4—94no—
99D3×C22⋊C4—181no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32—41no—
104C23×C32D1211yescharacteristic
105C22×C3⋊D3—28no—
106C22×C3⋊D3—22no—
107C22×C3⋊D3—24no—
108C22×C3⋊D3—24no—
109C22×C3⋊D3C4×D312yes—
110C22×C3⋊D3D1211yescharacteristic
111C22×C3⋊D3#6712yes—
112C22×C3⋊D3C22×D311yescharacteristic
113C22×C3×D3—62no—
114C22×C3×D3—61no—
115C22×C3×D3—62no—
116C22×C3×D3—61no—
117C22×C3×D3—61no—
118C22×C3×D3—61no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—68no—
122C22×C3×D3—68no—
123C22×C3×D3—68no—
124C22×C3×D3—62no—
125C22×C3×D3—64no—
126C22×C3×D3—64no—
127C22×C3×D3—62no—
128C22×C3×D3—124no—
129C22×C3×D3—124no—
130C22×C3×D3—122no—
131C22×C3×D3—128no—
132C2×D32—34no—
133C2×D32—38no—
134C2×D32—64no—
135C2×D32—68no—
136C2×D32—616no—
137C2×D32—632no—
138C2×D32—68no—
139C2×D32—64no—
140C2×D32—68no—
141C2×D32—632no—
142C2×D32—616no—
143C2×D32—68no—
144C2×D32—616no—
145C2×D32—64no—
146C2×D32—64no—
147C2×D32—62no—
148C2×#157—24no—
149C2×#157—22no—
150C2×C3⋊Dic3—22no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—62no—
153C2×C3×Dic3—122no—
154#624—64no—
155D3×Dic3—124no—
156#379—271noSylow
157C2×C33—22no—
158C2×C33C2×C22⋊C411yescharacteristic
159C2×C33#12012yes—
160C3×C3⋊D3—22no—
161C3×C3⋊D3—28no—
162C3×C3⋊D3C2×C22⋊C412yes—
163C3×C3⋊D3#12012yes—
164C24×C3—91no—
165C24×C3—181no—
166C23×D3—61no—
167C23×D3—61no—
168C23×D3—61no—
169C23×D3—61no—
170C23×D3—92no—
171C23×D3—94no—
172C23×D3—121no—
173C23×D3—182no—
174C23×D3—184no—
175C23×D3—188no—
176C23×D3—181no—
177C23×D3—182no—
178C23×D3—182no—
179C23×D3—184no—
180C23×D3—1816no—
181C22×C3×C4—92no—
182C22×Dic3—92no—
183C2×C4×D3—182no—
184C3×C22⋊C4—181no—
185C3×C22⋊C4—182no—
186#217—61no—
187#217—61no—
188#217—121no—
189#217—181no—
190(C2×D3)⋊C4—182no—
191(C2×D3)⋊C4—188no—
192C22×C32—24no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—22no—
197C22×C32—22no—
198C22×C32—42no—
199C22×C32—41no—
200C22×C32—41no—
201C22×C32—42no—
202C22×C32—42no—
203C22×C32—44no—
204C22×C32—82no—
205C22×C32(C2×D3)⋊C412yescomplemented
206C22×C32C2×D1211yescharacteristic
207C2×C3⋊D3—216no—
208C2×C3⋊D3—24no—
209C2×C3⋊D3—28no—
210C2×C3⋊D3—22no—
211C2×C3⋊D3—24no—
212C2×C3⋊D3—24no—
213C2×C3⋊D3(C2×D3)⋊C412yescomplemented
214C2×C3⋊D3C2×C4×D311yescharacteristic
215C2×C3⋊D3D3×D414yes—
216C2×C3⋊D3C2×#6711yescharacteristic
217C2×C3×D3—62no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—61no—
223C2×C3×D3—61no—
224C2×C3×D3—62no—
225C2×C3×D3—68no—
226C2×C3×D3—68no—
227C2×C3×D3—616no—
228C2×C3×D3—616no—
229C2×C3×D3—64no—
230C2×C3×D3—64no—
231C2×C3×D3—68no—
232C2×C3×D3—68no—
233C2×C3×D3—64no—
234C2×C3×D3—124no—
235C2×C3×D3—124no—
236C2×C3×D3—122no—
237C2×C3×D3—124no—
238C2×C3×D3—128no—
239C2×C3×D3—1216no—
240D32—38no—
241D32—68no—
242D32—68no—
243D32—616no—
244D32—632no—
245D32—616no—
246D32—64no—
247D32—68no—
248#157—42no—
249C3⋊Dic3—42no—
250C3×Dic3—122no—
251C3×Dic3—122no—
252C3×Dic3—242no—
253C25—271no—
254C2×C22⋊C4—91no—
255C2×C22⋊C4—271no—
256C2×C22⋊C4—274no—
257C33#37911yescharacteristic, complemented, Sylow
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—91no—
263C23×C3—91no—
264C23×C3—92no—
265C23×C3—92no—
266C23×C3—184no—
267C23×C3—188no—
268C23×C3—182no—
269C23×C3—184no—
270C23×C3GO+(4,𝔽2)11yescharacteristic
271C22×D3—62no—
272C22×D3—61no—
273C22×D3—62no—
274C22×D3—61no—
275C22×D3—61no—
276C22×D3—62no—
277C22×D3—61no—
278C22×D3—62no—
279C22×D3—68no—
280C22×D3—68no—
281C22×D3—68no—
282C22×D3—62no—
283C22×D3—64no—
284C22×D3—62no—
285C22×D3—64no—
286C22×D3—98no—
287C22×D3—94no—
288C22×D3—124no—
289C22×D3—128no—
290C22×D3—122no—
291C22×D3—124no—
292C22×D3—188no—
293C22×D3—184no—
294C22×D3—1816no—
295C22×D3—182no—
296C22×D3—184no—
297C22×D3—184no—
298C22×D3—188no—
299C22×D3—188no—
300C22×D3—184no—
301C22×D3—1832no—
302C22×D3—1816no—
303C22×D3—1832no—
304C22×D3—1816no—
305C22×D3—188no—
306C2×C3×C4—182no—
307C2×C3×C4—184no—
308C2×C3×C4—182no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—122no—
312C2×Dic3—184no—
313C4×D3—364no—
314C2×C32—24no—
315C2×C32—22no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—22no—
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—42no—
323C2×C32—42no—
324C2×C32—42no—
325C2×C32—44no—
326C2×C32—82no—
327C2×C32#88212yes—
328C2×C32C2×(C2×D3)⋊C411yescharacteristic
329C3⋊D3—28no—
330C3⋊D3—22no—
331C3⋊D3D3×C22⋊C412yes—
332C3⋊D3#93712yes—
333C3×D3—62no—
334C3×D3—62no—
335C3×D3—62no—
336C3×D3—62no—
337C3×D3—64no—
338C3×D3—68no—
339C3×D3—68no—
340C3×D3—122no—
341C3×D3—128no—
342C24—91no—
343C24—181no—
344C24—272no—
345C24—274no—
346C24—541no—
347C24—542no—
348C24—548no—
349C22×C4—92no—
350C22×C4—272no—
351C22⋊C4—182no—
352C22⋊C4—181no—
353C22⋊C4—541no—
354C22⋊C4—548no—
355C22×C3—24no—
356C22×C3—22no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—42no—
362C22×C3—44no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—82no—
368C22×C3—91no—
369C22×C3—91no—
370C22×C3—94no—
371C22×C3—92no—
372C22×C3—184no—
373C22×C3—184no—
374C22×C3—1816no—
375C22×C3—184no—
376C22×C3—188no—
377C22×C3—182no—
378C22×C3D32⋊C412yescomplemented
379C22×C3C2×GO+(4,𝔽2)11yescharacteristic
380C2×D3—64no—
381C2×D3—62no—
382C2×D3—61no—
383C2×D3—64no—
384C2×D3—61no—
385C2×D3—61no—
386C2×D3—61no—
387C2×D3—62no—
388C2×D3—68no—
389C2×D3—68no—
390C2×D3—616no—
391C2×D3—616no—
392C2×D3—64no—
393C2×D3—64no—
394C2×D3—68no—
395C2×D3—64no—
396C2×D3—68no—
397C2×D3—98no—
398C2×D3—124no—
399C2×D3—124no—
400C2×D3—122no—
401C2×D3—1216no—
402C2×D3—124no—
403C2×D3—128no—
404C2×D3—188no—
405C2×D3—188no—
406C2×D3—184no—
407C2×D3—188no—
408C2×D3—1816no—
409C2×D3—1832no—
410C2×D3—1816no—
411C3×C4—362no—
412C3×C4—362no—
413Dic3—122no—
414Dic3—122no—
415Dic3—242no—
416C32—21no—
417C32—21no—
418C32—41no—
419C32—41no—
420C32#495511yescharacteristic, complemented
421C23—91no—
422C23—91no—
423C23—92no—
424C23—92no—
425C23—184no—
426C23—188no—
427C23—182no—
428C23—184no—
429C23—278no—
430C23—274no—
431C23—542no—
432C23—544no—
433C23—544no—
434C23—5416no—
435C23—5416no—
436C23—548no—
437C23#157⋊D311yescharacteristic
438C2×C4—182no—
439C2×C4—184no—
440C2×C4—182no—
441C2×C4—544no—
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—42no—
449C2×C3—44no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—42no—
453C2×C3—42no—
454C2×C3—82no—
455C2×C3—92no—
456C2×C3—92no—
457C2×C3—188no—
458C2×C3—182no—
459C2×C3C2×D32⋊C411yescharacteristic
460C2×C3#6424112yes—
461D3—62no—
462D3—62no—
463D3—62no—
464D3—62no—
465D3—64no—
466D3—68no—
467D3—68no—
468D3—122no—
469D3—128no—
470C22—22no—
471C22—91no—
472C22—91no—
473C22—94no—
474C22—92no—
475C22—184no—
476C22—184no—
477C22—1816no—
478C22—184no—
479C22—182no—
480C22—188no—
481C22—278no—
482C22—548no—
483C22—544no—
484C22—5416no—
485C22#8893912yescomplemented
486C22C2×#157⋊D311yescharacteristic
487C4—362no—
488C4—362no—
489C3—21no—
490C3—21no—
491C3—41no—
492C3—41no—
493C3#1059351211yescharacteristic, complemented
494C2—22no—
495C2—92no—
496C2—92no—
497C2—182no—
498C2—188no—
499C2C2×#8893911yescharacteristic
500C2#1172435412yes—
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC23×#1113C2
cosocleC22×C3×C3⋊D3C23
hypercenterC23#157⋊D3
hypocenterC33#379
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×C3×C22⋊#157C2semidirect
3. C2×#88939C2semidirect
4. #11723723C2semidirect
11. C22×#1113C4semidirect
13. C22×C3×#157C22semidirect
14. C2×#6294C22semidirect
21. C2×C22⋊#157D3semidirect
64. C22×#157C2×D3semidirect
76. C22×C33C22⋊C4semidirect
84. C2×C3×C3⋊D3C22⋊C4semidirect
91. #1113C22⋊C4semidirect
205. C22×C32(C2×D3)⋊C4semidirect
213. C2×C3⋊D3(C2×D3)⋊C4semidirect
257. C33#379semidirect
378. C22×C3D32⋊C4semidirect
420. C32#4955semidirect
485. C22#88939semidirect
493. C3#10593512semidirect

Maximal subgroups

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

subgroupindex
C23×#11132
C2×C3×C22⋊#1572
C2×#889392
#117237232
#105935123
#49559