Finite Group Databasebeta

← #10831718

Subgroups — #10831718

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5543C211yesmaximal, characteristic, complemented
2C2×#5014C211yesmaximal, characteristic
3C2×#4846C211yesmaximal, characteristic, complemented
4#74190C211yesmaximal, characteristic, complemented
5C3×#1804C211yesmaximal, characteristic, complemented
6#70473C211yesmaximal, characteristic, complemented
7#70172C212yesmaximal
8#70168C212yesmaximal, complemented
9#69555C211yesmaximal, characteristic, complemented
10#68899C212yesmaximal, complemented
11#68829C212yesmaximal, complemented
12#24339—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×D4×Dic3C2212yes—
15C22×#213C2211yescharacteristic
16C22×C4⋊Dic3—21no—
17C2×#924—21no—
18#5543—24no—
19#5543—24no—
20C2×#890C2212yes—
21C2×#878—22no—
22C2×C4×D12C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#102C2212yes—
27C3×#382C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#821C2212yes—
30C2×#818C2212yes—
31#5014—24no—
32#4988C2212yes—
33#4984C2212yes—
34C2×#809C2211yescharacteristic
35C2×#808C2211yescharacteristic
36C2×#803C2211yescharacteristic
37#4846—24no—
38#4677—21no—
39#4673—21no—
40#4668—22no—
41C3×#340C2212yes—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#442—31no—
49#2580—31no—
50#2059—32no—
51#2025—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1862—32no—
56#1804D311yescharacteristic, complemented
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×Dic3—22no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×Dic6—21no—
63C22×Dic6D411yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4×D4—22no—
66C3×C4×D4D414yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72D4×Dic3—24no—
73D4×Dic3—24no—
74C2×#213—22no—
75C2×#213—22no—
76C2×#213Q814yescomplemented
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—28no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2312yes—
87#924—28no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4D412yes—
90C2×C4×Dic3C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93#890—24no—
94#890—24no—
95#878—216no—
96C4×D12—22no—
97C4×D12—22no—
98#869—22no—
99#869D414yes—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C3×C4⋊C8—22no—
103C3×#102—24no—
104C3×#95—21no—
105C3×#95—21no—
106#832—22no—
107C3⋊#95—22no—
108#821—24no—
109#818—24no—
110#809—22no—
111#808—22no—
112#803—22no—
113C22×D8—31no—
114#554—64no—
115#554—64no—
116C2×C4⋊Q8—31no—
117C2×#125—32no—
118C2×C22⋊Q8—62no—
119C2×C22⋊Q8—61no—
120C2×C4×Q8—31no—
121C2×C4×D4—32no—
122C2×C4×D4—31no—
123C2×C4×D4C2×D311yescharacteristic
124C22×C4⋊C4—61no—
125#474—64no—
126#442—64no—
127C2×#107—32no—
128C2×#106—31no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#102—31no—
131C2×#102—32no—
132C2×#102—31no—
133C2×#102C2×D312yes—
134C2×#97—31no—
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382C2×D311yescharacteristic
141#340—32no—
142#340—32no—
143#340C2×D312yes—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C23×D3—21no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4C2×D412yes—
152C2×C3×D4C2×Q812yes—
153C2×C3×D4#4112yes—
154C22×C3×C4—21no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—44no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—22no—
162C22×Dic3—28no—
163C22×Dic3—21no—
164C22×Dic3—82no—
165C22×Dic3—82no—
166C22×Dic3C2×D412yes—
167C22×Dic3C2×D412yes—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12C2×Q812yes—
174C2×C4×D3—42no—
175C2×C4×D3—82no—
176C2×Dic6—22no—
177C2×Dic6—24no—
178C2×Dic6—42no—
179C2×Dic6—82no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4C2×D412yes—
187C3×C22⋊C4—24no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190#217—28no—
191#213—28no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—42no—
194(C2×D3)⋊C4—82no—
195C4⋊Dic3—28no—
196C4⋊Dic3—22no—
197C4⋊Dic3—24no—
198C4⋊Dic3—22no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201C4⋊Dic3—82no—
202C4⋊Dic3C2×D414yes—
203Dic3⋊C4—28no—
204Dic3⋊C4—24no—
205C4×Dic3—22no—
206C4×Dic3—22no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8C2×Q812yes—
213C22×Q8—31no—
214C22×Q8—61no—
215C22×D4—31no—
216C22×D4C22×D311yescharacteristic
217C23×C4—21no—
218C23×C4—62no—
219C23×C4—61no—
220C2×D8—34no—
221C2×D8—62no—
222C2×D8—62no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8C22×D312yes—
226C4⋊Q8—34no—
227C4⋊Q8—62no—
228#125—64no—
229#125—64no—
230C22⋊Q8—616no—
231C22⋊Q8—68no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×D4—22no—
235C4×D4—64no—
236C4×D4—64no—
237C4×D4—62no—
238C4×D4—62no—
239C4×D4D1214yes—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—68no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4C22×D312yes—
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—61no—
255C2×C22⋊C4—62no—
256C2×C42—32no—
257C2×C42C22×D311yescharacteristic
258#107—64no—
259#106—62no—
260C4⋊C8—22no—
261#102—24no—
262#102—64no—
263#102—62no—
264#102—62no—
265#97—62no—
266#95—21no—
267#95—21no—
268#95—62no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3C22×D411yescharacteristic
275C22×D3—22no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—42no—
279C22×D3—42no—
280C3×D4—22no—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4C22⋊Q814yes—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—48no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4#12012yes—
293C2×C3×C4#12012yes—
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×Q811yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C42−1+411yescharacteristic
299C2×C3×C42−1+411yescharacteristic
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—24no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—42no—
311C2×Dic3—88no—
312D12—22no—
313D12—22no—
314D12—41no—
315D12—41no—
316C4×D3—84no—
317Dic6—44no—
318Dic6—84no—
319C3×C8—42no—
320C3⋊C8—22no—
321C3⋊C8—41no—
322C24—21no—
323C24—61no—
324C2×Q8—62no—
325C2×Q8—64no—
326C2×Q8—122no—
327C2×Q8—242no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—32no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4C2×D1212yes—
338C2×D4#23712yes—
339C2×D4D3×Q812yes—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—32no—
344C22×C4—32no—
345C22×C4—44no—
346C22×C4—62no—
347C22×C4—68no—
348C22×C4—61no—
349C22×C4—122no—
350C22×C4—242no—
351C22×C4—242no—
352C22×C4—242no—
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4C23×D311yescharacteristic
356D8—68no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—32no—
361C2×C8—61no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8—61no—
365C2×C8—61no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—34no—
370C4⋊C4—68no—
371C4⋊C4—68no—
372C4⋊C4—62no—
373C4⋊C4—64no—
374C4⋊C4—64no—
375C4⋊C4—62no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C4⋊C4—242no—
379C4⋊C4C2×D1212yes—
380C22⋊C4—24no—
381C22⋊C4—68no—
382C22⋊C4—124no—
383C22⋊C4—122no—
384C22⋊C4—242no—
385C42—62no—
386C42—62no—
387C42C2×D1212yes—
388C42D3×D412yes—
389C22×C3—24no—
390C22×C3—44no—
391C22×C3—44no—
392C22×C3—42no—
393C22×C3—42no—
394C22×C3C2×#12011yescharacteristic
395C22×C3#54811yescharacteristic
396C22×C3D4×Q811yescharacteristic
397C22×C3#57512yes—
398C22×C3#57612yes—
399C2×D3—44no—
400C2×D3—42no—
401C2×D3—82no—
402C2×D3—81no—
403C2×D3—81no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—82no—
408C3×C4C2×C22⋊Q811yescharacteristic
409C3×C4#53611yescharacteristic
410C3×C4#54811yescharacteristic
411C3×C4D4×Q811yescharacteristic
412Dic3—42no—
413Dic3—44no—
414Dic3—42no—
415Dic3—81no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—42no—
420C23—42no—
421C23—62no—
422C23—121no—
423C23—122no—
424C23—122no—
425C23—121no—
426C23C2×D3×D411yescharacteristic
427Q8—124no—
428Q8—244no—
429D4—22no—
430D4—22no—
431D4—22no—
432D4—62no—
433D4—62no—
434D4—121no—
435D4—121no—
436D4#89714yes—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—64no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—122no—
453C2×C4—122no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—244no—
457C2×C4—248no—
458C2×C4#88212yes—
459C2×C4#88212yes—
460C2×C4C22×D1211yescharacteristic
461C2×C4C2×D3×D411yescharacteristic
462C2×C4C2×#23711yescharacteristic
463C2×C4C2×D3×Q811yescharacteristic
464C2×C4#101011yescharacteristic
465C2×C4#101011yescharacteristic
466C8—42no—
467C8—62no—
468C8—121no—
469C2×C3—44no—
470C2×C3#240911yescharacteristic
471C2×C3#300511yescharacteristic
472C2×C3#300811yescharacteristic
473C2×C3#337012yes—
474C2×C3#338912yes—
475D3—82no—
476C22—24no—
477C22—44no—
478C22—42no—
479C22—44no—
480C22—42no—
481C22—124no—
482C22—122no—
483C22—241no—
484C22—242no—
485C22—241no—
486C22C2×#88211yescharacteristic
487C22#559811yescharacteristic
488C22#563211yescharacteristic
489C22#576912yes—
490C22#577312yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—82no—
495C4—122no—
496C4—122no—
497C4—124no—
498C4—241no—
499C4C2×#89711yescharacteristic
500C4#552611yescharacteristic
501C4#559811yescharacteristic
502C4#563211yescharacteristic
503C3#2433911yescharacteristic, complemented, Sylow
504C2—44no—
505C2—242no—
506C2#7401711yescharacteristic
507C2#7883411yescharacteristic
508C2#7892711yescharacteristic
509C2#8052012yes—
510C2#8116312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1804C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#1804D3
hypocenterC3#24339
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1804

Complements

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

NH ≅ G/Ntype
1. C2×#5543C2semidirect
3. C2×#4846C2semidirect
4. #74190C2semidirect
5. C3×#1804C2semidirect
6. #70473C2semidirect
8. #70168C2semidirect
9. #69555C2semidirect
10. #68899C2semidirect
11. #68829C2semidirect
56. #1804D3semidirect
76. C2×#213Q8semidirect
503. C3#24339semidirect

Maximal subgroups

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

subgroupindex
C2×#55432
C2×#50142
C2×#48462
#741902
C3×#18042
#704732
#701722
#701682
#695552
#688992
#688292
#243393