Finite Group Databasebeta

← #13155900

Subgroups — #13155900

Order 1728.

800 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic
2#11723435C211yesmaximal, characteristic, complemented
3C2×C3×#3719C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5#11722040C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic, complemented
8#10591656—31nomaximal
9#10590966—31nomaximal
10#10590960—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6295—21no—
16C2×#6294—21no—
17#89041—22no—
18#89041—22no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88806—21no—
23#88803—21no—
24C3×#3719—22no—
25C3×#3717—22no—
26C2×#3799—31no—
27C2×#3796—31no—
28C2×#3795—31no—
29#64155—31no—
30C2×C3×#216—31no—
31C2×C3×#215—31no—
32C2×C3×Dic12—31no—
33C2×C3×#204—31no—
34#63856—31no—
35#63856—31no—
36#63856—61no—
37C2×#3719D311yescharacteristic, complemented
38C2×#3717D311yescharacteristic, complemented
39#63648—31no—
40#63643—31no—
41C3×(C2×D3)⋊C8—31no—
42C3×(C2×D3)⋊C8—31no—
43#63566—31no—
44#63566—31no—
45#63566—61no—
46#63563—31no—
47#63563—31no—
48#63563—61no—
49#63557D311yescharacteristic, complemented
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)D412yes—
58C3×C4×C3⋊D3—21no—
59C3×C4×C3⋊D3—21no—
60C3×#634—21no—
61C3×#634D412yes—
62C2×C32×Dic3—21no—
63C32×Dic6—21no—
64C32×Dic6—21no—
65#6295—24no—
66#6294—24no—
67C32×C3⋊C8—21no—
68C32×C3⋊C8—21no—
69#5187—91no—
70#5173—91no—
71#4733—91no—
72C2×C32×Q8—31no—
73C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×C4×D3—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—61no—
82C2×C3×Dic6—21no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×C3×Dic6C2×D311yescharacteristic
87C2×C3×Dic6C2×D311yescharacteristic
88C2×#626—61no—
89C2×#626—61no—
90C2×#625—61no—
91C2×#624—61no—
92C2×D3×Dic3—61no—
93C2×D3×Dic3—61no—
94#3799—62no—
95#3799—62no—
96#3796—62no—
97#3796—62no—
98#3795—62no—
99#3795—62no—
100C2×C32×C8—31no—
101C2×C32×C8—31no—
102#3752—61no—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—31no—
105C4⋊(C3⋊Dic3)—31no—
106C4⋊(C3⋊Dic3)—61no—
107C4⋊(C3⋊Dic3)—61no—
108C4⋊(C3⋊Dic3)—61no—
109C4⋊(C3⋊Dic3)—121no—
110#3750—61no—
111C3×#216—62no—
112C3×#215—62no—
113C2×C3×C3⋊C8—21no—
114C2×C3×C3⋊C8—21no—
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C2×C3×C3⋊C8C2×D311yescharacteristic
117C2×C3×C3⋊C8C2×D311yescharacteristic
118C2×C3×C3⋊C8C2×D311yescharacteristic
119C3×Dic12—62no—
120C3×#204—62no—
121#3724—61no—
122#3724—61no—
123#3724—62no—
124(C2×D3)⋊Dic3—61no—
125(C2×D3)⋊Dic3—61no—
126(C2×D3)⋊Dic3—62no—
127#3719—22no—
128#3717—22no—
129C22×C33C2×D411yescharacteristic
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—41no—
135C2×C3×C3⋊D3—41no—
136C33×C4C2×D411yescharacteristic
137C33×C4C2×D411yescharacteristic
138C3⋊(C3⋊Dic3)—41no—
139C3×C3⋊Dic3—21no—
140C3×C3⋊Dic3—22no—
141C32×Dic3—41no—
142C2×#239—91no—
143C2×#237—91no—
144C2×#235—91no—
145C2×C3×Q16—91no—
146C2×C3×SD16—91no—
147#946—91no—
148#946—91no—
149#946—181no—
150C2×#216—31no—
151C2×#215—31no—
152#924—91no—
153C2×Dic12—31no—
154C2×#204—31no—
155C3×C22⋊C8—91no—
156#835—91no—
157#832—91no—
158#821—91no—
159#821—91no—
160#821—181no—
161(C2×D3)⋊C8—31no—
162(C2×D3)⋊C8—31no—
163#816—91no—
164#816—91no—
165#816—181no—
166C22×C3⋊D3—21no—
167C22×C3⋊D3D1211yescharacteristic
168C22×C3×D3—31no—
169C22×C3×D3—31no—
170C22×C3×D3—61no—
171C22×C3×D3—61no—
172C22×C3×D3—61no—
173C22×C3×D3—121no—
174C32×Q8—61no—
175C32×Q8—61no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—41no—
180C2×C32×C4—61no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3—61no—
186C2×C3⋊Dic3—62no—
187C2×C3⋊Dic3—61no—
188C2×C3⋊Dic3—61no—
189C2×C3⋊Dic3—121no—
190C2×C3⋊Dic3—121no—
191C2×C3⋊Dic3—122no—
192C2×C3⋊Dic3D1211yescharacteristic
193C4⋊(C3⋊D3)—21no—
194C4⋊(C3⋊D3)#6712yes—
195C4×C3⋊D3—21no—
196C4×C3⋊D3—21no—
197#634—21no—
198#634#6712yes—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—121no—
208C3×D12—32no—
209C3×D12—32no—
210C3×D12—62no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—121no—
214C3×C4×D3—61no—
215C3×C4×D3—61no—
216C3×C4×D3—61no—
217C3×C4×D3—61no—
218C3×C4×D3—121no—
219C3×C4×D3—121no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—21no—
224C3×Dic6—32no—
225C3×Dic6—32no—
226C3×Dic6—41no—
227C3×Dic6—41no—
228C3×Dic6—62no—
229C3×Dic6—61no—
230C3×Dic6—61no—
231C3×Dic6—121no—
232#626—64no—
233#626—64no—
234#625—64no—
235#624—64no—
236D3×Dic3—64no—
237D3×Dic3—64no—
238C32×C8—61no—
239C32×C8—61no—
240C3×C3⋊C8—21no—
241C3×C3⋊C8—21no—
242C3×C3⋊C8—21no—
243C3×C3⋊C8—21no—
244C3×C3⋊C8—41no—
245C3×C3⋊C8—41no—
246#452—271noSylow
247C2×C33#12011yescharacteristic
248C2×C33#13511yescharacteristic
249C2×C33#13711yescharacteristic
250C3×C3⋊D3—41no—
251C3×C3⋊D3—42no—
252C2×C3×Q8—31no—
253C2×C3×Q8—31no—
254C2×C3×Q8—61no—
255C2×C3×Q8—91no—
256C2×C3×D4—91no—
257C22×C3×C4—91no—
258C2×#67—181no—
259C2×#67—181no—
260C22×Dic3—181no—
261#239—182no—
262#239—182no—
263#237—182no—
264#237—182no—
265#235—182no—
266#235—182no—
267C2×D12—31no—
268C2×D12—31no—
269C2×D12—61no—
270C2×D12—181no—
271C2×C4×D3—31no—
272C2×C4×D3—31no—
273C2×C4×D3—61no—
274C2×C4×D3—181no—
275C2×C4×D3—181no—
276C2×Dic6—31no—
277C2×Dic6—31no—
278C2×Dic6—61no—
279C2×Dic6D3211yescharacteristic
280C3×Q16—182no—
281C3×SD16—182no—
282C2×C3×C8—31no—
283C2×C3×C8—31no—
284C2×C3×C8—31no—
285C2×C3×C8—31no—
286C2×C3×C8—61no—
287C2×C3×C8—61no—
288#217—181no—
289#216—62no—
290#215—62no—
291(C2×D3)⋊C4—181no—
292(C2×D3)⋊C4—181no—
293(C2×D3)⋊C4—182no—
294C4⋊Dic3—91no—
295C4⋊Dic3—91no—
296C4⋊Dic3—91no—
297C4⋊Dic3—181no—
298C4⋊Dic3—181no—
299C4⋊Dic3—181no—
300C4⋊Dic3—361no—
301Dic3⋊C4—181no—
302Dic3⋊C4—181no—
303Dic3⋊C4—181no—
304Dic3⋊C4—182no—
305C2×C3⋊C8D3211yescharacteristic
306C2×C3⋊C8D3211yescharacteristic
307Dic12—62no—
308#204—62no—
309C22×C32—21no—
310C22×C32—21no—
311C22×C32—21no—
312C22×C32—41no—
313C22×C32C2×D1211yescharacteristic
314C22×C32D3×D411yescharacteristic
315C22×C32D3×D411yescharacteristic
316C2×C3⋊D3—21no—
317C2×C3⋊D3—21no—
318C2×C3⋊D3—21no—
319C2×C3⋊D3—22no—
320C2×C3⋊D3—41no—
321C2×C3⋊D3—41no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—61no—
325C2×C3×D3—61no—
326C2×C3×D3—61no—
327C2×C3×D3—61no—
328C2×C3×D3—62no—
329C2×C3×D3—61no—
330C2×C3×D3—61no—
331C2×C3×D3—121no—
332C2×C3×D3—121no—
333C2×C3×D3—122no—
334C2×C3×D3—121no—
335C2×C3×D3—121no—
336C2×C3×D3—121no—
337C2×C3×D3—121no—
338C2×C3×D3—241no—
339C2×C3×D3—241no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—41no—
347C32×C4—41no—
348C32×C4—121no—
349C32×C4D3×D411yescharacteristic
350C32×C4D3×D411yescharacteristic
351C32×C4D3×D411yescharacteristic
352C32×C4D3×D411yescharacteristic
353C32×C4C2×#6711yescharacteristic
354C32×C4C2×#6711yescharacteristic
355C3⋊Dic3—21no—
356C3⋊Dic3—22no—
357C3⋊Dic3—121no—
358C3⋊Dic3—122no—
359C3⋊Dic3—121no—
360C3⋊Dic3—121no—
361C3⋊Dic3—241no—
362C3⋊Dic3—241no—
363C3⋊Dic3—242no—
364C3×Dic3—42no—
365C3×Dic3—41no—
366C3×Dic3—41no—
367C3×Dic3—62no—
368C3×Dic3—61no—
369C3×Dic3—61no—
370C3×Dic3—62no—
371C3×Dic3—122no—
372C3×Dic3—121no—
373C2×#41—271no—
374C2×Q16—91no—
375C2×SD16—91no—
376C22⋊Q8—271no—
377#103—271no—
378#102—271no—
379C22⋊C8—91no—
380C33#45211yescharacteristic, complemented, Sylow
381C23×C3—91no—
382C23×C3—181no—
383C22×D3—31no—
384C22×D3—31no—
385C22×D3—61no—
386C22×D3—61no—
387C22×D3—61no—
388C22×D3—121no—
389C3×Q8—61no—
390C3×Q8—61no—
391C3×Q8—61no—
392C3×Q8—61no—
393C3×Q8—92no—
394C3×Q8—121no—
395C3×Q8—121no—
396C3×Q8—181no—
397C3×D4—92no—
398C3×D4—181no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—41no—
403C2×C3×C4—62no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—91no—
407C2×C3×C4—181no—
408C2×C3×C4—181no—
409C2×C3×C4—181no—
410C2×C3×C4C2×D3211yescharacteristic
411C2×C3×C4C2×D3211yescharacteristic
412C2×C3×C4C2×D3211yescharacteristic
413#67—184no—
414#67—184no—
415C2×Dic3—21no—
416C2×Dic3—31no—
417C2×Dic3—31no—
418C2×Dic3—61no—
419C2×Dic3—61no—
420C2×Dic3—61no—
421C2×Dic3—121no—
422C2×Dic3—184no—
423C2×Dic3—182no—
424C2×Dic3—181no—
425C2×Dic3—181no—
426C2×Dic3—181no—
427C2×Dic3—361no—
428C2×Dic3—361no—
429C2×Dic3—362no—
430D12—32no—
431D12—32no—
432D12—62no—
433D12—61no—
434D12—61no—
435D12—121no—
436D12—184no—
437C4×D3—61no—
438C4×D3—61no—
439C4×D3—61no—
440C4×D3—61no—
441C4×D3—121no—
442C4×D3—121no—
443C4×D3—184no—
444C4×D3—184no—
445Dic6—21no—
446Dic6—21no—
447Dic6—32no—
448Dic6—32no—
449Dic6—62no—
450Dic6—61no—
451Dic6—61no—
452Dic6—121no—
453C3×C8—61no—
454C3×C8—61no—
455C3×C8—61no—
456C3×C8—61no—
457C3×C8—121no—
458C3×C8—121no—
459C3⋊C8—21no—
460C3⋊C8—21no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—21no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—41no—
471C2×C32—41no—
472C2×C32—41no—
473C2×C32#88211yescharacteristic
474C2×C32#90411yescharacteristic
475C2×C32#90911yescharacteristic
476C2×C32#91511yescharacteristic
477C2×C32#91611yescharacteristic
478C2×C32#91811yescharacteristic
479C2×C32#91911yescharacteristic
480C2×C32#93711yescharacteristic
481C2×C32#93711yescharacteristic
482C3⋊D3—41no—
483C3⋊D3—42no—
484C3×D3—122no—
485C3×D3—121no—
486C3×D3—121no—
487C3×D3—122no—
488C3×D3—122no—
489C3×D3—242no—
490#41—542no—
491#41—542no—
492C2×Q8—31no—
493C2×Q8—91no—
494C2×D4—91no—
495C2×D4—541no—
496C22×C4—91no—
497C22×C4—541no—
498Q16—182no—
499SD16—182no—
500C2×C8—31no—
501C2×C8—31no—
502C4⋊C4—271no—
503C4⋊C4—541no—
504C22⋊C4—541no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—21no—
508C22×C3—41no—
509C22×C3—182no—
510C22×C3—181no—
511C22×C3—181no—
512C22×C3—181no—
513C22×C3—361no—
514C22×C3—361no—
515C22×C3D3×D1211yescharacteristic
516C22×C3D3×D1211yescharacteristic
517C22×C3#381111yescharacteristic
518C2×D3—62no—
519C2×D3—61no—
520C2×D3—61no—
521C2×D3—61no—
522C2×D3—61no—
523C2×D3—61no—
524C2×D3—61no—
525C2×D3—62no—
526C2×D3—62no—
527C2×D3—121no—
528C2×D3—121no—
529C2×D3—122no—
530C2×D3—121no—
531C2×D3—121no—
532C2×D3—121no—
533C2×D3—121no—
534C2×D3—241no—
535C2×D3—241no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—41no—
543C3×C4—41no—
544C3×C4—122no—
545C3×C4—121no—
546C3×C4—121no—
547C3×C4—182no—
548C3×C4—181no—
549C3×C4#380211yescharacteristic
550C3×C4#380211yescharacteristic
551C3×C4D3×#6711yescharacteristic
552C3×C4D3×#6711yescharacteristic
553C3×C4D3×#6711yescharacteristic
554C3×C4D3×#6711yescharacteristic
555Dic3—41no—
556Dic3—62no—
557Dic3—61no—
558Dic3—61no—
559Dic3—62no—
560Dic3—122no—
561Dic3—121no—
562Dic3—362no—
563Dic3—361no—
564Dic3—361no—
565Dic3—361no—
566Dic3—721no—
567Dic3—721no—
568Dic3—722no—
569C32—21no—
570C32—21no—
571C32—21no—
572C32—41no—
573C32#473311yescharacteristic, complemented
574C32#517311yescharacteristic, complemented
575C32#518711yescharacteristic, complemented
576C23—91no—
577C23—181no—
578Q8—61no—
579Q8—61no—
580Q8—92no—
581Q8—181no—
582D4—92no—
583D4—181no—
584D4—544no—
585C2×C4—61no—
586C2×C4—91no—
587C2×C4—181no—
588C2×C4—181no—
589C2×C4—181no—
590C2×C4—544no—
591C2×C4—541no—
592C2×C4D3311yescharacteristic
593C8—61no—
594C8—61no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—41no—
605C2×C3—41no—
606C2×C3—41no—
607C2×C3—361no—
608C2×C3—362no—
609C2×C3#6380011yescharacteristic
610C2×C3#6380111yescharacteristic
611C2×C3#6380611yescharacteristic
612C2×C3#6380811yescharacteristic
613C2×C3#6392211yescharacteristic
614C2×C3#6392211yescharacteristic
615C2×C3#6395011yescharacteristic
616C2×C3#6395111yescharacteristic
617C2×C3#6398311yescharacteristic
618D3—122no—
619D3—122no—
620D3—121no—
621D3—122no—
622D3—121no—
623D3—242no—
624C22—181no—
625C22—181no—
626C22—181no—
627C22—182no—
628C22—361no—
629C22—361no—
630C22#8895511yescharacteristic
631C4—121no—
632C4—181no—
633C4—182no—
634C4—1081no—
635C4#8895311yescharacteristic
636C4#8895311yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059096011yescharacteristic, complemented
642C3#1059096611yescharacteristic, complemented
643C3#1059165611yescharacteristic, complemented
644C2—361no—
645C2—362no—
646C2#1172286511yescharacteristic
647C2#1172289611yescharacteristic
648C2#1172295111yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#89041C2
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723435C2semidirect
3. C2×C3×#3719C2semidirect
4. C2×C3×#3717C2semidirect
5. #11722040C2semidirect
6. #11722034C2semidirect
7. C3×#63557C2semidirect
37. C2×#3719D3semidirect
38. C2×#3717D3semidirect
49. #63557D3semidirect
380. C33#452semidirect
573. C32#4733semidirect
574. C32#5173semidirect
575. C32#5187semidirect
641. C3#10590960semidirect
642. C3#10590966semidirect
643. C3#10591656semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
#117234352
C2×C3×#37192
C2×C3×#37172
#117220402
#117220342
C3×#635572
#105916563
#105909663
#105909603