Finite Group Databasebeta

← #13161752

Subgroups — #13161752

Order 1728.

830 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C211yesmaximal, characteristic
2C2×C3×#3743C211yesmaximal, characteristic, complemented
3#11723689C211yesmaximal, characteristic, complemented
4C3×#63648C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6#11722076C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic, complemented
8#10590957—31nomaximal
9#10590671—31nomaximal
10#10590670—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×#6295—21no—
15C2×#6294—21no—
16C2×#6294—21no—
17#89041—22no—
18#89041C414yescomplemented
19C2×C33×C8C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C3×#3743—22no—
24#88815—21no—
25C2×#3799—31no—
26C2×#3796—31no—
27C2×#3795—31no—
28C2×#3743D311yescharacteristic, complemented
29C2×C3×ΓL(1,𝔽25)—31no—
30C2×C3×ΓL(1,𝔽25)—31no—
31C2×C3×ΓL(1,𝔽25)—61no—
32#63900—31no—
33D3×C4⋊Dic3—31no—
34D3×C4⋊Dic3—31no—
35#63648D311yescharacteristic, complemented
36#63643D311yescharacteristic, complemented
37C3×#821—31no—
38C3×#821—31no—
39C3×#821—61no—
40C3×#816—31no—
41C3×#816—31no—
42C3×#816—61no—
43#63567—31no—
44#63566—31no—
45#63565—31no—
46#63564—31no—
47#63563—31no—
48#63562—31no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3D411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)C2×C412yescomplemented
58C3×C4×C3⋊D3C2×C412yes—
59C3×C4×C3⋊D3D412yes—
60C3×#634—21no—
61C3×#634C2×C412yescomplemented
62#6295—24no—
63#6294—24no—
64#6294—42no—
65C33×C8—21no—
66C3×C3⋊(C3⋊C8)—21no—
67#5133—91no—
68#4803—91no—
69#4771—91no—
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C4×C3⋊D3C2×D311yescharacteristic
72C2×#634C2×D311yescharacteristic
73C2×C3×D12—31no—
74C2×C3×D12—31no—
75C2×C3×D12—61no—
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—31no—
78C2×C3×C4×D3—61no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×#626—61no—
83C2×#626—61no—
84C2×#625—61no—
85C2×#624—61no—
86C2×#624—61no—
87C2×D3×Dic3—61no—
88C2×D3×Dic3—61no—
89C2×D3×Dic3—61no—
90C2×D3×Dic3—61no—
91#3799—34no—
92#3799—62no—
93#3796—34no—
94#3796—62no—
95#3795—34no—
96#3795—62no—
97C2×C32×C8—21no—
98C2×C32×C8—21no—
99C2×C32×C8—21no—
100C2×C32×C8—41no—
101C2×C32×C8C2×D311yescharacteristic
102C2×C32×C8C2×D311yescharacteristic
103C2×C32×C8C2×D311yescharacteristic
104C4⋊(C3⋊Dic3)C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
107#3743—22no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—31no—
110C3×C4⋊Dic3—31no—
111C3×C4⋊Dic3—31no—
112C3×C4⋊Dic3—61no—
113C3×C4⋊Dic3—61no—
114C2×C3×C3⋊C8—31no—
115C2×C3×C3⋊C8—31no—
116C2×C3×C3⋊C8—61no—
117C3×ΓL(1,𝔽25)—62no—
118C3×ΓL(1,𝔽25)—62no—
119C3×ΓL(1,𝔽25)—64no—
120#3724—61no—
121Dic3⋊Dic3—61no—
122Dic3⋊Dic3—61no—
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—21no—
125C2×C3×C3⋊D3—21no—
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—41no—
128C2×C3×C3⋊D3—41no—
129C2×C3×C3⋊D3C22⋊C412yes—
130C33×C4C22×C411yescharacteristic
131C33×C4C2×D411yescharacteristic
132C3×C3⋊Dic3—22no—
133C3×C3⋊Dic3—41no—
134C3×C3⋊Dic3—41no—
135C3×C3⋊Dic3C22⋊C412yes—
136C2×#239—91no—
137C2×#237—91no—
138C2×#235—91no—
139C2×C3×M16—91no—
140C2×C4⋊Dic3—91no—
141C2×ΓL(1,𝔽25)—31no—
142C2×ΓL(1,𝔽25)—31no—
143C2×ΓL(1,𝔽25)—61no—
144D3×C4⋊C4—91no—
145D3×C4⋊C4—91no—
146C3×#103—91no—
147C3×#102—91no—
148#835—91no—
149#832—91no—
150#821—31no—
151#821—31no—
152#821—61no—
153#821—91no—
154#816—31no—
155#816—31no—
156#816—61no—
157#816—91no—
158#810—91no—
159#809—91no—
160C22×C3⋊D3—21no—
161C22×C3⋊D3D1211yescharacteristic
162C22×C3×D3—31no—
163C22×C3×D3—31no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C22×C3×D3—121no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—41no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3D1211yescharacteristic
179C4⋊(C3⋊D3)—21no—
180C4⋊(C3⋊D3)C4×D312yescomplemented
181C4×C3⋊D3C4×D312yes—
182C4×C3⋊D3#6712yes—
183#634—21no—
184#634C4×D312yescomplemented
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3—121no—
196C2×C3×Dic3—121no—
197C3×D12—32no—
198C3×D12—32no—
199C3×D12—62no—
200C3×D12—61no—
201C3×D12—61no—
202C3×D12—121no—
203C3×C4×D3—32no—
204C3×C4×D3—32no—
205C3×C4×D3—32no—
206C3×C4×D3—32no—
207C3×C4×D3—62no—
208C3×C4×D3—62no—
209C3×Dic6—32no—
210C3×Dic6—32no—
211C3×Dic6—62no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—121no—
215#626—64no—
216#626—64no—
217#625—64no—
218#624—64no—
219#624—122no—
220D3×Dic3—64no—
221D3×Dic3—64no—
222D3×Dic3—122no—
223D3×Dic3—122no—
224C32×C8—21no—
225C32×C8—22no—
226C32×C8—21no—
227C32×C8—21no—
228C32×C8—41no—
229C32×C8—41no—
230C32×C8—42no—
231C3⋊(C3⋊C8)—21no—
232C3×C3⋊C8—62no—
233C3×C3⋊C8—61no—
234C3×C3⋊C8—61no—
235#417—271noSylow
236C2×C33C2×C22⋊C411yescharacteristic
237C2×C33AGL(1,ℤ/8)11yescharacteristic
238C2×C33#13711yescharacteristic
239C3×C3⋊D3—22no—
240C3×C3⋊D3—42no—
241C2×C3×Q8—91no—
242C2×C3×D4—91no—
243C22×C3×C4—91no—
244C2×#67—181no—
245C2×#67—181no—
246C22×Dic3—181no—
247C22×Dic3—181no—
248#239—94no—
249#239—182no—
250#237—94no—
251#237—182no—
252#235—94no—
253#235—182no—
254C2×D12—31no—
255C2×D12—31no—
256C2×D12—61no—
257C2×D12—181no—
258C2×C4×D3—31no—
259C2×C4×D3—31no—
260C2×C4×D3—61no—
261C2×C4×D3—181no—
262C2×C4×D3—181no—
263C2×C4×D3—181no—
264C2×C4×D3—181no—
265C2×Dic6—31no—
266C2×Dic6—31no—
267C2×Dic6—61no—
268C3×M16—182no—
269C2×C3×C8—21no—
270C2×C3×C8—21no—
271C2×C3×C8—21no—
272C2×C3×C8—41no—
273C2×C3×C8—91no—
274C2×C3×C8D3211yescharacteristic
275C2×C3×C8D3211yescharacteristic
276C2×C3×C8D3211yescharacteristic
277C3×C4⋊C4—91no—
278C3×C4⋊C4—91no—
279C4⋊Dic3—31no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—61no—
284C4⋊Dic3—61no—
285C4⋊Dic3—181no—
286Dic3⋊C4—181no—
287Dic3⋊C4—181no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291ΓL(1,𝔽25)—62no—
292ΓL(1,𝔽25)—62no—
293ΓL(1,𝔽25)—64no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—41no—
298C22×C32C2×D1211yescharacteristic
299C22×C32D3×D411yescharacteristic
300C22×C32D3×D411yescharacteristic
301C2×C3⋊D3—21no—
302C2×C3⋊D3—21no—
303C2×C3⋊D3—22no—
304C2×C3⋊D3—41no—
305C2×C3⋊D3—41no—
306C2×C3⋊D3(C2×D3)⋊C412yes—
307C2×C3×D3—32no—
308C2×C3×D3—32no—
309C2×C3×D3—62no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
314C2×C3×D3—62no—
315C2×C3×D3—62no—
316C2×C3×D3—121no—
317C2×C3×D3—121no—
318C2×C3×D3—122no—
319C2×C3×D3—121no—
320C2×C3×D3—121no—
321C2×C3×D3—121no—
322C2×C3×D3—121no—
323C2×C3×D3—241no—
324C2×C3×D3—241no—
325C32×C4—21no—
326C32×C4—21no—
327C32×C4—21no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—41no—
332C32×C4—41no—
333C32×C4C2×C4×D311yescharacteristic
334C32×C4C2×C4×D311yescharacteristic
335C32×C4C2×C4×D311yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4C2×#6711yescharacteristic
339C3⋊Dic3—22no—
340C3⋊Dic3—41no—
341C3⋊Dic3—41no—
342C3⋊Dic3(C2×D3)⋊C412yes—
343C3×Dic3—32no—
344C3×Dic3—32no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—122no—
349C3×Dic3—121no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C3×Dic3—121no—
353C3×Dic3—241no—
354C3×Dic3—241no—
355C2×#41—271no—
356C2×M16—91no—
357C2×C4⋊C4—271no—
358#103—91no—
359#103—271no—
360#102—91no—
361#102—271no—
362C33#41711yescharacteristic, complemented, Sylow
363C23×C3—91no—
364C23×C3—181no—
365C22×D3—31no—
366C22×D3—31no—
367C22×D3—61no—
368C22×D3—61no—
369C22×D3—61no—
370C22×D3—121no—
371C3×Q8—92no—
372C3×Q8—181no—
373C3×D4—92no—
374C3×D4—181no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—41no—
379C2×C3×C4—91no—
380C2×C3×C4—92no—
381C2×C3×C4—92no—
382C2×C3×C4—181no—
383C2×C3×C4—181no—
384C2×C3×C4—181no—
385C2×C3×C4C2×D3211yescharacteristic
386C2×C3×C4C2×D3211yescharacteristic
387C2×C3×C4C2×D3211yescharacteristic
388#67—184no—
389#67—184no—
390C2×Dic3—31no—
391C2×Dic3—31no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—121no—
400C2×Dic3—121no—
401C2×Dic3—121no—
402C2×Dic3—184no—
403C2×Dic3—362no—
404D12—32no—
405D12—32no—
406D12—62no—
407D12—61no—
408D12—61no—
409D12—121no—
410D12—184no—
411C4×D3—32no—
412C4×D3—32no—
413C4×D3—32no—
414C4×D3—32no—
415C4×D3—62no—
416C4×D3—62no—
417C4×D3—184no—
418C4×D3—184no—
419C4×D3—362no—
420C4×D3—362no—
421Dic6—32no—
422Dic6—32no—
423Dic6—62no—
424Dic6—61no—
425Dic6—61no—
426Dic6—121no—
427C3×C8—22no—
428C3×C8—21no—
429C3×C8—21no—
430C3×C8—21no—
431C3×C8—41no—
432C3×C8—41no—
433C3×C8—42no—
434C3×C8—181no—
435C3⋊C8—62no—
436C3⋊C8—61no—
437C3⋊C8—61no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—41no—
450C2×C32D3×C22⋊C411yescharacteristic
451C2×C32D3×C22⋊C411yescharacteristic
452C2×C32#90811yescharacteristic
453C2×C32#90911yescharacteristic
454C2×C32#91111yescharacteristic
455C2×C32#91411yescharacteristic
456C2×C32#91511yescharacteristic
457C2×C32#91811yescharacteristic
458C2×C32C2×(C2×D3)⋊C411yescharacteristic
459C3⋊D3—22no—
460C3⋊D3—42no—
461C3×D3—62no—
462C3×D3—62no—
463C3×D3—122no—
464C3×D3—122no—
465C3×D3—122no—
466C3×D3—242no—
467#41—274no—
468#41—542no—
469C2×Q8—91no—
470C2×D4—91no—
471C2×D4—541no—
472C22×C4—91no—
473C22×C4—541no—
474C22×C4—541no—
475M16—182no—
476C2×C8—91no—
477C2×C8#111311yescharacteristic
478C4⋊C4—91no—
479C4⋊C4—91no—
480C4⋊C4—541no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—21no—
484C22×C3—41no—
485C22×C3—92no—
486C22×C3—181no—
487C22×C3—181no—
488C22×C3—182no—
489C22×C3—361no—
490C22×C3—361no—
491C22×C3D3×D1211yescharacteristic
492C22×C3D3×D1211yescharacteristic
493C22×C3#380211yescharacteristic
494C2×D3—32no—
495C2×D3—32no—
496C2×D3—62no—
497C2×D3—61no—
498C2×D3—61no—
499C2×D3—61no—
500C2×D3—61no—
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—122no—
506C2×D3—121no—
507C2×D3—121no—
508C2×D3—121no—
509C2×D3—121no—
510C2×D3—241no—
511C2×D3—241no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—41no—
519C3×C4—41no—
520C3×C4—92no—
521C3×C4—182no—
522C3×C4—361no—
523C3×C4—361no—
524C3×C4C4×D3211yescharacteristic
525C3×C4C4×D3211yescharacteristic
526C3×C4C4×D3211yescharacteristic
527C3×C4D3×#6711yescharacteristic
528C3×C4D3×#6711yescharacteristic
529C3×C4#381111yescharacteristic
530Dic3—32no—
531Dic3—32no—
532Dic3—62no—
533Dic3—62no—
534Dic3—62no—
535Dic3—122no—
536Dic3—121no—
537Dic3—121no—
538Dic3—121no—
539Dic3—121no—
540Dic3—241no—
541Dic3—241no—
542C32—21no—
543C32—21no—
544C32—21no—
545C32—41no—
546C32#477111yescharacteristic, complemented
547C32#480311yescharacteristic, complemented
548C32#513311yescharacteristic, complemented
549C23—91no—
550C23—181no—
551Q8—92no—
552Q8—181no—
553D4—92no—
554D4—181no—
555D4—544no—
556C2×C4—91no—
557C2×C4—92no—
558C2×C4—92no—
559C2×C4—181no—
560C2×C4—181no—
561C2×C4—181no—
562C2×C4—544no—
563C2×C4—1082no—
564C2×C4C2×#111311yescharacteristic
565C8—21no—
566C8—181no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—41no—
577C2×C3—41no—
578C2×C3—41no—
579C2×C3—182no—
580C2×C3—362no—
581C2×C3#6379411yescharacteristic
582C2×C3#6379511yescharacteristic
583C2×C3#6379811yescharacteristic
584C2×C3#6380011yescharacteristic
585C2×C3#6380111yescharacteristic
586C2×C3#6381111yescharacteristic
587C2×C3D3×(C2×D3)⋊C411yescharacteristic
588C2×C3D3×(C2×D3)⋊C411yescharacteristic
589C2×C3#6392111yescharacteristic
590D3—62no—
591D3—62no—
592D3—122no—
593D3—122no—
594D3—122no—
595D3—242no—
596C22—92no—
597C22—181no—
598C22—181no—
599C22—182no—
600C22—361no—
601C22—361no—
602C22C4⋊#111311yescharacteristic
603C4—92no—
604C4—182no—
605C4—361no—
606C4—361no—
607C4C4×#111311yescharacteristic
608C4#8904911yescharacteristic
609C3—21no—
610C3—21no—
611C3—21no—
612C3—41no—
613C3#1059067011yescharacteristic, complemented
614C3#1059067111yescharacteristic, complemented
615C3#1059095711yescharacteristic, complemented
616C2—182no—
617C2—362no—
618C2#1172364811yescharacteristic
619C2#1172365011yescharacteristic
620C2#1172370011yescharacteristic
subgroupG/H
centerC22C4⋊#1113
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89041C2
cosocleC2×C33×C4C23
hypercenterC2×C8#1113
hypocenterC33#417
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3743C2semidirect
3. #11723689C2semidirect
4. C3×#63648C2semidirect
5. C3×#63643C2semidirect
6. #11722076C2semidirect
7. #11722075C2semidirect
18. #89041C4semidirect
28. C2×#3743D3semidirect
35. #63648D3semidirect
36. #63643D3semidirect
57. C3×C4⋊(C3⋊D3)C2×C4semidirect
61. C3×#634C2×C4semidirect
180. C4⋊(C3⋊D3)C4×D3semidirect
184. #634C4×D3semidirect
362. C33#417semidirect
546. C32#4771semidirect
547. C32#4803semidirect
548. C32#5133semidirect
613. C3#10590670semidirect
614. C3#10590671semidirect
615. C3#10590957semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C2×C3×#37432
#117236892
C3×#636482
C3×#636432
#117220762
#117220752
#105909573
#105906713
#105906703