Finite Group Databasebeta

← #13140109

Subgroups — #13140109

Order 1728.

727 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88542C211yesmaximal, characteristic
2C2×#88451C211yesmaximal, characteristic
3C2×#88451C211yesmaximal, characteristic
4C2×#88448C211yesmaximal, characteristic, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#63658⋊C3C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591299—31nomaximal
9#10591296—31nomaximal
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#88542—22no—
18#88451—22no—
19#88451—22no—
20#88448—22no—
21C2×C3×(C2×D3)⋊C4—31no—
22C2×C3×(C2×D3)⋊C4—31no—
23C2×C3×(C2×D3)⋊C4—61no—
24C2×#3724—31no—
25C2×Dic3⋊Dic3—31no—
26C2×(C2×D3)⋊Dic3—31no—
27C2×(C2×D3)⋊Dic3—31no—
28C2×(C2×D3)⋊Dic3—31no—
29C2×(C2×D3)⋊Dic3—31no—
30#63658D311yescharacteristic, complemented
31#63658D311yescharacteristic, complemented
32C3×C3⋊#95—31no—
33C3×C3⋊#95—31no—
34#63579—31no—
35#63579—31no—
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×#272—21no—
38C22×#272—21no—
39C22×#272—21no—
40C22×#272—21no—
41C22×#272—21no—
42C22×#272—21no—
43C22×#272D412yes—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×#1084—21no—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3—21no—
53C2×(C3⋊Dic3)⋊C3—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)⋊C3—21no—
61C2×(C3⋊Dic3)⋊C3D412yes—
62#4952—91no—
63#4675—91no—
64#4674—91no—
65C23×C3×D3—31no—
66C23×C3×D3—31no—
67C23×C3×D3—61no—
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⋊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—61no—
82C3×(C2×D3)⋊C4—62no—
83C3×(C2×D3)⋊C4—62no—
84C3×(C2×D3)⋊C4—64no—
85#3724—62no—
86Dic3⋊Dic3—62no—
87(C2×D3)⋊Dic3—62no—
88(C2×D3)⋊Dic3—62no—
89(C2×D3)⋊Dic3—62no—
90(C2×D3)⋊Dic3—62no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C22×H(3,𝔽3)#4111yescharacteristic
94C22×H(3,𝔽3)#4111yescharacteristic
95C22×H(3,𝔽3)#4111yescharacteristic
96C22×H(3,𝔽3)#4111yescharacteristic
97C22×H(3,𝔽3)#4111yescharacteristic
98C2×#272—22no—
99C2×#272—22no—
100C2×#272—22no—
101C2×#272—41no—
102C2×#272—41no—
103C2×#272—41no—
104C2×#272—41no—
105C4×H(3,𝔽3)—41no—
106#1084—41no—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3—41no—
109(C3⋊Dic3)⋊C3—41no—
110(C3⋊Dic3)⋊C3—41no—
111C2×C3×C22⋊C4—91no—
112C2×#217—91no—
113C2×#217—91no—
114C2×(C2×D3)⋊C4—31no—
115C2×(C2×D3)⋊C4—31no—
116C2×(C2×D3)⋊C4—61no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×(C2×D3)⋊C4—91no—
121C2×C4⋊Dic3—91no—
122C2×Dic3⋊C4—91no—
123C2×Dic3⋊C4—91no—
124C3×#95—91no—
125C3×#95—91no—
126C3⋊#95—31no—
127C3⋊#95—31no—
128C3⋊#95—91no—
129C3⋊#95—91no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C3⋊#95—91no—
133C23×C32—21no—
134C23×C32C22×D311yescharacteristic
135C23×C32C22×D311yescharacteristic
136C22×C3×D3—32no—
137C22×C3×D3—32no—
138C22×C3×D3—62no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C22×C3×D3—61no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C22×C3×D3—121no—
154C22×C3×D3—121no—
155C22×C3×D3—121no—
156C22×C3×D3—121no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—41no—
165C2×C32×C4—41no—
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⋊Dic3—21no—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3D1212yes—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—62no—
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—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—121no—
198C2×C3×Dic3—121no—
199#397—271noSylow
200C2×H(3,𝔽3)#12111yescharacteristic
201C2×H(3,𝔽3)#12311yescharacteristic
202C2×H(3,𝔽3)#12311yescharacteristic
203C2×H(3,𝔽3)#12311yescharacteristic
204C2×H(3,𝔽3)#12411yescharacteristic
205C2×H(3,𝔽3)#12611yescharacteristic
206C2×H(3,𝔽3)#12611yescharacteristic
207#272—42no—
208C24×C3—91no—
209C23×D3—31no—
210C23×D3—31no—
211C23×D3—61no—
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—61no—
228C22×Dic3—91no—
229C22×Dic3—91no—
230C22×Dic3—91no—
231C22×Dic3—91no—
232C3×C22⋊C4—182no—
233#217—182no—
234#217—182no—
235(C2×D3)⋊C4—62no—
236(C2×D3)⋊C4—62no—
237(C2×D3)⋊C4—64no—
238(C2×D3)⋊C4—182no—
239(C2×D3)⋊C4—182no—
240(C2×D3)⋊C4—182no—
241(C2×D3)⋊C4—182no—
242C4⋊Dic3—182no—
243Dic3⋊C4—182no—
244Dic3⋊C4—182no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32C2×D1211yescharacteristic
253C22×C32#23511yescharacteristic
254C22×C32#23511yescharacteristic
255C22×C32#23511yescharacteristic
256C22×C32#23511yescharacteristic
257C22×C32#23511yescharacteristic
258C22×C32D3×D411yescharacteristic
259C22×C32D3×D411yescharacteristic
260C22×C32D3×D411yescharacteristic
261C22×C32#23711yescharacteristic
262C22×C32#23711yescharacteristic
263C22×C32#23711yescharacteristic
264C22×C32#23911yescharacteristic
265C22×C32#23911yescharacteristic
266C2×C3×D3—64no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—122no—
274C2×C3×D3—122no—
275C2×C3×D3—122no—
276C2×C3×D3—122no—
277C2×C3×D3—122no—
278C2×C3×D3—122no—
279C2×C3×D3—121no—
280C2×C3×D3—121no—
281C2×C3×D3—121no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C32×C4—42no—
288C32×C4—41no—
289C32×C4—41no—
290C3⋊Dic3—22no—
291C3⋊Dic3—41no—
292C3⋊Dic3—41no—
293C3⋊Dic3—41no—
294C3×Dic3—62no—
295C3×Dic3—122no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—121no—
301C2×C4⋊C4—271no—
302C2×C22⋊C4—91no—
303C2×C22⋊C4—271no—
304C2×C22⋊C4—271no—
305#95—91no—
306#95—91no—
307#95—271no—
308H(3,𝔽3)#39711yescharacteristic, 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—32no—
322C22×D3—62no—
323C22×D3—61no—
324C22×D3—61no—
325C22×D3—61no—
326C22×D3—61no—
327C22×D3—61no—
328C22×D3—61no—
329C22×D3—61no—
330C22×D3—61no—
331C22×D3—61no—
332C22×D3—61no—
333C22×D3—61no—
334C22×D3—61no—
335C22×D3—62no—
336C22×D3—62no—
337C22×D3—121no—
338C22×D3—121no—
339C22×D3—121no—
340C22×D3—121no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—62no—
351C2×C3×C4—92no—
352C2×C3×C4—121no—
353C2×C3×C4—121no—
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×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×Dic3—32no—
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—62no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—92no—
388C2×Dic3—121no—
389C2×Dic3—121no—
390C2×Dic3—181no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—181no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—181no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32#87511yescharacteristic
409C2×C32#87611yescharacteristic
410C2×C32#88311yescharacteristic
411C2×C32#88311yescharacteristic
412C2×C32#88311yescharacteristic
413C2×C32#88411yescharacteristic
414C2×C32#88511yescharacteristic
415C2×C32#88611yescharacteristic
416C2×C32#89411yescharacteristic
417C2×C32#89411yescharacteristic
418C2×C32#89511yescharacteristic
419C2×C32#89811yescharacteristic
420C2×C32#89811yescharacteristic
421C2×C32#89811yescharacteristic
422C3×D3—124no—
423C3×D3—122no—
424C3×D3—122no—
425C24—91no—
426C22×C4—91no—
427C22×C4—91no—
428C22×C4—91no—
429C22×C4—91no—
430C22×C4—91no—
431C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C4⋊C4—542no—
433C22⋊C4—182no—
434C22⋊C4—542no—
435C22⋊C4—542no—
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—31no—
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—61no—
455C22×C3—61no—
456C22×C3—61no—
457C22×C3—182no—
458C22×C3—182no—
459C22×C3—182no—
460C22×C3—361no—
461C22×C3—361no—
462C22×C3—361no—
463C22×C3—361no—
464C22×C3#379611yescharacteristic
465C22×C3#379811yescharacteristic
466C22×C3#379911yescharacteristic
467C22×C3D3×D1211yescharacteristic
468C22×C3#380411yescharacteristic
469C22×C3#380411yescharacteristic
470C22×C3#381111yescharacteristic
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—62no—
477C2×D3—64no—
478C2×D3—122no—
479C2×D3—122no—
480C2×D3—121no—
481C2×D3—121no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—122no—
489C2×D3—122no—
490C2×D3—122no—
491C2×D3—122no—
492C3×C4—41no—
493C3×C4—121no—
494C3×C4—121no—
495C3×C4—122no—
496C3×C4—182no—
497C3×C4—361no—
498C3×C4—361no—
499C3×C4—361no—
500C3×C4—361no—
501Dic3—62no—
502Dic3—121no—
503Dic3—121no—
504Dic3—122no—
505Dic3—121no—
506Dic3—121no—
507Dic3—121no—
508Dic3—182no—
509Dic3—361no—
510Dic3—361no—
511Dic3—361no—
512C32—21no—
513C32#467411yescharacteristic, complemented
514C32#467511yescharacteristic, complemented
515C23—92no—
516C23—181no—
517C23—181no—
518C23—181no—
519C23—181no—
520C23—181no—
521C23—181no—
522C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
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×C4—181no—
540C2×C4—181no—
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—31no—
554C2×C3—31no—
555C2×C3—61no—
556C2×C3—61no—
557C2×C3—61no—
558C2×C3—61no—
559C2×C3—61no—
560C2×C3—61no—
561C2×C3—61no—
562C2×C3—362no—
563C2×C3#6384811yescharacteristic
564C2×C3#6385211yescharacteristic
565C2×C3#6385311yescharacteristic
566C2×C3#6386811yescharacteristic
567C2×C3#6389011yescharacteristic
568C2×C3#6391011yescharacteristic
569C2×C3#6397511yescharacteristic
570D3—122no—
571D3—122no—
572D3—124no—
573C22—182no—
574C22—182no—
575C22—182no—
576C22—361no—
577C22—361no—
578C22—361no—
579C22—361no—
580C22#8864911yescharacteristic
581C22#8865011yescharacteristic
582C22#8865211yescharacteristic
583C22#8865411yescharacteristic
584C22#8867111yescharacteristic
585C22#8867111yescharacteristic
586C22#8867711yescharacteristic
587C4—41no—
588C4—182no—
589C4—361no—
590C4—361no—
591C4—361no—
592C4—361no—
593C3—31no—
594C3—31no—
595C3—61no—
596C3#1059120811yescharacteristic
597C2—362no—
598C2#1172108811yescharacteristic
599C2#1172109211yescharacteristic
600C2#1172110311yescharacteristic
601C2#1172110411yescharacteristic
602C2#1172113011yescharacteristic
603C2#1172115711yescharacteristic
604C2#1172131911yescharacteristic
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×#272C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#397
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
4. C2×#88448C2semidirect
5. #63658⋊C3C2semidirect
6. #63658⋊C3C2semidirect
7. #11720334C2semidirect
30. #63658D3semidirect
31. #63658D3semidirect
308. H(3,𝔽3)#397semidirect
513. C32#4674semidirect
514. C32#4675semidirect

Maximal subgroups

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

subgroupindex
C2×#885422
C2×#884512
C2×#884482
#63658⋊C32
#117203342
#105912993
#105912963