Finite Group Databasebeta

← #13155594

Subgroups — #13155594

Order 1728.

690 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
1C2×#88787C211yesmaximal, characteristic, complemented
2C2×#88786C211yesmaximal, characteristic, complemented
3C3×C4×#626C211yesmaximal, characteristic
4#11722072C211yesmaximal, characteristic, complemented
5Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591067—31nomaximal
9#10590732—31nomaximal
10#10590339—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×#626C411yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88787—22no—
20#88786—22no—
21C3×Dic3⋊Dic3C411yescharacteristic
22C3×#3722C411yescharacteristic
23C3×(C2×D3)⋊Dic3C411yescharacteristic
24C2×#3748—31no—
25C2×#3743—31no—
26C3×C4×#67—31no—
27C3×C4×D12—31no—
28C4×#626D311yescharacteristic
29C2×#3711—31no—
30C2×#3711—31no—
31C2×#3711—31no—
32C2×#3711—31no—
33C2×#3711—61no—
34C2×#3711—61no—
35C4×C3⋊(C3⋊C8)—31no—
36C3×#830—31no—
37C3×(C2×D3)⋊C8—31no—
38C3×Dic3⋊C8—31no—
39C3×C4⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
41#63557—31no—
42(C2×D3)⋊(C3⋊C8)—31no—
43(C2×D3)⋊(C3⋊C8)—31no—
44(C2×D3)⋊(C3⋊C8)D311yescharacteristic, complemented
45Dic3×C3⋊C8—31no—
46Dic3×C3⋊C8—31no—
47Dic3×C3⋊C8—61no—
48C22×C3×C3⋊D3C2×C411yescharacteristic
49C22×C32×D3C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C2×C32×Dic3C2×C411yescharacteristic
55C2×C32×Dic3C2×C411yescharacteristic
56C2×C32×Dic3C2×C411yescharacteristic
57C32×C4×D3—21no—
58C32×C4×D3—21no—
59C3×#626—22no—
60C3⋊(C3⋊(C3⋊C8))D412yescomplemented
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5012—91no—
65#4838—91no—
66#4729—91no—
67C22×C32×C4—31no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×#67—31no—
70C2×C3×D12—31no—
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—31no—
74C2×C3×C4×D3—61no—
75C2×C3×C4×D3C2×D311yescharacteristic
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×#626Dic311yescharacteristic
78C2×C32×C8—31no—
79C32×C42—31no—
80#3748—62no—
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—61no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—121no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90#3743—62no—
91C3×#217—31no—
92C3×(C2×D3)⋊C4—31no—
93C3×(C2×D3)⋊C4—31no—
94C3×(C2×D3)⋊C4—31no—
95C3×C4⋊Dic3—31no—
96C3×Dic3⋊C4—31no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109Dic3⋊Dic3Dic311yescharacteristic
110#3722Dic311yescharacteristic
111(C2×D3)⋊Dic3Dic311yescharacteristic
112#3711—62no—
113#3711—62no—
114#3711—64no—
115#3711—62no—
116#3711—62no—
117#3711—64no—
118C22×C33C22×C411yescharacteristic
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C32×D3—21no—
123C2×C32×D3—21no—
124C2×C32×D3—21no—
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—21no—
128C32×Dic3—21no—
129C32×Dic3—21no—
130C32×Dic3M1612yescomplemented
131C3×C4×D4—91no—
132C4×#67—31no—
133C2×#208—91no—
134C2×#208—91no—
135C2×#208—91no—
136C2×#208—181no—
137C2×ΓL(1,𝔽25)—91no—
138C2×ΓL(1,𝔽25)—91no—
139C2×ΓL(1,𝔽25)—91no—
140C2×ΓL(1,𝔽25)—181no—
141C4×D12—31no—
142C3×C4⋊C8—91no—
143C3×C22⋊C8—91no—
144#830—31no—
145#830—91no—
146(C2×D3)⋊C8—31no—
147(C2×D3)⋊C8—91no—
148(C2×D3)⋊C8—91no—
149Dic3⋊C8—31no—
150C8×Dic3—91no—
151C4⋊(C3⋊C8)—31no—
152C4×C3⋊C8—91no—
153C4×C3⋊C8—91no—
154C4×C3⋊C8—181no—
155C23×C32—31no—
156C22×C3⋊D3C2×Dic311yescharacteristic
157C22×C3×D3—21no—
158C22×C3×D3—31no—
159C22×C3×D3—31no—
160C22×C3×D3—61no—
161C22×C3×D3C4×D311yescharacteristic
162C22×C3×D3C2×Dic311yescharacteristic
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—31no—
167C2×C32×C4—31no—
168C2×C32×C4—31no—
169C2×C32×C4—41no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3C2×Dic311yescharacteristic
176C4×C3⋊D3—21no—
177C4×C3⋊D3—21no—
178C3×#67—62no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3C4×D311yescharacteristic
186C2×C3×Dic3C4×D311yescharacteristic
187C2×C3×Dic3C4×D311yescharacteristic
188C2×C3×Dic3C2×Dic311yescharacteristic
189C2×C3×Dic3C2×Dic311yescharacteristic
190C2×C3×Dic3C2×Dic311yescharacteristic
191C3×D12—62no—
192C3×C4×D3—22no—
193C3×C4×D3—22no—
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—21no—
197C3×C4×D3—21no—
198C3×C4×D3—62no—
199C3×C4×D3—62no—
200C3×C4×D3—61no—
201C3×C4×D3—61no—
202C3×C4×D3—61no—
203C3×C4×D3—61no—
204#626—22no—
205C32×C8—61no—
206C3⋊(C3⋊C8)—21no—
207C3⋊(C3⋊C8)—21no—
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—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—62no—
219C3×C3⋊C8—62no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224#436—271noSylow
225C2×C33C4×D411yescharacteristic
226C2×C33C2×M1611yescharacteristic
227C2×C33#13111yescharacteristic
228C3×C3⋊D3—41no—
229C32×D3—41no—
230C2×C3×D4—91no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—61no—
234C22×C3×C4—91no—
235C2×#67—31no—
236C2×D12—31no—
237C2×C4×D3—31no—
238C2×C4×D3—31no—
239C2×C4×D3—61no—
240C2×C4×D3D3211yescharacteristic
241C2×C3×C8—31no—
242C2×C3×C8—31no—
243C2×C3×C8—61no—
244C2×C3×C8—91no—
245C2×C3×C8—91no—
246C3×C4⋊C4—91no—
247C3×C22⋊C4—91no—
248C3×C22⋊C4—91no—
249C3×C42—31no—
250C3×C42—31no—
251C3×C42—61no—
252#217—31no—
253(C2×D3)⋊C4—31no—
254(C2×D3)⋊C4—31no—
255(C2×D3)⋊C4—31no—
256C4⋊Dic3—31no—
257Dic3⋊C4—31no—
258C4×Dic3D3211yescharacteristic
259#208—182no—
260#208—182no—
261#208—182no—
262#208—184no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—61no—
269C2×C3⋊C8—91no—
270C2×C3⋊C8—91no—
271C2×C3⋊C8—91no—
272C2×C3⋊C8—181no—
273C2×C3⋊C8—181no—
274C2×C3⋊C8—181no—
275C2×C3⋊C8—361no—
276C2×C3⋊C8D3211yescharacteristic
277ΓL(1,𝔽25)—182no—
278ΓL(1,𝔽25)—182no—
279ΓL(1,𝔽25)—182no—
280ΓL(1,𝔽25)—184no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32—61no—
286C22×C32—61no—
287C22×C32—61no—
288C22×C32C2×C4×D311yescharacteristic
289C22×C32C2×C4×D311yescharacteristic
290C22×C32C22×Dic311yescharacteristic
291C2×C3⋊D3—21no—
292C2×C3⋊D3—21no—
293C2×C3⋊D3—21no—
294C2×C3×D3—22no—
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D3—41no—
302C2×C3×D3—41no—
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—32no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—61no—
322C32×C4—61no—
323C32×C4D3×D411yescharacteristic
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4#23711yescharacteristic
327C32×C4#23711yescharacteristic
328C32×C4#23911yescharacteristic
329C3⋊Dic3—21no—
330C3×Dic3—22no—
331C3×Dic3—22no—
332C3×Dic3—21no—
333C3×Dic3—21no—
334C3×Dic3—21no—
335C3×Dic3—21no—
336C3×Dic3—22no—
337C3×Dic3—62no—
338C3×Dic3—61no—
339C3×Dic3—61no—
340C3×Dic3ΓL(1,𝔽25)12yescomplemented
341C3×Dic3#20812yescomplemented
342C2×M16—271no—
343C2×M16—271no—
344C4×D4—91no—
345C4⋊C8—91no—
346C22⋊C8—91no—
347C22⋊C8—271no—
348C4×C8—271no—
349C33#43611yescharacteristic, complemented, Sylow
350C23×C3—31no—
351C23×C3—31no—
352C23×C3—61no—
353C23×C3—91no—
354C22×D3—31no—
355C22×D3—31no—
356C22×D3—61no—
357C22×D3D3×Dic311yescharacteristic
358C3×D4—182no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—31no—
368C2×C3×C4—41no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—62no—
373C2×C3×C4—62no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—91no—
379C2×C3×C4—181no—
380C2×C3×C4—181no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384#67—62no—
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—61no—
388C2×Dic3D3×Dic311yescharacteristic
389C2×Dic3D3×Dic311yescharacteristic
390C2×Dic3D3×Dic311yescharacteristic
391D12—62no—
392C4×D3—21no—
393C4×D3—21no—
394C4×D3—62no—
395C4×D3—62no—
396C4×D3—61no—
397C4×D3—61no—
398C4×D3—61no—
399C4×D3—61no—
400C3×C8—62no—
401C3×C8—61no—
402C3×C8—61no—
403C3×C8—181no—
404C3×C8—181no—
405C3⋊C8—21no—
406C3⋊C8—62no—
407C3⋊C8—62no—
408C3⋊C8—61no—
409C3⋊C8—61no—
410C3⋊C8—61no—
411C3⋊C8—61no—
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—121no—
432C2×C32#67⋊C411yescharacteristic
433C2×C32#89311yescharacteristic
434C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
435C2×C32#90111yescharacteristic
436C2×C32D3×M1611yescharacteristic
437C2×C32#90711yescharacteristic
438C2×C32C2×#20811yescharacteristic
439C2×C32D4×Dic311yescharacteristic
440C2×C32#94811yescharacteristic
441C3⋊D3—41no—
442C3×D3—42no—
443C3×D3—41no—
444C3×D3—41no—
445C3×D3—122no—
446C3×D3—121no—
447C3×D3—121no—
448C2×D4—91no—
449C22×C4—31no—
450C22×C4—91no—
451M16—542no—
452M16—542no—
453C2×C8—31no—
454C2×C8—91no—
455C2×C8—91no—
456C2×C8—271no—
457C4⋊C4—91no—
458C22⋊C4—91no—
459C22⋊C4—91no—
460C42—31no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—41no—
465C22×C3—62no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—61no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—181no—
475C22×C3—181no—
476C22×C3—181no—
477C22×C3C4×D3211yescharacteristic
478C22×C3C2×D3×Dic311yescharacteristic
479C22×C3C2×D3×Dic311yescharacteristic
480C2×D3—21no—
481C2×D3—21no—
482C2×D3—21no—
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—32no—
499C3×C4—32no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—62no—
503C3×C4—62no—
504C3×C4—62no—
505C3×C4—61no—
506C3×C4—61no—
507C3×C4—61no—
508C3×C4—61no—
509C3×C4—181no—
510C3×C4#379611yescharacteristic
511C3×C4#379611yescharacteristic
512C3×C4#380211yescharacteristic
513C3×C4#380511yescharacteristic
514C3×C4#381111yescharacteristic
515C3×C4#381111yescharacteristic
516Dic3—21no—
517Dic3—21no—
518Dic3—62no—
519Dic3—61no—
520Dic3—61no—
521Dic3#371112yescomplemented
522C32—21no—
523C32—21no—
524C32—21no—
525C32—41no—
526C32#472911yescharacteristic, complemented
527C32#483811yescharacteristic, complemented
528C32#501211yescharacteristic, complemented
529C23—31no—
530C23—91no—
531D4—182no—
532C2×C4—31no—
533C2×C4—31no—
534C2×C4—31no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—91no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4D3311yescharacteristic
541C8—61no—
542C8—181no—
543C8—181no—
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—122no—
558C2×C3—121no—
559C2×C3—121no—
560C2×C3—361no—
561C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
562C2×C3#6380411yescharacteristic
563C2×C3D3×#20811yescharacteristic
564C2×C3#6381411yescharacteristic
565C2×C3#6381511yescharacteristic
566C2×C3C2×#371111yescharacteristic
567C2×C3#6388111yescharacteristic
568C2×C3#6389811yescharacteristic
569C2×C3#67⋊Dic311yescharacteristic
570D3—41no—
571D3—122no—
572D3—121no—
573D3—121no—
574C22—61no—
575C22—61no—
576C22—61no—
577C22—181no—
578C22—181no—
579C22—181no—
580C22D32×Dic311yescharacteristic
581C4—32no—
582C4—61no—
583C4—61no—
584C4—181no—
585C4#8895411yescharacteristic
586C4#8896411yescharacteristic
587C3—21no—
588C3—21no—
589C3—21no—
590C3—41no—
591C3#1059033911yescharacteristic, complemented
592C3#1059073211yescharacteristic, complemented
593C3#1059106711yescharacteristic, complemented
594C2—121no—
595C2—361no—
596C2D3×#371111yescharacteristic
597C2#1172282211yescharacteristic
598C2#1172292911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC2×C3×#626C4
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88787C2semidirect
2. C2×#88786C2semidirect
4. #11722072C2semidirect
5. Dic3×C3⋊(C3⋊C8)C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
40. Dic3⋊(C3⋊C8)D3semidirect
44. (C2×D3)⋊(C3⋊C8)D3semidirect
60. C3⋊(C3⋊(C3⋊C8))D4semidirect
130. C32×Dic3M16semidirect
340. C3×Dic3ΓL(1,𝔽25)semidirect
341. C3×Dic3#208semidirect
349. C33#436semidirect
521. Dic3#3711semidirect
526. C32#4729semidirect
527. C32#4838semidirect
528. C32#5012semidirect
591. C3#10590339semidirect
592. C3#10590732semidirect
593. C3#10591067semidirect

Maximal subgroups

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

subgroupindex
C2×#887872
C2×#887862
C3×C4×#6262
#117220722
Dic3×C3⋊(C3⋊C8)2
C3×Dic3⋊(C3⋊C8)2
C3×(C2×D3)⋊(C3⋊C8)2
#105910673
#105907323
#105903393