Finite Group Databasebeta

← #13155896

Subgroups — #13155896

Order 1728.

944 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89018C211yesmaximal, characteristic
2#11723674C211yesmaximal, characteristic, complemented
3C2×#88797C211yesmaximal, characteristic, complemented
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic, complemented
8#10591602—31nomaximal
9#10590970—31nomaximal
10#10590966—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6292—21no—
16C2×#6289—21no—
17#89018—22no—
18#89018—22no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88815—21no—
23#88814—21no—
24#88797—22no—
25C3×#3719—22no—
26C2×#3832—31no—
27C2×#3799—31no—
28C2×#3799—31no—
29C2×#3799—61no—
30C2×#3744—31no—
31C2×C3×#216—31no—
32C2×C3×Dic12—31no—
33#63871—31no—
34#63857—31no—
35#63856—31no—
36C2×#3719D311yescharacteristic, complemented
37C2×#3717—31no—
38C2×#3717—31no—
39C2×#3717—61no—
40#63648—31no—
41C3×#835—31no—
42C3×(C2×D3)⋊C8—31no—
43C3×(C2×D3)⋊C8—31no—
44C3×#816—31no—
45#63566D311yescharacteristic, complemented
46#63564—31no—
47#63564—31no—
48#63564—61no—
49#63557D311yescharacteristic, complemented
50C22×C3⋊(C3⋊D3)—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))D412yes—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3D411yescharacteristic
58C3×C4×C3⋊D3—21no—
59C3×C4×C3⋊D3—21no—
60C3×#634—21no—
61C3×#634—21no—
62C2×C32×Dic3—21no—
63C32×Dic6—21no—
64C32×Dic6D412yes—
65#6292—24no—
66#6289—24no—
67C32×C3⋊C8—21no—
68C32×C3⋊C8—21no—
69#5187—91no—
70#4808—91no—
71#4733—91no—
72C2×C32×Q8—31no—
73#3832—62no—
74#3832—62no—
75C2×C4⋊(C3⋊D3)—31no—
76C2×C4⋊(C3⋊D3)—31no—
77C2×C4⋊(C3⋊D3)—31no—
78C2×C4⋊(C3⋊D3)—61no—
79C2×C4⋊(C3⋊D3)—61no—
80C2×C4⋊(C3⋊D3)—61no—
81C2×C4⋊(C3⋊D3)—61no—
82C2×C4⋊(C3⋊D3)—121no—
83C2×C4×C3⋊D3—61no—
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×#634C2×D311yescharacteristic
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—61no—
89C2×C3×Dic6—21no—
90C2×C3×Dic6—31no—
91C2×C3×Dic6—31no—
92C2×C3×Dic6—61no—
93C2×C3×Dic6C2×D311yescharacteristic
94C2×C3×Dic6C2×D311yescharacteristic
95C2×#626—61no—
96C2×#626—61no—
97C2×#626—62no—
98C2×#624—61no—
99C2×#624—61no—
100C2×#624—62no—
101#3799—62no—
102#3799—62no—
103#3799—64no—
104#3799—64no—
105#3799—62no—
106#3799—62no—
107C2×C32×C8—31no—
108C2×C32×C8—31no—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110#3744—62no—
111C3×#216—62no—
112C3×C4⋊Dic3—31no—
113C3×C4⋊Dic3—31no—
114C3×C4⋊Dic3—61no—
115C2×C3×C3⋊C8—21no—
116C2×C3×C3⋊C8—21no—
117C2×C3×C3⋊C8C2×D311yescharacteristic
118C2×C3×C3⋊C8C2×D311yescharacteristic
119C2×C3×C3⋊C8C2×D311yescharacteristic
120C2×C3×C3⋊C8C2×D311yescharacteristic
121C3×Dic12—62no—
122#3724—61no—
123Dic3⋊Dic3—61no—
124Dic3⋊Dic3—61no—
125#3722—61no—
126(C2×D3)⋊Dic3—61no—
127(C2×D3)⋊Dic3—61no—
128#3719—22no—
129#3717—62no—
130#3717—62no—
131#3717—64no—
132C22×C33C2×D411yescharacteristic
133C2×C3⋊(C3⋊D3)—22no—
134C2×C3⋊(C3⋊D3)—41no—
135C2×C3⋊(C3⋊D3)—41no—
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—21no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D411yescharacteristic
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—41no—
143C3×C3⋊Dic3—41no—
144C32×Dic3—22no—
145C2×#239—91no—
146C2×#239—91no—
147C2×#239—181no—
148C2×#235—91no—
149C2×C3×Q16—91no—
150#946—91no—
151C2×#216—31no—
152C2×#215—91no—
153#924—91no—
154C2×Dic12—31no—
155C2×#204—91no—
156C2×#204—91no—
157C2×#204—181no—
158#897—91no—
159C3×#103—91no—
160C3×C22⋊C8—91no—
161#835—31no—
162#821—91no—
163#821—91no—
164#821—181no—
165(C2×D3)⋊C8—31no—
166(C2×D3)⋊C8—31no—
167#816—31no—
168#809—91no—
169C22×C3⋊D3—61no—
170C22×C3⋊D3—62no—
171C22×C3⋊D3—61no—
172C22×C3⋊D3—61no—
173C22×C3⋊D3—121no—
174C22×C3⋊D3—121no—
175C22×C3⋊D3—122no—
176C22×C3⋊D3D1211yescharacteristic
177C22×C3×D3—31no—
178C22×C3×D3—31no—
179C22×C3×D3—61no—
180C32×Q8—32no—
181C32×Q8—61no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4—61no—
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3—21no—
191C2×C3⋊Dic3—21no—
192C2×C3⋊Dic3D1211yescharacteristic
193C4⋊(C3⋊D3)—32no—
194C4⋊(C3⋊D3)—32no—
195C4⋊(C3⋊D3)—32no—
196C4⋊(C3⋊D3)—62no—
197C4⋊(C3⋊D3)—62no—
198C4⋊(C3⋊D3)—62no—
199C4⋊(C3⋊D3)—64no—
200C4⋊(C3⋊D3)—61no—
201C4⋊(C3⋊D3)—62no—
202C4⋊(C3⋊D3)—61no—
203C4⋊(C3⋊D3)—61no—
204C4⋊(C3⋊D3)—122no—
205C4⋊(C3⋊D3)—121no—
206C4⋊(C3⋊D3)—121no—
207C4⋊(C3⋊D3)—122no—
208C4×C3⋊D3—21no—
209C4×C3⋊D3—21no—
210C4×C3⋊D3—64no—
211#634—21no—
212#634—21no—
213C2×C3×Dic3—22no—
214C2×C3×Dic3—21no—
215C2×C3×Dic3—21no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—61no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—121no—
224C2×C3×Dic3—121no—
225C3×C4×D3—62no—
226C3×C4×D3—62no—
227C3×C4×D3—61no—
228C3×C4×D3—61no—
229C3×C4×D3—61no—
230C3×C4×D3—61no—
231C3×Dic6—22no—
232C3×Dic6—21no—
233C3×Dic6—21no—
234C3×Dic6—22no—
235C3×Dic6—62no—
236C3×Dic6—61no—
237C3×Dic6—61no—
238C3×Dic6—61no—
239C3×Dic6—61no—
240C3×Dic6—121no—
241C3×Dic6D1212yes—
242C3×Dic6#6712yes—
243#626—68no—
244#626—64no—
245#626—64no—
246#624—68no—
247#624—64no—
248#624—64no—
249C32×C8—61no—
250C32×C8—61no—
251C3×C3⋊C8—22no—
252C3×C3⋊C8—21no—
253C3×C3⋊C8—21no—
254C3×C3⋊C8—21no—
255C3×C3⋊C8—21no—
256C3×C3⋊C8—41no—
257#452—271noSylow
258C2×C33#12011yescharacteristic
259C2×C33#13511yescharacteristic
260C2×C33#13711yescharacteristic
261C3⋊(C3⋊D3)—42no—
262C3×C3⋊D3—41no—
263C2×C3×Q8—31no—
264C2×C3×Q8—31no—
265C2×C3×Q8—61no—
266C2×C3×Q8—91no—
267C22×C3×C4—91no—
268C2×#67—181no—
269#239—182no—
270#239—182no—
271#239—182no—
272#239—182no—
273#239—184no—
274#239—184no—
275#235—182no—
276#235—182no—
277C2×D12—91no—
278C2×D12—91no—
279C2×D12—91no—
280C2×D12—181no—
281C2×D12—181no—
282C2×D12—181no—
283C2×D12—181no—
284C2×D12—181no—
285C2×D12—182no—
286C2×D12—361no—
287C2×C4×D3—31no—
288C2×C4×D3—31no—
289C2×C4×D3—61no—
290C2×C4×D3—181no—
291C2×C4×D3—181no—
292C2×C4×D3—181no—
293C2×C4×D3—182no—
294C2×Dic6—31no—
295C2×Dic6—31no—
296C2×Dic6—61no—
297C2×Dic6D3211yescharacteristic
298C3×Q16—182no—
299C2×C3×C8—31no—
300C2×C3×C8—31no—
301C2×C3×C8—31no—
302C2×C3×C8—31no—
303C2×C3×C8—61no—
304C2×C3×C8—61no—
305C3×C4⋊C4—91no—
306#217—181no—
307#216—62no—
308#215—182no—
309(C2×D3)⋊C4—181no—
310(C2×D3)⋊C4—181no—
311C4⋊Dic3—31no—
312C4⋊Dic3—31no—
313C4⋊Dic3—61no—
314C4⋊Dic3—181no—
315Dic3⋊C4—181no—
316Dic3⋊C4—181no—
317C2×C3⋊C8D3211yescharacteristic
318C2×C3⋊C8D3211yescharacteristic
319Dic12—62no—
320#204—182no—
321#204—182no—
322#204—184no—
323C22×C32—21no—
324C22×C32—21no—
325C22×C32—21no—
326C22×C32—41no—
327C22×C32C2×D1211yescharacteristic
328C22×C32D3×D411yescharacteristic
329C22×C32D3×D411yescharacteristic
330C2×C3⋊D3—21no—
331C2×C3⋊D3—21no—
332C2×C3⋊D3—21no—
333C2×C3⋊D3—62no—
334C2×C3⋊D3—64no—
335C2×C3⋊D3—62no—
336C2×C3⋊D3—62no—
337C2×C3⋊D3—122no—
338C2×C3⋊D3—122no—
339C2×C3⋊D3—124no—
340C2×C3⋊D3—121no—
341C2×C3⋊D3—121no—
342C2×C3⋊D3—122no—
343C2×C3⋊D3—122no—
344C2×C3⋊D3—121no—
345C2×C3⋊D3—121no—
346C2×C3⋊D3—121no—
347C2×C3⋊D3—121no—
348C2×C3⋊D3—241no—
349C2×C3⋊D3—241no—
350C2×C3⋊D3—241no—
351C2×C3⋊D3—241no—
352C2×C3⋊D3—242no—
353C2×C3⋊D3—242no—
354C2×C3×D3—62no—
355C2×C3×D3—61no—
356C2×C3×D3—61no—
357C2×C3×D3—61no—
358C2×C3×D3—61no—
359C2×C3×D3—61no—
360C2×C3×D3—61no—
361C2×C3×D3—121no—
362C2×C3×D3—121no—
363C32×C4—21no—
364C32×C4—21no—
365C32×C4—21no—
366C32×C4—21no—
367C32×C4—21no—
368C32×C4—21no—
369C32×C4—41no—
370C32×C4—41no—
371C32×C4—62no—
372C32×C4C2×D1211yescharacteristic
373C32×C4D3×D411yescharacteristic
374C32×C4D3×D411yescharacteristic
375C32×C4D3×D411yescharacteristic
376C32×C4D3×D411yescharacteristic
377C32×C4C2×#6711yescharacteristic
378C3⋊Dic3—21no—
379C3⋊Dic3—41no—
380C3⋊Dic3—41no—
381C3×Dic3—24no—
382C3×Dic3—22no—
383C3×Dic3—22no—
384C3×Dic3—62no—
385C3×Dic3—61no—
386C3×Dic3—61no—
387C3×Dic3—122no—
388C3×Dic3—121no—
389C3×Dic3—121no—
390C3×Dic3—121no—
391C3×Dic3—121no—
392C3×Dic3—241no—
393C2×#41—271no—
394C2×Q16—91no—
395C2×SD16—271no—
396C22⋊Q8—271no—
397#103—91no—
398#102—271no—
399C22⋊C8—91no—
400C33#45211yescharacteristic, complemented, Sylow
401C23×C3—91no—
402C22×D3—31no—
403C22×D3—31no—
404C22×D3—61no—
405C22×D3—182no—
406C22×D3—181no—
407C22×D3—181no—
408C22×D3—181no—
409C22×D3—361no—
410C22×D3—361no—
411C22×D3—362no—
412C3×Q8—32no—
413C3×Q8—32no—
414C3×Q8—62no—
415C3×Q8—62no—
416C3×Q8—61no—
417C3×Q8—61no—
418C3×Q8—181no—
419C3×Q8—181no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—41no—
424C2×C3×C4—62no—
425C2×C3×C4—61no—
426C2×C3×C4—61no—
427C2×C3×C4—91no—
428C2×C3×C4—181no—
429C2×C3×C4—181no—
430C2×C3×C4—181no—
431C2×C3×C4—181no—
432C2×C3×C4C2×D3211yescharacteristic
433C2×C3×C4C2×D3211yescharacteristic
434C2×C3×C4C2×D3211yescharacteristic
435#67—184no—
436C2×Dic3—21no—
437C2×Dic3—31no—
438C2×Dic3—31no—
439C2×Dic3—61no—
440C2×Dic3—61no—
441C2×Dic3—61no—
442C2×Dic3—61no—
443C2×Dic3—61no—
444C2×Dic3—121no—
445C2×Dic3—121no—
446D12—92no—
447D12—92no—
448D12—92no—
449D12—182no—
450D12—182no—
451D12—182no—
452D12—182no—
453D12—188no—
454D12—184no—
455D12—184no—
456D12—181no—
457D12—181no—
458D12—181no—
459D12—362no—
460D12—361no—
461D12—361no—
462D12—362no—
463C4×D3—62no—
464C4×D3—62no—
465C4×D3—61no—
466C4×D3—61no—
467C4×D3—61no—
468C4×D3—61no—
469C4×D3—184no—
470C4×D3—188no—
471C4×D3—184no—
472C4×D3—184no—
473Dic6—21no—
474Dic6—62no—
475Dic6—61no—
476Dic6—61no—
477Dic6—61no—
478Dic6—61no—
479Dic6—121no—
480Dic6#62612yes—
481C3×C8—62no—
482C3×C8—61no—
483C3×C8—61no—
484C3×C8—61no—
485C3×C8—61no—
486C3×C8—121no—
487C3⋊C8—21no—
488C3⋊C8—21no—
489C2×C32—21no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—21no—
493C2×C32—21no—
494C2×C32—21no—
495C2×C32—21no—
496C2×C32—21no—
497C2×C32—21no—
498C2×C32—41no—
499C2×C32—41no—
500C2×C32—41no—
501C2×C32#88211yescharacteristic
502C2×C32#88211yescharacteristic
503C2×C32#90411yescharacteristic
504C2×C32#90911yescharacteristic
505C2×C32#91611yescharacteristic
506C2×C32#91811yescharacteristic
507C2×C32#91811yescharacteristic
508C2×C32#91911yescharacteristic
509C2×C32#93711yescharacteristic
510C3⋊D3—41no—
511C3⋊D3—122no—
512C3⋊D3—124no—
513C3⋊D3—122no—
514C3⋊D3—122no—
515C3⋊D3—242no—
516C3⋊D3—242no—
517C3⋊D3—244no—
518C3×D3—122no—
519C3×D3—121no—
520C3×D3—121no—
521#41—542no—
522#41—542no—
523C2×Q8—31no—
524C2×Q8—91no—
525C2×D4—271no—
526C2×D4—541no—
527C22×C4—91no—
528C22×C4—541no—
529Q16—182no—
530SD16—542no—
531C2×C8—31no—
532C2×C8—31no—
533C4⋊C4—91no—
534C4⋊C4—541no—
535C22⋊C4—541no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—41no—
540C22×C3—181no—
541C22×C3—181no—
542C22×C3—181no—
543C22×C3D3×D1211yescharacteristic
544C22×C3D3×D1211yescharacteristic
545C22×C3#381111yescharacteristic
546C2×D3—62no—
547C2×D3—61no—
548C2×D3—61no—
549C2×D3—61no—
550C2×D3—61no—
551C2×D3—61no—
552C2×D3—61no—
553C2×D3—121no—
554C2×D3—121no—
555C2×D3—184no—
556C2×D3—182no—
557C2×D3—182no—
558C2×D3—182no—
559C2×D3—362no—
560C2×D3—362no—
561C2×D3—364no—
562C2×D3—362no—
563C2×D3—362no—
564C2×D3—361no—
565C2×D3—361no—
566C2×D3—361no—
567C2×D3—361no—
568C2×D3—361no—
569C2×D3—361no—
570C2×D3—721no—
571C2×D3—721no—
572C2×D3—721no—
573C2×D3—721no—
574C2×D3—722no—
575C2×D3—722no—
576C3×C4—21no—
577C3×C4—21no—
578C3×C4—21no—
579C3×C4—21no—
580C3×C4—21no—
581C3×C4—21no—
582C3×C4—41no—
583C3×C4—41no—
584C3×C4—64no—
585C3×C4—62no—
586C3×C4—62no—
587C3×C4—181no—
588C3×C4—361no—
589C3×C4—361no—
590C3×C4D3×D1211yescharacteristic
591C3×C4#380211yescharacteristic
592C3×C4C2×#62611yescharacteristic
593C3×C4D3×#6711yescharacteristic
594C3×C4#381111yescharacteristic
595C3×C4#381111yescharacteristic
596Dic3—22no—
597Dic3—62no—
598Dic3—61no—
599Dic3—61no—
600Dic3—122no—
601Dic3—121no—
602Dic3—121no—
603Dic3—121no—
604Dic3—121no—
605Dic3—241no—
606C32—21no—
607C32—21no—
608C32—21no—
609C32—41no—
610C32#473311yescharacteristic, complemented
611C32#480811yescharacteristic, complemented
612C32#518711yescharacteristic, complemented
613C23—91no—
614C23—541no—
615Q8—32no—
616Q8—61no—
617Q8—181no—
618Q8—181no—
619D4—272no—
620D4—544no—
621D4—541no—
622C2×C4—61no—
623C2×C4—91no—
624C2×C4—181no—
625C2×C4—181no—
626C2×C4—181no—
627C2×C4—181no—
628C2×C4—544no—
629C2×C4D3311yescharacteristic
630C8—61no—
631C8—61no—
632C2×C3—21no—
633C2×C3—21no—
634C2×C3—21no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—21no—
641C2×C3—41no—
642C2×C3—41no—
643C2×C3—41no—
644C2×C3—361no—
645C2×C3#6380111yescharacteristic
646C2×C3#6380111yescharacteristic
647C2×C3#6380611yescharacteristic
648C2×C3#6380811yescharacteristic
649C2×C3#6392211yescharacteristic
650C2×C3#6392311yescharacteristic
651C2×C3#6394511yescharacteristic
652C2×C3#6395011yescharacteristic
653C2×C3#6398111yescharacteristic
654D3—122no—
655D3—121no—
656D3—121no—
657D3—364no—
658D3—362no—
659D3—362no—
660D3—362no—
661D3—722no—
662D3—722no—
663D3—724no—
664C22—181no—
665C22—181no—
666C22—181no—
667C22—542no—
668C22—1081no—
669C22—1081no—
670C22#8895511yescharacteristic
671C4—62no—
672C4—181no—
673C4—361no—
674C4—361no—
675C4D3×#62611yescharacteristic
676C4#8895411yescharacteristic
677C3—21no—
678C3—21no—
679C3—21no—
680C3—41no—
681C3#1059096611yescharacteristic, complemented
682C3#1059097011yescharacteristic, complemented
683C3#1059160211yescharacteristic, complemented
684C2—361no—
685C2—1082no—
686C2#1172286511yescharacteristic
687C2#1172289711yescharacteristic
688C2#1172295011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#89018C2
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. #11723674C2semidirect
3. C2×#88797C2semidirect
4. C2×C3×#3719C2semidirect
5. #11722036C2semidirect
6. C3×#63566C2semidirect
7. C3×#63557C2semidirect
36. C2×#3719D3semidirect
45. #63566D3semidirect
49. #63557D3semidirect
400. C33#452semidirect
610. C32#4733semidirect
611. C32#4808semidirect
612. C32#5187semidirect
681. C3#10590966semidirect
682. C3#10590970semidirect
683. C3#10591602semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
#117236742
C2×#887972
C2×C3×#37192
#117220362
C3×#635662
C3×#635572
#105916023
#105909703
#105909663