Finite Group Databasebeta

← #13155888

Subgroups — #13155888

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×#89022C211yesmaximal, characteristic
2C2×#88795C211yesmaximal, characteristic, complemented
3C3×#63856C211yesmaximal, characteristic, complemented
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5#11722076C211yesmaximal, characteristic, complemented
6#11722036C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591657—31nomaximal
9#10591184—31nomaximal
10#10590966—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6291—21no—
16C2×#6289—21no—
17#89022—22no—
18#89022—22no—
19C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22#88795—22no—
23C3×#3724—21no—
24C3×(C2×D3)⋊Dic3—21no—
25C3×#3719—22no—
26C2×#3828—31no—
27C2×#3799—31no—
28C2×#3799—31no—
29C2×#3799—61no—
30C2×#3755—31no—
31C3×#946—31no—
32C2×C3×#216—31no—
33C3×#924—31no—
34C2×C3×Dic12—31no—
35#63856D311yescharacteristic, complemented
36C2×#3719D311yescharacteristic, complemented
37C2×#3717—31no—
38C2×#3717—31no—
39C2×#3717—61no—
40#63648—31no—
41C3×#830—31no—
42C3×(C2×D3)⋊C8—31no—
43#63567—31no—
44#63566—31no—
45#63565—31no—
46#63564—31no—
47#63564—31no—
48#63564—61no—
49(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
50C22×C3⋊(C3⋊D3)—21no—
51C22×C32×D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C4⋊(C3⋊(C3⋊D3))—21no—
54C4⋊(C3⋊(C3⋊D3))D412yes—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C3×#634—21no—
58C3×#634D412yes—
59C2×C32×Dic3—21no—
60C2×C32×Dic3D411yescharacteristic
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6—21no—
65#6291—24no—
66#6289—24no—
67C3×C3⋊(C3⋊C8)—21no—
68C32×C3⋊C8—21no—
69#5187—91no—
70#5047—91no—
71#4733—91no—
72C2×C32×Q8—31no—
73C22×C32×C4—31no—
74C2×#638—61no—
75#3828—62no—
76#3828—62no—
77C2×C4⋊(C3⋊D3)—31no—
78C2×C4⋊(C3⋊D3)—31no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—61no—
81C2×C4⋊(C3⋊D3)—61no—
82C2×C4⋊(C3⋊D3)—61no—
83C2×C4⋊(C3⋊D3)—121no—
84C2×C4×C3⋊D3—61no—
85C2×#634C2×D311yescharacteristic
86C2×C3×C4×D3—21no—
87C2×C3×C4×D3C2×D311yescharacteristic
88C2×C3×C4×D3C2×D311yescharacteristic
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—62no—
96C2×#626—61no—
97C2×#626—61no—
98C2×#624—62no—
99C2×#624—61no—
100C2×#624—61no—
101#3799—62no—
102#3799—62no—
103#3799—62no—
104#3799—62no—
105#3799—64no—
106#3799—64no—
107C2×C32×C8—31no—
108#3755—62no—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111C3×#217—61no—
112C3×#216—62no—
113C3×(C2×D3)⋊C4—61no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C3×Dic3⋊C4—61no—
118C3×Dic3⋊C4—61no—
119C2×C3×C3⋊C8—21no—
120C2×C3×C3⋊C8—31no—
121C2×C3×C3⋊C8—31no—
122C2×C3×C3⋊C8—61no—
123C2×C3×C3⋊C8C2×D311yescharacteristic
124C2×C3×C3⋊C8C2×D311yescharacteristic
125C3×Dic12—62no—
126#3724—21no—
127(C2×D3)⋊Dic3—21no—
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×C32×D3—21no—
137C2×C32×D3—21no—
138C2×C32×D3—21no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D411yescharacteristic
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—41no—
143C32×Dic3—21no—
144C32×Dic3—41no—
145C2×#239—91no—
146C2×#235—91no—
147C2×#235—91no—
148C2×#235—181no—
149C2×C3×Q16—91no—
150C3×C22⋊Q8—91no—
151#946—31no—
152C2×#216—31no—
153C2×#215—91no—
154C2×#215—91no—
155C2×#215—181no—
156#924—31no—
157C2×Dic12—31no—
158C2×#204—91no—
159C3×C22⋊C8—91no—
160#835—91no—
161#830—31no—
162#821—91no—
163#821—91no—
164#821—181no—
165(C2×D3)⋊C8—31no—
166#816—91no—
167#810—91no—
168#809—91no—
169C23×C32—31no—
170C22×C3⋊D3—61no—
171C22×C3⋊D3—62no—
172C22×C3⋊D3—61no—
173C22×C3⋊D3—61no—
174C22×C3⋊D3—121no—
175C22×C3⋊D3—121no—
176C22×C3⋊D3—122no—
177C22×C3×D3—21no—
178C22×C3×D3D1211yescharacteristic
179C22×C3×D3#6711yescharacteristic
180C32×Q8—61no—
181C32×Q8—61no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—31no—
186C2×C32×C4—41no—
187C2×C32×C4—61no—
188C2×C32×C4—61no—
189C2×C32×C4—61no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193#638—64no—
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—21no—
196C4⋊(C3⋊D3)—32no—
197C4⋊(C3⋊D3)—32no—
198C4⋊(C3⋊D3)—32no—
199C4⋊(C3⋊D3)—62no—
200C4⋊(C3⋊D3)—62no—
201C4⋊(C3⋊D3)—62no—
202C4⋊(C3⋊D3)—61no—
203C4⋊(C3⋊D3)—62no—
204C4⋊(C3⋊D3)—61no—
205C4⋊(C3⋊D3)—61no—
206C4⋊(C3⋊D3)—122no—
207C4⋊(C3⋊D3)—121no—
208C4⋊(C3⋊D3)—121no—
209C4⋊(C3⋊D3)—122no—
210C4×C3⋊D3—64no—
211#634—21no—
212#634#6712yes—
213C2×C3×Dic3—21no—
214C2×C3×Dic3—21no—
215C2×C3×Dic3—21no—
216C2×C3×Dic3—41no—
217C2×C3×Dic3—62no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—61no—
222C2×C3×Dic3—121no—
223C2×C3×Dic3D1211yescharacteristic
224C2×C3×Dic3#6711yescharacteristic
225C3×C4×D3—22no—
226C3×C4×D3—22no—
227C3×C4×D3—21no—
228C3×C4×D3—21no—
229C3×C4×D3—21no—
230C3×C4×D3—21no—
231C3×Dic6—22no—
232C3×Dic6—21no—
233C3×Dic6—21no—
234C3×Dic6—21no—
235C3×Dic6—21no—
236C3×Dic6—32no—
237C3×Dic6—32no—
238C3×Dic6—41no—
239C3×Dic6—62no—
240C3×Dic6—62no—
241C3×Dic6—61no—
242C3×Dic6—61no—
243#626—68no—
244#626—64no—
245#626—64no—
246#624—68no—
247#624—64no—
248#624—64no—
249C32×C8—61no—
250C3⋊(C3⋊C8)—21no—
251C3×C3⋊C8—21no—
252C3×C3⋊C8—21no—
253C3×C3⋊C8—41no—
254C3×C3⋊C8—62no—
255C3×C3⋊C8—61no—
256C3×C3⋊C8—61no—
257#452—271noSylow
258C2×C33#12011yescharacteristic
259C2×C33#13511yescharacteristic
260C2×C33#13711yescharacteristic
261C3⋊(C3⋊D3)—42no—
262C32×D3—41no—
263C2×C3×Q8—31no—
264C2×C3×Q8—31no—
265C2×C3×Q8—61no—
266C2×C3×Q8—91no—
267C22×C3×C4—31no—
268C22×C3×C4—31no—
269C22×C3×C4—61no—
270C2×#67—181no—
271C2×#67—181no—
272C2×#67—182no—
273#239—182no—
274#239—182no—
275#235—182no—
276#235—182no—
277#235—182no—
278#235—182no—
279#235—184no—
280#235—184no—
281C2×D12—91no—
282C2×D12—91no—
283C2×D12—91no—
284C2×D12—181no—
285C2×D12—181no—
286C2×D12—181no—
287C2×D12—181no—
288C2×D12—361no—
289C2×C4×D3—181no—
290C2×C4×D3—181no—
291C2×C4×D3—182no—
292C2×C4×D3—181no—
293C2×C4×D3D3211yescharacteristic
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—61no—
302C2×C3×C8—91no—
303C3×C4⋊C4—91no—
304C3×C4⋊C4—181no—
305C3×C22⋊C4—181no—
306#217—61no—
307#216—62no—
308#215—182no—
309#215—182no—
310#215—184no—
311(C2×D3)⋊C4—61no—
312C4⋊Dic3—31no—
313C4⋊Dic3—31no—
314C4⋊Dic3—61no—
315Dic3⋊C4—61no—
316Dic3⋊C4—61no—
317C2×C3⋊C8—31no—
318C2×C3⋊C8—31no—
319C2×C3⋊C8—61no—
320C2×C3⋊C8D3211yescharacteristic
321Dic12—62no—
322#204—182no—
323C22×C32—21no—
324C22×C32—21no—
325C22×C32—21no—
326C22×C32—41no—
327C22×C32—61no—
328C22×C32—61no—
329C22×C32—61no—
330C22×C32C2×D1211yescharacteristic
331C22×C32D3×D411yescharacteristic
332C22×C32C2×#6711yescharacteristic
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—22no—
355C2×C3×D3—21no—
356C2×C3×D3—21no—
357C2×C3×D3—21no—
358C2×C3×D3—21no—
359C2×C3×D3—21no—
360C2×C3×D3—21no—
361C2×C3×D3—41no—
362C2×C3×D3—41no—
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—61no—
372C32×C4—121no—
373C32×C4D3×D411yescharacteristic
374C32×C4D3×D411yescharacteristic
375C32×C4D3×D411yescharacteristic
376C32×C4D3×D411yescharacteristic
377C32×C4D3×D411yescharacteristic
378C32×C4C2×#6711yescharacteristic
379C3⋊Dic3—22no—
380C3⋊Dic3—41no—
381C3×Dic3—22no—
382C3×Dic3—21no—
383C3×Dic3—21no—
384C3×Dic3—42no—
385C3×Dic3—41no—
386C3×Dic3—41no—
387C3×Dic3—64no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—121no—
391C3×Dic3—121no—
392C3×Dic3—241no—
393C2×#41—271no—
394C2×Q16—91no—
395C2×SD16—271no—
396C22⋊Q8—91no—
397#103—271no—
398#102—271no—
399C22⋊C8—91no—
400C33#45211yescharacteristic, complemented, Sylow
401C23×C3—31no—
402C23×C3—31no—
403C23×C3—61no—
404C22×D3—182no—
405C22×D3—181no—
406C22×D3—181no—
407C22×D3—181no—
408C22×D3—361no—
409C22×D3—361no—
410C22×D3—362no—
411C22×D3#62611yescharacteristic
412C3×Q8—62no—
413C3×Q8—61no—
414C3×Q8—61no—
415C3×Q8—61no—
416C3×Q8—61no—
417C3×Q8—92no—
418C3×Q8—121no—
419C3×Q8—181no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—21no—
423C2×C3×C4—31no—
424C2×C3×C4—31no—
425C2×C3×C4—41no—
426C2×C3×C4—61no—
427C2×C3×C4—62no—
428C2×C3×C4—62no—
429C2×C3×C4—61no—
430C2×C3×C4—61no—
431C2×C3×C4—61no—
432C2×C3×C4—61no—
433C2×C3×C4—61no—
434C2×C3×C4—61no—
435C2×C3×C4—121no—
436C2×C3×C4—181no—
437C2×C3×C4—181no—
438C2×C3×C4C2×D3211yescharacteristic
439C2×C3×C4C2×D3211yescharacteristic
440C2×C3×C4C2×D3211yescharacteristic
441#67—188no—
442#67—184no—
443#67—184no—
444C2×Dic3—21no—
445C2×Dic3—62no—
446C2×Dic3—61no—
447C2×Dic3—61no—
448C2×Dic3—61no—
449C2×Dic3—61no—
450C2×Dic3—121no—
451C2×Dic3#62611yescharacteristic
452D12—92no—
453D12—92no—
454D12—92no—
455D12—182no—
456D12—182no—
457D12—182no—
458D12—182no—
459D12—184no—
460D12—181no—
461D12—181no—
462D12—181no—
463D12—362no—
464D12—361no—
465D12—361no—
466D12—362no—
467C4×D3—21no—
468C4×D3—21no—
469C4×D3—184no—
470C4×D3—188no—
471C4×D3—184no—
472C4×D3—184no—
473Dic6—21no—
474Dic6—21no—
475Dic6—32no—
476Dic6—32no—
477Dic6—62no—
478Dic6—62no—
479Dic6—61no—
480Dic6—61no—
481C3×C8—61no—
482C3×C8—61no—
483C3×C8—121no—
484C3×C8—181no—
485C3⋊C8—21no—
486C3⋊C8—62no—
487C3⋊C8—61no—
488C3⋊C8—61no—
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—121no—
502C2×C32#88211yescharacteristic
503C2×C32#90411yescharacteristic
504C2×C32#90911yescharacteristic
505C2×C32#91811yescharacteristic
506C2×C32#91911yescharacteristic
507C2×C32#93711yescharacteristic
508C2×C32#93711yescharacteristic
509C2×C32#94211yescharacteristic
510C2×C32#95011yescharacteristic
511C3⋊D3—122no—
512C3⋊D3—124no—
513C3⋊D3—122no—
514C3⋊D3—122no—
515C3⋊D3—242no—
516C3⋊D3—242no—
517C3⋊D3—244no—
518C3×D3—42no—
519C3×D3—41no—
520C3×D3—41no—
521#41—542no—
522#41—542no—
523C2×Q8—31no—
524C2×Q8—91no—
525C2×D4—271no—
526C2×D4—541no—
527C22×C4—31no—
528C22×C4—541no—
529Q16—182no—
530SD16—542no—
531C2×C8—31no—
532C2×C8—91no—
533C4⋊C4—91no—
534C4⋊C4—181no—
535C22⋊C4—181no—
536C22×C3—21no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—41no—
540C22×C3—62no—
541C22×C3—61no—
542C22×C3—61no—
543C22×C3—61no—
544C22×C3—61no—
545C22×C3—61no—
546C22×C3—61no—
547C22×C3—121no—
548C22×C3—121no—
549C22×C3D3×D1211yescharacteristic
550C22×C3C2×#62611yescharacteristic
551C22×C3D3×#6711yescharacteristic
552C2×D3—21no—
553C2×D3—21no—
554C2×D3—21no—
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—62no—
585C3×C4—61no—
586C3×C4—61no—
587C3×C4—122no—
588C3×C4—121no—
589C3×C4—121no—
590C3×C4—182no—
591C3×C4—361no—
592C3×C4#380211yescharacteristic
593C3×C4D3×#6711yescharacteristic
594C3×C4D3×#6711yescharacteristic
595C3×C4#381111yescharacteristic
596C3×C4#381111yescharacteristic
597C3×C4#381111yescharacteristic
598Dic3—21no—
599Dic3—41no—
600Dic3—64no—
601Dic3—62no—
602Dic3—62no—
603Dic3—121no—
604Dic3—121no—
605Dic3—241no—
606C32—21no—
607C32—21no—
608C32—21no—
609C32—41no—
610C32#473311yescharacteristic, complemented
611C32#504711yescharacteristic, complemented
612C32#518711yescharacteristic, complemented
613C23—31no—
614C23—541no—
615Q8—61no—
616Q8—61no—
617Q8—92no—
618Q8—181no—
619D4—272no—
620D4—544no—
621D4—541no—
622C2×C4—31no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—181no—
627C2×C4—181no—
628C2×C4—544no—
629C2×C4D3311yescharacteristic
630C8—61no—
631C8—181no—
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—122no—
645C2×C3—121no—
646C2×C3—121no—
647C2×C3#6380111yescharacteristic
648C2×C3#6380811yescharacteristic
649C2×C3#6382911yescharacteristic
650C2×C3#6383311yescharacteristic
651C2×C3#6392211yescharacteristic
652C2×C3#6394811yescharacteristic
653C2×C3#6394911yescharacteristic
654C2×C3#6398111yescharacteristic
655C2×C3#6398311yescharacteristic
656D3—41no—
657D3—364no—
658D3—362no—
659D3—362no—
660D3—362no—
661D3—722no—
662D3—722no—
663D3—724no—
664C22—61no—
665C22—61no—
666C22—61no—
667C22—542no—
668C22—1081no—
669C22—1081no—
670C22D3×#62611yescharacteristic
671C4—61no—
672C4—121no—
673C4—182no—
674C4—361no—
675C4#8895211yescharacteristic
676C4#8895411yescharacteristic
677C3—21no—
678C3—21no—
679C3—21no—
680C3—41no—
681C3#1059096611yescharacteristic, complemented
682C3#1059118411yescharacteristic, complemented
683C3#1059165711yescharacteristic, complemented
684C2—121no—
685C2—1082no—
686C2#1172285611yescharacteristic
687C2#1172288511yescharacteristic
688C2#1172294911yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×#89022C2
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. C2×#88795C2semidirect
3. C3×#63856C2semidirect
4. C2×C3×#3719C2semidirect
5. #11722076C2semidirect
6. #11722036C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
35. #63856D3semidirect
36. C2×#3719D3semidirect
49. (C2×D3)⋊(C3⋊C8)D3semidirect
400. C33#452semidirect
610. C32#4733semidirect
611. C32#5047semidirect
612. C32#5187semidirect
681. C3#10590966semidirect
682. C3#10591184semidirect
683. C3#10591657semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
C2×#887952
C3×#638562
C2×C3×#37192
#117220762
#117220362
C3×(C2×D3)⋊(C3⋊C8)2
#105916573
#105911843
#105909663