Finite Group Databasebeta

← #13140117

Subgroups — #13140117

Order 1728.

671 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3752⋊C3C211yesmaximal, characteristic
2C2×#88450C211yesmaximal, characteristic
3C2×#88450C211yesmaximal, characteristic
4C2×#88448C211yesmaximal, characteristic, complemented
5#11720574C211yesmaximal, characteristic, complemented
6#63658⋊C3C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591297—31nomaximal
9#10591216—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#88450—22no—
19#88450—22no—
20#88448—22no—
21C2×#3752D311yescharacteristic
22C2×C3×(C2×D3)⋊C4—31no—
23C2×#3724—31no—
24C2×Dic3⋊Dic3—31no—
25C2×#3722—31no—
26C2×#3722—31no—
27C2×(C2×D3)⋊Dic3—31no—
28C2×(C2×D3)⋊Dic3—31no—
29#63658D311yescharacteristic, complemented
30C3×C3⋊#95—31no—
31C3×C3⋊#95—31no—
32C3×C3⋊#95—31no—
33C3×C3⋊#95—61no—
34#63579—31no—
35#63579—31no—
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
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)⋊C3D412yes—
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×#1084—21no—
51C2×#1084—21no—
52C2×#1084—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#4953—91no—
63#4675—91no—
64#4675—91no—
65C23×C3⋊D3C2×D311yescharacteristic
66C23×C3×D3—31no—
67C22×C32×C4—21no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3C2×D311yescharacteristic
71C22×C3⋊Dic3C2×D311yescharacteristic
72C22×C3⋊Dic3C2×D311yescharacteristic
73C22×C3×Dic3—31no—
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—61no—
81C22×C3×Dic3—61no—
82#3752—22no—
83C3×(C2×D3)⋊C4—62no—
84#3724—62no—
85Dic3⋊Dic3—62no—
86#3722—62no—
87#3722—62no—
88(C2×D3)⋊Dic3—62no—
89(C2×D3)⋊Dic3—62no—
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C22×H(3,𝔽3)#4111yescharacteristic
93C22×H(3,𝔽3)#4111yescharacteristic
94C22×H(3,𝔽3)#4111yescharacteristic
95C22×H(3,𝔽3)#4111yescharacteristic
96C22×H(3,𝔽3)#4111yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—22no—
100C2×(C3⋊D3)⋊C3—41no—
101C2×(C3⋊D3)⋊C3—41no—
102C2×(C3⋊D3)⋊C3—41no—
103C2×(C3⋊D3)⋊C3—41no—
104C4×H(3,𝔽3)—41no—
105#1084—41no—
106#1084—41no—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3—41no—
109(C3⋊Dic3)⋊C3—41no—
110C2×C3×C22⋊C4—91no—
111C2×#217—91no—
112C2×#217—91no—
113C2×(C2×D3)⋊C4—31no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—91no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×C4⋊Dic3—91no—
120C2×Dic3⋊C4—91no—
121C2×Dic3⋊C4—91no—
122C3×#95—91no—
123C3×#95—91no—
124C3⋊#95—31no—
125C3⋊#95—31no—
126C3⋊#95—31no—
127C3⋊#95—61no—
128C3⋊#95—91no—
129C3⋊#95—91no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C23×C32—21no—
133C23×C32C22×D311yescharacteristic
134C23×C32C22×D311yescharacteristic
135C22×C3⋊D3—21no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3—21no—
138C22×C3⋊D3—21no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3—21no—
141C22×C3⋊D3D1212yes—
142C22×C3×D3—32no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—61no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4—41no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3#6712yes—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—61no—
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—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—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—121no—
191C2×C3×Dic3—121no—
192C2×C3×Dic3—121no—
193C2×C3×Dic3—121no—
194#397—271noSylow
195C2×H(3,𝔽3)#12111yescharacteristic
196C2×H(3,𝔽3)#12311yescharacteristic
197C2×H(3,𝔽3)#12311yescharacteristic
198C2×H(3,𝔽3)#12311yescharacteristic
199C2×H(3,𝔽3)#12411yescharacteristic
200C2×H(3,𝔽3)#12611yescharacteristic
201C2×H(3,𝔽3)#12611yescharacteristic
202(C3⋊D3)⋊C3—42no—
203C24×C3—91no—
204C23×D3—31no—
205C23×D3—91no—
206C22×C3×C4—31no—
207C22×C3×C4—31no—
208C22×C3×C4—61no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4D3211yescharacteristic
215C22×Dic3—31no—
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—31no—
219C22×Dic3—31no—
220C22×Dic3—31no—
221C22×Dic3—31no—
222C22×Dic3—61no—
223C22×Dic3—61no—
224C22×Dic3—91no—
225C22×Dic3—91no—
226C22×Dic3—91no—
227C3×C22⋊C4—182no—
228#217—182no—
229#217—182no—
230(C2×D3)⋊C4—62no—
231(C2×D3)⋊C4—182no—
232(C2×D3)⋊C4—182no—
233(C2×D3)⋊C4—182no—
234(C2×D3)⋊C4—182no—
235(C2×D3)⋊C4—182no—
236C4⋊Dic3—182no—
237Dic3⋊C4—182no—
238Dic3⋊C4—182no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32C2×D1211yescharacteristic
247C22×C32#23511yescharacteristic
248C22×C32#23511yescharacteristic
249C22×C32#23511yescharacteristic
250C22×C32#23511yescharacteristic
251C22×C32D3×D411yescharacteristic
252C22×C32D3×D411yescharacteristic
253C22×C32#23711yescharacteristic
254C22×C32#23711yescharacteristic
255C22×C32#23711yescharacteristic
256C22×C32#23711yescharacteristic
257C22×C32#23711yescharacteristic
258C22×C32#23911yescharacteristic
259C22×C32C2×#6711yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3—22no—
262C2×C3⋊D3—22no—
263C2×C3⋊D3—41no—
264C2×C3⋊D3—41no—
265C2×C3⋊D3—41no—
266C2×C3⋊D3—41no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—121no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C32×C4—42no—
275C32×C4—41no—
276C32×C4—41no—
277C3⋊Dic3—22no—
278C3⋊Dic3—41no—
279C3⋊Dic3—41no—
280C3×Dic3—62no—
281C3×Dic3—122no—
282C3×Dic3—122no—
283C3×Dic3—121no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—121no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289C2×C4⋊C4—271no—
290C2×C22⋊C4—91no—
291C2×C22⋊C4—271no—
292C2×C22⋊C4—271no—
293#95—91no—
294#95—91no—
295#95—271no—
296H(3,𝔽3)#39711yescharacteristic, complemented, Sylow
297C23×C3—31no—
298C23×C3—31no—
299C23×C3—61no—
300C23×C3—92no—
301C23×C3—181no—
302C23×C3—181no—
303C23×C3—181no—
304C23×C3—181no—
305C23×C3—181no—
306C23×C3—181no—
307C23×C3C2×D3211yescharacteristic
308C22×D3—32no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—61no—
313C22×D3—61no—
314C22×D3—61no—
315C22×D3—92no—
316C22×D3—181no—
317C22×D3—181no—
318C22×D3—181no—
319C22×D3—181no—
320C22×D3—181no—
321C22×D3—181no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—62no—
332C2×C3×C4—92no—
333C2×C3×C4—121no—
334C2×C3×C4—121no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4—181no—
339C2×C3×C4—181no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
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×Dic3—32no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—62no—
365C2×Dic3—62no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—92no—
373C2×Dic3—121no—
374C2×Dic3—121no—
375C2×Dic3—121no—
376C2×Dic3—121no—
377C2×Dic3—181no—
378C2×Dic3—181no—
379C2×Dic3—181no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383C2×Dic3—181no—
384C2×Dic3—181no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32#87611yescharacteristic
393C2×C32#87911yescharacteristic
394C2×C32#87911yescharacteristic
395C2×C32#88311yescharacteristic
396C2×C32#88411yescharacteristic
397C2×C32#88511yescharacteristic
398C2×C32#88611yescharacteristic
399C2×C32#88611yescharacteristic
400C2×C32#89411yescharacteristic
401C2×C32#89411yescharacteristic
402C2×C32#89811yescharacteristic
403C2×C32#93011yescharacteristic
404C2×C32#93511yescharacteristic
405C2×C32#93611yescharacteristic
406C3⋊D3—42no—
407C3×D3—122no—
408C24—91no—
409C22×C4—91no—
410C22×C4—91no—
411C22×C4—91no—
412C22×C4—91no—
413C22×C4—91no—
414C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
415C4⋊C4—542no—
416C22⋊C4—182no—
417C22⋊C4—542no—
418C22⋊C4—542no—
419C22×C3—31no—
420C22×C3—31no—
421C22×C3—31no—
422C22×C3—31no—
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—31no—
428C22×C3—31no—
429C22×C3—31no—
430C22×C3—31no—
431C22×C3—31no—
432C22×C3—31no—
433C22×C3—61no—
434C22×C3—61no—
435C22×C3—61no—
436C22×C3—61no—
437C22×C3—61no—
438C22×C3—61no—
439C22×C3—61no—
440C22×C3—182no—
441C22×C3—182no—
442C22×C3—182no—
443C22×C3—361no—
444C22×C3—361no—
445C22×C3—361no—
446C22×C3—361no—
447C22×C3#379511yescharacteristic
448C22×C3#379611yescharacteristic
449C22×C3#379811yescharacteristic
450C22×C3D3×D1211yescharacteristic
451C22×C3#380411yescharacteristic
452C22×C3#380511yescharacteristic
453C22×C3D3×#6711yescharacteristic
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—62no—
457C2×D3—121no—
458C2×D3—121no—
459C2×D3—121no—
460C2×D3—121no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—182no—
464C2×D3—361no—
465C2×D3—361no—
466C2×D3—361no—
467C2×D3—361no—
468C3×C4—41no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—122no—
472C3×C4—182no—
473C3×C4—361no—
474C3×C4—361no—
475C3×C4—361no—
476C3×C4—361no—
477Dic3—62no—
478Dic3—121no—
479Dic3—121no—
480Dic3—121no—
481Dic3—121no—
482Dic3—122no—
483Dic3—122no—
484Dic3—121no—
485Dic3—121no—
486Dic3—182no—
487Dic3—361no—
488Dic3—361no—
489C32—21no—
490C32#467511yescharacteristic, complemented
491C32#495311yescharacteristic, complemented
492C23—92no—
493C23—181no—
494C23—181no—
495C23—181no—
496C23—181no—
497C23—181no—
498C23—181no—
499C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—92no—
504C2×C4—181no—
505C2×C4—181no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4—181no—
509C2×C4—181no—
510C2×C4—181no—
511C2×C4—181no—
512C2×C4—181no—
513C2×C4—181no—
514C2×C4—181no—
515C2×C4—181no—
516C2×C4—181no—
517C2×C4—181no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—61no—
533C2×C3—61no—
534C2×C3—61no—
535C2×C3—61no—
536C2×C3—61no—
537C2×C3—61no—
538C2×C3—61no—
539C2×C3—362no—
540C2×C3#6385911yescharacteristic
541C2×C3#6386211yescharacteristic
542C2×C3#6388411yescharacteristic
543C2×C3#6389011yescharacteristic
544C2×C3#6389111yescharacteristic
545C2×C3#6390611yescharacteristic
546C2×C3#6395911yescharacteristic
547D3—122no—
548D3—362no—
549C22—182no—
550C22—182no—
551C22—182no—
552C22—361no—
553C22—361no—
554C22—361no—
555C22—361no—
556C22#8864811yescharacteristic
557C22#8864911yescharacteristic
558C22#8865011yescharacteristic
559C22#8865411yescharacteristic
560C22#8867111yescharacteristic
561C22#8867211yescharacteristic
562C22#8867611yescharacteristic
563C4—41no—
564C4—182no—
565C4—361no—
566C4—361no—
567C4—361no—
568C4—361no—
569C3—31no—
570C3—31no—
571C3—61no—
572C3#1059129711yescharacteristic
573C2—362no—
574C2#1172110911yescharacteristic
575C2#1172111211yescharacteristic
576C2#1172112511yescharacteristic
577C2#1172113011yescharacteristic
578C2#1172113111yescharacteristic
579C2#1172114011yescharacteristic
580C2#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)#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. #11720574C2semidirect
6. #63658⋊C3C2semidirect
7. #11720334C2semidirect
29. #63658D3semidirect
296. H(3,𝔽3)#397semidirect
490. C32#4675semidirect
491. C32#4953semidirect

Maximal subgroups

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

subgroupindex
C2×#3752⋊C32
C2×#884502
C2×#884482
#117205742
#63658⋊C32
#117203342
#105912973
#105912163