Finite Group Databasebeta

← #13167585

Subgroups — #13167585

Order 1728.

1152 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 99 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×#89049C211yesmaximal, characteristic
2C22×#6294C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊#157C211yesmaximal, characteristic, complemented
4C2×#88939C212yesmaximal, complemented
5#11722791C212yesmaximal, complemented
6#10593514—31nomaximal
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×#1113C2211yescharacteristic
9C22×C3×#157C2212yes—
10C2×C3×#638—21no—
11C22×C3×C3⋊Dic3—21no—
12#89049—22no—
13#89049C414yescomplemented
14C2×#6295—21no—
15C2×#6294—21no—
16C2×#6294—24no—
17C2×#6294C2211yescharacteristic
18C2×#6294C2212yes—
19C3×C22⋊#157—22no—
20#88939—24no—
21C2×#3811—31no—
22C2×D3×#67—61no—
23C22×#624—31no—
24C22×D3×Dic3—61no—
25C2×C22⋊#157D311yescharacteristic, complemented
26C2×D32⋊C4—32no—
27#63737—32no—
28C23×C33D411yescharacteristic
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—22no—
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3C2×C412yes—
33C22×C3×C3⋊D3D411yescharacteristic
34C22×C3×C3⋊D3D412yes—
35C22×C3×C3⋊D3C2311yescharacteristic
36C2×#1113—22no—
37C2×#1113—22no—
38C2×#1113—21no—
39C2×#1113C2×C412yescomplemented
40C2×C3×#157—22no—
41C2×C3×#157—22no—
42C2×C3×#157—22no—
43C3×#638—24no—
44C2×C3×C3⋊Dic3—24no—
45C2×C3×C3⋊Dic3—22no—
46C2×C3×C3⋊Dic3—21no—
47#6295—24no—
48#6294—24no—
49#6294—22no—
50#6294—21no—
51#6294C2×C412yescomplemented
52#6294D414yescomplemented
53#4958—91nomaximal
54C23×C3⋊D3C2×D311yescharacteristic
55C23×C3×D3—61no—
56C23×C3×D3—61no—
57C22×D32—31no—
58C22×D32—61no—
59C22×#157C2×D312yes—
60C2×#638—21no—
61C22×C3⋊Dic3—21no—
62C2×C3×#67—61no—
63C2×C3×#67—61no—
64C2×C3×#67—121no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3—61no—
67C22×C3×Dic3—121no—
68#3811—34no—
69#3811—62no—
70D3×#67—64no—
71D3×#67—64no—
72C2×#626—61no—
73C2×#626—61no—
74C2×#625—61no—
75C2×#624—32no—
76C2×#624—31no—
77C2×#624—64no—
78C2×#624—61no—
79C2×D3×Dic3—62no—
80C2×D3×Dic3—61no—
81C2×D3×Dic3—68no—
82C2×D3×Dic3—62no—
83C22⋊#157—22no—
84D32⋊C4—64no—
85C22×C33—21no—
86C22×C33—21no—
87C22×C33C22⋊C412yescomplemented
88C22×C33C2×D411yescharacteristic
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3—24no—
92C2×C3×C3⋊D3—24no—
93C2×C3×C3⋊D3—22no—
94C2×C3×C3⋊D3—41no—
95C2×C3×C3⋊D3—41no—
96C2×C3×C3⋊D3C22⋊C412yescomplemented
97C2×C3×C3⋊D3C22×C411yescharacteristic
98C2×C3×C3⋊D3C2×D412yes—
99C2×C3×C3⋊D3C2×D411yescharacteristic
100C2×C3×C3⋊D3#4112yes—
101#1113—24no—
102#1113—42no—
103C3×#157—42no—
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—24no—
106C22×#67—91no—
107C23×Dic3—91no—
108C2×D3×D4—181no—
109C22×C4×D3—181no—
110C2×C3×C22⋊C4—91no—
111C2×(C2×D3)⋊C4—92no—
112C3⋊#95—92no—
113C23×C32—21no—
114C23×C32—21no—
115C23×C32—41no—
116C23×C32—41no—
117C23×C32D1211yescharacteristic
118C22×C3⋊D3—22no—
119C22×C3⋊D3—22no—
120C22×C3⋊D3—21no—
121C22×C3⋊D3C4×D312yes—
122C22×C3⋊D3D1211yescharacteristic
123C22×C3⋊D3#6712yes—
124C22×C3⋊D3C22×D311yescharacteristic
125C22×C3×D3—61no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—61no—
129C22×C3×D3—61no—
130C22×C3×D3—62no—
131C22×C3×D3—61no—
132C22×C3×D3—62no—
133C22×C3×D3—64no—
134C22×C3×D3—64no—
135C22×C3×D3—61no—
136C22×C3×D3—61no—
137C22×C3×D3—122no—
138C22×C3×D3—122no—
139C22×C3×D3—121no—
140C2×D32—32no—
141C2×D32—62no—
142C2×D32—62no—
143C2×D32—61no—
144C2×D32—62no—
145C2×D32—62no—
146C2×D32—122no—
147C2×D32—122no—
148C2×#157—22no—
149C2×#157—22no—
150C2×#157—22no—
151#638—24no—
152C2×C3⋊Dic3—24no—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3—21no—
155C3×#67—64no—
156C3×#67—64no—
157C3×#67—124no—
158C2×C3×Dic3—64no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—61no—
161C2×C3×Dic3—64no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—122no—
165C2×C3×Dic3—124no—
166C2×C3×Dic3—121no—
167#626—64no—
168#626—64no—
169#625—64no—
170#624—32no—
171#624—34no—
172#624—64no—
173#624—62no—
174#624—61no—
175D3×Dic3—62no—
176D3×Dic3—64no—
177D3×Dic3—64no—
178D3×Dic3—68no—
179D3×Dic3—62no—
180#386—271noSylow
181C2×C33—22no—
182C2×C33C2×C22⋊C411yescharacteristic
183C2×C33#12011yescharacteristic
184C2×C33#12311yescharacteristic
185C3×C3⋊D3—22no—
186C3×C3⋊D3—42no—
187C3×C3⋊D3C4×D412yes—
188C3×C3⋊D3#12112yes—
189C24×C3—91no—
190C23×D3—61no—
191C23×D3—61no—
192C23×D3—91no—
193C23×D3—181no—
194C2×C3×D4—181no—
195C22×C3×C4—92no—
196C22×C3×C4—181no—
197C2×#67—61no—
198C2×#67—61no—
199C2×#67—94no—
200C2×#67—121no—
201C2×#67—181no—
202C2×#67—181no—
203C2×#67—182no—
204C22×Dic3—61no—
205C22×Dic3—61no—
206C22×Dic3—91no—
207C22×Dic3—92no—
208C22×Dic3—121no—
209C22×Dic3—184no—
210C22×Dic3—181no—
211D3×D4—184no—
212D3×D4—184no—
213C2×D12—181no—
214C2×C4×D3—182no—
215C2×C4×D3—181no—
216C2×C4×D3—188no—
217C2×C4×D3—182no—
218C3×C22⋊C4—182no—
219(C2×D3)⋊C4—184no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—42no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—42no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—42no—
235C22×C32—42no—
236C22×C32—81no—
237C22×C32—81no—
238C22×C32(C2×D3)⋊C412yescomplemented
239C22×C32C2×D1211yescharacteristic
240C2×C3⋊D3—21no—
241C2×C3⋊D3—21no—
242C2×C3⋊D3—24no—
243C2×C3⋊D3—24no—
244C2×C3⋊D3—22no—
245C2×C3⋊D3—41no—
246C2×C3⋊D3—41no—
247C2×C3⋊D3(C2×D3)⋊C412yescomplemented
248C2×C3⋊D3C2×C4×D311yescharacteristic
249C2×C3⋊D3D3×D412yes—
250C2×C3⋊D3#23712yes—
251C2×C3⋊D3C2×#6711yescharacteristic
252C2×C3×D3—62no—
253C2×C3×D3—61no—
254C2×C3×D3—61no—
255C2×C3×D3—62no—
256C2×C3×D3—61no—
257C2×C3×D3—62no—
258C2×C3×D3—61no—
259C2×C3×D3—62no—
260C2×C3×D3—62no—
261C2×C3×D3—62no—
262C2×C3×D3—62no—
263C2×C3×D3—68no—
264C2×C3×D3—68no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—124no—
269C2×C3×D3—124no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—122no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C2×C3×D3—121no—
277C2×C3×D3—241no—
278C2×C3×D3—241no—
279D32—64no—
280D32—124no—
281D32—124no—
282D32—122no—
283#157—42no—
284C3⋊Dic3—24no—
285C3⋊Dic3—22no—
286C3×Dic3—64no—
287C3×Dic3—62no—
288C3×Dic3—64no—
289C3×Dic3—62no—
290C3×Dic3—124no—
291C3×Dic3—122no—
292C22×D4—271no—
293C23×C4—271no—
294C2×C22⋊C4—91no—
295C2×C22⋊C4—272no—
296#95—272no—
297C33#38611yescharacteristic, complemented, Sylow
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—91no—
303C23×C3—91no—
304C23×C3—92no—
305C23×C3—92no—
306C23×C3—182no—
307C23×C3—182no—
308C23×C3—181no—
309C23×C3GO+(4,𝔽2)11yescharacteristic
310C22×D3—62no—
311C22×D3—61no—
312C22×D3—62no—
313C22×D3—61no—
314C22×D3—61no—
315C22×D3—62no—
316C22×D3—61no—
317C22×D3—62no—
318C22×D3—64no—
319C22×D3—64no—
320C22×D3—61no—
321C22×D3—61no—
322C22×D3—92no—
323C22×D3—122no—
324C22×D3—122no—
325C22×D3—121no—
326C22×D3—182no—
327C22×D3—181no—
328C22×D3—182no—
329C22×D3—182no—
330C22×D3—182no—
331C22×D3—362no—
332C22×D3—362no—
333C3×D4—184no—
334C2×C3×C4—184no—
335C2×C3×C4—182no—
336C2×C3×C4—181no—
337C2×C3×C4—182no—
338C2×C3×C4—182no—
339C2×C3×C4—182no—
340#67—64no—
341#67—64no—
342#67—124no—
343#67—184no—
344#67—184no—
345C2×Dic3—64no—
346C2×Dic3—62no—
347C2×Dic3—61no—
348C2×Dic3—64no—
349C2×Dic3—62no—
350C2×Dic3—61no—
351C2×Dic3—94no—
352C2×Dic3—92no—
353C2×Dic3—124no—
354C2×Dic3—122no—
355C2×Dic3—121no—
356C2×Dic3—182no—
357C2×Dic3—181no—
358C2×Dic3—184no—
359D12—184no—
360C4×D3—184no—
361C4×D3—182no—
362C4×D3—184no—
363C4×D3—188no—
364C4×D3—182no—
365C2×C32—24no—
366C2×C32—22no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32—41no—
378C2×C32—41no—
379C2×C32—42no—
380C2×C32—44no—
381C2×C32—82no—
382C2×C32#88211yescharacteristic
383C2×C32#88611yescharacteristic
384C2×C32C2×(C2×D3)⋊C411yescharacteristic
385C3⋊D3—22no—
386C3⋊D3—42no—
387C3⋊D3#67⋊C412yes—
388C3⋊D3#93912yes—
389C3×D3—62no—
390C3×D3—62no—
391C3×D3—62no—
392C3×D3—62no—
393C3×D3—64no—
394C3×D3—122no—
395C3×D3—122no—
396C3×D3—122no—
397C3×D3—242no—
398C24—91no—
399C24—271no—
400C2×D4—181no—
401C2×D4—274no—
402C2×D4—541no—
403C2×D4—542no—
404C22×C4—92no—
405C22×C4—181no—
406C22×C4—271no—
407C22×C4—272no—
408C22×C4—544no—
409C22×C4—541no—
410C22⋊C4—182no—
411C22⋊C4—544no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—91no—
431C22×C3—91no—
432C22×C3—92no—
433C22×C3—92no—
434C22×C3—92no—
435C22×C3—184no—
436C22×C3—184no—
437C22×C3—182no—
438C22×C3—181no—
439C22×C3—181no—
440C22×C3—361no—
441C22×C3—361no—
442C22×C3D32⋊C412yescomplemented
443C22×C3C2×GO+(4,𝔽2)11yescharacteristic
444C2×D3—62no—
445C2×D3—61no—
446C2×D3—61no—
447C2×D3—62no—
448C2×D3—61no—
449C2×D3—62no—
450C2×D3—61no—
451C2×D3—62no—
452C2×D3—62no—
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—68no—
456C2×D3—68no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—62no—
460C2×D3—124no—
461C2×D3—124no—
462C2×D3—121no—
463C2×D3—121no—
464C2×D3—122no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—121no—
469C2×D3—184no—
470C2×D3—241no—
471C2×D3—241no—
472C2×D3—364no—
473C2×D3—364no—
474C2×D3—362no—
475C3×C4—184no—
476C3×C4—182no—
477C3×C4—362no—
478Dic3—64no—
479Dic3—62no—
480Dic3—62no—
481Dic3—64no—
482Dic3—124no—
483Dic3—122no—
484C32—21no—
485C32—21no—
486C32—41no—
487C32—41no—
488C32#495811yescharacteristic, complemented
489C23—91no—
490C23—91no—
491C23—92no—
492C23—92no—
493C23—182no—
494C23—182no—
495C23—181no—
496C23—272no—
497C23—541no—
498C23—542no—
499C23—542no—
500C23#157⋊D311yescharacteristic
501D4—184no—
502D4—544no—
503C2×C4—184no—
504C2×C4—182no—
505C2×C4—181no—
506C2×C4—182no—
507C2×C4—182no—
508C2×C4—182no—
509C2×C4—274no—
510C2×C4—272no—
511C2×C4—542no—
512C2×C4—541no—
513C2×C4—544no—
514C2×C3—24no—
515C2×C3—22no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—42no—
523C2×C3—44no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—82no—
531C2×C3—92no—
532C2×C3—92no—
533C2×C3—182no—
534C2×C3—362no—
535C2×C3C2×D32⋊C411yescharacteristic
536C2×C3#6423311yescharacteristic
537C2×C3#6424111yescharacteristic
538D3—62no—
539D3—62no—
540D3—62no—
541D3—62no—
542D3—64no—
543D3—122no—
544D3—122no—
545D3—122no—
546D3—242no—
547C22—21no—
548C22—21no—
549C22—91no—
550C22—91no—
551C22—92no—
552C22—92no—
553C22—92no—
554C22—184no—
555C22—184no—
556C22—181no—
557C22—181no—
558C22—182no—
559C22—361no—
560C22—361no—
561C22—544no—
562C22—1082no—
563C22#8893912yescomplemented
564C22C2×#157⋊D311yescharacteristic
565C4—182no—
566C4—184no—
567C4—362no—
568C3—21no—
569C3—21no—
570C3—41no—
571C3—41no—
572C3#1059351411yescharacteristic, complemented
573C2—22no—
574C2—92no—
575C2—92no—
576C2—182no—
577C2—362no—
578C2C2×#8893911yescharacteristic
579C2#1172434611yescharacteristic
580C2#1172435411yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutatorC2×C3×C3⋊D3C22×C4
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×#89049C2
cosocleC22×C3×C3⋊D3C23
hypercenterC23#157⋊D3
hypocenterC33#386
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×#6294C2semidirect
3. C2×C3×C22⋊#157C2semidirect
4. C2×#88939C2semidirect
5. #11722791C2semidirect
13. #89049C4semidirect
25. C2×C22⋊#157D3semidirect
39. C2×#1113C2×C4semidirect
51. #6294C2×C4semidirect
52. #6294D4semidirect
87. C22×C33C22⋊C4semidirect
96. C2×C3×C3⋊D3C22⋊C4semidirect
238. C22×C32(C2×D3)⋊C4semidirect
247. C2×C3⋊D3(C2×D3)⋊C4semidirect
297. C33#386semidirect
442. C22×C3D32⋊C4semidirect
488. C32#4958semidirect
563. C22#88939semidirect
572. C3#10593514semidirect

Maximal subgroups

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

subgroupindex
C2×#890492
C22×#62942
C2×C3×C22⋊#1572
C2×#889392
#117227912
#105935143
#49589