Finite Group Databasebeta

← #13140168

Subgroups — #13140168

Order 1728.

741 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3752⋊C3C211yesmaximal, characteristic
2C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×#88450C211yesmaximal, characteristic
4C2×#88450C211yesmaximal, characteristic
5C2×#88448C211yesmaximal, characteristic, complemented
6#11720574C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591286—31nomaximal
9#10591202—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⋊C3C414yescomplemented
18C4×(C3⋊Dic3)⋊C3—22no—
19#88450—22no—
20#88450—22no—
21#88448—22no—
22C2×#3752D311yescharacteristic
23C2×C4×C3⋊Dic3D311yescharacteristic, complemented
24C2×C3×(C2×D3)⋊C4—31no—
25C2×C3×C4×Dic3—31no—
26C2×#3724—31no—
27C2×Dic3⋊Dic3—31no—
28C2×#3722—31no—
29C2×#3722—31no—
30C2×(C2×D3)⋊Dic3—31no—
31C2×(C2×D3)⋊Dic3—31no—
32C3×C3⋊#95—31no—
33C3×C3⋊#95—31no—
34C3×C3⋊#95—61no—
35#63579—31no—
36#63579—31no—
37C23×H(3,𝔽3)C2311yescharacteristic
38C22×(C3⋊D3)⋊C3—21no—
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)⋊C3C2×C412yes—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)C2×C412yes—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084—21no—
53C2×#1084—21no—
54C2×(C3⋊Dic3)⋊C3—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)⋊C3C2×C412yes—
61C2×(C3⋊Dic3)⋊C3D412yes—
62C2×(C3⋊Dic3)⋊C3D412yes—
63#4951—91no—
64#4940—91no—
65#4669—91no—
66C23×C3⋊D3C2×D311yescharacteristic
67C23×C3×D3—31no—
68C22×C32×C4—21no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3×Dic3—31no—
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—61no—
82C22×C3×Dic3—61no—
83#3752Dic314yescomplemented
84C4×C3⋊Dic3—22no—
85C3×(C2×D3)⋊C4—34no—
86C3×C4×Dic3—62no—
87#3724—62no—
88Dic3⋊Dic3—62no—
89#3722—62no—
90#3722—62no—
91(C2×D3)⋊Dic3—62no—
92(C2×D3)⋊Dic3—62no—
93C22×H(3,𝔽3)C22×C411yescharacteristic
94C22×H(3,𝔽3)C2×D411yescharacteristic
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C22×H(3,𝔽3)#4111yescharacteristic
97C22×H(3,𝔽3)#4111yescharacteristic
98C22×H(3,𝔽3)#4111yescharacteristic
99C22×H(3,𝔽3)#4111yescharacteristic
100C2×(C3⋊D3)⋊C3—22no—
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—41no—
104C2×(C3⋊D3)⋊C3—41no—
105C2×(C3⋊D3)⋊C3—41no—
106C2×(C3⋊D3)⋊C3—41no—
107C4×H(3,𝔽3)—22no—
108#1084—41no—
109#1084—41no—
110(C3⋊Dic3)⋊C3—22no—
111(C3⋊Dic3)⋊C3—22no—
112(C3⋊Dic3)⋊C3—22no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C42—91no—
115C2×#217—91no—
116C2×#217—91no—
117C2×(C2×D3)⋊C4—31no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×(C2×D3)⋊C4—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×C4⋊Dic3—91no—
124C2×Dic3⋊C4—91no—
125C2×Dic3⋊C4—91no—
126C2×C4×Dic3—31no—
127C2×C4×Dic3—91no—
128C3×#95—91no—
129C3⋊#95—31no—
130C3⋊#95—31no—
131C3⋊#95—61no—
132C3⋊#95—91no—
133C3⋊#95—91no—
134C3⋊#95—91no—
135C23×C32—21no—
136C23×C32C22×D311yescharacteristic
137C23×C32C22×D311yescharacteristic
138C22×C3⋊D3—21no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3—21no—
141C22×C3⋊D3—21no—
142C22×C3⋊D3—21no—
143C22×C3⋊D3—21no—
144C22×C3⋊D3C2×Dic312yes—
145C22×C3×D3—32no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—41no—
158C2×C32×C4—41no—
159C2×C32×C4C4×D312yes—
160C2×C32×C4C2×Dic312yes—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3D1212yes—
168C2×C3⋊Dic3C2×Dic312yes—
169C2×C3⋊Dic3#6712yes—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
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—121no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3—121no—
196C2×C3×Dic3—121no—
197#388—271noSylow
198C2×H(3,𝔽3)#11711yescharacteristic
199C2×H(3,𝔽3)C4×D411yescharacteristic
200C2×H(3,𝔽3)C4×D411yescharacteristic
201C2×H(3,𝔽3)#12111yescharacteristic
202C2×H(3,𝔽3)#12311yescharacteristic
203C2×H(3,𝔽3)#12411yescharacteristic
204C2×H(3,𝔽3)#12611yescharacteristic
205(C3⋊D3)⋊C3—42no—
206C24×C3—91no—
207C23×D3—31no—
208C23×D3—91no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—61no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C22×C3×C4—91no—
217C22×C3×C4D3211yescharacteristic
218C22×Dic3—31no—
219C22×Dic3—31no—
220C22×Dic3—31no—
221C22×Dic3—31no—
222C22×Dic3—31no—
223C22×Dic3—31no—
224C22×Dic3—31no—
225C22×Dic3—61no—
226C22×Dic3—61no—
227C22×Dic3—91no—
228C22×Dic3—91no—
229C22×Dic3—91no—
230C3×C22⋊C4—94no—
231C3×C42—182no—
232#217—182no—
233#217—182no—
234(C2×D3)⋊C4—34no—
235(C2×D3)⋊C4—94no—
236(C2×D3)⋊C4—182no—
237(C2×D3)⋊C4—182no—
238(C2×D3)⋊C4—182no—
239(C2×D3)⋊C4—182no—
240C4⋊Dic3—182no—
241Dic3⋊C4—182no—
242Dic3⋊C4—182no—
243C4×Dic3—62no—
244C4×Dic3—182no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32C2×C4×D311yescharacteristic
253C22×C32C2×D1211yescharacteristic
254C22×C32#23511yescharacteristic
255C22×C32#23511yescharacteristic
256C22×C32#23511yescharacteristic
257C22×C32#23511yescharacteristic
258C22×C32#23511yescharacteristic
259C22×C32#23511yescharacteristic
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32#23711yescharacteristic
263C22×C32#23711yescharacteristic
264C22×C32C22×Dic311yescharacteristic
265C22×C32C2×#6711yescharacteristic
266C2×C3⋊D3—22no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3—41no—
270C2×C3⋊D3—41no—
271C2×C3⋊D3—41no—
272C2×C3⋊D3—41no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C32×C4—22no—
281C32×C4—22no—
282C32×C4—42no—
283C3⋊Dic3—22no—
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3×Dic3—62no—
287C3×Dic3—62no—
288C3×Dic3—62no—
289C3×Dic3—122no—
290C3×Dic3—122no—
291C3×Dic3—121no—
292C3×Dic3—121no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295C2×C4⋊C4—271no—
296C2×C22⋊C4—91no—
297C2×C22⋊C4—271no—
298C2×C22⋊C4—271no—
299C2×C42—91no—
300#95—91no—
301#95—271no—
302H(3,𝔽3)#38811yescharacteristic, complemented, Sylow
303C23×C3—31no—
304C23×C3—31no—
305C23×C3—61no—
306C23×C3—92no—
307C23×C3—181no—
308C23×C3—181no—
309C23×C3—181no—
310C23×C3—181no—
311C23×C3—181no—
312C23×C3—181no—
313C23×C3C2×D3211yescharacteristic
314C22×D3—32no—
315C22×D3—61no—
316C22×D3—61no—
317C22×D3—61no—
318C22×D3—61no—
319C22×D3—61no—
320C22×D3—61no—
321C22×D3—92no—
322C22×D3—181no—
323C22×D3—181no—
324C22×D3—181no—
325C22×D3—181no—
326C22×D3—181no—
327C22×D3—181no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—32no—
331C2×C3×C4—32no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—62no—
337C2×C3×C4—92no—
338C2×C3×C4—92no—
339C2×C3×C4—92no—
340C2×C3×C4—121no—
341C2×C3×C4—121no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4D3×Dic312yes—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—32no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
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—62no—
377C2×Dic3—62no—
378C2×Dic3—92no—
379C2×Dic3—92no—
380C2×Dic3—92no—
381C2×Dic3—121no—
382C2×Dic3—121no—
383C2×Dic3—121no—
384C2×Dic3—121no—
385C2×Dic3—181no—
386C2×Dic3—181no—
387C2×Dic3—181no—
388C2×Dic3—181no—
389C2×Dic3—181no—
390C2×Dic3—181no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32#87211yescharacteristic
399C2×C32C4×D1211yescharacteristic
400C2×C32#87511yescharacteristic
401C2×C32#87611yescharacteristic
402C2×C32#87911yescharacteristic
403C2×C32#88311yescharacteristic
404C2×C32#88411yescharacteristic
405C2×C32#88511yescharacteristic
406C2×C32#92611yescharacteristic
407C2×C32C4×#6711yescharacteristic
408C2×C32#92911yescharacteristic
409C2×C32#93011yescharacteristic
410C2×C32D4×Dic311yescharacteristic
411C2×C32D4×Dic311yescharacteristic
412C3⋊D3—42no—
413C3×D3—122no—
414C24—91no—
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—91no—
419C22×C4—91no—
420C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
421C4⋊C4—542no—
422C22⋊C4—94no—
423C22⋊C4—542no—
424C22⋊C4—542no—
425C42—182no—
426C22×C3—31no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—31no—
430C22×C3—31no—
431C22×C3—31no—
432C22×C3—31no—
433C22×C3—31no—
434C22×C3—31no—
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—31no—
440C22×C3—61no—
441C22×C3—61no—
442C22×C3—61no—
443C22×C3—61no—
444C22×C3—61no—
445C22×C3—61no—
446C22×C3—61no—
447C22×C3—182no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3—361no—
451C22×C3—361no—
452C22×C3—361no—
453C22×C3—361no—
454C22×C3#379511yescharacteristic
455C22×C3#379811yescharacteristic
456C22×C3#379811yescharacteristic
457C22×C3D3×D1211yescharacteristic
458C22×C3C2×D3×Dic311yescharacteristic
459C22×C3#380411yescharacteristic
460C22×C3D3×#6711yescharacteristic
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—121no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—182no—
469C2×D3—182no—
470C2×D3—182no—
471C2×D3—361no—
472C2×D3—361no—
473C2×D3—361no—
474C2×D3—361no—
475C3×C4—22no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—122no—
479C3×C4—182no—
480C3×C4—182no—
481C3×C4—182no—
482C3×C4—361no—
483C3×C4—361no—
484Dic3—62no—
485Dic3—62no—
486Dic3—62no—
487Dic3—121no—
488Dic3—121no—
489Dic3—121no—
490Dic3—121no—
491Dic3—122no—
492Dic3—122no—
493Dic3—182no—
494Dic3—182no—
495Dic3—182no—
496C32—21no—
497C32#494011yescharacteristic, complemented
498C32#495111yescharacteristic, complemented
499C23—92no—
500C23—181no—
501C23—181no—
502C23—181no—
503C23—181no—
504C23—181no—
505C23—181no—
506C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—92no—
510C2×C4—92no—
511C2×C4—92no—
512C2×C4—181no—
513C2×C4—181no—
514C2×C4—181no—
515C2×C4—181no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C4—181no—
519C2×C4—181no—
520C2×C4—181no—
521C2×C4—181no—
522C2×C4—181no—
523C2×C4—181no—
524C2×C4((C3⋊D3)⋊C3)⋊C412yes—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—31no—
535C2×C3—31no—
536C2×C3—31no—
537C2×C3—31no—
538C2×C3—31no—
539C2×C3—61no—
540C2×C3—61no—
541C2×C3—61no—
542C2×C3—61no—
543C2×C3—61no—
544C2×C3—61no—
545C2×C3—61no—
546C2×C3—362no—
547C2×C3#6385211yescharacteristic
548C2×C3#6385511yescharacteristic
549C2×C3#6385911yescharacteristic
550C2×C3Dic3×D1211yescharacteristic
551C2×C3#6390511yescharacteristic
552C2×C3#6390611yescharacteristic
553C2×C3Dic3×#6711yescharacteristic
554D3—122no—
555D3—362no—
556C22—182no—
557C22—182no—
558C22—182no—
559C22—361no—
560C22—361no—
561C22—361no—
562C22—361no—
563C22#8864811yescharacteristic
564C22#8865011yescharacteristic
565C22#8865011yescharacteristic
566C22#8865411yescharacteristic
567C22C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
568C22#8867111yescharacteristic
569C22#8867611yescharacteristic
570C4—22no—
571C4—182no—
572C4—182no—
573C4—182no—
574C4—361no—
575C4—361no—
576C3—31no—
577C3—31no—
578C3—61no—
579C3#1059130511yescharacteristic
580C2—362no—
581C2#1172110311yescharacteristic
582C2#1172110511yescharacteristic
583C2#1172110911yescharacteristic
584C2#1172113211yescharacteristic
585C2#1172113911yescharacteristic
586C2#1172114011yescharacteristic
587C2#1172131411yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC22×H(3,𝔽3)C22×C4
Frattini, socleC23×C3C2×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#388
LCSG ▷ C22×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×C4×(C3⋊Dic3)⋊C3C2semidirect
5. C2×#88448C2semidirect
6. #11720574C2semidirect
7. #11720334C2semidirect
17. #3752⋊C3C4semidirect
23. C2×C4×C3⋊Dic3D3semidirect
83. #3752Dic3semidirect
302. H(3,𝔽3)#388semidirect
497. C32#4940semidirect
498. C32#4951semidirect

Maximal subgroups

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

subgroupindex
C2×#3752⋊C32
C2×C4×(C3⋊Dic3)⋊C32
C2×#884502
C2×#884482
#117205742
#117203342
#105912863
#105912023