Finite Group Databasebeta

← #13140236

Subgroups — #13140236

Order 1728.

839 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
1C22×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
2C2×#3752⋊C3C211yesmaximal, characteristic
3C2×#88450C211yesmaximal, characteristic
4#11720574C211yesmaximal, characteristic, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#11720334C211yesmaximal, characteristic, complemented
7#11720334C211yesmaximal, characteristic, complemented
8#10591371—31nomaximal
9#10591362—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×((C3⋊D3)⋊C3)⋊C4—22no—
18C2×((C3⋊D3)⋊C3)⋊C4—22no—
19C2×((C3⋊D3)⋊C3)⋊C4C414yescomplemented
20#3752⋊C3—22no—
21#88450—22no—
22C22×#624—31no—
23C22×D3×Dic3—31no—
24C2×#3752D311yescharacteristic
25C2×C3×(C2×D3)⋊C4—31no—
26C2×#3722—31no—
27C2×(C2×D3)⋊Dic3—31no—
28#63658D311yescharacteristic, complemented
29C3×C3⋊#95—31no—
30C3×C3⋊#95—31no—
31C3×C3⋊#95—31no—
32C3×C3⋊#95—61no—
33#63579—31no—
34#63579—31no—
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)⋊C3C2×C412yes—
43C22×(C3⋊D3)⋊C3D412yes—
44C22×(C3⋊D3)⋊C3D412yes—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—21no—
47C2×C4×H(3,𝔽3)—21no—
48C2×#1084—21no—
49C2×#1084—21no—
50C2×#1084—21no—
51C2×#1084—21no—
52C2×#1084—21no—
53C2×#1084C2×C412yes—
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)⋊C3—21no—
62C2×(C3⋊Dic3)⋊C3C2×C412yes—
63((C3⋊D3)⋊C3)⋊C4—28no—
64#4943—91no—
65#4664—91no—
66#4664—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—
84C2×#624—34no—
85C2×#624—62no—
86C2×#624—62no—
87C2×D3×Dic3—34no—
88C2×D3×Dic3—62no—
89C2×D3×Dic3—62no—
90#3752—22no—
91C3×(C2×D3)⋊C4—62no—
92#3722—62no—
93(C2×D3)⋊Dic3—62no—
94C22×H(3,𝔽3)C22×C411yescharacteristic
95C22×H(3,𝔽3)C2×D411yescharacteristic
96C22×H(3,𝔽3)C2×D411yescharacteristic
97C22×H(3,𝔽3)#4111yescharacteristic
98C22×H(3,𝔽3)#4111yescharacteristic
99C22×H(3,𝔽3)#4111yescharacteristic
100C22×H(3,𝔽3)#4111yescharacteristic
101C2×(C3⋊D3)⋊C3—22no—
102C2×(C3⋊D3)⋊C3—22no—
103C2×(C3⋊D3)⋊C3—22no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—22no—
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3C22⋊C414yes—
108C4×H(3,𝔽3)—41no—
109#1084—22no—
110#1084—41no—
111(C3⋊Dic3)⋊C3—22no—
112(C3⋊Dic3)⋊C3—41no—
113(C3⋊Dic3)⋊C3—41no—
114C23×Dic3—91no—
115C22×C4×D3—91no—
116C22×C4×D3—91no—
117C2×C3×C22⋊C4—91no—
118C2×#217—91no—
119C2×(C2×D3)⋊C4—31no—
120C2×(C2×D3)⋊C4—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C3×#95—91no—
124C3×#95—91no—
125C3⋊#95—31no—
126C3⋊#95—31no—
127C3⋊#95—31no—
128C3⋊#95—61no—
129C3⋊#95—91no—
130C3⋊#95—91no—
131C3⋊#95—91no—
132C3⋊#95—91no—
133C3⋊#95—91no—
134C3⋊#95—91no—
135C3⋊#95—91no—
136C23×C32—21no—
137C23×C32C22×D311yescharacteristic
138C23×C32C22×D311yescharacteristic
139C22×C3⋊D3—21no—
140C22×C3⋊D3—21no—
141C22×C3⋊D3—21no—
142C22×C3⋊D3—21no—
143C22×C3⋊D3C4×D312yes—
144C22×C3⋊D3D1212yes—
145C22×C3⋊D3#6712yes—
146C22×C3×D3—32no—
147C22×C3×D3—32no—
148C22×C3×D3—32no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C2×C32×C4—22no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—41no—
161C2×C32×C4—41no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
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⋊Dic3C4×D312yes—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—62no—
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—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—121no—
195C2×C3×Dic3—121no—
196C2×C3×Dic3—121no—
197C2×C3×Dic3—121no—
198#624—68no—
199D3×Dic3—68no—
200#381—271noSylow
201C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
202C2×H(3,𝔽3)#11711yescharacteristic
203C2×H(3,𝔽3)#11711yescharacteristic
204C2×H(3,𝔽3)#12311yescharacteristic
205C2×H(3,𝔽3)#12311yescharacteristic
206C2×H(3,𝔽3)#12311yescharacteristic
207C2×H(3,𝔽3)#12311yescharacteristic
208(C3⋊D3)⋊C3—24no—
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—
233C22×Dic3—94no—
234C22×Dic3—182no—
235C22×Dic3—182no—
236C2×C4×D3—94no—
237C2×C4×D3—94no—
238C2×C4×D3—182no—
239C2×C4×D3—182no—
240C2×C4×D3—182no—
241C2×C4×D3—182no—
242C3×C22⋊C4—182no—
243#217—182no—
244(C2×D3)⋊C4—62no—
245(C2×D3)⋊C4—182no—
246(C2×D3)⋊C4—182no—
247(C2×D3)⋊C4—182no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32C2×C4×D311yescharacteristic
256C22×C32C2×C4×D311yescharacteristic
257C22×C32C2×D1211yescharacteristic
258C22×C32#23511yescharacteristic
259C22×C32#23511yescharacteristic
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32#23711yescharacteristic
263C22×C32#23711yescharacteristic
264C22×C32#23711yescharacteristic
265C22×C32#23711yescharacteristic
266C22×C32#23711yescharacteristic
267C22×C32#23911yescharacteristic
268C22×C32C2×#6711yescharacteristic
269C2×C3⋊D3—22no—
270C2×C3⋊D3—22no—
271C2×C3⋊D3—22no—
272C2×C3⋊D3—22no—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3(C2×D3)⋊C414yes—
276C2×C3×D3—34no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C32×C4—42no—
284C32×C4—41no—
285C32×C4—41no—
286C3⋊Dic3—22no—
287C3⋊Dic3—41no—
288C3⋊Dic3—41no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C3×Dic3—62no—
292C3×Dic3—122no—
293C3×Dic3—122no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3—121no—
297C3×Dic3—121no—
298C23×C4—271no—
299C2×C22⋊C4—91no—
300C2×C22⋊C4—271no—
301#95—91no—
302#95—91no—
303#95—271no—
304#95—271no—
305H(3,𝔽3)#38111yescharacteristic, complemented, Sylow
306C23×C3—31no—
307C23×C3—31no—
308C23×C3—61no—
309C23×C3—92no—
310C23×C3—92no—
311C23×C3—92no—
312C23×C3—181no—
313C23×C3—181no—
314C23×C3—181no—
315C23×C3—181no—
316C23×C3C2×D3211yescharacteristic
317C22×D3—32no—
318C22×D3—32no—
319C22×D3—32no—
320C22×D3—61no—
321C22×D3—61no—
322C22×D3—61no—
323C22×D3—61no—
324C22×D3—92no—
325C22×D3—92no—
326C22×D3—92no—
327C22×D3—181no—
328C22×D3—181no—
329C22×D3—181no—
330C22×D3—181no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—62no—
341C2×C3×C4—92no—
342C2×C3×C4—92no—
343C2×C3×C4—121no—
344C2×C3×C4—121no—
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×C4—181no—
355C2×C3×C4—181no—
356C2×C3×C4—181no—
357C2×C3×C4—181no—
358C2×Dic3—32no—
359C2×Dic3—32no—
360C2×Dic3—32no—
361C2×Dic3—62no—
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—62no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—92no—
382C2×Dic3—121no—
383C2×Dic3—121no—
384C2×Dic3—121no—
385C2×Dic3—121no—
386C2×Dic3—181no—
387C2×Dic3—181no—
388C2×Dic3—181no—
389C2×Dic3—181no—
390C2×Dic3—181no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—188no—
395C4×D3—188no—
396C4×D3—188no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32#87211yescharacteristic
405C2×C32#87711yescharacteristic
406C2×C32#87711yescharacteristic
407C2×C32D3×C22⋊C411yescharacteristic
408C2×C32#88311yescharacteristic
409C2×C32#88311yescharacteristic
410C2×C32#88611yescharacteristic
411C2×C32#88611yescharacteristic
412C2×C32#89211yescharacteristic
413C2×C32#89411yescharacteristic
414C2×C32#89411yescharacteristic
415C2×C32C2×(C2×D3)⋊C411yescharacteristic
416C2×C32#93511yescharacteristic
417C2×C32#93511yescharacteristic
418C3⋊D3—24no—
419C3×D3—64no—
420C24—91no—
421C22×C4—91no—
422C22×C4—91no—
423C22×C4—91no—
424C22×C4—91no—
425C22×C4—91no—
426C22×C4—274no—
427C22×C4—542no—
428C22×C4—542no—
429C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
430C22⋊C4—182no—
431C22⋊C4—542no—
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—31no—
441C22×C3—31no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—94no—
454C22×C3—182no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3—182no—
458C22×C3—182no—
459C22×C3—182no—
460C22×C3#379511yescharacteristic
461C22×C3#379611yescharacteristic
462C22×C3C4×D3211yescharacteristic
463C22×C3D3×D1211yescharacteristic
464C22×C3#380411yescharacteristic
465C22×C3#380511yescharacteristic
466C22×C3D3×#6711yescharacteristic
467C2×D3—34no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—94no—
475C2×D3—182no—
476C2×D3—182no—
477C2×D3—182no—
478C2×D3—182no—
479C2×D3—182no—
480C2×D3—182no—
481C3×C4—41no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—122no—
485C3×C4—182no—
486C3×C4—182no—
487C3×C4—361no—
488C3×C4—361no—
489C3×C4—361no—
490Dic3—62no—
491Dic3—62no—
492Dic3—62no—
493Dic3—122no—
494Dic3—121no—
495Dic3—121no—
496Dic3—122no—
497Dic3—121no—
498Dic3—121no—
499Dic3—182no—
500Dic3—361no—
501Dic3—361no—
502C32—21no—
503C32#466411yescharacteristic, complemented
504C32#494311yescharacteristic, complemented
505C23—92no—
506C23—92no—
507C23—92no—
508C23—181no—
509C23—181no—
510C23—181no—
511C23—181no—
512C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—92no—
517C2×C4—92no—
518C2×C4—181no—
519C2×C4—181no—
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—548no—
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—184no—
554C2×C3#6387711yescharacteristic
555C2×C3#6387811yescharacteristic
556C2×C3#6388411yescharacteristic
557C2×C3#6389011yescharacteristic
558C2×C3#6389111yescharacteristic
559C2×C3D3×(C2×D3)⋊C411yescharacteristic
560C2×C3#6396911yescharacteristic
561D3—64no—
562D3—184no—
563C22—94no—
564C22—182no—
565C22—182no—
566C22—182no—
567C22—182no—
568C22—182no—
569C22—182no—
570C22#8864811yescharacteristic
571C22#8864911yescharacteristic
572C22C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
573C22#8865411yescharacteristic
574C22#8867111yescharacteristic
575C22#8867211yescharacteristic
576C22#8867611yescharacteristic
577C4—41no—
578C4—182no—
579C4—182no—
580C4—361no—
581C4—361no—
582C4—361no—
583C3—31no—
584C3—31no—
585C3—61no—
586C3#1059136211yescharacteristic
587C2—184no—
588C2#1172112011yescharacteristic
589C2#1172112111yescharacteristic
590C2#1172112511yescharacteristic
591C2#1172113011yescharacteristic
592C2#1172113111yescharacteristic
593C2#1172116011yescharacteristic
594C2#1172131111yescharacteristic
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)#381
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
4. #11720574C2semidirect
5. #63658⋊C3C2semidirect
6. #11720334C2semidirect
7. #11720334C2semidirect
19. C2×((C3⋊D3)⋊C3)⋊C4C4semidirect
28. #63658D3semidirect
305. H(3,𝔽3)#381semidirect
503. C32#4664semidirect
504. C32#4943semidirect

Maximal subgroups

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

subgroupindex
C22×((C3⋊D3)⋊C3)⋊C42
C2×#3752⋊C32
C2×#884502
#117205742
#63658⋊C32
#117203342
#105913713
#105913623