Finite Group Databasebeta

← #13140175

Subgroups — #13140175

Order 1728.

727 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3752⋊C3C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic, complemented
3C2×#88450C211yesmaximal, characteristic
4C2×#88450C211yesmaximal, characteristic
5C2×#88448C211yesmaximal, characteristic, complemented
6C2×#88448C211yesmaximal, characteristic, complemented
7#11720574C211yesmaximal, characteristic, complemented
8#10591347—31nomaximal
9#10591309—31nomaximal
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#3752⋊C3—22no—
18#3750⋊C3—22no—
19#88450—22no—
20#88450—22no—
21#88448—22no—
22#88448—22no—
23C2×#3752D311yescharacteristic
24C2×#3750D311yescharacteristic, complemented
25C2×C3×(C2×D3)⋊C4—31no—
26C2×C3×Dic3⋊C4—31no—
27C2×#3724—31no—
28C2×#3724—31no—
29C2×Dic3⋊Dic3—31no—
30C2×Dic3⋊Dic3—31no—
31C2×#3722—31no—
32C2×#3722—31no—
33C2×(C2×D3)⋊Dic3—31no—
34C2×(C2×D3)⋊Dic3—31no—
35C3×C3⋊#95—31no—
36C3×C3⋊#95—31no—
37C3×C3⋊#95—61no—
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×(C3⋊D3)⋊C3—21no—
40C22×(C3⋊D3)⋊C3—21no—
41C22×(C3⋊D3)⋊C3—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—21no—
44C22×(C3⋊D3)⋊C3—21no—
45C22×(C3⋊D3)⋊C3Q812yes—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×C4×H(3,𝔽3)—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084—21no—
53C2×#1084—21no—
54C2×#1084—21no—
55C2×(C3⋊Dic3)⋊C3—21no—
56C2×(C3⋊Dic3)⋊C3—21no—
57C2×(C3⋊Dic3)⋊C3—21no—
58C2×(C3⋊Dic3)⋊C3—21no—
59C2×(C3⋊Dic3)⋊C3—21no—
60C2×(C3⋊Dic3)⋊C3—21no—
61C2×(C3⋊Dic3)⋊C3D412yes—
62C2×(C3⋊Dic3)⋊C3D412yes—
63C2×(C3⋊Dic3)⋊C3D412yes—
64#4994—91no—
65#4948—91no—
66#4676—91no—
67C23×C3⋊D3C2×D311yescharacteristic
68C23×C3×D3—31no—
69C22×C32×C4—21no—
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—61no—
83C22×C3×Dic3—61no—
84#3752—22no—
85#3750—22no—
86C3×(C2×D3)⋊C4—62no—
87C3×Dic3⋊C4—62no—
88#3724—62no—
89#3724—62no—
90Dic3⋊Dic3—62no—
91Dic3⋊Dic3—62no—
92#3722—62no—
93#3722—62no—
94(C2×D3)⋊Dic3—62no—
95(C2×D3)⋊Dic3—62no—
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C22×H(3,𝔽3)C2×Q811yescharacteristic
100C22×H(3,𝔽3)#4111yescharacteristic
101C22×H(3,𝔽3)#4111yescharacteristic
102C22×H(3,𝔽3)#4111yescharacteristic
103C2×(C3⋊D3)⋊C3—22no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3—41no—
109C2×(C3⋊D3)⋊C3—41no—
110C4×H(3,𝔽3)—41no—
111#1084—41no—
112#1084—41no—
113(C3⋊Dic3)⋊C3—22no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3—22no—
116C2×C3×C4⋊C4—91no—
117C2×C3×C22⋊C4—91no—
118C2×#217—91no—
119C2×#217—91no—
120C2×(C2×D3)⋊C4—31no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×(C2×D3)⋊C4—91no—
125C2×(C2×D3)⋊C4—91no—
126C2×C4⋊Dic3—91no—
127C2×C4⋊Dic3—91no—
128C2×Dic3⋊C4—31no—
129C2×Dic3⋊C4—91no—
130C2×Dic3⋊C4—91no—
131C2×Dic3⋊C4—91no—
132C2×Dic3⋊C4—91no—
133C2×Dic3⋊C4—91no—
134C3×#95—91no—
135C3⋊#95—31no—
136C3⋊#95—31no—
137C3⋊#95—61no—
138C23×C32—21no—
139C23×C32C22×D311yescharacteristic
140C23×C32C22×D311yescharacteristic
141C22×C3⋊D3—21no—
142C22×C3⋊D3—21no—
143C22×C3⋊D3—21no—
144C22×C3⋊D3—21no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3—21no—
147C22×C3⋊D3Dic612yes—
148C22×C3×D3—32no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—41no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3D1212yes—
171C2×C3⋊Dic3#6712yes—
172C2×C3⋊Dic3#6712yes—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
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—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—121no—
197C2×C3×Dic3—121no—
198C2×C3×Dic3—121no—
199C2×C3×Dic3—121no—
200#396—271noSylow
201C2×H(3,𝔽3)#12111yescharacteristic
202C2×H(3,𝔽3)#12111yescharacteristic
203C2×H(3,𝔽3)#12111yescharacteristic
204C2×H(3,𝔽3)C22⋊Q811yescharacteristic
205C2×H(3,𝔽3)C22⋊Q811yescharacteristic
206C2×H(3,𝔽3)C22⋊Q811yescharacteristic
207C2×H(3,𝔽3)#12611yescharacteristic
208(C3⋊D3)⋊C3—42no—
209C24×C3—91no—
210C23×D3—31no—
211C23×D3—91no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—61no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4—91no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C22×C3×C4D3211yescharacteristic
221C22×Dic3—31no—
222C22×Dic3—31no—
223C22×Dic3—31no—
224C22×Dic3—31no—
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—31no—
228C22×Dic3—61no—
229C22×Dic3—61no—
230C22×Dic3—91no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C3×C4⋊C4—182no—
234C3×C22⋊C4—182no—
235#217—182no—
236#217—182no—
237(C2×D3)⋊C4—62no—
238(C2×D3)⋊C4—182no—
239(C2×D3)⋊C4—182no—
240(C2×D3)⋊C4—182no—
241(C2×D3)⋊C4—182no—
242(C2×D3)⋊C4—182no—
243C4⋊Dic3—182no—
244C4⋊Dic3—182no—
245Dic3⋊C4—62no—
246Dic3⋊C4—182no—
247Dic3⋊C4—182no—
248Dic3⋊C4—182no—
249Dic3⋊C4—182no—
250Dic3⋊C4—182no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32C2×Dic611yescharacteristic
259C22×C32C2×D1211yescharacteristic
260C22×C32#23511yescharacteristic
261C22×C32#23511yescharacteristic
262C22×C32D3×D411yescharacteristic
263C22×C32D3×D411yescharacteristic
264C22×C32D3×D411yescharacteristic
265C22×C32#23711yescharacteristic
266C22×C32#23711yescharacteristic
267C22×C32#23711yescharacteristic
268C22×C32D3×Q811yescharacteristic
269C22×C32#23911yescharacteristic
270C22×C32C2×#6711yescharacteristic
271C22×C32C2×#6711yescharacteristic
272C2×C3⋊D3—22no—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3—41no—
276C2×C3⋊D3—41no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C32×C4—42no—
287C32×C4—41no—
288C32×C4—41no—
289C3⋊Dic3—22no—
290C3⋊Dic3—22no—
291C3⋊Dic3—22no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—122no—
296C3×Dic3—122no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—121no—
301C2×C4⋊C4—91no—
302C2×C4⋊C4—271no—
303C2×C4⋊C4—271no—
304C2×C22⋊C4—91no—
305C2×C22⋊C4—271no—
306C2×C22⋊C4—271no—
307#95—91no—
308H(3,𝔽3)#39611yescharacteristic, complemented, Sylow
309C23×C3—31no—
310C23×C3—31no—
311C23×C3—61no—
312C23×C3—92no—
313C23×C3—181no—
314C23×C3—181no—
315C23×C3—181no—
316C23×C3—181no—
317C23×C3—181no—
318C23×C3—181no—
319C23×C3C2×D3211yescharacteristic
320C22×D3—32no—
321C22×D3—61no—
322C22×D3—61no—
323C22×D3—61no—
324C22×D3—61no—
325C22×D3—61no—
326C22×D3—61no—
327C22×D3—92no—
328C22×D3—181no—
329C22×D3—181no—
330C22×D3—181no—
331C22×D3—181no—
332C22×D3—181no—
333C22×D3—181no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—62no—
344C2×C3×C4—92no—
345C2×C3×C4—92no—
346C2×C3×C4—92no—
347C2×C3×C4—121no—
348C2×C3×C4—121no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4—181no—
355C2×C3×C4—181no—
356C2×C3×C4—181no—
357C2×C3×C4—181no—
358C2×C3×C4—181no—
359C2×C3×C4—181no—
360C2×C3×C4—181no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—92no—
385C2×Dic3—92no—
386C2×Dic3—92no—
387C2×Dic3—121no—
388C2×Dic3—121no—
389C2×Dic3—121no—
390C2×Dic3—121no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—181no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32#87811yescharacteristic
405C2×C32#87811yescharacteristic
406C2×C32#87911yescharacteristic
407C2×C32#88411yescharacteristic
408C2×C32#89511yescharacteristic
409C2×C32#89611yescharacteristic
410C2×C32#89711yescharacteristic
411C2×C32#89811yescharacteristic
412C2×C32#92411yescharacteristic
413C2×C32#93011yescharacteristic
414C2×C32#93811yescharacteristic
415C2×C32#93911yescharacteristic
416C2×C32#93911yescharacteristic
417C2×C32#94611yescharacteristic
418C3⋊D3—42no—
419C3×D3—122no—
420C24—91no—
421C22×C4—91no—
422C22×C4—91no—
423C22×C4—91no—
424C22×C4—91no—
425C22×C4—91no—
426C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
427C4⋊C4—182no—
428C4⋊C4—542no—
429C4⋊C4—542no—
430C22⋊C4—182no—
431C22⋊C4—542no—
432C22⋊C4—542no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—31no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—182no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3—361no—
458C22×C3—361no—
459C22×C3—361no—
460C22×C3—361no—
461C22×C3D3×Dic611yescharacteristic
462C22×C3#379611yescharacteristic
463C22×C3#379811yescharacteristic
464C22×C3D3×D1211yescharacteristic
465C22×C3#380511yescharacteristic
466C22×C3D3×#6711yescharacteristic
467C22×C3D3×#6711yescharacteristic
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—121no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
475C2×D3—182no—
476C2×D3—182no—
477C2×D3—182no—
478C2×D3—361no—
479C2×D3—361no—
480C2×D3—361no—
481C2×D3—361no—
482C3×C4—41no—
483C3×C4—121no—
484C3×C4—121no—
485C3×C4—122no—
486C3×C4—182no—
487C3×C4—182no—
488C3×C4—182no—
489C3×C4—361no—
490C3×C4—361no—
491Dic3—62no—
492Dic3—62no—
493Dic3—62no—
494Dic3—121no—
495Dic3—121no—
496Dic3—121no—
497Dic3—121no—
498Dic3—122no—
499Dic3—122no—
500Dic3—182no—
501Dic3—182no—
502Dic3—182no—
503C32—21no—
504C32#494811yescharacteristic, complemented
505C32#499411yescharacteristic, complemented
506C23—92no—
507C23—181no—
508C23—181no—
509C23—181no—
510C23—181no—
511C23—181no—
512C23—181no—
513C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—92no—
518C2×C4—92no—
519C2×C4—92no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—181no—
523C2×C4—181no—
524C2×C4—181no—
525C2×C4—181no—
526C2×C4—181no—
527C2×C4—181no—
528C2×C4—181no—
529C2×C4—181no—
530C2×C4—181no—
531C2×C4—181no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—61no—
547C2×C3—61no—
548C2×C3—61no—
549C2×C3—61no—
550C2×C3—61no—
551C2×C3—61no—
552C2×C3—61no—
553C2×C3—362no—
554C2×C3#6385111yescharacteristic
555C2×C3#6386211yescharacteristic
556C2×C3#6388611yescharacteristic
557C2×C3#6388711yescharacteristic
558C2×C3#6389311yescharacteristic
559C2×C3#6390611yescharacteristic
560C2×C3#6397011yescharacteristic
561D3—122no—
562D3—362no—
563C22—182no—
564C22—182no—
565C22—182no—
566C22—361no—
567C22—361no—
568C22—361no—
569C22—361no—
570C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
571C22#8864911yescharacteristic
572C22#8865011yescharacteristic
573C22#8865411yescharacteristic
574C22#8867211yescharacteristic
575C22#8867611yescharacteristic
576C22#8867611yescharacteristic
577C4—41no—
578C4—182no—
579C4—182no—
580C4—182no—
581C4—361no—
582C4—361no—
583C3—31no—
584C3—31no—
585C3—61no—
586C3#1059130611yescharacteristic
587C2—362no—
588C2#1172110111yescharacteristic
589C2#1172111211yescharacteristic
590C2#1172112711yescharacteristic
591C2#1172112811yescharacteristic
592C2#1172113311yescharacteristic
593C2#1172114011yescharacteristic
594C2#1172131311yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC23×H(3,𝔽3)C23
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#396
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#3750⋊C3C2semidirect
5. C2×#88448C2semidirect
6. C2×#88448C2semidirect
7. #11720574C2semidirect
24. C2×#3750D3semidirect
308. H(3,𝔽3)#396semidirect
504. C32#4948semidirect
505. C32#4994semidirect

Maximal subgroups

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

subgroupindex
C2×#3752⋊C32
C2×#3750⋊C32
C2×#884502
C2×#884482
#117205742
#105913473
#105913093