Finite Group Databasebeta

← #10877267

Subgroups — #10877267

Order 768.

882 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4977C211yesmaximal, characteristic, complemented
2#73865C211yesmaximal, characteristic
3C3×#1852C211yesmaximal, characteristic, complemented
4#71560C211yesmaximal, characteristic
5#70640C212yesmaximal, complemented
6#70616C212yesmaximal, complemented
7#69932C211yesmaximal, characteristic
8#69908C211yesmaximal, characteristic, complemented
9#69505C211yesmaximal, characteristic, complemented
10#68893C212yesmaximal, complemented
11#68844C212yesmaximal, complemented
12#27684—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4×Dic3C2211yescharacteristic
17C2×#893C2211yescharacteristic
18C2×#67⋊C4—21no—
19C2×C3×#107C2212yes—
20C2×C3×C22⋊C8C2211yescharacteristic
21C3×#380C2211yescharacteristic
22C2×C3⋊#95—21no—
23C2×C3⋊#95—21no—
24C2×#830C2211yescharacteristic
25C2×#821C2212yes—
26C2×Dic3⋊C8C2212yes—
27#4988C2211yescharacteristic
28#4977—21no—
29#4977—21no—
30#4977C2212yes—
31#4977C2212yes—
32Dic3×C4⋊C4C2211yescharacteristic
33#4972C2211yescharacteristic
34C2×#809C2212yes—
35C2×#808C2212yes—
36#4958—21no—
37#4669—21no—
38#4666—21no—
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×#121—31no—
45C2×#382—31no—
46#2101—32no—
47#2092—32no—
48#1949—32no—
49#1874—31no—
50#1874—31no—
51#1853—31no—
52#1852D311yescharacteristic, complemented
53#1804—32no—
54#1797—31no—
55C23×C3×C4—21no—
56C23×C3×C4C2311yescharacteristic
57C22×#67—21no—
58C23×Dic3—21no—
59C23×Dic3D411yescharacteristic
60C23×Dic3D411yescharacteristic
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×C3×C8C2312yes—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70#930—22no—
71#930Q814yes—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—42no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×C4×Dic3—24no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3C2311yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2312yes—
88#893—24no—
89#67⋊C4—44no—
90C3×#107—42no—
91C3×C22⋊C8—22no—
92C3×#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#830—22no—
100#821—42no—
101Dic3⋊C8—24no—
102#809—42no—
103#808—42no—
104C2×#121—31no—
105C2×C4×D4—31no—
106C2×C4×D4—61no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4C2×D311yescharacteristic
109C22×C42—31no—
110C2×#107C2×D312yes—
111C2×#106—32no—
112C2×C4⋊C8—32no—
113C2×#102—32no—
114C2×#102—32no—
115C2×C22⋊C8—31no—
116C2×C22⋊C8C2×D311yescharacteristic
117#390—31no—
118#389—31no—
119#388—61no—
120#386—61no—
121#382—32no—
122#382—32no—
123#382—61no—
124#382—61no—
125#380C2×D311yescharacteristic
126C4×C4⋊C4—31no—
127C4×C22⋊C4—61no—
128C2×#95—61no—
129C2×#95—61no—
130#340—32no—
131#340—32no—
132#340—32no—
133#340C2×D312yes—
134#336—32no—
135C24×C3C2×D411yescharacteristic
136C23×D3—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×Q812yes—
147C22×C3×C4#4112yes—
148C22×C3×C4C2411yescharacteristic
149C2×#67—24no—
150C2×#67—42no—
151C2×#67—42no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—21no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C22×Dic3—41no—
159C22×Dic3—41no—
160C22×Dic3—42no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3C2×D411yescharacteristic
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166C22×Dic3C2×D411yescharacteristic
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—21no—
170C2×D12C2×Q812yes—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3—42no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C22⋊C4—42no—
183(C2×D3)⋊C4—24no—
184(C2×D3)⋊C4—42no—
185(C2×D3)⋊C4—42no—
186(C2×D3)⋊C4—42no—
187C4⋊Dic3—21no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—42no—
191C4⋊Dic3C2×Q812yes—
192Dic3⋊C4—42no—
193C4×Dic3—22no—
194C4×Dic3—22no—
195C4×Dic3—22no—
196C4×Dic3—42no—
197C4×Dic3#4114yes—
198C2×C3⋊C8—22no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C22×D4—31no—
202C22×D4—61no—
203C23×C4—21no—
204C23×C4—31no—
205C23×C4—31no—
206C23×C4—61no—
207C23×C4—61no—
208C23×C4C22×D311yescharacteristic
209C22×C8—32no—
210C22×C8C22×D312yes—
211#121—34no—
212#121—62no—
213C4×D4—64no—
214C4×D4—124no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—122no—
222C2×C4⋊C4C22×D312yes—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—31no—
231C2×C42—31no—
232C2×C42—31no—
233C2×C42—31no—
234C2×C42—64no—
235C2×C42—61no—
236#107—42no—
237#106—122no—
238C4⋊C8—64no—
239#102—122no—
240#102—122no—
241C22⋊C8—22no—
242C22⋊C8—62no—
243#95—21no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—62no—
248#95—61no—
249#95—61no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3C22⋊Q812yes—
255C23×C3#13512yes—
256C23×C3AGL(1,ℤ/8)12yes—
257C23×C3C22×D411yescharacteristic
258C22×D3—22no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—84no—
279C2×C3×C4C22⋊Q812yes—
280C2×C3×C4C22×Q811yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4112yes—
283C2×C3×C4H(5,𝔽2)12yes—
284#67—48no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—22no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—44no—
297C2×Dic3—44no—
298C2×Dic3—41no—
299C2×Dic3—41no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—84no—
304C2×Dic3#12312yes—
305C2×Dic3#12312yes—
306D12—24no—
307D12—42no—
308C4×D3—84no—
309C3×C8—42no—
310C3⋊C8—42no—
311C24—61no—
312C24D3×D411yescharacteristic
313C2×D4—32no—
314C2×D4—64no—
315C2×D4—61no—
316C2×D4—62no—
317C2×D4—62no—
318C2×D4—122no—
319C2×D4—122no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—61no—
337C22×C4—62no—
338C22×C4—122no—
339C22×C4—122no—
340C22×C4—122no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4—122no—
344C22×C4—122no—
345C22×C4—122no—
346C22×C4#23512yes—
347C22×C4D3×D411yescharacteristic
348C22×C4D3×Q812yes—
349C22×C4C23×D311yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—62no—
354C2×C8—62no—
355C2×C8—62no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—32no—
361C4⋊C4—42no—
362C4⋊C4—61no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—122no—
366C4⋊C4—122no—
367C22⋊C4—42no—
368C22⋊C4—64no—
369C22⋊C4—122no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C42—34no—
373C42—62no—
374C42—62no—
375C42—62no—
376C42—122no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3#47814yes—
384C22×C3C2×C22⋊Q811yescharacteristic
385C22×C3C2×#12311yescharacteristic
386C22×C3#53511yescharacteristic
387C22×C3C2×#13511yescharacteristic
388C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
389C22×C3#57511yescharacteristic
390C22×C3#57511yescharacteristic
391C2×D3—42no—
392C2×D3—42no—
393C2×D3—42no—
394C2×D3—81no—
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C3×C4—22no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4—82no—
402C3×C4#55312yes—
403C3×C4#55412yes—
404Dic3—22no—
405Dic3—22no—
406Dic3—42no—
407Dic3—41no—
408Dic3—41no—
409Dic3—82no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—62no—
415C23—121no—
416C23—121no—
417C23—121no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23#89612yes—
422C23#91112yes—
423C23#91912yes—
424C23C2×D3×D411yescharacteristic
425D4—64no—
426D4—128no—
427D4—122no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—32no—
433C2×C4—32no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—62no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—84no—
455C2×C4—124no—
456C2×C4—124no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—244no—
463C2×C4—244no—
464C2×C4#89612yes—
465C2×C4C2×#23511yescharacteristic
466C2×C4#100412yes—
467C2×C4C2×D3×Q811yescharacteristic
468C2×C4D3×#4112yes—
469C8—42no—
470C8—122no—
471C2×C3—24no—
472C2×C3#248711yescharacteristic
473C2×C3C2×#47811yescharacteristic
474C2×C3#308011yescharacteristic
475C2×C3#309111yescharacteristic
476C2×C3#309411yescharacteristic
477C2×C3#321111yescharacteristic
478C2×C3#321311yescharacteristic
479D3—82no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—122no—
487C22—122no—
488C22—122no—
489C22—241no—
490C22—241no—
491C22—241no—
492C22—241no—
493C22#489114yes—
494C22C2×#89611yescharacteristic
495C22#559211yescharacteristic
496C22D3×#12311yescharacteristic
497C22C2×#91111yescharacteristic
498C22C2×#91911yescharacteristic
499C22#577311yescharacteristic
500C22#577711yescharacteristic
501C4—22no—
502C4—62no—
503C4—62no—
504C4—82no—
505C4—81no—
506C4—81no—
507C4—122no—
508C4—121no—
509C4—121no—
510C4—242no—
511C4#554012yes—
512C4#563612yes—
513C3#2768411yescharacteristic, complemented, Sylow
514C2—24no—
515C2—242no—
516C2#7445711yescharacteristic
517C2C2×#489111yescharacteristic
518C2#7924111yescharacteristic
519C2#7999611yescharacteristic
520C2#8020111yescharacteristic
521C2#8128311yescharacteristic
522C2#8146711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1852C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1852D3
hypocenterC3#27684
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C2×#4977C2semidirect
3. C3×#1852C2semidirect
5. #70640C2semidirect
6. #70616C2semidirect
8. #69908C2semidirect
9. #69505C2semidirect
10. #68893C2semidirect
11. #68844C2semidirect
52. #1852D3semidirect
513. C3#27684semidirect

Maximal subgroups

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

subgroupindex
C2×#49772
#738652
C3×#18522
#715602
#706402
#706162
#699322
#699082
#695052
#688932
#688442
#276843