Finite Group Databasebeta

← #10945033

Subgroups — #10945033

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5504C211yesmaximal, characteristic
2C2×#4970C211yesmaximal, characteristic, complemented
3C2×#4733C211yesmaximal, characteristic
4#73951C211yesmaximal, characteristic, complemented
5C3×#1823C211yesmaximal, characteristic, complemented
6#69911C211yesmaximal, characteristic, complemented
7#69522C211yesmaximal, characteristic
8#68908C212yesmaximal, complemented
9#68906C212yesmaximal, complemented
10#68896C212yesmaximal, complemented
11#68894C212yesmaximal, complemented
12#33017—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C2×C4×#67—22no—
16C2×#926C2211yescharacteristic
17C2×#924C2211yescharacteristic
18C22×Dic12C2211yescharacteristic
19C22×#204C2211yescharacteristic
20#5504—24no—
21#5504—24no—
22C2×#893C2212yes—
23C2×#892—21no—
24C2×D3×C4⋊C4—21no—
25C2×#887C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×#380C2211yescharacteristic
28C2×#830C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#816C2211yescharacteristic
31#4984C2212yes—
32#4973C2212yes—
33#4970—24no—
34C2×#810C2212yes—
35C2×#810C2212yes—
36C2×#809C2212yes—
37C2×#807C2212yes—
38#4953—22no—
39#4733—24no—
40#4661—21no—
41#4659—21no—
42C3×#340C2212yes—
43C3×#340C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#520—31no—
47C2×#484—31no—
48C2×#452—31no—
49#2625—31no—
50#1878—32no—
51#1877—32no—
52#1863—32no—
53#1862—32no—
54#1823D311yescharacteristic, complemented
55#1818—31no—
56#1788—31no—
57C23×C3×C4—21no—
58C23×C3×C4C2311yescharacteristic
59C22×#67—21no—
60C2×#235—22no—
61C2×#235—24no—
62C2×#235—24no—
63C2×#235D414yes—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×C4×D3—22no—
67C22×Dic6C2311yescharacteristic
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D414yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×#217—21no—
78C2×#217D412yes—
79C4×#67—216no—
80C2×(C2×D3)⋊C4—22no—
81#926—22no—
82#926—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3D411yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87#924—22no—
88#924—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3C2312yes—
93C22×C3⋊C8C2312yes—
94C2×Dic12—22no—
95C2×Dic12—24no—
96C2×#204—22no—
97C2×#204—24no—
98#893—24no—
99#893—24no—
100#892—28no—
101D3×C4⋊C4—28no—
102#887—24no—
103#887—24no—
104C3×C22⋊C8—22no—
105C3×#95—21no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109#830—22no—
110#821—22no—
111#816—22no—
112#810—24no—
113#810—24no—
114#809—24no—
115#807—24no—
116C22×#41—31no—
117C22×Q16—31no—
118C22×SD16—31no—
119C2×C22⋊Q8—31no—
120#520—64no—
121#520—64no—
122C2×C4×Q8—32no—
123C2×C4×D4—32no—
124C2×C4×D4—62no—
125C2×#117—31no—
126C2×#117—61no—
127C22×C4⋊C4—61no—
128C22×C4⋊C4C2×D311yescharacteristic
129#484—64no—
130#452—64no—
131C2×#107—32no—
132C2×#103—31no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#102—31no—
136C2×#102—32no—
137C2×C22⋊C8—31no—
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380C2×D311yescharacteristic
145#340C2×D312yes—
146#340C2×D312yes—
147#336—32no—
148#336—32no—
149C24×C3C2×D411yescharacteristic
150C23×D3—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—24no—
155C22×C3×C4—22no—
156C22×C3×C4—81no—
157C22×C3×C4—81no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2411yescharacteristic
162C2×#67—28no—
163C2×#67—24no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3C2×D412yes—
168C22×Dic3C2×D412yes—
169#235—216no—
170#235—28no—
171#235—28no—
172C2×D12—21no—
173C2×D12—22no—
174C2×D12C2×D412yes—
175C2×D12#4114yes—
176C2×C4×D3—28no—
177C2×C4×D3—24no—
178C2×C4×D3—28no—
179C2×C4×D3—82no—
180C2×C4×D3—82no—
181C2×Dic6—21no—
182C2×Dic6—22no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6C2×D412yes—
188C2×Dic6#4114yes—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4C2×D414yes—
198C3×C22⋊C4—81no—
199#217—24no—
200#217—42no—
201(C2×D3)⋊C4—28no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—82no—
207Dic3⋊C4—28no—
208Dic3⋊C4—42no—
209C4×Dic3—28no—
210C4×Dic3—22no—
211C4×Dic3—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215Dic12—44no—
216#204—44no—
217C2×#41—34no—
218C2×#41—64no—
219C2×#41—64no—
220C2×#41—62no—
221C22×Q8—31no—
222C22×Q8—31no—
223C22×D4—31no—
224C22×D4—61no—
225C23×C4—21no—
226C23×C4—62no—
227C23×C4—61no—
228C23×C4C22×D311yescharacteristic
229C2×Q16—64no—
230C2×Q16—62no—
231C2×SD16—64no—
232C2×SD16—62no—
233C22×C8—32no—
234C22×C8C22×D311yescharacteristic
235C22×C8C22×D311yescharacteristic
236C22⋊Q8—62no—
237C22⋊Q8—62no—
238C4×Q8—64no—
239C4×Q8—64no—
240C4×D4—64no—
241C4×D4—616no—
242C4×D4—64no—
243#117—68no—
244#117—62no—
245#117—62no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—68no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4D1214yes—
256C2×C4⋊C4C22×D312yes—
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—61no—
262C2×C42—32no—
263C2×C42—62no—
264#107—64no—
265#103—64no—
266#103—62no—
267#103—64no—
268#102—64no—
269#102—62no—
270C22⋊C8—22no—
271C22⋊C8—62no—
272#95—21no—
273#95—62no—
274#95—62no—
275#95—62no—
276C23×C3—22no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—22no—
280C23×C3#12012yes—
281C23×C3C22×D411yescharacteristic
282C22×D3—22no—
283C22×D3—24no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—42no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—84no—
298C2×C3×C4#12012yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×C3×C4C2×#4112yes—
303#67—216no—
304C2×Dic3—22no—
305C2×Dic3—24no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—24no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316D12—22no—
317D12—24no—
318D12#12314yes—
319C4×D3—216no—
320C4×D3—88no—
321Dic6—22no—
322Dic6—24no—
323Dic6—41no—
324Dic6—41no—
325Dic6—42no—
326Dic6#12314yes—
327C3×C8—41no—
328C3×C8—41no—
329C3⋊C8—42no—
330C24—61no—
331C24D3×D411yescharacteristic
332#41—68no—
333#41—68no—
334#41—616no—
335C2×Q8—32no—
336C2×Q8—34no—
337C2×Q8—61no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—62no—
342C2×Q8—62no—
343C2×D4—32no—
344C2×D4—34no—
345C2×D4—68no—
346C2×D4—64no—
347C2×D4—61no—
348C2×D4—62no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—32no—
355C22×C4—32no—
356C22×C4—68no—
357C22×C4—68no—
358C22×C4—61no—
359C22×C4—64no—
360C22×C4—61no—
361C22×C4—61no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—242no—
365C22×C4—242no—
366C22×C4C2×D1212yes—
367C22×C4D3×D411yescharacteristic
368C22×C4D3×D412yes—
369C22×C4C23×D311yescharacteristic
370Q16—124no—
371SD16—124no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—68no—
384C4⋊C4—62no—
385C4⋊C4—62no—
386C4⋊C4—61no—
387C4⋊C4—61no—
388C4⋊C4—122no—
389C4⋊C4—242no—
390C4⋊C4C2×D1214yes—
391C22⋊C4—68no—
392C22⋊C4—64no—
393C22⋊C4—81no—
394C22⋊C4—122no—
395C42—68no—
396C42—62no—
397C42—62no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#52512yes—
405C22×C3#57611yescharacteristic
406C22×C3#57711yescharacteristic
407C22×C3#57812yes—
408C2×D3—24no—
409C2×D3—44no—
410C2×D3—44no—
411C2×D3—42no—
412C2×D3—42no—
413C3×C4—22no—
414C3×C4—44no—
415C3×C4—82no—
416C3×C4C2×#12311yescharacteristic
417C3×C4C2×#12311yescharacteristic
418C3×C4#52512yes—
419Dic3—24no—
420Dic3—42no—
421Dic3—42no—
422Dic3—81no—
423Dic3—81no—
424C23—22no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—64no—
429C23—62no—
430C23—121no—
431C23—121no—
432C23—122no—
433C23#88212yes—
434C23C2×D3×D411yescharacteristic
435Q8—34no—
436Q8—62no—
437Q8—64no—
438Q8—121no—
439Q8—121no—
440Q8—122no—
441D4—34no—
442D4—616no—
443D4—62no—
444D4—64no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—616no—
456C2×C4—64no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—64no—
463C2×C4—84no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—248no—
470C2×C4#88212yes—
471C2×C4C22×D1211yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4D3×#4112yes—
474C2×C4D3×#4112yes—
475C8—41no—
476C8—41no—
477C8—122no—
478C2×C3—42no—
479C2×C3#284911yescharacteristic
480C2×C3#301111yescharacteristic
481C2×C3#301211yescharacteristic
482C2×C3#312012yes—
483C2×C3#312412yes—
484D3—44no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—64no—
491C22—124no—
492C22—122no—
493C22—124no—
494C22—122no—
495C22C2×#88211yescharacteristic
496C22#565612yes—
497C22#575711yescharacteristic
498C22#576211yescharacteristic
499C22#577912yes—
500C4—22no—
501C4—44no—
502C4—64no—
503C4—82no—
504C4—122no—
505C4—122no—
506C4—241no—
507C4—241no—
508C4#552412yes—
509C4D3×#12311yescharacteristic
510C4D3×#12311yescharacteristic
511C3#3301711yescharacteristic, complemented, Sylow
512C2—42no—
513C2—124no—
514C2#7414611yescharacteristic
515C2#7874711yescharacteristic
516C2#7896111yescharacteristic
517C2#8127312yes—
518C2#8141612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1823C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#1823D3
hypocenterC3#33017
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
2. C2×#4970C2semidirect
4. #73951C2semidirect
5. C3×#1823C2semidirect
6. #69911C2semidirect
8. #68908C2semidirect
9. #68906C2semidirect
10. #68896C2semidirect
11. #68894C2semidirect
54. #1823D3semidirect
511. C3#33017semidirect

Maximal subgroups

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

subgroupindex
C2×#55042
C2×#49702
C2×#47332
#739512
C3×#18232
#699112
#695222
#689082
#689062
#688962
#688942
#330173