Finite Group Databasebeta

← #13155588

Subgroups — #13155588

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88809C211yesmaximal, characteristic, complemented
2C4×#6292C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic, complemented
4#11722024C211yesmaximal, characteristic, complemented
5C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×#63557C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590739—31nomaximal
9#10590737—31nomaximal
10#10590328—31nomaximal
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6292C411yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88809—22no—
20Dic3⋊(C3⋊Dic3)C411yescharacteristic
21#88804C411yescharacteristic
22#88803C411yescharacteristic
23#88785—22no—
24C2×C8×C3⋊D3—31no—
25C4×C4⋊(C3⋊D3)—31no—
26C4×#626—31no—
27C4×#626—31no—
28C4×#626—61no—
29C2×#3712—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32C2×#3710—31no—
33C2×#3710—31no—
34C2×#3710—61no—
35#63647—31no—
36C3×(C2×D3)⋊C8—31no—
37C3×(C2×D3)⋊C8—31no—
38C3×#815—31no—
39C3×Dic3⋊C8—31no—
40C3×C4⋊(C3⋊C8)—31no—
41C3×#803—31no—
42Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
43#63557—31no—
44#63557—31no—
45#63557—61no—
46#63557D311yescharacteristic, complemented
47#63554D311yescharacteristic, complemented
48C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
49C22×C3×C3⋊D3C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
52C4×C3⋊(C3⋊D3)—21no—
53C4×C3⋊(C3⋊D3)—21no—
54C2×C3×C3⋊Dic3C2×C411yescharacteristic
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C2×C32×Dic3C2×C411yescharacteristic
58C2×C32×Dic3C2×C411yescharacteristic
59#6292—22no—
60C3×C3⋊(C3⋊C8)—21no—
61C3×C3⋊(C3⋊C8)D412yescomplemented
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64#4836—91no—
65#4728—91no—
66#4706—91no—
67C2×C4⋊(C3⋊D3)—31no—
68C2×C4×C3⋊D3—31no—
69C2×C4×C3⋊D3—31no—
70C2×C4×C3⋊D3—31no—
71C2×C4×C3⋊D3—61no—
72C2×C4×C3⋊D3—61no—
73C2×C4×C3⋊D3—61no—
74C2×C4×C3⋊D3—121no—
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×C4×D3—31no—
77C2×C3×C4×D3—31no—
78C2×C3×C4×D3—61no—
79C2×#626—31no—
80C2×#626—31no—
81C2×#626—61no—
82C2×C32×C8—31no—
83C2×C32×C8—31no—
84C32×C42—31no—
85#3752—31no—
86#3752—31no—
87C4⋊(C3⋊Dic3)—31no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C8×C3⋊D3—62no—
91C3×C4×Dic3—21no—
92C3×C4×Dic3C2×D311yescharacteristic
93C3×C4×Dic3C2×D311yescharacteristic
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106Dic3⋊Dic3—31no—
107Dic3⋊Dic3—31no—
108Dic3⋊Dic3—61no—
109#3722—31no—
110#3722—31no—
111#3722—61no—
112(C2×D3)⋊Dic3—31no—
113(C2×D3)⋊Dic3—31no—
114(C2×D3)⋊Dic3—61no—
115#3712—62no—
116#3711—62no—
117#3711—62no—
118#3710—62no—
119#3710—62no—
120#3710—64no—
121C22×C33C22×C411yescharacteristic
122C2×C3⋊(C3⋊D3)—21no—
123C2×C3⋊(C3⋊D3)—21no—
124C2×C3⋊(C3⋊D3)M1612yescomplemented
125C2×C3×C3⋊D3—21no—
126C2×C3×C3⋊D3—21no—
127C2×C3×C3⋊D3—21no—
128C33×C4C2×D411yescharacteristic
129C33×C4#4111yescharacteristic
130C3⋊(C3⋊Dic3)—21no—
131C3×C3⋊Dic3—21no—
132C32×Dic3—21no—
133C32×Dic3—21no—
134C4×#67—91no—
135C2×#208—91no—
136C2×ΓL(1,𝔽25)—91no—
137C2×ΓL(1,𝔽25)—91no—
138C2×D3×C8—91no—
139C2×D3×C8—91no—
140C2×D3×C8—91no—
141C2×D3×C8—181no—
142C4×D12—91no—
143C4×D12—91no—
144C4×D12—181no—
145C3×C4⋊C8—91no—
146C3×C22⋊C8—91no—
147C3×#97—91no—
148(C2×D3)⋊C8—31no—
149(C2×D3)⋊C8—31no—
150(C2×D3)⋊C8—91no—
151(C2×D3)⋊C8—91no—
152(C2×D3)⋊C8—91no—
153(C2×D3)⋊C8—181no—
154#815—31no—
155Dic3⋊C8—31no—
156C4⋊(C3⋊C8)—31no—
157#803—31no—
158C22×C3⋊D3—31no—
159C22×C3⋊D3—31no—
160C22×C3⋊D3—31no—
161C22×C3⋊D3—61no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—61no—
164C22×C3⋊D3—121no—
165C22×C3⋊D3C4×D311yescharacteristic
166C22×C3×D3—31no—
167C22×C3×D3—31no—
168C22×C3×D3—61no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—31no—
173C2×C32×C4—31no—
174C2×C32×C4—41no—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—61no—
183C2×C3⋊Dic3—61no—
184C2×C3⋊Dic3—121no—
185C2×C3⋊Dic3C4×D311yescharacteristic
186C4⋊(C3⋊D3)—62no—
187C4×C3⋊D3—21no—
188C4×C3⋊D3—21no—
189C4×C3⋊D3—61no—
190C4×C3⋊D3—61no—
191C4×C3⋊D3—62no—
192C4×C3⋊D3—62no—
193C4×C3⋊D3—61no—
194C4×C3⋊D3—61no—
195C4×C3⋊D3—61no—
196C4×C3⋊D3—61no—
197C4×C3⋊D3—121no—
198C4×C3⋊D3—121no—
199C4×C3⋊D3—121no—
200C4×C3⋊D3—121no—
201C4×C3⋊D3—122no—
202C4×C3⋊D3—122no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—21no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3C4×D311yescharacteristic
209C2×C3×Dic3C4×D311yescharacteristic
210C2×C3×Dic3C4×D311yescharacteristic
211C2×C3×Dic3C4×D311yescharacteristic
212C3×C4×D3—61no—
213C3×C4×D3—61no—
214C3×C4×D3—61no—
215C3×C4×D3—61no—
216C3×C4×D3—121no—
217C3×C4×D3—121no—
218#626—62no—
219#626—62no—
220#626—64no—
221C32×C8—61no—
222C32×C8—61no—
223C3⋊(C3⋊C8)—21no—
224C3⋊(C3⋊C8)D1212yescomplemented
225C3×C3⋊C8—22no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—21no—
229C3×C3⋊C8—21no—
230C3×C3⋊C8—32no—
231C3×C3⋊C8—32no—
232C3×C3⋊C8—41no—
233C3×C3⋊C8—62no—
234C3×C3⋊C8—61no—
235C3×C3⋊C8—61no—
236C3×C3⋊C8—121no—
237#435—271noSylow
238C2×C33C4×D411yescharacteristic
239C2×C33C2×M1611yescharacteristic
240C2×C33#13111yescharacteristic
241C3⋊(C3⋊D3)—22no—
242C3×C3⋊D3—41no—
243C22×C3×C4—91no—
244C2×#67—91no—
245C2×D12—91no—
246C2×D12—91no—
247C2×D12—181no—
248C2×C4×D3—31no—
249C2×C4×D3—31no—
250C2×C4×D3—61no—
251C2×C4×D3—91no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C2×C4×D3—181no—
255C2×C4×D3—181no—
256C2×C4×D3—181no—
257C2×C4×D3—361no—
258C2×C3×C8—31no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—31no—
262C2×C3×C8—61no—
263C2×C3×C8—61no—
264C2×C3×C8—91no—
265C2×C3×C8—91no—
266C3×C42—31no—
267C3×C42—31no—
268C3×C42—61no—
269#217—91no—
270(C2×D3)⋊C4—91no—
271(C2×D3)⋊C4—91no—
272(C2×D3)⋊C4—91no—
273(C2×D3)⋊C4—91no—
274(C2×D3)⋊C4—91no—
275(C2×D3)⋊C4—181no—
276(C2×D3)⋊C4—181no—
277C4⋊Dic3—91no—
278C4⋊Dic3—91no—
279C4⋊Dic3—181no—
280Dic3⋊C4—91no—
281C4×Dic3D3211yescharacteristic
282#208—182no—
283C2×C3⋊C8—31no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8—61no—
289C2×C3⋊C8D3211yescharacteristic
290C2×C3⋊C8D3211yescharacteristic
291ΓL(1,𝔽25)—182no—
292ΓL(1,𝔽25)—182no—
293D3×C8—182no—
294D3×C8—182no—
295D3×C8—182no—
296D3×C8—184no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—41no—
301C22×C32C2×C4×D311yescharacteristic
302C22×C32C2×C4×D311yescharacteristic
303C22×C32C2×C4×D311yescharacteristic
304C2×C3⋊D3—21no—
305C2×C3⋊D3—21no—
306C2×C3⋊D3—21no—
307C2×C3⋊D3—32no—
308C2×C3⋊D3—32no—
309C2×C3⋊D3—32no—
310C2×C3⋊D3—62no—
311C2×C3⋊D3—62no—
312C2×C3⋊D3—62no—
313C2×C3⋊D3—61no—
314C2×C3⋊D3—61no—
315C2×C3⋊D3—62no—
316C2×C3⋊D3—62no—
317C2×C3⋊D3—61no—
318C2×C3⋊D3—61no—
319C2×C3⋊D3—61no—
320C2×C3⋊D3—61no—
321C2×C3⋊D3—122no—
322C2×C3⋊D3—121no—
323C2×C3⋊D3—121no—
324C2×C3⋊D3—121no—
325C2×C3⋊D3—121no—
326C2×C3⋊D3—122no—
327C2×C3⋊D3—122no—
328C2×C3×D3—62no—
329C2×C3×D3—61no—
330C2×C3×D3—61no—
331C2×C3×D3—61no—
332C2×C3×D3—61no—
333C2×C3×D3—61no—
334C2×C3×D3—61no—
335C2×C3×D3—121no—
336C2×C3×D3—121no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—21no—
340C32×C4—21no—
341C32×C4—21no—
342C32×C4—21no—
343C32×C4—41no—
344C32×C4—41no—
345C32×C4—61no—
346C32×C4—61no—
347C32×C4C2×D1211yescharacteristic
348C32×C4#23511yescharacteristic
349C32×C4D3×D411yescharacteristic
350C32×C4D3×D411yescharacteristic
351C32×C4#23711yescharacteristic
352C32×C4#23911yescharacteristic
353C3⋊Dic3—21no—
354C3⋊Dic3—61no—
355C3⋊Dic3—62no—
356C3⋊Dic3—61no—
357C3⋊Dic3—61no—
358C3⋊Dic3—121no—
359C3⋊Dic3—121no—
360C3⋊Dic3—122no—
361C3×Dic3—22no—
362C3×Dic3—21no—
363C3×Dic3—21no—
364C3×Dic3—21no—
365C3×Dic3—21no—
366C3×Dic3—41no—
367C3×Dic3—61no—
368C3×Dic3—61no—
369C3×Dic3—121no—
370C2×M16—271no—
371C22×C8—271no—
372C4×D4—271no—
373C4⋊C8—91no—
374C22⋊C8—91no—
375C22⋊C8—271no—
376#97—91no—
377C33#43511yescharacteristic, complemented, Sylow
378C23×C3—91no—
379C22×D3—31no—
380C22×D3—31no—
381C22×D3—61no—
382C22×D3—91no—
383C22×D3—91no—
384C22×D3—91no—
385C22×D3—181no—
386C22×D3—181no—
387C22×D3—181no—
388C22×D3—361no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—31no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—41no—
397C2×C3×C4—61no—
398C2×C3×C4—61no—
399C2×C3×C4—91no—
400C2×C3×C4—181no—
401C2×C3×C4—181no—
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C2×D3211yescharacteristic
404C2×C3×C4C2×D3211yescharacteristic
405#67—182no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—61no—
409C2×Dic3—91no—
410C2×Dic3—91no—
411C2×Dic3—91no—
412C2×Dic3—181no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—361no—
416C2×Dic3#62411yescharacteristic
417C2×Dic3#62411yescharacteristic
418D12—182no—
419D12—182no—
420D12—184no—
421C4×D3—61no—
422C4×D3—61no—
423C4×D3—61no—
424C4×D3—61no—
425C4×D3—121no—
426C4×D3—121no—
427C4×D3—182no—
428C4×D3—182no—
429C4×D3—181no—
430C4×D3—181no—
431C4×D3—181no—
432C4×D3—181no—
433C4×D3—181no—
434C4×D3—181no—
435C4×D3—361no—
436C4×D3—361no—
437C4×D3—361no—
438C4×D3—361no—
439C4×D3—362no—
440C4×D3—362no—
441C3×C8—62no—
442C3×C8—61no—
443C3×C8—61no—
444C3×C8—61no—
445C3×C8—61no—
446C3×C8—92no—
447C3×C8—121no—
448C3×C8—181no—
449C3⋊C8—21no—
450C3⋊C8—21no—
451C3⋊C8—32no—
452C3⋊C8—32no—
453C3⋊C8—62no—
454C3⋊C8—61no—
455C3⋊C8—61no—
456C3⋊C8—121no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—21no—
460C2×C32—21no—
461C2×C32—21no—
462C2×C32—21no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—21no—
466C2×C32—41no—
467C2×C32—41no—
468C2×C32—41no—
469C2×C32C4×D1211yescharacteristic
470C2×C32#67⋊C411yescharacteristic
471C2×C32#89311yescharacteristic
472C2×C32#90111yescharacteristic
473C2×C32#90111yescharacteristic
474C2×C32D3×M1611yescharacteristic
475C2×C32D3×M1611yescharacteristic
476C2×C32D3×M1611yescharacteristic
477C2×C32#90711yescharacteristic
478C3⋊D3—41no—
479C3⋊D3—62no—
480C3⋊D3—64no—
481C3⋊D3—62no—
482C3⋊D3—62no—
483C3⋊D3—122no—
484C3⋊D3—122no—
485C3⋊D3—124no—
486C3×D3—122no—
487C3×D3—121no—
488C3×D3—121no—
489C2×D4—271no—
490C22×C4—91no—
491C22×C4—271no—
492M16—542no—
493C2×C8—31no—
494C2×C8—31no—
495C2×C8—91no—
496C2×C8—91no—
497C2×C8—542no—
498C4⋊C4—271no—
499C22⋊C4—271no—
500C22⋊C4—271no—
501C42—31no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—41no—
506C22×C3—181no—
507C22×C3—181no—
508C22×C3—181no—
509C22×C3C4×D3211yescharacteristic
510C22×C3C4×D3211yescharacteristic
511C22×C3C2×#62411yescharacteristic
512C2×D3—62no—
513C2×D3—61no—
514C2×D3—61no—
515C2×D3—61no—
516C2×D3—61no—
517C2×D3—61no—
518C2×D3—61no—
519C2×D3—92no—
520C2×D3—92no—
521C2×D3—92no—
522C2×D3—121no—
523C2×D3—121no—
524C2×D3—182no—
525C2×D3—182no—
526C2×D3—182no—
527C2×D3—182no—
528C2×D3—182no—
529C2×D3—181no—
530C2×D3—181no—
531C2×D3—181no—
532C2×D3—181no—
533C2×D3—181no—
534C2×D3—181no—
535C2×D3—362no—
536C2×D3—361no—
537C2×D3—361no—
538C2×D3—361no—
539C2×D3—361no—
540C2×D3—362no—
541C2×D3—362no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—21no—
545C3×C4—21no—
546C3×C4—21no—
547C3×C4—21no—
548C3×C4—41no—
549C3×C4—41no—
550C3×C4—62no—
551C3×C4—61no—
552C3×C4—61no—
553C3×C4—61no—
554C3×C4—61no—
555C3×C4—121no—
556C3×C4—181no—
557C3×C4#379611yescharacteristic
558C3×C4#379911yescharacteristic
559C3×C4D3×D1211yescharacteristic
560C3×C4D3×D1211yescharacteristic
561C3×C4#380411yescharacteristic
562C3×C4#381111yescharacteristic
563Dic3—21no—
564Dic3—21no—
565Dic3—61no—
566Dic3—61no—
567Dic3—121no—
568Dic3—182no—
569Dic3—181no—
570Dic3—181no—
571Dic3—181no—
572Dic3—361no—
573Dic3—361no—
574Dic3—362no—
575C32—21no—
576C32—21no—
577C32—21no—
578C32—41no—
579C32#470611yescharacteristic, complemented
580C32#472811yescharacteristic, complemented
581C32#483611yescharacteristic, complemented
582C23—91no—
583C23—271no—
584D4—542no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—91no—
588C2×C4—181no—
589C2×C4—181no—
590C2×C4—271no—
591C2×C4—541no—
592C2×C4—541no—
593C2×C4D3311yescharacteristic
594C8—61no—
595C8—61no—
596C8—92no—
597C8—181no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—41no—
608C2×C3—41no—
609C2×C3—41no—
610C2×C3—361no—
611C2×C3#6379111yescharacteristic
612C2×C3#6379111yescharacteristic
613C2×C3#6380311yescharacteristic
614C2×C3#6380411yescharacteristic
615C2×C3#6381711yescharacteristic
616C2×C3#6381811yescharacteristic
617C2×C3#6388011yescharacteristic
618C2×C3#6388111yescharacteristic
619C2×C3#6389411yescharacteristic
620D3—122no—
621D3—121no—
622D3—121no—
623D3—184no—
624D3—182no—
625D3—182no—
626D3—182no—
627D3—362no—
628D3—362no—
629D3—364no—
630C22—181no—
631C22—181no—
632C22—181no—
633C22—272no—
634C22—541no—
635C22—541no—
636C22D3×#62411yescharacteristic
637C4—61no—
638C4—61no—
639C4—181no—
640C4—541no—
641C4#8895511yescharacteristic
642C4#8896211yescharacteristic
643C3—21no—
644C3—21no—
645C3—21no—
646C3—41no—
647C3#1059032811yescharacteristic, complemented
648C3#1059073711yescharacteristic, complemented
649C3#1059073911yescharacteristic, complemented
650C2—361no—
651C2—542no—
652C2#1172281911yescharacteristic
653C2#1172282411yescharacteristic
654C2#1172293711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×#6292C4
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88809C2semidirect
3. C2×#88785C2semidirect
4. #11722024C2semidirect
5. C3×Dic3⋊(C3⋊C8)C2semidirect
6. C3×#63557C2semidirect
7. C3×#63554C2semidirect
42. Dic3⋊(C3⋊C8)D3semidirect
46. #63557D3semidirect
47. #63554D3semidirect
61. C3×C3⋊(C3⋊C8)D4semidirect
124. C2×C3⋊(C3⋊D3)M16semidirect
224. C3⋊(C3⋊C8)D12semidirect
377. C33#435semidirect
579. C32#4706semidirect
580. C32#4728semidirect
581. C32#4836semidirect
647. C3#10590328semidirect
648. C3#10590737semidirect
649. C3#10590739semidirect

Maximal subgroups

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

subgroupindex
C2×#888092
C4×#62922
C2×#887852
#117220242
C3×Dic3⋊(C3⋊C8)2
C3×#635572
C3×#635542
#105907393
#105907373
#105903283