Finite Group Databasebeta

← #10823939

Subgroups — #10823939

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4772C212yesmaximal
2#76290C211yesmaximal, characteristic
3#74318C211yesmaximal, characteristic, complemented
4C3×#2071C211yesmaximal, characteristic, complemented
5#70166C212yesmaximal, complemented
6#69640C211yesmaximal, characteristic, complemented
7#69589C211yesmaximal, characteristic, complemented
8#68915C212yesmaximal, complemented
9#68866C212yesmaximal, complemented
10D3×#336C211yesmaximal, characteristic, complemented
11#68636C211yesmaximal, characteristic, complemented
12#23627—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C22×D3×C8C2212yes—
15C2×#892C2211yescharacteristic
16C2×#892C2212yes—
17C2×C42×D3C2211yescharacteristic
18C2×C3×#102C2212yes—
19C3×#390C2211yescharacteristic
20C3×#384C2211yescharacteristic
21#5221C2211yescharacteristic
22C2×#832C2212yes—
23C2×#816C2212yes—
24#4987—21no—
25#4987—21no—
26#4977C2212yes—
27Dic3×C4⋊C4—21no—
28Dic3×C4⋊C4—21no—
29#4973C2211yescharacteristic
30C2×#810C2212yes—
31#4956—21no—
32#4956—21no—
33Dic3×C22⋊C4—21no—
34Dic3×C22⋊C4—21no—
35#4935C2211yescharacteristic
36#4929C2211yescharacteristic
37#4772—28no—
38C3×#340C2212yes—
39C3×#336C2211yescharacteristic
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2211yescharacteristic
45C2×#416—32no—
46#2297—31no—
47#2285—31no—
48#2074—32no—
49#2073—32no—
50#2072—31no—
51#2072—31no—
52#2071—31no—
53#2071D311yescharacteristic, complemented
54#1735—32no—
55C2×#336—31no—
56C22×C3×D4C2311yescharacteristic
57C22×#67—21no—
58C23×Dic3—21no—
59C2×#237—24no—
60C2×#237—24no—
61C2×#237—22no—
62C2×#237D414yes—
63C22×C4×D3D411yescharacteristic
64C22×C4×D3D411yescharacteristic
65C22×C4×D3C2311yescharacteristic
66C22×Dic6C2311yescharacteristic
67C22×C3×C8C2312yes—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C42C2311yescharacteristic
75C2×#217—21no—
76C2×#217—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×C4×Dic3—21no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89C2×D3×C8—24no—
90C2×D3×C8—24no—
91C2×D3×C8—24no—
92#892—24no—
93#892—24no—
94#892—24no—
95C42×D3—24no—
96C3×#102—24no—
97#832—24no—
98C3⋊#95—21no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C3⋊#95—22no—
103C3⋊#95—22no—
104#816—24no—
105#810—24no—
106C22×#41—31no—
107C23×C8—32no—
108C2×#117—32no—
109C2×#117—31no—
110C22×C42—31no—
111#416—68no—
112C2×#103—32no—
113C2×#103—32no—
114C2×#102—32no—
115C2×#102C2×D312yes—
116#391—31no—
117#391—31no—
118#390—31no—
119#390C2×D311yescharacteristic
120#386—61no—
121#386—61no—
122#384—31no—
123#384C2×D311yescharacteristic
124#382—32no—
125#380—61no—
126#380—61no—
127C4×C4⋊C4—61no—
128C4×C4⋊C4—61no—
129C4×C22⋊C4—61no—
130C4×C22⋊C4—61no—
131#340—32no—
132#340—32no—
133#340—32no—
134#340C2×D312yes—
135#336—32no—
136#336—31no—
137#336C2×D311yescharacteristic
138C24×C3—21no—
139C23×D3C2×D411yescharacteristic
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4C2×D412yes—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C2×#67—24no—
150C2×#67—42no—
151C2×#67—42no—
152C22×Dic3—21no—
153C22×Dic3—24no—
154C22×Dic3—22no—
155C22×Dic3—21no—
156C22×Dic3—42no—
157C22×Dic3—42no—
158C22×Dic3C2×D411yescharacteristic
159C22×Dic3C2×D411yescharacteristic
160C22×Dic3C2×D411yescharacteristic
161#237—216no—
162#237—48no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—41no—
168C2×C4×D3—41no—
169C2×C4×D3C2×D412yes—
170C2×C4×D3#4114yes—
171C2×C4×D3#4114yes—
172C2×C4×D3#4112yes—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—21no—
176C2×Dic6C2×D412yes—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—42no—
187C3×C42—22no—
188#217—42no—
189#217—42no—
190(C2×D3)⋊C4—44no—
191(C2×D3)⋊C4—42no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—42no—
196C4⋊Dic3—42no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—22no—
200C4×Dic3—44no—
201C4×Dic3—42no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C2×C3⋊C8—22no—
205D3×C8—48no—
206C2×#41—34no—
207C2×#41—62no—
208C2×#41—64no—
209C2×#41—64no—
210C22×Q8—31no—
211C22×D4—61no—
212C22×D4C22×D311yescharacteristic
213C23×C4—31no—
214C23×C4—31no—
215C23×C4—31no—
216C23×C4—61no—
217C22×C8—32no—
218C22×C8—64no—
219C22×C8—64no—
220C22×C8—64no—
221C22×C8C22×D312yes—
222#117—64no—
223#117—64no—
224#117—64no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—61no—
231C2×C4⋊C4C22×D311yescharacteristic
232C2×C4⋊C4C22×D312yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—62no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C42—31no—
240C2×C42—31no—
241C2×C42—31no—
242C2×C42—64no—
243C2×C42—61no—
244C2×C42—62no—
245C2×C42C22×D311yescharacteristic
246#103—64no—
247#103—64no—
248#102—24no—
249#102—64no—
250#95—61no—
251#95—61no—
252#95—62no—
253#95—62no—
254#95—61no—
255#95—61no—
256C23×C3—22no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3C22×D411yescharacteristic
264C22×D3—21no—
265C22×D3—21no—
266C22×D3—21no—
267C22×D3—21no—
268C22×D3#12112yes—
269C22×D3#12312yes—
270C22×D3#12412yes—
271C3×D4—24no—
272C3×D4—42no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4#13512yes—
284C2×C3×C4#13512yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4112yes—
287C2×C3×C4C2×#4112yes—
288C2×C3×C4C2×#4111yescharacteristic
289#67—48no—
290C2×Dic3—22no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—21no—
296C2×Dic3—21no—
297C2×Dic3—41no—
298C2×Dic3—41no—
299C2×Dic3—48no—
300C2×Dic3—42no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3#12112yes—
307C2×Dic3#12312yes—
308C2×Dic3#12412yes—
309C4×D3—44no—
310C4×D3—44no—
311C4×D3#12318yes—
312C4×D3#12618yes—
313Dic6—24no—
314Dic6—42no—
315C3×C8—42no—
316C3⋊C8—42no—
317C24—21no—
318C24—31no—
319#41—616no—
320#41—128no—
321C2×Q8—32no—
322C2×Q8—61no—
323C2×Q8—62no—
324C2×Q8—62no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—64no—
329C2×D4—122no—
330C2×D4—122no—
331C2×D4D3×D412yes—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—31no—
335C22×C4—32no—
336C22×C4—34no—
337C22×C4—32no—
338C22×C4—34no—
339C22×C4—31no—
340C22×C4—31no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—64no—
344C22×C4—61no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—61no—
348C22×C4—62no—
349C22×C4—121no—
350C22×C4—121no—
351C22×C4—122no—
352C22×C4—122no—
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4C23×D311yescharacteristic
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—62no—
360C2×C8—62no—
361C2×C8—62no—
362C2×C8—128no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—122no—
372C4⋊C4—122no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—122no—
376C22⋊C4—124no—
377C22⋊C4—122no—
378C22⋊C4—122no—
379C42—22no—
380C42—62no—
381C42—62no—
382C42—62no—
383C42—122no—
384C42—124no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3C2×#12111yescharacteristic
393C22×C3C2×#12311yescharacteristic
394C22×C3C2×#12411yescharacteristic
395C22×C3C2×#13511yescharacteristic
396C22×C3C2×#13511yescharacteristic
397C22×C3#57511yescharacteristic
398C22×C3#57511yescharacteristic
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—22no—
405C2×D3#39714yes—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—81no—
409C3×C4—82no—
410C3×C4C2×#12312yes—
411C3×C4C2×#12612yes—
412Dic3—41no—
413Dic3—41no—
414Dic3—42no—
415Dic3—81no—
416Dic3—82no—
417Dic3#39714yes—
418C23—22no—
419C23—32no—
420C23—32no—
421C23—32no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—61no—
429C23—61no—
430C23—61no—
431C23—61no—
432C23C2×D3×D411yescharacteristic
433Q8—64no—
434Q8—122no—
435D4—24no—
436D4—42no—
437D4—128no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—32no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—38no—
446C2×C4—38no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—62no—
460C2×C4—124no—
461C2×C4—124no—
462C2×C4—128no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4#91212yes—
472C2×C4#91612yes—
473C2×C4C2×D3×D411yescharacteristic
474C2×C4D3×#4112yes—
475C2×C4D3×#4112yes—
476C2×C4D3×#4111yescharacteristic
477C8—42no—
478C8—122no—
479C2×C3—82no—
480C2×C3C2×#39711yescharacteristic
481C2×C3#303511yescharacteristic
482C2×C3#303511yescharacteristic
483C2×C3#309111yescharacteristic
484C2×C3#309111yescharacteristic
485C2×C3#313111yescharacteristic
486C2×C3#313511yescharacteristic
487D3—24no—
488C22—34no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—62no—
493C22—62no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—62no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22D3×#12111yescharacteristic
503C22D3×#12311yescharacteristic
504C22D3×#12411yescharacteristic
505C22C2×#91211yescharacteristic
506C22C2×#91611yescharacteristic
507C22#577311yescharacteristic
508C22#577711yescharacteristic
509C4—22no—
510C4—22no—
511C4—34no—
512C4—81no—
513C4—82no—
514C4—122no—
515C4—121no—
516C4—121no—
517C4—241no—
518C4—242no—
519C4D3×#12312yes—
520C4D3×#12612yes—
521C3#2362711yescharacteristic, complemented, Sylow
522C2—64no—
523C2—82no—
524C2D3×#39711yescharacteristic
525C2#8033111yescharacteristic
526C2#8042711yescharacteristic
527C2#8128111yescharacteristic
528C2#8132411yescharacteristic
529C2#8156211yescharacteristic
530C2#8167911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radicalC22×#237C22
hypercenter#2071D3
hypocenterC3#23627
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
3. #74318C2semidirect
4. C3×#2071C2semidirect
5. #70166C2semidirect
6. #69640C2semidirect
7. #69589C2semidirect
8. #68915C2semidirect
9. #68866C2semidirect
10. D3×#336C2semidirect
11. #68636C2semidirect
53. #2071D3semidirect
521. C3#23627semidirect

Maximal subgroups

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

subgroupindex
C2×#47722
#762902
#743182
C3×#20712
#701662
#696402
#695892
#689152
#688662
D3×#3362
#686362
#236273