Finite Group Databasebeta

← #13172185

Subgroups — #13172185

Order 1728.

1510 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89046C211yesmaximal, characteristic
2#11723626C212yesmaximal, complemented
3C2×#88807C211yesmaximal, characteristic, complemented
4C3×#63957C211yesmaximal, characteristic, complemented
5C3×#63952C212yesmaximal, complemented
6#11722944C212yesmaximal, complemented
7#11722934C212yesmaximal, complemented
8#11722922C212yesmaximal, complemented
9#11722910C212yesmaximal, complemented
10#10594876—31nomaximal
11#10594049—32nomaximal
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3×#638C2212yes—
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C2×#6296C2211yescharacteristic
17C2×#6295C2211yescharacteristic
18#89046C418yescomplemented
19C2×#6294C2212yes—
20C2×Dic3×C3⋊D3C2212yes—
21C2×C3×#624C2212yes—
22C32×#217C2212yes—
23#88807C2212yes—
24#88807C2212yes—
25#88807C2212yes—
26#88803C2212yes—
27Dic3×C3⋊Dic3C2212yes—
28C3×#3724C2212yes—
29C3×#3724C2211yescharacteristic
30C3×#3724C2211yescharacteristic
31C3×#3722C2212yes—
32C3×Dic32C2212yes—
33C3×Dic32C2212yes—
34C2×#3805—31no—
35C2×#3804—32no—
36C2×#3750—31no—
37#64090—32no—
38C3×C4×#67—32no—
39C3×#67⋊C4—32no—
40C3×#877—32no—
41#67⋊Dic3—32no—
42#67⋊Dic3—32no—
43#67⋊Dic3—62no—
44C2×Dic3⋊Dic3—32no—
45C2×Dic3⋊Dic3—61no—
46#63957D311yescharacteristic, complemented
47#63952D312yescomplemented
48C4×#627—32no—
49C4×#626—32no—
50#63883—32no—
51#63880—32no—
52#63879—32no—
53#63878—32no—
54#63877—32no—
55D3×Dic3⋊C4—32no—
56D3×Dic3⋊C4—32no—
57#63837—32no—
58#63837—32no—
59#63836—32no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C2×C3⋊(C3⋊Dic3)—41no—
63C2×C3⋊(C3⋊Dic3)C2312yes—
64C3×#638C2×C418yescomplemented
65C2×C3×C3⋊Dic3C2×C414yescomplemented
66C2×C3×C3⋊Dic3C2311yescharacteristic
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C32×Dic3—41no—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2312yes—
72C2×C32×Dic3C2312yes—
73#6296C2×C414yescomplemented
74#6295C2×C414yescomplemented
75#6294C2×C418yescomplemented
76Dic3×C3⋊D3—42no—
77C3×#624—42no—
78#5546—92no—
79#5523—91no—
80C22×C32×C4—31no—
81C2×#638C2×D312yes—
82C22×C3⋊Dic3—32no—
83C22×C3⋊Dic3—31no—
84C22×C3⋊Dic3—61no—
85C22×C3⋊Dic3—62no—
86C22×C3⋊Dic3—121no—
87C22×C3⋊Dic3C2×D311yescharacteristic
88C2×C4×C3⋊D3—32no—
89C2×C3×#67—32no—
90C2×C3×#67—32no—
91C2×C3×#67—62no—
92C22×C3×Dic3—21no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—61no—
95C22×C3×Dic3C2×D312yes—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—32no—
98C2×#627—32no—
99C2×#627—31no—
100C2×#626—32no—
101C2×#625—31no—
102C2×#624—32no—
103C2×#624C2×D312yes—
104#3805—38no—
105#3804—316no—
106C2×D3×Dic3—32no—
107C2×D3×Dic3—32no—
108C2×D3×Dic3—32no—
109C2×D3×Dic3—32no—
110C2×D3×Dic3—62no—
111C32×C22⋊C4—32no—
112#3752—32no—
113#3750—32no—
114#3750—32no—
115#3750—32no—
116C4×C3⋊Dic3—32no—
117C3×#217—22no—
118C3×#217C2×D312yes—
119C3×#217C2×D312yes—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—32no—
130#3724C2×D311yescharacteristic
131#3724C2×D312yes—
132#3724C2×D311yescharacteristic
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—62no—
140Dic3⋊Dic3—62no—
141Dic3⋊Dic3—62no—
142#3722C2×D312yes—
143(C2×D3)⋊Dic3—32no—
144(C2×D3)⋊Dic3—32no—
145(C2×D3)⋊Dic3—62no—
146Dic32—32no—
147Dic32—32no—
148Dic32—62no—
149Dic32C2×D312yes—
150Dic32C2×D312yes—
151C22×C33—21no—
152C22×C33—21no—
153C22×C33C22×C412yes—
154C22×C33C2411yescharacteristic
155C2×C3×C3⋊D3—22no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3C22×C414yes—
158C3⋊(C3⋊Dic3)—22no—
159C3×C3⋊Dic3C22×C412yes—
160C3×C3⋊Dic3C22×C414yes—
161C3×C3⋊Dic3C22×C412yes—
162C32×Dic3—22no—
163C32×Dic3—22no—
164C32×Dic3—22no—
165C2×#237—92no—
166C2×#235—91no—
167C3×C4×D4—92no—
168C3×#117—91no—
169D4×Dic3—92no—
170C4×#67—32no—
171C4×#67—92no—
172C2×C4⋊Dic3—91no—
173C2×Dic3⋊C4—92no—
174C2×Dic3⋊C4—92no—
175C2×Dic3⋊C4—181no—
176#892—92no—
177D3×C4⋊C4—92no—
178D3×C4⋊C4—92no—
179#887—92no—
180#67⋊C4—32no—
181#67⋊C4—92no—
182#67⋊C4—92no—
183#67⋊C4—92no—
184#67⋊C4—182no—
185#877—32no—
186#877—92no—
187C4×D12—92no—
188#872—92no—
189C4×Dic6—92no—
190C4×Dic6—92no—
191C23×C32—21no—
192C23×C32—22no—
193C23×C32—41no—
194C23×C32C22×D311yescharacteristic
195C23×C32C22×D312yes—
196C22×C3⋊D3C22×D312yes—
197C22×C3×D3—32no—
198C22×C3×D3—32no—
199C22×C3×D3—62no—
200C2×C32×C4—32no—
201C2×C32×C4—32no—
202C2×C32×C4—32no—
203C2×C32×C4—121no—
204#638C4×D318yescomplemented
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—32no—
207C2×C3⋊Dic3—32no—
208C2×C3⋊Dic3—62no—
209C2×C3⋊Dic3—62no—
210C2×C3⋊Dic3—62no—
211C2×C3⋊Dic3—122no—
212C2×C3⋊Dic3—122no—
213C2×C3⋊Dic3—121no—
214C2×C3⋊Dic3—124no—
215C2×C3⋊Dic3—122no—
216C2×C3⋊Dic3—124no—
217C2×C3⋊Dic3C4×D314yescomplemented
218C2×C3⋊Dic3C22×D311yescharacteristic
219C2×C3⋊Dic3C22×D312yes—
220C2×C3⋊Dic3C22×D311yescharacteristic
221C4×C3⋊D3—122no—
222C3×#67—38no—
223C3×#67—38no—
224C3×#67—68no—
225C2×C3×Dic3—22no—
226C2×C3×Dic3—22no—
227C2×C3×Dic3—22no—
228C2×C3×Dic3—38no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—42no—
234C2×C3×Dic3—42no—
235C2×C3×Dic3—62no—
236C2×C3×Dic3—61no—
237C2×C3×Dic3—61no—
238C2×C3×Dic3—64no—
239C2×C3×Dic3C22×D312yes—
240C2×C3×Dic3C22×D312yes—
241C2×C3×Dic3C22×D312yes—
242C2×C3×Dic3C22×D312yes—
243C2×C3×Dic3C22×D312yes—
244C2×C3×Dic3C22×D312yes—
245C3×C4×D3—122no—
246C3×C4×D3—122no—
247#627—38no—
248#627—34no—
249#626—38no—
250#625—34no—
251#624—38no—
252#624—42no—
253D3×Dic3—38no—
254D3×Dic3—38no—
255D3×Dic3—122no—
256D3×Dic3—122no—
257D3×Dic3—124no—
258#520—271noSylow
259C2×C33—22no—
260C2×C33C23×C411yescharacteristic
261C2×C33H(5,𝔽2)11yescharacteristic
262C2×C332−1+411yescharacteristic
263C3×C3⋊D3—24no—
264C2×C3×D4—92no—
265C22×C3×C4—32no—
266C22×C3×C4—61no—
267C22×C3×C4—92no—
268C22×C3×C4—91no—
269C2×#67—32no—
270C2×#67—32no—
271C2×#67—62no—
272C2×#67—92no—
273C22×Dic3—32no—
274C22×Dic3—61no—
275C22×Dic3—92no—
276C22×Dic3—91no—
277C22×Dic3—92no—
278C22×Dic3—92no—
279C22×Dic3—182no—
280C22×Dic3—181no—
281C22×Dic3—361no—
282C22×Dic3D3211yescharacteristic
283#237—916no—
284#235—98no—
285C2×D12—91no—
286C2×C4×D3—32no—
287C2×C4×D3—32no—
288C2×C4×D3—92no—
289C2×C4×D3—92no—
290C2×C4×D3—92no—
291C2×C4×D3—92no—
292C2×C4×D3—182no—
293C2×Dic6—91no—
294C2×Dic6—92no—
295C3×C4⋊C4—92no—
296C3×C4⋊C4—91no—
297C3×C4⋊C4—91no—
298C3×C22⋊C4—32no—
299C3×C22⋊C4—32no—
300C3×C22⋊C4—62no—
301C3×C22⋊C4—92no—
302C3×C42—92no—
303C3×C42—92no—
304#217—92no—
305#217D3212yes—
306(C2×D3)⋊C4—32no—
307(C2×D3)⋊C4—32no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4—92no—
310(C2×D3)⋊C4—182no—
311C4⋊Dic3—92no—
312C4⋊Dic3—92no—
313C4⋊Dic3—92no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—32no—
317Dic3⋊C4—32no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—182no—
325Dic3⋊C4—182no—
326Dic3⋊C4—182no—
327C4×Dic3—32no—
328C4×Dic3—32no—
329C4×Dic3—32no—
330C4×Dic3—32no—
331C4×Dic3—92no—
332C4×Dic3—92no—
333C4×Dic3—92no—
334C4×Dic3—182no—
335C22×C32—21no—
336C22×C32—21no—
337C22×C32—22no—
338C22×C32—22no—
339C22×C32—22no—
340C22×C32—22no—
341C22×C32—21no—
342C22×C32—22no—
343C22×C32—22no—
344C22×C32—24no—
345C22×C32—42no—
346C22×C32—41no—
347C22×C32—42no—
348C22×C32—42no—
349C22×C32—42no—
350C22×C32—42no—
351C22×C32C2×C4×D314yes—
352C22×C32C2×C4×D312yes—
353C22×C32C23×D312yes—
354C22×C32C23×D311yescharacteristic
355C2×C3⋊D3—22no—
356C2×C3⋊D3—22no—
357C2×C3⋊D3C2×C4×D314yes—
358C2×C3×D3—34no—
359C2×C3×D3—34no—
360C2×C3×D3—64no—
361C2×C3×D3—62no—
362C2×C3×D3—62no—
363C2×C3×D3—62no—
364C2×C3×D3—62no—
365C2×C3×D3—122no—
366C2×C3×D3—122no—
367C32×C4—62no—
368C32×C4—62no—
369C32×C4—62no—
370C3⋊Dic3—62no—
371C3⋊Dic3—64no—
372C3⋊Dic3—62no—
373C3⋊Dic3—62no—
374C3⋊Dic3—122no—
375C3⋊Dic3—122no—
376C3⋊Dic3—124no—
377C3⋊Dic3C2×C4×D312yes—
378C3⋊Dic3C2×C4×D314yes—
379C3⋊Dic3C2×C4×D312yes—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—22no—
383C3×Dic3—22no—
384C3×Dic3—22no—
385C3×Dic3—22no—
386C3×Dic3—34no—
387C3×Dic3—34no—
388C3×Dic3—34no—
389C3×Dic3—34no—
390C3×Dic3—42no—
391C3×Dic3—42no—
392C3×Dic3—42no—
393C3×Dic3—62no—
394C3×Dic3—62no—
395C3×Dic3—64no—
396C2×#41—271no—
397C4×Q8—272no—
398C4×D4—92no—
399C4×D4—272no—
400C4×D4—272no—
401#117—91no—
402#117—272no—
403C2×C4⋊C4—272no—
404C2×C4⋊C4—271no—
405C33#52011yescharacteristic, complemented, Sylow
406C23×C3—21no—
407C23×C3—22no—
408C23×C3—41no—
409C23×C3—92no—
410C23×C3C2×D3212yes—
411C23×C3C2×D3211yescharacteristic
412C22×D3—32no—
413C22×D3—32no—
414C22×D3—62no—
415C3×D4—98no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—32no—
420C2×C3×C4—32no—
421C2×C3×C4—32no—
422C2×C3×C4—62no—
423C2×C3×C4—62no—
424C2×C3×C4—62no—
425C2×C3×C4—92no—
426C2×C3×C4—91no—
427C2×C3×C4—91no—
428C2×C3×C4—94no—
429C2×C3×C4—122no—
430C2×C3×C4—122no—
431C2×C3×C4—362no—
432#67—38no—
433#67—38no—
434#67—68no—
435#67—98no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—38no—
441C2×Dic3—41no—
442C2×Dic3—64no—
443C2×Dic3—62no—
444C2×Dic3—61no—
445C2×Dic3—61no—
446C2×Dic3—98no—
447C2×Dic3—92no—
448C2×Dic3—92no—
449C2×Dic3—92no—
450C2×Dic3—182no—
451C2×Dic3—182no—
452C2×Dic3—182no—
453C2×Dic3—362no—
454C2×Dic3—362no—
455C2×Dic3—362no—
456C2×Dic3—361no—
457C2×Dic3—364no—
458C2×Dic3—362no—
459C2×Dic3—364no—
460C2×Dic3C2×D3212yes—
461C2×Dic3C2×D3212yes—
462C2×Dic3C2×D3212yes—
463D12—94no—
464C4×D3—98no—
465C4×D3—98no—
466C4×D3—122no—
467C4×D3—122no—
468C4×D3—362no—
469C4×D3—362no—
470C4×D3—364no—
471Dic6—94no—
472Dic6—98no—
473C2×C32—22no—
474C2×C32—24no—
475C2×C32—24no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—22no—
479C2×C32—22no—
480C2×C32—22no—
481C2×C32—21no—
482C2×C32—41no—
483C2×C32—41no—
484C2×C32—41no—
485C2×C32—44no—
486C2×C32—44no—
487C2×C32C22×C4×D311yescharacteristic
488C2×C32C22×C4×D312yes—
489C2×C32#100412yes—
490C2×C32#100911yescharacteristic
491C2×C32#101012yes—
492C2×C32#101011yescharacteristic
493C3⋊D3—24no—
494C3×D3—64no—
495C3×D3—64no—
496C3×D3—124no—
497#41—278no—
498C2×Q8—271no—
499C2×D4—92no—
500C2×D4—271no—
501C22×C4—31no—
502C22×C4—92no—
503C22×C4—91no—
504C22×C4—272no—
505C22×C4—272no—
506C22×C4—271no—
507C4⋊C4—91no—
508C4⋊C4—92no—
509C4⋊C4—91no—
510C4⋊C4—272no—
511C4⋊C4—272no—
512C4⋊C4—272no—
513C22⋊C4—32no—
514C22⋊C4—92no—
515C22⋊C4—272no—
516C42—92no—
517C42—92no—
518C42—272no—
519C22×C3—22no—
520C22×C3—22no—
521C22×C3—21no—
522C22×C3—22no—
523C22×C3—22no—
524C22×C3—21no—
525C22×C3—21no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—24no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—42no—
532C22×C3—42no—
533C22×C3—41no—
534C22×C3—42no—
535C22×C3—94no—
536C22×C3—182no—
537C22×C3—182no—
538C22×C3C4×D3214yes—
539C22×C3C4×D3212yes—
540C22×C3C22×D3211yescharacteristic
541C22×C3C22×D3212yes—
542C2×D3—34no—
543C2×D3—34no—
544C2×D3—64no—
545C2×D3—62no—
546C2×D3—62no—
547C2×D3—62no—
548C2×D3—62no—
549C2×D3—122no—
550C2×D3—122no—
551C3×C4—62no—
552C3×C4—62no—
553C3×C4—62no—
554C3×C4—62no—
555C3×C4—62no—
556C3×C4—62no—
557C3×C4—94no—
558C3×C4—92no—
559C3×C4—92no—
560C3×C4—122no—
561C3×C4—122no—
562C3×C4—122no—
563Dic3—22no—
564Dic3—22no—
565Dic3—22no—
566Dic3—34no—
567Dic3—34no—
568Dic3—34no—
569Dic3—34no—
570Dic3—62no—
571Dic3—62no—
572Dic3—64no—
573Dic3—184no—
574Dic3—182no—
575Dic3—182no—
576Dic3—182no—
577Dic3—362no—
578Dic3—362no—
579Dic3—364no—
580C32—22no—
581C32—21no—
582C32—41no—
583C32#552311yescharacteristic, complemented
584C32#554612yescomplemented
585C23—92no—
586C23D3311yescharacteristic
587Q8—274no—
588D4—98no—
589D4—274no—
590C2×C4—32no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—91no—
594C2×C4—92no—
595C2×C4—91no—
596C2×C4—94no—
597C2×C4—121no—
598C2×C4—278no—
599C2×C4—272no—
600C2×C4—362no—
601C2×C4—1082no—
602C2×C4—1081no—
603C2×C3—24no—
604C2×C3—24no—
605C2×C3—22no—
606C2×C3—21no—
607C2×C3—22no—
608C2×C3—22no—
609C2×C3—21no—
610C2×C3—22no—
611C2×C3—21no—
612C2×C3—44no—
613C2×C3—44no—
614C2×C3—41no—
615C2×C3—41no—
616C2×C3—41no—
617C2×C3—184no—
618C2×C3#6429811yescharacteristic
619C2×C3C2×C4×D3212yes—
620C2×C3C2×C4×D3211yescharacteristic
621C2×C3#6430912yes—
622C2×C3#6431412yes—
623C2×C3#6433011yescharacteristic
624D3—64no—
625D3—64no—
626D3—124no—
627C22—21no—
628C22—21no—
629C22—94no—
630C22—182no—
631C22—182no—
632C22#8894912yes—
633C22C2×D3311yescharacteristic
634C4—62no—
635C4—62no—
636C4—62no—
637C4—92no—
638C4—92no—
639C4—94no—
640C4—542no—
641C3—22no—
642C3—21no—
643C3—41no—
644C3#1059404912yescomplemented
645C3#1059487611yescharacteristic, complemented
646C2—22no—
647C2—184no—
648C2C2×#8894911yescharacteristic
649C2#1172441911yescharacteristic
650C2#1172442711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC23×C33C23
socle, cosocleC22×C33C24
prime radicalC2×#89046C2
hypercenterC23D33
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11723626C2semidirect
3. C2×#88807C2semidirect
4. C3×#63957C2semidirect
5. C3×#63952C2semidirect
6. #11722944C2semidirect
7. #11722934C2semidirect
8. #11722922C2semidirect
9. #11722910C2semidirect
18. #89046C4semidirect
46. #63957D3semidirect
47. #63952D3semidirect
64. C3×#638C2×C4semidirect
65. C2×C3×C3⋊Dic3C2×C4semidirect
73. #6296C2×C4semidirect
74. #6295C2×C4semidirect
75. #6294C2×C4semidirect
204. #638C4×D3semidirect
217. C2×C3⋊Dic3C4×D3semidirect
405. C33#520semidirect
583. C32#5523semidirect
584. C32#5546semidirect
644. C3#10594049semidirect
645. C3#10594876semidirect

Maximal subgroups

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

subgroupindex
C2×#890462
#117236262
C2×#888072
C3×#639572
C3×#639522
#117229442
#117229342
#117229222
#117229102
#105948763
#105940493