Finite Group Databasebeta

← #10723733

Subgroups — #10723733

Order 768.

1450 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5212C211yesmaximal, characteristic
2C3×C22⋊#95C211yesmaximal, characteristic
3#72127C211yesmaximal, characteristic
4#71547C212yesmaximal
5#71545C212yesmaximal
6#69418C212yesmaximal, complemented
7#69411C212yesmaximal, complemented
8#69407C212yesmaximal, complemented
9#69405C212yesmaximal, complemented
10#14453—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C22⋊C4C2212yes—
13C22×#217C2211yescharacteristic
14C22×#217C2212yes—
15C22×#217C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C22×Dic3⋊C4C2212yes—
19C22×C4×Dic3C2212yes—
20C2×C3×#95C2212yes—
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2211yescharacteristic
23#5212C418yescomplemented
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2212yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×#380—31no—
34#1443—32no—
35#1441—32no—
36#1439—32no—
37#1437—32no—
38#1436—32no—
39#1436—32no—
40C22⋊#95—31no—
41C22⋊#95D311yescharacteristic
42C25×C3C2311yescharacteristic
43C23×C3×C4—24no—
44C23×C3×C4—42no—
45C23×C3×C4—41no—
46C23×C3×C4—41no—
47C23×C3×C4C2×C414yes—
48C23×C3×C4C2312yes—
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2312yes—
55C23×Dic3C2312yes—
56C2×C3×C22⋊C4—24no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—42no—
59C2×#217—22no—
60C2×#217—42no—
61C2×#217—42no—
62C2×#217—41no—
63C2×#217—41no—
64C2×#217—41no—
65C2×#217—41no—
66C2×#217C2×C418yes—
67C2×#217Q814yes—
68C2×C4⋊Dic3—41no—
69C2×C4⋊Dic3—42no—
70C2×C4⋊Dic3—41no—
71C2×C4⋊Dic3C2×C414yes—
72C2×C4⋊Dic3C2×C414yes—
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—42no—
75C2×Dic3⋊C4—42no—
76C2×C4×Dic3—42no—
77C2×C4×Dic3—42no—
78C2×C4×Dic3—42no—
79C3×#95—24no—
80C3×#95—41no—
81C3×#95—41no—
82C3⋊#95—24no—
83C3⋊#95—24no—
84C3⋊#95—42no—
85C3⋊#95—42no—
86C3⋊#95—42no—
87C3⋊#95—42no—
88C3⋊#95—42no—
89C3⋊#95—42no—
90C3⋊#95C2×C418yes—
91C24×C4C2×D311yescharacteristic
92C22×C4⋊C4—31no—
93C22×C4⋊C4—31no—
94C22×C4⋊C4—32no—
95C22×C22⋊C4—31no—
96C22×C22⋊C4—32no—
97C22×C22⋊C4—31no—
98C22×C22⋊C4C2×D312yes—
99C22×C42—32no—
100#380—38no—
101C2×#95—32no—
102C2×#95—32no—
103C2×#95—32no—
104C2×#95—32no—
105C2×#95—32no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—32no—
110C2×#95C2×D312yes—
111C2×#95C2×D311yescharacteristic
112C2×#95C2×D311yescharacteristic
113C24×C3—22no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3—21no—
117C24×C3—21no—
118C24×C3—21no—
119C24×C3—22no—
120C24×C3—21no—
121C24×C3C22×C412yes—
122C24×C3C2×Q812yes—
123C24×C3#4112yes—
124C24×C3C2411yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—28no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4—48no—
149C22×C3×C4—48no—
150C22×C3×C4C22×C412yes—
151C22×C3×C4C22×C412yes—
152C22×C3×C4#4114yes—
153C22×C3×C4#4114yes—
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—22no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—22no—
168C22×Dic3—22no—
169C22×Dic3—22no—
170C22×Dic3—22no—
171C22×Dic3—22no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3C22×C414yes—
180C22×Dic3C22×C414yes—
181C22×Dic3C2×Q814yes—
182C3×C22⋊C4—48no—
183#217—48no—
184#217—44no—
185#217—44no—
186C4⋊Dic3—44no—
187C4⋊Dic3—44no—
188Dic3⋊C4—48no—
189C4×Dic3—48no—
190C25C22×D311yescharacteristic
191C23×C4—24no—
192C23×C4—32no—
193C23×C4—32no—
194C23×C4—32no—
195C23×C4—32no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—42no—
199C23×C4C4×D314yes—
200C23×C4C22×D312yes—
201C23×C4C22×D311yescharacteristic
202C23×C4C22×D312yes—
203C23×C4C22×D311yescharacteristic
204C2×C4⋊C4—34no—
205C2×C4⋊C4—34no—
206C2×C4⋊C4—64no—
207C2×C4⋊C4—122no—
208C2×C4⋊C4—122no—
209C2×C4⋊C4—121no—
210C2×C4⋊C4—122no—
211C2×C4⋊C4—121no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—38no—
215C2×C22⋊C4—34no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—121no—
219C2×C22⋊C4—122no—
220C2×C22⋊C4—122no—
221C2×C22⋊C4—121no—
222C2×C22⋊C4—121no—
223C2×C22⋊C4—121no—
224C2×C42—122no—
225C2×C42—122no—
226C2×C42—122no—
227#95—24no—
228#95—38no—
229#95—41no—
230#95—41no—
231#95—64no—
232#95—64no—
233#95—122no—
234#95—122no—
235#95—122no—
236#95—122no—
237#95—122no—
238#95—122no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—24no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3C4×Q814yes—
262C23×C3C23×C411yescharacteristic
263C23×C3C22×Q811yescharacteristic
264C23×C3C2×#4112yes—
265C23×C3C2×#4112yes—
266C23×C3C2×#4111yescharacteristic
267C23×C3H(5,𝔽2)11yescharacteristic
268C23×C3H(5,𝔽2)11yescharacteristic
269C23×C3H(5,𝔽2)12yes—
270C23×C3H(5,𝔽2)12yes—
271C23×C3H(5,𝔽2)11yescharacteristic
272C23×C32−1+411yescharacteristic
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—416no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4#11714yes—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3—24no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—21no—
334C24C2×C4×D312yes—
335C24#23712yes—
336C24D3×Q812yes—
337C24C23×D311yescharacteristic
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—34no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—62no—
387C22×C4—62no—
388C22×C4—62no—
389C22×C4—62no—
390C22×C4—62no—
391C22×C4C2×C4×D312yes—
392C22×C4C2×C4×D312yes—
393C22×C4#23514yes—
394C22×C4#23714yes—
395C4⋊C4—124no—
396C4⋊C4—124no—
397C4⋊C4—128no—
398C22⋊C4—48no—
399C22⋊C4—124no—
400C22⋊C4—128no—
401C22⋊C4—124no—
402C42—128no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3C2×#11711yescharacteristic
415C22×C3C2×C4×Q811yescharacteristic
416C22×C3C4×#4111yescharacteristic
417C22×C3#51811yescharacteristic
418C22×C3#51811yescharacteristic
419C22×C3#51811yescharacteristic
420C22×C3#52011yescharacteristic
421C22×C3#53812yes—
422C22×C3#53812yes—
423C22×C3#53812yes—
424C22×C3#53811yescharacteristic
425C22×C3#53911yescharacteristic
426C22×C3#54012yes—
427C22×C3#54311yescharacteristic
428C22×C3#54411yescharacteristic
429C22×C3#55112yes—
430C22×C3#55312yes—
431C22×C3#55312yes—
432C22×C3#55412yes—
433C22×C3#55412yes—
434C22×C3#55612yes—
435C22×C3#56012yes—
436C22×C3#56212yes—
437C3×C4—24no—
438C3×C4—42no—
439C3×C4—44no—
440C3×C4—82no—
441Dic3—44no—
442Dic3—44no—
443Dic3—44no—
444Dic3—82no—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23#88714yes—
468C23C22×C4×D311yescharacteristic
469C23C2×#23512yes—
470C23C2×#23712yes—
471C23C2×#23711yescharacteristic
472C23#100411yescharacteristic
473C23#100412yes—
474C23#100411yescharacteristic
475C23C2×D3×Q811yescharacteristic
476C23#100911yescharacteristic
477C23#100912yes—
478C23#101011yescharacteristic
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—416no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—64no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4#87714yes—
532C2×C3—44no—
533C2×C3#231211yescharacteristic
534C2×C3#232911yescharacteristic
535C2×C3#233311yescharacteristic
536C2×C3#234511yescharacteristic
537C2×C3#234711yescharacteristic
538C2×C3#236811yescharacteristic
539C2×C3#237711yescharacteristic
540C2×C3#277812yes—
541C2×C3#278012yes—
542C2×C3#278612yes—
543C2×C3#278812yes—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22C4×#23511yescharacteristic
556C22C2×#87711yescharacteristic
557C22#548611yescharacteristic
558C22#548611yescharacteristic
559C22#549512yes—
560C22C2×#88711yescharacteristic
561C22#551112yes—
562C22#552111yescharacteristic
563C22#552311yescharacteristic
564C22#554212yes—
565C22#556012yes—
566C22#556212yes—
567C22#556512yes—
568C22#559312yes—
569C22#559312yes—
570C22#560211yescharacteristic
571C22#561711yescharacteristic
572C22#561812yes—
573C22#562311yescharacteristic
574C22#563411yescharacteristic
575C22#563512yes—
576C22#566612yes—
577C22#566712yes—
578C4—24no—
579C4—42no—
580C4—44no—
581C4—82no—
582C4—124no—
583C4—124no—
584C4—124no—
585C4—242no—
586C3#1445311yescharacteristic, complemented, Sylow
587C2—44no—
588C2#7333911yescharacteristic
589C2#7344911yescharacteristic
590C2#7346011yescharacteristic
591C2#7382011yescharacteristic
592C2#7387411yescharacteristic
593C2#7387911yescharacteristic
594C2#7412411yescharacteristic
595C2#7456412yes—
596C2#7457312yes—
597C2#7485312yes—
598C2#7486212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C22⋊#95C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC22⋊#95D3
hypocenterC3#14453
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. #69418C2semidirect
7. #69411C2semidirect
8. #69407C2semidirect
9. #69405C2semidirect
23. #5212C4semidirect
586. C3#14453semidirect

Maximal subgroups

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

subgroupindex
C2×#52122
C3×C22⋊#952
#721272
#715472
#715452
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