Finite Group Databasebeta

← #13155590

Subgroups — #13155590

Order 1728.

702 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6295C211yesmaximal, characteristic
2C2×#88787C211yesmaximal, characteristic, complemented
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4#11722020C211yesmaximal, characteristic, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#63557C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic, complemented
8#10590907—31nomaximal
9#10590737—31nomaximal
10#10590725—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6295C411yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815C411yescharacteristic
20#88814C411yescharacteristic
21#88814C411yescharacteristic
22#88787—22no—
23C3⋊D3×C3⋊C8—22no—
24C2×#3743—31no—
25C2×C8×C3⋊D3—31no—
26C4×#626—31no—
27C4×#626—31no—
28C4×#625—31no—
29C2×#3711—31no—
30C2×#3711—31no—
31C2×#3711—61no—
32C2×D3×C3⋊C8—31no—
33C2×D3×C3⋊C8—31no—
34C2×D3×C3⋊C8—61no—
35#63642—31no—
36C3×(C2×D3)⋊C8—31no—
37C3×(C2×D3)⋊C8—31no—
38C3×(C2×D3)⋊C8—31no—
39C3×(C2×D3)⋊C8—31no—
40C3×Dic3⋊C8—31no—
41C3×Dic3⋊C8—31no—
42#63572D311yescharacteristic, complemented
43#63557D311yescharacteristic, complemented
44#63557D311yescharacteristic, complemented
45#63554—31no—
46#63554—31no—
47#63554—61no—
48C22×C3×C3⋊D3C2×C411yescharacteristic
49C22×C3×C3⋊D3C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C2×C3×C3⋊Dic3C2×C411yescharacteristic
53C2×C3×C3⋊Dic3C2×C411yescharacteristic
54C2×C3×C3⋊Dic3C2×C411yescharacteristic
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C3×C4×C3⋊D3—21no—
59#6295—22no—
60C3⋊(C3⋊(C3⋊C8))D412yescomplemented
61C32×C3⋊C8—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64#4836—91no—
65#4730—91no—
66#4728—91no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—61no—
75C2×#626—31no—
76C2×#626—31no—
77C2×#625—31no—
78C2×C32×C8—31no—
79C2×C32×C8—31no—
80C2×C32×C8—31no—
81C4×C3⋊Dic3C2×D311yescharacteristic
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—121no—
89#3743—62no—
90C8×C3⋊D3—62no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—61no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103#3724—31no—
104Dic3⋊Dic3—31no—
105Dic3⋊Dic3—31no—
106#3722—31no—
107#3722—31no—
108(C2×D3)⋊Dic3—31no—
109(C2×D3)⋊Dic3—31no—
110(C2×D3)⋊Dic3—31no—
111(C2×D3)⋊Dic3—31no—
112#3711—62no—
113#3711—62no—
114#3711—64no—
115D3×C3⋊C8—62no—
116D3×C3⋊C8—62no—
117D3×C3⋊C8—64no—
118C22×C33C22×C411yescharacteristic
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3—21no—
123C2×C3×C3⋊D3—21no—
124C2×C3×C3⋊D3M1612yescomplemented
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3—21no—
131C4×#67—91no—
132C4×#67—91no—
133C2×#208—91no—
134C22×C3⋊C8—91no—
135C2×ΓL(1,𝔽25)—91no—
136C2×ΓL(1,𝔽25)—91no—
137C2×ΓL(1,𝔽25)—181no—
138C2×D3×C8—91no—
139C2×D3×C8—91no—
140C2×D3×C8—181no—
141C4×D12—91no—
142C3×C4⋊C8—91no—
143C3×C22⋊C8—91no—
144C3×C22⋊C8—91no—
145(C2×D3)⋊C8—31no—
146(C2×D3)⋊C8—31no—
147(C2×D3)⋊C8—31no—
148(C2×D3)⋊C8—31no—
149#815—91no—
150#815—91no—
151#815—181no—
152Dic3⋊C8—31no—
153Dic3⋊C8—31no—
154#803—91no—
155C22×C3⋊D3C4×D311yescharacteristic
156C22×C3⋊D3C4×D311yescharacteristic
157C22×C3×D3—31no—
158C22×C3×D3—31no—
159C22×C3×D3—31no—
160C22×C3×D3—31no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3C4×D311yescharacteristic
171C2×C3⋊Dic3C4×D311yescharacteristic
172C2×C3⋊Dic3C4×D311yescharacteristic
173C2×C3⋊Dic3C4×D311yescharacteristic
174C4×C3⋊D3—21no—
175C4×C3⋊D3—21no—
176C4×C3⋊D3—21no—
177C4×C3⋊D3—21no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C3×C4×D3—62no—
191C3×C4×D3—62no—
192C3×C4×D3—62no—
193C3×C4×D3—62no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197C3×C4×D3—61no—
198C3×C4×D3—61no—
199C3×C4×D3—61no—
200C3×C4×D3—61no—
201C3×C4×D3—61no—
202#626—62no—
203#626—62no—
204#625—62no—
205C32×C8—61no—
206C32×C8—61no—
207C32×C8—61no—
208C3⋊(C3⋊C8)—32no—
209C3⋊(C3⋊C8)—32no—
210C3⋊(C3⋊C8)—32no—
211C3⋊(C3⋊C8)—62no—
212C3⋊(C3⋊C8)—62no—
213C3⋊(C3⋊C8)—62no—
214C3⋊(C3⋊C8)—122no—
215C3×C3⋊C8—22no—
216C3×C3⋊C8—22no—
217C3×C3⋊C8—22no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224#435—271noSylow
225C2×C33C4×D411yescharacteristic
226C2×C33C2×M1611yescharacteristic
227C2×C33#13111yescharacteristic
228C3×C3⋊D3—22no—
229C3×C3⋊D3—41no—
230C22×C3×C4—91no—
231C22×C3×C4—91no—
232C2×#67—91no—
233C2×#67—91no—
234C2×D12—91no—
235C2×C4×D3—31no—
236C2×C4×D3—31no—
237C2×C4×D3—31no—
238C2×C4×D3—31no—
239C2×C4×D3—61no—
240C2×C4×D3—61no—
241C2×C3×C8—31no—
242C2×C3×C8—31no—
243C2×C3×C8—31no—
244C2×C3×C8—31no—
245C2×C3×C8—31no—
246C2×C3×C8—31no—
247C2×C3×C8—61no—
248C2×C3×C8—61no—
249C2×C3×C8—61no—
250C3×C42—91no—
251#217—91no—
252#217—91no—
253(C2×D3)⋊C4—91no—
254(C2×D3)⋊C4—91no—
255(C2×D3)⋊C4—91no—
256(C2×D3)⋊C4—91no—
257C4⋊Dic3—91no—
258Dic3⋊C4—91no—
259Dic3⋊C4—91no—
260C4×Dic3—31no—
261C4×Dic3—31no—
262C4×Dic3—61no—
263#208—182no—
264C2×C3⋊C8—91no—
265C2×C3⋊C8—91no—
266C2×C3⋊C8—91no—
267C2×C3⋊C8—181no—
268C2×C3⋊C8—181no—
269C2×C3⋊C8—181no—
270C2×C3⋊C8—182no—
271C2×C3⋊C8—361no—
272C2×C3⋊C8D3211yescharacteristic
273C2×C3⋊C8D3211yescharacteristic
274C2×C3⋊C8D3211yescharacteristic
275ΓL(1,𝔽25)—182no—
276ΓL(1,𝔽25)—182no—
277ΓL(1,𝔽25)—184no—
278D3×C8—182no—
279D3×C8—182no—
280D3×C8—184no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32C2×C4×D311yescharacteristic
286C22×C32C2×C4×D311yescharacteristic
287C22×C32C2×C4×D311yescharacteristic
288C2×C3⋊D3—21no—
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3⋊D3—21no—
293C2×C3⋊D3ΓL(1,𝔽25)12yescomplemented
294C2×C3×D3—32no—
295C2×C3×D3—32no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—62no—
304C2×C3×D3—61no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4D3×D411yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4D3×D411yescharacteristic
323C32×C4#23711yescharacteristic
324C32×C4#23711yescharacteristic
325C32×C4#23911yescharacteristic
326C3⋊Dic3—21no—
327C3⋊Dic3—21no—
328C3⋊Dic3—21no—
329C3⋊Dic3—21no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3—121no—
341C3×Dic3—121no—
342C2×M16—271no—
343C22×C8—271no—
344C4×D4—271no—
345C4⋊C8—91no—
346C22⋊C8—91no—
347C22⋊C8—91no—
348#97—271no—
349C33#43511yescharacteristic, complemented, Sylow
350C23×C3—91no—
351C23×C3—91no—
352C22×D3—31no—
353C22×D3—31no—
354C22×D3—31no—
355C22×D3—31no—
356C22×D3—61no—
357C22×D3—61no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—91no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4—181no—
369C2×C3×C4—181no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3211yescharacteristic
373#67—182no—
374#67—182no—
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—31no—
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387D12—182no—
388C4×D3—62no—
389C4×D3—62no—
390C4×D3—62no—
391C4×D3—62no—
392C4×D3—61no—
393C4×D3—61no—
394C4×D3—61no—
395C4×D3—61no—
396C4×D3—61no—
397C4×D3—61no—
398C4×D3—61no—
399C4×D3—61no—
400C3×C8—62no—
401C3×C8—62no—
402C3×C8—62no—
403C3×C8—61no—
404C3×C8—61no—
405C3×C8—61no—
406C3×C8—61no—
407C3×C8—61no—
408C3×C8—61no—
409C3⋊C8—21no—
410C3⋊C8—21no—
411C3⋊C8—21no—
412C3⋊C8—92no—
413C3⋊C8—92no—
414C3⋊C8—92no—
415C3⋊C8—182no—
416C3⋊C8—182no—
417C3⋊C8—182no—
418C3⋊C8—362no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32#67⋊C411yescharacteristic
432C2×C32#67⋊C411yescharacteristic
433C2×C32#89311yescharacteristic
434C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
435C2×C32#90111yescharacteristic
436C2×C32#90111yescharacteristic
437C2×C32D3×M1611yescharacteristic
438C2×C32D3×M1611yescharacteristic
439C2×C32#90711yescharacteristic
440C3⋊D3—22no—
441C3⋊D3—41no—
442C3×D3—64no—
443C3×D3—62no—
444C3×D3—62no—
445C3×D3—122no—
446C3×D3—121no—
447C3×D3—121no—
448C2×D4—271no—
449C22×C4—91no—
450C22×C4—91no—
451M16—542no—
452C2×C8—31no—
453C2×C8—31no—
454C2×C8—31no—
455C2×C8—271no—
456C2×C8—542no—
457C4⋊C4—271no—
458C22⋊C4—271no—
459C22⋊C4—271no—
460C42—91no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—41no—
465C22×C3—92no—
466C22×C3—181no—
467C22×C3—181no—
468C22×C3—181no—
469C22×C3—181no—
470C22×C3—181no—
471C22×C3C4×D3211yescharacteristic
472C22×C3C4×D3211yescharacteristic
473C22×C3C4×D3211yescharacteristic
474C2×D3—32no—
475C2×D3—32no—
476C2×D3—62no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—61no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—62no—
484C2×D3—61no—
485C2×D3—61no—
486C2×D3—61no—
487C2×D3—61no—
488C2×D3—61no—
489C2×D3—61no—
490C2×D3—121no—
491C2×D3—121no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—181no—
501C3×C4—181no—
502C3×C4—181no—
503C3×C4—181no—
504C3×C4#379611yescharacteristic
505C3×C4#379611yescharacteristic
506C3×C4#380211yescharacteristic
507C3×C4#380511yescharacteristic
508C3×C4#381111yescharacteristic
509C3×C4#381111yescharacteristic
510Dic3—62no—
511Dic3—62no—
512Dic3—61no—
513Dic3—61no—
514Dic3—61no—
515Dic3—61no—
516Dic3—61no—
517Dic3—61no—
518Dic3—61no—
519Dic3—61no—
520Dic3—121no—
521Dic3—121no—
522C32—21no—
523C32—21no—
524C32—21no—
525C32—41no—
526C32#472811yescharacteristic, complemented
527C32#473011yescharacteristic, complemented
528C32#483611yescharacteristic, complemented
529C23—91no—
530C23—91no—
531D4—542no—
532C2×C4—91no—
533C2×C4—91no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4D3311yescharacteristic
541C8—61no—
542C8—61no—
543C8—61no—
544C8—272no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—41no—
555C2×C3—41no—
556C2×C3—41no—
557C2×C3—182no—
558C2×C3—361no—
559C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
560C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
561C2×C3#6379111yescharacteristic
562C2×C3#6380311yescharacteristic
563C2×C3#6380411yescharacteristic
564C2×C3#6380411yescharacteristic
565C2×C3#6388111yescharacteristic
566C2×C3#6388111yescharacteristic
567C2×C3#6390211yescharacteristic
568D3—64no—
569D3—62no—
570D3—62no—
571D3—122no—
572D3—121no—
573D3—121no—
574C22—92no—
575C22—181no—
576C22—181no—
577C22—181no—
578C22—181no—
579C22—181no—
580C22#8894911yescharacteristic
581C4—181no—
582C4—181no—
583C4—181no—
584C4—181no—
585C4#8895411yescharacteristic
586C4#8896411yescharacteristic
587C3—21no—
588C3—21no—
589C3—21no—
590C3—41no—
591C3#1059072511yescharacteristic, complemented
592C3#1059073711yescharacteristic, complemented
593C3#1059090711yescharacteristic, complemented
594C2—182no—
595C2—361no—
596C2#1172282011yescharacteristic
597C2#1172282611yescharacteristic
598C2#1172293811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×#6295C4
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88787C2semidirect
3. C2×C3⋊D3×C3⋊C8C2semidirect
4. #11722020C2semidirect
5. C3×#63572C2semidirect
6. C3×#63557C2semidirect
7. C3×#63557C2semidirect
42. #63572D3semidirect
43. #63557D3semidirect
44. #63557D3semidirect
60. C3⋊(C3⋊(C3⋊C8))D4semidirect
124. C2×C3×C3⋊D3M16semidirect
293. C2×C3⋊D3ΓL(1,𝔽25)semidirect
349. C33#435semidirect
526. C32#4728semidirect
527. C32#4730semidirect
528. C32#4836semidirect
591. C3#10590725semidirect
592. C3#10590737semidirect
593. C3#10590907semidirect

Maximal subgroups

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

subgroupindex
C4×#62952
C2×#887872
C2×C3⋊D3×C3⋊C82
#117220202
C3×#635722
C3×#635572
#105909073
#105907373
#105907253