Finite Group Databasebeta

← #13165382

Subgroups — #13165382

Order 1728.

876 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 94 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×D32×Dic3C211yesmaximal, characteristic
2C2×#88934C211yesmaximal, characteristic, complemented
3C2×D32⋊Dic3C211yesmaximal, characteristic
4C2×C3×D32⋊C4C211yesmaximal, characteristic
5#11722910C211yesmaximal, characteristic, complemented
6#11722791C211yesmaximal, characteristic, complemented
7#11722722C211yesmaximal, characteristic, complemented
8#10593398—31nomaximal
9C22×C3×D32C2211yescharacteristic
10C22×C3⋊#157C2211yescharacteristic
11C22×C3×#157C2211yescharacteristic
12C2×#6294C2211yescharacteristic
13C2×#6294C2211yescharacteristic
14C2×Dic3×C3⋊D3C2211yescharacteristic
15C2×D3×C3⋊Dic3—21no—
16C2×C3×#624C2211yescharacteristic
17C2×C3×D3×Dic3—21no—
18D32×Dic3—22no—
19D32×Dic3—22no—
20#88934—22no—
21D32⋊Dic3C414yescomplemented
22C3×D32⋊C4—22no—
23#88807—21no—
24C3×#3724—21no—
25C22×D3×Dic3—61no—
26C2×C4×D32—31no—
27C2×#3773—31no—
28C2×D32⋊C4—31no—
29C2×D32⋊C4D311yescharacteristic
30#63883—31no—
31D3×Dic3⋊C4—61no—
32#63781—31no—
33#63737—31no—
34C22×C3×C3⋊D3C2311yescharacteristic
35C22×C32×D3—21no—
36C2×C3×D32—21no—
37C2×C3×D32—21no—
38C2×C3×D32—21no—
39C2×C3×D32C2×C412yes—
40C2×C3×D32D412yes—
41C2×C3×D32Q812yes—
42C2×C3⋊#157—21no—
43C2×C3⋊#157—21no—
44C2×C3⋊#157C2×C412yescomplemented
45C2×C3×#157—21no—
46C2×C3×#157—21no—
47C2×C3×#157—21no—
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53#6294—21no—
54#6294—21no—
55#6294C2×C412yescomplemented
56#6294D412yes—
57Dic3×C3⋊D3—21no—
58Dic3×C3⋊D3—21no—
59D3×C3⋊Dic3—24no—
60C3×#624—21no—
61C3×#624—21no—
62C3×D3×Dic3—24no—
63#4946—91nomaximal
64C23×C3×D3—61no—
65C22×D32C2×D311yescharacteristic
66C22×#157—31no—
67C22×#157C2×D311yescharacteristic
68C22×C3⋊Dic3—61no—
69C2×C4×C3⋊D3—31no—
70C22×C3×Dic3—61no—
71C2×C3×C4×D3—61no—
72C2×C3×C4×D3—61no—
73C2×#624—31no—
74C2×#624—31no—
75C2×#624C2×D311yescharacteristic
76C2×D3×Dic3—21no—
77C2×D3×Dic3—64no—
78C2×D3×Dic3—64no—
79C2×D3×Dic3—61no—
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—61no—
83C2×D3×Dic3—61no—
84C2×D3×Dic3—61no—
85C2×D3×Dic3—121no—
86C4×D32—62no—
87C4×D32—62no—
88#3773—62no—
89D32⋊C4—22no—
90D32⋊C4—34no—
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⋊D3C22×C411yescharacteristic
100C2×C3×C3⋊D3C2×D411yescharacteristic
101C2×C3×C3⋊D3C2×D411yescharacteristic
102C2×C3×C3⋊D3C2×Q811yescharacteristic
103C2×C3×C3⋊D3#4111yescharacteristic
104C2×C3×C3⋊D3#4111yescharacteristic
105C2×C32×D3—22no—
106C2×C32×D3—22no—
107C2×C32×D3—22no—
108C3×D32—22no—
109C3×D32—22no—
110C3×D32—22no—
111C3×D32C4⋊C414yescomplemented
112C3⋊#157—22no—
113C3×#157—41no—
114C3⋊(C3⋊Dic3)—21no—
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3—22no—
117C32×Dic3—21no—
118C32×Dic3—41no—
119C23×Dic3—91no—
120C22×C4×D3—181no—
121C2×C3×C22⋊C4—91no—
122C2×#217—91no—
123C2×Dic3⋊C4—91no—
124C2×Dic3⋊C4—91no—
125D3×C4⋊C4—181no—
126C3⋊#95—91no—
127C3⋊#95—91no—
128C23×C32—61no—
129C22×C3⋊D3C22×D311yescharacteristic
130C22×C3×D3—21no—
131C22×C3×D3—21no—
132C22×C3×D3—41no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—62no—
136C22×C3×D3—61no—
137C22×C3×D3—61no—
138C22×C3×D3—62no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C2×D32—21no—
142C2×D32—21no—
143C2×D32—21no—
144C2×D32Dic612yes—
145C2×D32C4×D312yes—
146C2×D32#6712yes—
147C2×#157—21no—
148C2×#157—21no—
149C2×#157—21no—
150C2×#157—32no—
151C2×#157—61no—
152C2×#157—61no—
153C2×C32×C4—31no—
154C2×C32×C4—61no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—64no—
161C2×C3⋊Dic3—121no—
162C2×C3⋊Dic3—121no—
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—41no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—121no—
176C2×C3×Dic3—121no—
177C3×C4×D3—64no—
178C3×C4×D3—121no—
179C3×C4×D3—121no—
180#624—21no—
181#624—21no—
182#624—32no—
183#624—32no—
184#624—61no—
185#624—61no—
186D3×Dic3—24no—
187D3×Dic3—62no—
188D3×Dic3—62no—
189D3×Dic3—62no—
190D3×Dic3—62no—
191D3×Dic3—62no—
192D3×Dic3—62no—
193D3×Dic3—64no—
194D3×Dic3—64no—
195D3×Dic3—124no—
196D3×Dic3—121no—
197D3×Dic3—121no—
198#385—271noSylow
199C2×C33C4×D411yescharacteristic
200C2×C33#12011yescharacteristic
201C2×C33C22⋊Q811yescharacteristic
202C3×C3⋊D3C2×C4⋊C411yescharacteristic
203C3×C3⋊D3C4×D411yescharacteristic
204C3×C3⋊D3C22⋊Q811yescharacteristic
205C3×C3⋊D3#12311yescharacteristic
206C32×D3—24no—
207C24×C3—91no—
208C23×D3—61no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4—181no—
212C22×Dic3—61no—
213C22×Dic3—91no—
214C22×Dic3—91no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C22×Dic3—182no—
218C22×Dic3—182no—
219C22×Dic3—181no—
220C22×Dic3—181no—
221C22×Dic3—361no—
222C2×C4×D3—61no—
223C2×C4×D3—61no—
224C2×C4×D3—181no—
225C2×C4×D3—181no—
226C2×C4×D3—181no—
227C2×C4×D3—181no—
228C2×C4×D3—184no—
229C2×C4×D3—184no—
230C2×C4×D3—181no—
231C3×C4⋊C4—181no—
232C3×C22⋊C4—182no—
233#217—94no—
234C4⋊Dic3—181no—
235Dic3⋊C4—61no—
236Dic3⋊C4—61no—
237Dic3⋊C4—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—182no—
240Dic3⋊C4—361no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32—41no—
245C22×C32—62no—
246C22×C32—62no—
247C22×C32—62no—
248C22×C32D3×D411yescharacteristic
249C2×C3⋊D3C2×Dic611yescharacteristic
250C2×C3⋊D3C2×C4×D311yescharacteristic
251C2×C3⋊D3D3×D411yescharacteristic
252C2×C3⋊D3#23711yescharacteristic
253C2×C3⋊D3#23711yescharacteristic
254C2×C3⋊D3C2×#6711yescharacteristic
255C2×C3×D3—22no—
256C2×C3×D3—22no—
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—22no—
260C2×C3×D3—22no—
261C2×C3×D3—42no—
262C2×C3×D3—42no—
263C2×C3×D3—42no—
264C2×C3×D3—64no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—61no—
274C2×C3×D3—61no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C2×C3×D3—64no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280D32—22no—
281D32—22no—
282D32—22no—
283D32Dic3⋊C414yescomplemented
284#157—41no—
285#157—62no—
286C32×C4—61no—
287C32×C4—121no—
288C3⋊Dic3—22no—
289C3⋊Dic3—22no—
290C3⋊Dic3—62no—
291C3⋊Dic3—61no—
292C3⋊Dic3—121no—
293C3⋊Dic3—122no—
294C3⋊Dic3—241no—
295C3×Dic3—22no—
296C3×Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—41no—
299C3×Dic3—62no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—62no—
303C3×Dic3—81no—
304C3×Dic3—122no—
305C3×Dic3—122no—
306C23×C4—271no—
307C2×C4⋊C4—271no—
308C2×C4⋊C4—271no—
309C2×C22⋊C4—91no—
310C2×C22⋊C4—271no—
311#95—271no—
312#95—271no—
313C33#38511yescharacteristic, complemented, Sylow
314C23×C3—61no—
315C23×C3—61no—
316C23×C3—92no—
317C23×C3—92no—
318C23×C3—92no—
319C23×C3—91no—
320C23×C3—121no—
321C23×C3—181no—
322C23×C3—181no—
323C23×C3—181no—
324C22×D3—21no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—62no—
328C22×D3—61no—
329C22×D3—61no—
330C22×D3—62no—
331C22×D3—62no—
332C22×D3—62no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—121no—
338C2×C3×C4—184no—
339C2×C3×C4—181no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×Dic3—21no—
347C2×Dic3—64no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—92no—
355C2×Dic3—91no—
356C2×Dic3—121no—
357C2×Dic3—121no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—184no—
365C2×Dic3—184no—
366C2×Dic3—181no—
367C2×Dic3—181no—
368C2×Dic3—361no—
369C2×Dic3—361no—
370C2×Dic3—364no—
371C2×Dic3GO+(4,𝔽2)11yescharacteristic
372C4×D3—64no—
373C4×D3—121no—
374C4×D3—121no—
375C4×D3—182no—
376C4×D3—182no—
377C4×D3—182no—
378C4×D3—182no—
379C4×D3—182no—
380C4×D3—182no—
381C4×D3—184no—
382C4×D3—361no—
383C4×D3—361no—
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—64no—
397C2×C32#87811yescharacteristic
398C2×C32#67⋊C411yescharacteristic
399C2×C32#93711yescharacteristic
400C3⋊D3#87811yescharacteristic
401C3⋊D3#67⋊C411yescharacteristic
402C3⋊D3C2×Dic3⋊C411yescharacteristic
403C3⋊D3#93511yescharacteristic
404C3×D3—24no—
405C3×D3—24no—
406C3×D3—44no—
407C3×D3—61no—
408C3×D3—61no—
409C3×D3—61no—
410C3×D3—61no—
411C3×D3—61no—
412C3×D3—61no—
413C3×D3—61no—
414C3×D3—61no—
415C24—91no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—181no—
419C22×C4—271no—
420C22×C4—271no—
421C22×C4—271no—
422C22×C4—271no—
423C22×C4—542no—
424C22×C4—542no—
425C22×C4—541no—
426C4⋊C4—181no—
427C4⋊C4—541no—
428C4⋊C4—542no—
429C22⋊C4—182no—
430C22⋊C4—274no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—62no—
436C22×C3—62no—
437C22×C3—62no—
438C22×C3—62no—
439C22×C3—62no—
440C22×C3—62no—
441C22×C3—94no—
442C22×C3—91no—
443C22×C3—91no—
444C22×C3—91no—
445C22×C3—91no—
446C22×C3—91no—
447C22×C3—91no—
448C22×C3—122no—
449C22×C3—122no—
450C22×C3—122no—
451C22×C3—182no—
452C22×C3—182no—
453C22×C3—182no—
454C22×C3C2×GO+(4,𝔽2)11yescharacteristic
455C2×D3—22no—
456C2×D3—22no—
457C2×D3—22no—
458C2×D3—64no—
459C2×D3—61no—
460C2×D3—61no—
461C2×D3—61no—
462C2×D3—61no—
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—61no—
467C2×D3—61no—
468C2×D3—61no—
469C2×D3—61no—
470C2×D3—61no—
471C2×D3—64no—
472C2×D3—64no—
473C2×D3—64no—
474C3×C4—62no—
475C3×C4—121no—
476C3×C4—121no—
477C3×C4—121no—
478C3×C4—182no—
479C3×C4—182no—
480C3×C4—241no—
481C3×C4—361no—
482Dic3—21no—
483Dic3—41no—
484Dic3—62no—
485Dic3—62no—
486Dic3—62no—
487Dic3—62no—
488Dic3—122no—
489Dic3—122no—
490Dic3—182no—
491Dic3—182no—
492Dic3—181no—
493Dic3—361no—
494Dic3—362no—
495Dic3—721no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32—41no—
500C32#494611yescharacteristic, complemented
501C23—61no—
502C23—92no—
503C23—92no—
504C23—92no—
505C23—91no—
506C23—181no—
507C23—181no—
508C23—181no—
509C2×C4—31no—
510C2×C4—61no—
511C2×C4—184no—
512C2×C4—181no—
513C2×C4—181no—
514C2×C4—181no—
515C2×C4—181no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C4—181no—
519C2×C4—272no—
520C2×C4—272no—
521C2×C4—272no—
522C2×C4—271no—
523C2×C4—541no—
524C2×C4—541no—
525C2×C4—541no—
526C2×C4—541no—
527C2×C4—544no—
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—64no—
543C2×C3—64no—
544C2×C3—91no—
545C2×C3—91no—
546C2×C3—91no—
547C2×C3—91no—
548C2×C3—124no—
549C2×C3D32⋊Q811yescharacteristic
550C2×C3C4×GO+(4,𝔽2)11yescharacteristic
551C2×C3#6424111yescharacteristic
552D3—24no—
553D3—61no—
554D3—61no—
555D3—61no—
556D3—61no—
557D3—61no—
558D3—61no—
559D3—61no—
560D3—61no—
561C22—62no—
562C22—62no—
563C22—62no—
564C22—94no—
565C22—91no—
566C22—91no—
567C22—91no—
568C22—91no—
569C22—91no—
570C22—91no—
571C22—182no—
572C22—182no—
573C22—182no—
574C22D3×GO+(4,𝔽2)11yescharacteristic
575C4—61no—
576C4—121no—
577C4—182no—
578C4—182no—
579C4—361no—
580C4—542no—
581C4—541no—
582C3—21no—
583C3—21no—
584C3—41no—
585C3—41no—
586C3#1059339811yescharacteristic, complemented
587C2—64no—
588C2—91no—
589C2—91no—
590C2—91no—
591C2—91no—
592C2(D32⋊D3)⋊C411yescharacteristic
593C2#1172411211yescharacteristic
594C2#1172411911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×GO+(4,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×D4
prime radicalC22×C3×D32C22
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
2. C2×#88934C2semidirect
5. #11722910C2semidirect
6. #11722791C2semidirect
7. #11722722C2semidirect
21. D32⋊Dic3C4semidirect
44. C2×C3⋊#157C2×C4semidirect
55. #6294C2×C4semidirect
111. C3×D32C4⋊C4semidirect
283. D32Dic3⋊C4semidirect
313. C33#385semidirect
500. C32#4946semidirect
586. C3#10593398semidirect

Maximal subgroups

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

subgroupindex
C2×D32×Dic32
C2×#889342
C2×D32⋊Dic32
C2×C3×D32⋊C42
#117229102
#117227912
#117227222
#105933983
#49469