Finite Group Databasebeta

← #10831733

Subgroups — #10831733

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic
3#82501C211yesmaximal, characteristic
4C2×#5026C211yesmaximal, characteristic
5#71792C212yesmaximal
6#71788C211yesmaximal, characteristic, complemented
7#71782C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71633C212yesmaximal
10#71628C212yesmaximal, complemented
11#71627C211yesmaximal, characteristic, complemented
12#24341—31nomaximal, Sylow
13C3×Q82—24no—
14C3×Q82—24no—
15C2×C3×C4⋊Q8—22no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C22×#215C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25#5231—21no—
26C2×#835C2211yescharacteristic
27C2×#835C2212yes—
28#5087—24no—
29#5026—24no—
30#4993—21no—
31#4989—22no—
32C2×#809C2211yescharacteristic
33C2×#809C2212yes—
34C2×#807C2211yescharacteristic
35C2×#807C2212yes—
36#4939C2212yes—
37#4936C2211yescharacteristic
38#4928C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×Q82D311yescharacteristic, complemented
47C2×#474—31no—
48C2×#440—31no—
49#2580—31no—
50#2057—32no—
51#2023—32no—
52#1993—31no—
53#1985—31no—
54#1880—32no—
55#1863—32no—
56#1805—31no—
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C3×C4⋊Q8—216no—
64C3×C4⋊Q8—28no—
65C3×C4⋊Q8—24no—
66C3×C4⋊Q8—24no—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yes—
69C3×C4×Q8—28no—
70C3×C4×Q8—24no—
71C3×C4×Q8—24no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×Q8D414yes—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—22no—
85C2×C3×C42—21no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C2×#215—22no—
89C2×#215—22no—
90C2×#215Q814yescomplemented
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96C2×Dic3⋊C4—22no—
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C4×D12—22no—
101C4×D12—22no—
102#835—22no—
103#835—24no—
104C3⋊#95—22no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107#809—22no—
108#809—24no—
109#807—22no—
110#807—24no—
111C4⋊(C3⋊C8)—22no—
112#803—22no—
113C22×SD16—31no—
114Q82—24no—
115Q82—24no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2×D312yes—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8C2×D312yes—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×Q8C2×D311yescharacteristic
124C2×C4×D4—31no—
125#474—64no—
126#440—64no—
127C2×#107—31no—
128C2×#107—32no—
129C2×C4⋊C8—31no—
130C2×#103—32no—
131C2×#103—31no—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97—31no—
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C23×D3—21no—
147C2×C3×Q8—28no—
148C2×C3×Q8—22no—
149C2×C3×Q8—24no—
150C2×C3×Q8—24no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—82no—
155C2×C3×Q8—82no—
156C2×C3×Q8—82no—
157C2×C3×Q8C2×D412yes—
158C2×C3×Q8C2×Q812yes—
159C2×C3×Q8#4112yes—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—22no—
169C22×Dic3—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12C2×Q812yes—
176C2×C4×D3—42no—
177C2×C4×D3—82no—
178C3×C4⋊C4—28no—
179C3×C4⋊C4—28no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4C2×D414yes—
187C3×C4⋊C4C2×D412yes—
188C3×C42—28no—
189C3×C42—24no—
190C3×C42—22no—
191C3×C42—22no—
192C3×C42C2×D412yes—
193C3×C42C2×D412yes—
194#215—28no—
195(C2×D3)⋊C4—42no—
196(C2×D3)⋊C4—44no—
197(C2×D3)⋊C4—82no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202Dic3⋊C4—82no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8C2×Q812yes—
212C22×Q8—22no—
213C22×Q8—21no—
214C22×Q8#6711yescharacteristic
215C22×Q8C22×D311yescharacteristic
216C22×D4—31no—
217C23×C4—61no—
218C2×SD16—34no—
219C2×SD16—62no—
220C2×SD16—62no—
221C22×C8—31no—
222C22×C8—31no—
223C22×C8—32no—
224C4⋊Q8—216no—
225C4⋊Q8—28no—
226C4⋊Q8—24no—
227C4⋊Q8—24no—
228C4⋊Q8—22no—
229C4⋊Q8#6714yes—
230C4×Q8—28no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8#6714yes—
237C4×D4—62no—
238C4×D4—62no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4#6712yes—
247C2×C4⋊C4#6711yescharacteristic
248C2×C4⋊C4C22×D312yes—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C22⋊C4—61no—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—62no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42C22×D312yes—
256C2×C42C22×D311yescharacteristic
257#107—62no—
258#107—64no—
259C4⋊C8—62no—
260#103—64no—
261#103—62no—
262#102—62no—
263#102—64no—
264#97—62no—
265#95—61no—
266#95—61no—
267#95—62no—
268C23×C3C22×D411yescharacteristic
269C22×D3—22no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—42no—
273C22×D3—42no—
274C3×Q8—28no—
275C3×Q8—22no—
276C3×Q8—22no—
277C3×Q8—22no—
278C3×Q8—84no—
279C3×Q8—88no—
280C3×Q8C22⋊Q814yes—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4#12012yes—
297C2×C3×C4#12012yes—
298C2×C3×C4#13714yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×Q811yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C42−1+411yescharacteristic
304C2×C3×C42−1+411yescharacteristic
305C2×Dic3—41no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310D12—22no—
311D12—22no—
312D12—41no—
313D12—41no—
314C4×D3—84no—
315C3⋊C8—22no—
316C3⋊C8—41no—
317C3⋊C8—42no—
318C24—61no—
319C2×Q8—28no—
320C2×Q8—24no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8D3×Q812yes—
330C2×Q8#23912yes—
331C2×Q8C2×#6712yes—
332C2×D4—32no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—61no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—62no—
342C22×C4—61no—
343C22×C4—122no—
344C22×C4—242no—
345C22×C4C2×#6712yes—
346C22×C4C2×#6712yes—
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C22×C4C23×D311yescharacteristic
350SD16—68no—
351C2×C8—32no—
352C2×C8—61no—
353C2×C8—61no—
354C2×C8—62no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—62no—
358C2×C8—61no—
359C2×C8—62no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—62no—
369C4⋊C4—62no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—242no—
373C4⋊C4C2×#6714yes—
374C4⋊C4C2×#6712yes—
375C22⋊C4—124no—
376C22⋊C4—122no—
377C22⋊C4—242no—
378C42—28no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42C2×#6712yes—
383C42C2×#6712yes—
384C22×C3C2×#12011yescharacteristic
385C22×C3#54811yescharacteristic
386C22×C3D4×Q811yescharacteristic
387C22×C3C2×#13712yes—
388C22×C3#57612yes—
389C2×D3—44no—
390C2×D3—42no—
391C2×D3—82no—
392C2×D3—81no—
393C2×D3—81no—
394C3×C4—24no—
395C3×C4—22no—
396C3×C4—24no—
397C3×C4—22no—
398C3×C4—42no—
399C3×C4—44no—
400C3×C4—42no—
401C3×C4C2×C22⋊Q811yescharacteristic
402C3×C4#53611yescharacteristic
403C3×C4#54811yescharacteristic
404C3×C4D4×Q811yescharacteristic
405Dic3—82no—
406Dic3—81no—
407C23—62no—
408C23—122no—
409C23—121no—
410C23—122no—
411C23—121no—
412C23C22×#6711yescharacteristic
413Q8—28no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—84no—
418Q8—88no—
419Q8#94614yes—
420D4—62no—
421D4—62no—
422D4—121no—
423D4—121no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—122no—
440C2×C4—122no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—121no—
444C2×C4—244no—
445C2×C4#94214yes—
446C2×C4#95312yes—
447C2×C4#95312yes—
448C2×C4C2×D3×Q811yescharacteristic
449C2×C4C2×#23911yescharacteristic
450C2×C4#100711yescharacteristic
451C2×C4#100711yescharacteristic
452C2×C4C22×#6711yescharacteristic
453C2×C4C22×#6711yescharacteristic
454C8—62no—
455C8—121no—
456C8—122no—
457C2×C3#240911yescharacteristic
458C2×C3#300411yescharacteristic
459C2×C3#300811yescharacteristic
460C2×C3#337512yes—
461C2×C3#338512yes—
462D3—82no—
463C22—122no—
464C22—124no—
465C22—241no—
466C22—241no—
467C22—242no—
468C22C2×#94212yes—
469C22Q8×#6711yescharacteristic
470C22#581711yescharacteristic
471C22#583512yes—
472C22C2×#95311yescharacteristic
473C4—24no—
474C4—22no—
475C4—24no—
476C4—22no—
477C4—42no—
478C4—44no—
479C4—42no—
480C4—241no—
481C4—242no—
482C4C2×#94611yescharacteristic
483C4Q8×#6711yescharacteristic
484C4#581611yescharacteristic
485C4#581711yescharacteristic
486C3#2434111yescharacteristic, complemented, Sylow
487C2—242no—
488C2#8370612yes—
489C2#8370812yes—
490C2#8405911yescharacteristic
491C2#8406811yescharacteristic
492C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×Q82C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×Q82D3
hypocenterC3#24341
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q82C2semidirect
6. #71788C2semidirect
7. #71782C2semidirect
8. #71635C2semidirect
10. #71628C2semidirect
11. #71627C2semidirect
46. C2×Q82D3semidirect
90. C2×#215Q8semidirect
486. C3#24341semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×#50872
#825012
C2×#50262
#717922
#717882
#717822
#716352
#716332
#716282
#716272
#243413