Finite Group Databasebeta

← #13140231

Subgroups — #13140231

Order 1728.

783 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6196C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#88450C211yesmaximal, characteristic
4C2×#88450C211yesmaximal, characteristic
5#11720574C211yesmaximal, characteristic, complemented
6#63658⋊C3C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591293—31nomaximal
9#10591205—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
17C2×#6196—22no—
18C2×#6196—22no—
19C2×#6196—22no—
20#3752⋊C3—22no—
21#88450—22no—
22#88450—22no—
23C22×#627—31no—
24C22×#627—31no—
25C2×#3752D311yescharacteristic
26C2×C3×(C2×D3)⋊C4—31no—
27C2×#3722—31no—
28C2×#3722—31no—
29C2×(C2×D3)⋊Dic3—31no—
30C2×(C2×D3)⋊Dic3—31no—
31#63658D311yescharacteristic, complemented
32C3×C3⋊#95—31no—
33C3×C3⋊#95—31no—
34C3×C3⋊#95—31no—
35C3×C3⋊#95—61no—
36#63579—31no—
37#63579—31no—
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)⋊C3Q812yescomplemented
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×#1084D412yes—
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)⋊C3—21no—
62C2×(C3⋊Dic3)⋊C3D412yes—
63C2×(C3⋊Dic3)⋊C3D412yes—
64#6196—28no—
65#4947—91no—
66#4673—91no—
67#4673—91no—
68C23×C3⋊D3C2×D311yescharacteristic
69C23×C3×D3—31no—
70C22×C32×C4—21no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C3⋊Dic3C2×D311yescharacteristic
74C22×C3⋊Dic3C2×D311yescharacteristic
75C22×C3⋊Dic3C2×D311yescharacteristic
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—31no—
83C22×C3×Dic3—61no—
84C22×C3×Dic3—61no—
85C2×#627—62no—
86C2×#627—62no—
87C2×#627—62no—
88C2×#627—62no—
89C2×#627—62no—
90C2×#627—62no—
91#3752—22no—
92C3×(C2×D3)⋊C4—62no—
93#3722—62no—
94#3722—62no—
95(C2×D3)⋊Dic3—62no—
96(C2×D3)⋊Dic3—62no—
97C22×H(3,𝔽3)C2×D411yescharacteristic
98C22×H(3,𝔽3)C2×D411yescharacteristic
99C22×H(3,𝔽3)C2×D411yescharacteristic
100C22×H(3,𝔽3)C2×Q811yescharacteristic
101C22×H(3,𝔽3)#4111yescharacteristic
102C22×H(3,𝔽3)#4111yescharacteristic
103C22×H(3,𝔽3)#4111yescharacteristic
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3—41no—
109C2×(C3⋊D3)⋊C3—41no—
110C2×(C3⋊D3)⋊C3—41no—
111C4×H(3,𝔽3)—41no—
112#1084—22no—
113#1084—41no—
114(C3⋊Dic3)⋊C3—22no—
115(C3⋊Dic3)⋊C3—22no—
116(C3⋊Dic3)⋊C3—41no—
117C22×Dic6—91no—
118C22×Dic6—91no—
119C22×Dic6—91no—
120C2×C3×C22⋊C4—91no—
121C2×#217—91no—
122C2×#217—91no—
123C2×(C2×D3)⋊C4—31no—
124C2×(C2×D3)⋊C4—91no—
125C2×(C2×D3)⋊C4—91no—
126C2×(C2×D3)⋊C4—91no—
127C2×(C2×D3)⋊C4—91no—
128C2×(C2×D3)⋊C4—91no—
129C3×#95—91no—
130C3×#95—91no—
131C3⋊#95—31no—
132C3⋊#95—31no—
133C3⋊#95—31no—
134C3⋊#95—61no—
135C3⋊#95—91no—
136C3⋊#95—91no—
137C3⋊#95—91no—
138C3⋊#95—91no—
139C23×C32—21no—
140C23×C32C22×D311yescharacteristic
141C23×C32C22×D311yescharacteristic
142C22×C3⋊D3—21no—
143C22×C3⋊D3—21no—
144C22×C3⋊D3—21no—
145C22×C3⋊D3—21no—
146C22×C3⋊D3—21no—
147C22×C3⋊D3—21no—
148C22×C3⋊D3Dic612yescomplemented
149C22×C3×D3—32no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C2×C32×C4—22no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3D1212yes—
173C2×C3⋊Dic3#6712yes—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
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—62no—
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—61no—
197C2×C3×Dic3—121no—
198C2×C3×Dic3—121no—
199C2×C3×Dic3—121no—
200C2×C3×Dic3—121no—
201#627—68no—
202#627—68no—
203#393—271noSylow
204C2×H(3,𝔽3)#12011yescharacteristic
205C2×H(3,𝔽3)C22⋊Q811yescharacteristic
206C2×H(3,𝔽3)C22⋊Q811yescharacteristic
207C2×H(3,𝔽3)C22⋊Q811yescharacteristic
208C2×H(3,𝔽3)#12411yescharacteristic
209C2×H(3,𝔽3)#12411yescharacteristic
210C2×H(3,𝔽3)#12411yescharacteristic
211(C3⋊D3)⋊C3—42no—
212C24×C3—91no—
213C23×D3—31no—
214C23×D3—91no—
215C22×C3×C4—31no—
216C22×C3×C4—31no—
217C22×C3×C4—61no—
218C22×C3×C4—91no—
219C22×C3×C4—91no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C22×C3×C4D3211yescharacteristic
224C22×Dic3—31no—
225C22×Dic3—31no—
226C22×Dic3—31no—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—31no—
230C22×Dic3—31no—
231C22×Dic3—61no—
232C22×Dic3—61no—
233C22×Dic3—91no—
234C22×Dic3—91no—
235C22×Dic3—91no—
236C2×Dic6—182no—
237C2×Dic6—182no—
238C2×Dic6—182no—
239C2×Dic6—182no—
240C2×Dic6—182no—
241C2×Dic6—182no—
242C2×Dic6—182no—
243C2×Dic6—182no—
244C2×Dic6—182no—
245C3×C22⋊C4—182no—
246#217—182no—
247#217—182no—
248(C2×D3)⋊C4—62no—
249(C2×D3)⋊C4—182no—
250(C2×D3)⋊C4—182no—
251(C2×D3)⋊C4—182no—
252(C2×D3)⋊C4—182no—
253(C2×D3)⋊C4—182no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32C2×Dic611yescharacteristic
262C22×C32C2×D1211yescharacteristic
263C22×C32#23511yescharacteristic
264C22×C32#23511yescharacteristic
265C22×C32#23511yescharacteristic
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32D3×D411yescharacteristic
269C22×C32D3×D411yescharacteristic
270C22×C32#23711yescharacteristic
271C22×C32#23711yescharacteristic
272C22×C32#23711yescharacteristic
273C22×C32D3×Q811yescharacteristic
274C22×C32C2×#6711yescharacteristic
275C2×C3⋊D3—22no—
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—41no—
279C2×C3⋊D3—41no—
280C2×C3⋊D3—41no—
281C2×C3⋊D3—41no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C32×C4—42no—
290C32×C4—41no—
291C32×C4—41no—
292C3⋊Dic3—22no—
293C3⋊Dic3—22no—
294C3⋊Dic3—41no—
295C3×Dic3—62no—
296C3×Dic3—62no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—122no—
300C3×Dic3—122no—
301C3×Dic3—121no—
302C3×Dic3—121no—
303C3×Dic3—121no—
304C22×Q8—271no—
305C2×C22⋊C4—91no—
306C2×C22⋊C4—271no—
307C2×C22⋊C4—271no—
308#95—91no—
309#95—91no—
310#95—271no—
311H(3,𝔽3)#39311yescharacteristic, complemented, Sylow
312C23×C3—31no—
313C23×C3—31no—
314C23×C3—61no—
315C23×C3—92no—
316C23×C3—181no—
317C23×C3—181no—
318C23×C3—181no—
319C23×C3—181no—
320C23×C3—181no—
321C23×C3—181no—
322C23×C3C2×D3211yescharacteristic
323C22×D3—32no—
324C22×D3—61no—
325C22×D3—61no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—61no—
329C22×D3—61no—
330C22×D3—92no—
331C22×D3—181no—
332C22×D3—181no—
333C22×D3—181no—
334C22×D3—181no—
335C22×D3—181no—
336C22×D3—181no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—62no—
347C2×C3×C4—92no—
348C2×C3×C4—92no—
349C2×C3×C4—92no—
350C2×C3×C4—121no—
351C2×C3×C4—121no—
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×C3×C4—181no—
362C2×C3×C4—181no—
363C2×C3×C4—181no—
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—62no—
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—61no—
383C2×Dic3—62no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—92no—
388C2×Dic3—92no—
389C2×Dic3—121no—
390C2×Dic3—121no—
391C2×Dic3—121no—
392C2×Dic3—121no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—181no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400Dic6—188no—
401Dic6—188no—
402Dic6—188no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32#87511yescharacteristic
411C2×C32#87811yescharacteristic
412C2×C32#87811yescharacteristic
413C2×C32#88211yescharacteristic
414C2×C32#88511yescharacteristic
415C2×C32#88511yescharacteristic
416C2×C32#88511yescharacteristic
417C2×C32#88511yescharacteristic
418C2×C32#89611yescharacteristic
419C2×C32#89611yescharacteristic
420C2×C32#89711yescharacteristic
421C2×C32#92411yescharacteristic
422C2×C32#93611yescharacteristic
423C2×C32#93711yescharacteristic
424C3⋊D3—42no—
425C3×D3—122no—
426C24—91no—
427C2×Q8—542no—
428C2×Q8—542no—
429C2×Q8—542no—
430C22×C4—91no—
431C22×C4—91no—
432C22×C4—91no—
433C22×C4—91no—
434C22×C4—91no—
435C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
436C22⋊C4—182no—
437C22⋊C4—542no—
438C22⋊C4—542no—
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—31no—
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—31no—
451C22×C3—31no—
452C22×C3—31no—
453C22×C3—61no—
454C22×C3—61no—
455C22×C3—61no—
456C22×C3—61no—
457C22×C3—61no—
458C22×C3—61no—
459C22×C3—61no—
460C22×C3—182no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3—361no—
464C22×C3—361no—
465C22×C3—361no—
466C22×C3—361no—
467C22×C3D3×Dic611yescharacteristic
468C22×C3#379811yescharacteristic
469C22×C3D3×D1211yescharacteristic
470C22×C3#380211yescharacteristic
471C22×C3#380411yescharacteristic
472C22×C3#380511yescharacteristic
473C22×C3D3×#6711yescharacteristic
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—121no—
480C2×D3—121no—
481C2×D3—182no—
482C2×D3—182no—
483C2×D3—182no—
484C2×D3—361no—
485C2×D3—361no—
486C2×D3—361no—
487C2×D3—361no—
488C3×C4—41no—
489C3×C4—121no—
490C3×C4—121no—
491C3×C4—122no—
492C3×C4—182no—
493C3×C4—182no—
494C3×C4—182no—
495C3×C4—361no—
496C3×C4—361no—
497Dic3—62no—
498Dic3—62no—
499Dic3—62no—
500Dic3—62no—
501Dic3—122no—
502Dic3—121no—
503Dic3—121no—
504Dic3—122no—
505Dic3—121no—
506Dic3—182no—
507Dic3—182no—
508Dic3—361no—
509C32—21no—
510C32#467311yescharacteristic, complemented
511C32#494711yescharacteristic, complemented
512C23—92no—
513C23—181no—
514C23—181no—
515C23—181no—
516C23—181no—
517C23—181no—
518C23—181no—
519C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
520Q8—548no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—92no—
525C2×C4—92no—
526C2×C4—92no—
527C2×C4—181no—
528C2×C4—181no—
529C2×C4—181no—
530C2×C4—181no—
531C2×C4—181no—
532C2×C4—181no—
533C2×C4—181no—
534C2×C4—181no—
535C2×C4—181no—
536C2×C4—181no—
537C2×C4—181no—
538C2×C4—181no—
539C2×C3—31no—
540C2×C3—31no—
541C2×C3—31no—
542C2×C3—31no—
543C2×C3—31no—
544C2×C3—31no—
545C2×C3—31no—
546C2×C3—31no—
547C2×C3—31no—
548C2×C3—31no—
549C2×C3—31no—
550C2×C3—31no—
551C2×C3—31no—
552C2×C3—31no—
553C2×C3—61no—
554C2×C3—61no—
555C2×C3—61no—
556C2×C3—61no—
557C2×C3—61no—
558C2×C3—61no—
559C2×C3—61no—
560C2×C3—362no—
561C2×C3#6385111yescharacteristic
562C2×C3#6385211yescharacteristic
563C2×C3#6389311yescharacteristic
564C2×C3#6389911yescharacteristic
565C2×C3#6391511yescharacteristic
566C2×C3#6392211yescharacteristic
567C2×C3#6395911yescharacteristic
568D3—122no—
569D3—362no—
570C22—182no—
571C22—182no—
572C22—182no—
573C22—361no—
574C22—361no—
575C22—361no—
576C22—361no—
577C22((C3⋊D3)⋊C3)⋊Q811yescharacteristic
578C22#8865011yescharacteristic
579C22#8865411yescharacteristic
580C22#8865511yescharacteristic
581C22#8867111yescharacteristic
582C22#8867211yescharacteristic
583C22#8867611yescharacteristic
584C4—41no—
585C4—182no—
586C4—182no—
587C4—182no—
588C4—361no—
589C4—361no—
590C3—31no—
591C3—31no—
592C3—61no—
593C3#1059129311yescharacteristic
594C2—362no—
595C2#1172110111yescharacteristic
596C2#1172110311yescharacteristic
597C2#1172113311yescharacteristic
598C2#1172113411yescharacteristic
599C2#1172114511yescharacteristic
600C2#1172116211yescharacteristic
601C2#1172130411yescharacteristic
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)#393
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
1. C22×#6196C2semidirect
5. #11720574C2semidirect
6. #63658⋊C3C2semidirect
7. #11720334C2semidirect
31. #63658D3semidirect
45. C22×(C3⋊D3)⋊C3Q8semidirect
148. C22×C3⋊D3Dic6semidirect
311. H(3,𝔽3)#393semidirect
510. C32#4673semidirect
511. C32#4947semidirect

Maximal subgroups

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

subgroupindex
C22×#61962
C2×#3752⋊C32
C2×#884502
#117205742
#63658⋊C32
#117203342
#105912933
#105912053