Finite Group Databasebeta

← #13165384

Subgroups — #13165384

Order 1728.

876 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 88 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×#88949C211yesmaximal, characteristic
2C2×#88939C211yesmaximal, characteristic
3C2×#88932C211yesmaximal, characteristic
4C2×C3×#3773C211yesmaximal, characteristic, complemented
5#11722910C211yesmaximal, characteristic, complemented
6#11722769C211yesmaximal, characteristic, complemented
7#11722722C211yesmaximal, characteristic, complemented
8#10593398—31nomaximal
9C22×#1113C2211yescharacteristic
10C22×C3⋊#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C2×#6294C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C2×Dic3×C3⋊D3C2211yescharacteristic
15C2×C3×#624—21no—
16C2×C3×#624C2211yescharacteristic
17C2×C3×#624C2211yescharacteristic
18#88949—22no—
19#88949—22no—
20#88939C414yescomplemented
21#88932—22no—
22C3×#3773—22no—
23#88807—21no—
24C3×#3724—21no—
25C2×C4×D32—31no—
26C2×C4×D32—61no—
27C2×#3773D311yescharacteristic, complemented
28C2×D32⋊C4—31no—
29C2×D32⋊C4—31no—
30#63883—31no—
31D3×Dic3⋊C4—61no—
32#63781—31no—
33#63737—31no—
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3C2311yescharacteristic
36C2×#1113—21no—
37C2×#1113—21no—
38C2×#1113—21no—
39C2×#1113C2×C412yes—
40C2×#1113D412yes—
41C2×#1113Q812yes—
42C2×C3⋊#157—21no—
43C2×C3⋊#157—21no—
44C2×C3⋊#157—21no—
45C2×C3×#157—21no—
46C2×C3×#157—21no—
47C2×C3×#157C2×C412yescomplemented
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53#6294—21no—
54#6294C2×C412yescomplemented
55Dic3×C3⋊D3—24no—
56Dic3×C3⋊D3—21no—
57Dic3×C3⋊D3—21no—
58C3×#624—24no—
59C3×#624—21no—
60C3×#624—21no—
61C3×#624—21no—
62C3×#624D412yes—
63#4989—91nomaximal
64C22×D32—31no—
65C22×D32—61no—
66C22×#157—31no—
67C22×#157C2×D311yescharacteristic
68C2×C4×C3⋊D3—31no—
69C2×C4×C3⋊D3—61no—
70C2×C3×C4×D3—61no—
71C2×C3×C4×D3—61no—
72C2×C3×C4×D3—61no—
73C2×C3×C4×D3—61no—
74C2×#624—21no—
75C2×#624—31no—
76C2×#624C2×D311yescharacteristic
77C2×#624C2×D311yescharacteristic
78C2×D3×Dic3—61no—
79C2×D3×Dic3—61no—
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—61no—
83C2×D3×Dic3—121no—
84C4×D32—62no—
85C4×D32—64no—
86C4×D32—64no—
87C4×D32—62no—
88#3773—22no—
89D32⋊C4—34no—
90D32⋊C4—62no—
91#3750—61no—
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94#3724—21no—
95Dic3⋊Dic3—61no—
96Dic3⋊Dic3—61no—
97Dic3⋊Dic3—121no—
98C22×C33C2×D411yescharacteristic
99C2×C3×C3⋊D3—22no—
100C2×C3×C3⋊D3—22no—
101C2×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3C22×C411yescharacteristic
103C2×C3×C3⋊D3C2×D411yescharacteristic
104C2×C3×C3⋊D3C2×D411yescharacteristic
105C2×C3×C3⋊D3C2×Q811yescharacteristic
106C2×C3×C3⋊D3#4111yescharacteristic
107C2×C3×C3⋊D3#4111yescharacteristic
108#1113—22no—
109#1113—22no—
110#1113—22no—
111#1113C4⋊C414yescomplemented
112C3⋊#157—41no—
113C3×#157—22no—
114C3⋊(C3⋊Dic3)—21no—
115C3×C3⋊Dic3—22no—
116C32×Dic3—21no—
117C32×Dic3—22no—
118C32×Dic3—41no—
119C22×C4×D3—91no—
120C22×C4×D3—181no—
121C2×C3×C4⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×Dic3⋊C4—91no—
125D3×C4⋊C4—181no—
126C3⋊#95—91no—
127C3⋊#95—91no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3C22×D311yescharacteristic
130C22×C3×D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3—61no—
134C22×C3×D3—121no—
135C2×D32—32no—
136C2×D32—32no—
137C2×D32—32no—
138C2×D32—62no—
139C2×D32—62no—
140C2×D32—62no—
141C2×D32—62no—
142C2×D32—62no—
143C2×D32—62no—
144C2×D32—61no—
145C2×D32—61no—
146C2×D32—61no—
147C2×#157—21no—
148C2×#157—21no—
149C2×#157—61no—
150C2×#157—61no—
151C2×#157—61no—
152C2×#157C4×D312yescomplemented
153C2×C32×C4—31no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—121no—
161C2×C3⋊Dic3—121no—
162C4×C3⋊D3—64no—
163C4×C3⋊D3—61no—
164C4×C3⋊D3—61no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—121no—
176C3×C4×D3—62no—
177C3×C4×D3—62no—
178C3×C4×D3—64no—
179C3×C4×D3—64no—
180C3×C4×D3—121no—
181C3×C4×D3—121no—
182#624—24no—
183#624—21no—
184#624—21no—
185#624—21no—
186#624—32no—
187#624—61no—
188#624#6712yes—
189D3×Dic3—62no—
190D3×Dic3—62no—
191D3×Dic3—62no—
192D3×Dic3—64no—
193D3×Dic3—64no—
194D3×Dic3—62no—
195D3×Dic3—121no—
196D3×Dic3—121no—
197D3×Dic3—124no—
198#385—271noSylow
199C2×C33C4×D411yescharacteristic
200C2×C33#12011yescharacteristic
201C2×C33C22⋊Q811yescharacteristic
202C3×C3⋊D3—24no—
203C3×C3⋊D3C2×C4⋊C411yescharacteristic
204C3×C3⋊D3C4×D411yescharacteristic
205C3×C3⋊D3C22⋊Q811yescharacteristic
206C3×C3⋊D3#12311yescharacteristic
207C23×D3—91no—
208C23×D3—181no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4—91no—
212C22×C3×C4—181no—
213C22×Dic3—91no—
214C22×Dic3—91no—
215C22×Dic3—91no—
216C22×Dic3—181no—
217C2×C4×D3—61no—
218C2×C4×D3—61no—
219C2×C4×D3—61no—
220C2×C4×D3—61no—
221C2×C4×D3—181no—
222C2×C4×D3—181no—
223C2×C4×D3—181no—
224C2×C4×D3—182no—
225C2×C4×D3—182no—
226C2×C4×D3—184no—
227C2×C4×D3—184no—
228C2×C4×D3—181no—
229C2×C4×D3—181no—
230C2×C4×D3—361no—
231C3×C4⋊C4—181no—
232C3×C4⋊C4—182no—
233(C2×D3)⋊C4—94no—
234(C2×D3)⋊C4—182no—
235C4⋊Dic3—181no—
236Dic3⋊C4—61no—
237Dic3⋊C4—61no—
238Dic3⋊C4—181no—
239Dic3⋊C4—181no—
240Dic3⋊C4—361no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32D3×D411yescharacteristic
246C2×C3⋊D3—22no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—22no—
249C2×C3⋊D3C2×C4×D311yescharacteristic
250C2×C3⋊D3#23511yescharacteristic
251C2×C3⋊D3D3×D411yescharacteristic
252C2×C3⋊D3D3×Q811yescharacteristic
253C2×C3⋊D3#23911yescharacteristic
254C2×C3⋊D3C2×#6711yescharacteristic
255C2×C3×D3—61no—
256C2×C3×D3—61no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—61no—
264C2×C3×D3—61no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—122no—
274C2×C3×D3—122no—
275C2×C3×D3—122no—
276D32—34no—
277D32—64no—
278D32—64no—
279D32—64no—
280D32—64no—
281D32—62no—
282D32—62no—
283D32—62no—
284#157—22no—
285#157—121no—
286C32×C4—62no—
287C32×C4—61no—
288C32×C4—121no—
289C3⋊Dic3—22no—
290C3⋊Dic3—62no—
291C3⋊Dic3—61no—
292C3⋊Dic3—121no—
293C3⋊Dic3—122no—
294C3⋊Dic3—241no—
295C3×Dic3—22no—
296C3×Dic3—22no—
297C3×Dic3—22no—
298C3×Dic3—41no—
299C3×Dic3—42no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3—81no—
305C3×Dic3—122no—
306C23×C4—271no—
307C2×C4⋊C4—91no—
308C2×C4⋊C4—271no—
309C2×C22⋊C4—271no—
310C2×C22⋊C4—271no—
311#95—271no—
312#95—271no—
313C33#38511yescharacteristic, complemented, Sylow
314C23×C3—91no—
315C23×C3—181no—
316C22×D3—61no—
317C22×D3—61no—
318C22×D3—61no—
319C22×D3—61no—
320C22×D3—92no—
321C22×D3—92no—
322C22×D3—92no—
323C22×D3—121no—
324C22×D3—182no—
325C22×D3—182no—
326C22×D3—182no—
327C22×D3—181no—
328C22×D3—181no—
329C22×D3—181no—
330C22×D3—182no—
331C22×D3—182no—
332C22×D3—182no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—92no—
340C2×C3×C4—92no—
341C2×C3×C4—121no—
342C2×C3×C4—121no—
343C2×C3×C4—181no—
344C2×C3×C4—184no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—92no—
355C2×Dic3—91no—
356C2×Dic3—121no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—184no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—361no—
367C2×Dic3—361no—
368C2×Dic3GO+(4,𝔽2)11yescharacteristic
369C4×D3—62no—
370C4×D3—62no—
371C4×D3—64no—
372C4×D3—64no—
373C4×D3—121no—
374C4×D3—121no—
375C4×D3—182no—
376C4×D3—182no—
377C4×D3—182no—
378C4×D3—182no—
379C4×D3—184no—
380C4×D3—184no—
381C4×D3—361no—
382C4×D3—361no—
383C4×D3—364no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32#89311yescharacteristic
397C2×C32#89611yescharacteristic
398C2×C32#93711yescharacteristic
399C3⋊D3—24no—
400C3⋊D3D3×C4⋊C411yescharacteristic
401C3⋊D3#89411yescharacteristic
402C3⋊D3C4×#6711yescharacteristic
403C3⋊D3#94611yescharacteristic
404C3×D3—61no—
405C3×D3—61no—
406C3×D3—61no—
407C3×D3—61no—
408C3×D3—61no—
409C3×D3—61no—
410C3×D3—61no—
411C3×D3—61no—
412C3×D3—64no—
413C3×D3—64no—
414C3×D3—124no—
415C24—271no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—91no—
419C22×C4—181no—
420C22×C4—271no—
421C22×C4—271no—
422C22×C4—271no—
423C22×C4—541no—
424C22×C4—542no—
425C22×C4—542no—
426C4⋊C4—181no—
427C4⋊C4—182no—
428C4⋊C4—541no—
429C22⋊C4—274no—
430C22⋊C4—542no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—91no—
436C22×C3—91no—
437C22×C3—91no—
438C22×C3—91no—
439C22×C3—91no—
440C22×C3—91no—
441C22×C3—182no—
442C22×C3—182no—
443C22×C3—182no—
444C22×C3C2×GO+(4,𝔽2)11yescharacteristic
445C2×D3—61no—
446C2×D3—61no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—94no—
464C2×D3—122no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—184no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—184no—
472C2×D3—184no—
473C2×D3—184no—
474C3×C4—62no—
475C3×C4—62no—
476C3×C4—62no—
477C3×C4—121no—
478C3×C4—122no—
479C3×C4—121no—
480C3×C4—121no—
481C3×C4—182no—
482C3×C4—182no—
483C3×C4—241no—
484Dic3—22no—
485Dic3—21no—
486Dic3—41no—
487Dic3—62no—
488Dic3—62no—
489Dic3—122no—
490Dic3—182no—
491Dic3—181no—
492Dic3—361no—
493Dic3—362no—
494Dic3—361no—
495Dic3—721no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32—41no—
500C32#498911yescharacteristic, complemented
501C23—91no—
502C23—181no—
503C23—272no—
504C23—272no—
505C23—272no—
506C23—541no—
507C23—541no—
508C23—541no—
509C2×C4—31no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—92no—
513C2×C4—92no—
514C2×C4—181no—
515C2×C4—184no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C4—181no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—272no—
522C2×C4—271no—
523C2×C4—541no—
524C2×C4—541no—
525C2×C4—544no—
526C2×C4—541no—
527C2×C4—541no—
528C2×C4—541no—
529C2×C4—541no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—41no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—91no—
543C2×C3—91no—
544C2×C3—91no—
545C2×C3—91no—
546C2×C3—184no—
547C2×C3D32⋊Q811yescharacteristic
548C2×C3C4×GO+(4,𝔽2)11yescharacteristic
549C2×C3#6424111yescharacteristic
550D3—61no—
551D3—61no—
552D3—61no—
553D3—61no—
554D3—61no—
555D3—61no—
556D3—61no—
557D3—61no—
558D3—64no—
559D3—64no—
560D3—124no—
561C22—91no—
562C22—91no—
563C22—91no—
564C22—91no—
565C22—91no—
566C22—91no—
567C22—182no—
568C22—182no—
569C22—182no—
570C22—274no—
571C22—542no—
572C22—542no—
573C22—542no—
574C22D3×GO+(4,𝔽2)11yescharacteristic
575C4—61no—
576C4—62no—
577C4—121no—
578C4—182no—
579C4—182no—
580C4—541no—
581C4—1081no—
582C3—21no—
583C3—21no—
584C3—41no—
585C3—41no—
586C3#1059339811yescharacteristic, complemented
587C2—91no—
588C2—91no—
589C2—91no—
590C2—91no—
591C2—184no—
592C2#1172411111yescharacteristic
593C2#1172411211yescharacteristic
594C2#1172412011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC22×#1113C22
cosocleC22×C3×C3⋊D3C23
hypocenterC33#385
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
4. C2×C3×#3773C2semidirect
5. #11722910C2semidirect
6. #11722769C2semidirect
7. #11722722C2semidirect
20. #88939C4semidirect
27. C2×#3773D3semidirect
47. C2×C3×#157C2×C4semidirect
54. #6294C2×C4semidirect
111. #1113C4⋊C4semidirect
152. C2×#157C4×D3semidirect
313. C33#385semidirect
500. C32#4989semidirect
586. C3#10593398semidirect

Maximal subgroups

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

subgroupindex
C2×#889492
C2×#889392
C2×#889322
C2×C3×#37732
#117229102
#117227692
#117227222
#105933983
#49899