Finite Group Databasebeta

← #10900972

Subgroups — #10900972

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic, complemented
2#82080C211yesmaximal, characteristic
3C2×#4737C212yesmaximal
4C3×#1817C211yesmaximal, characteristic, complemented
5#69926C212yesmaximal, complemented
6#68679C212yesmaximal, complemented
7#68678C211yesmaximal, characteristic, complemented
8#68678C211yesmaximal, characteristic, complemented
9#68634C212yesmaximal, complemented
10#68633C211yesmaximal, characteristic, complemented
11#68632C211yesmaximal, characteristic, complemented
12#29644—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C2×#929—21no—
16C2×#926C2212yes—
17C2×#924—21no—
18C2×#924—21no—
19#5470—28no—
20C2×#870C2212yes—
21C2×#869C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C4×Dic6—21no—
24C2×C4×Dic6—21no—
25C2×C3×C22⋊C8C2212yes—
26#5213—21no—
27#5212C2211yescharacteristic
28C2×#821C2212yes—
29C2×#821C2212yes—
30C2×#818C2212yes—
31C2×#817C2212yes—
32#4940—21no—
33#4940—21no—
34#4939C2211yescharacteristic
35#4939C2211yescharacteristic
36#4928C2211yescharacteristic
37#4928C2211yescharacteristic
38#4737—28no—
39C3×#336C2211yescharacteristic
40C3×#336C2211yescharacteristic
41C3×#336C2212yes—
42#4551C2212yes—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#533—31no—
47C2×#482—32no—
48#2854—31no—
49#2059—32no—
50#2058—31no—
51#2057—31no—
52#2025—32no—
53#2023—31no—
54#2023—31no—
55#1825—32no—
56#1817D311yescharacteristic, complemented
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—21no—
61C22×D12C2311yescharacteristic
62C22×Dic6—21no—
63C22×Dic6—21no—
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C2×C3×C42—24no—
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#217—21no—
73#930—24no—
74#929—44no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—21no—
80#926—24no—
81#926D418yes—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2312yes—
89#924—28no—
90#924—28no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—21no—
94C2×Dic3⋊C4—21no—
95C2×C4×Dic3—22no—
96#870—24no—
97#870—24no—
98#869—24no—
99#869—24no—
100C4×Dic6—28no—
101C4×Dic6—28no—
102C3×C22⋊C8—24no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106#821—24no—
107#821—24no—
108#818—24no—
109#817—24no—
110#533—68no—
111C2×C4⋊Q8—31no—
112C2×C4⋊Q8—31no—
113C2×#125—32no—
114C2×#123—61no—
115C2×C22⋊Q8—61no—
116C2×C22⋊Q8—61no—
117C2×#121—31no—
118C2×C4×Q8—61no—
119C2×C4×Q8—61no—
120C2×#117—32no—
121C22×C42C2×D311yescharacteristic
122#482—68no—
123C2×#107—32no—
124C2×#106—32no—
125C2×#102—32no—
126C2×#102—32no—
127C2×C22⋊C8C2×D312yes—
128#390—31no—
129#390—31no—
130#389—31no—
131#389—31no—
132#388—61no—
133#388—61no—
134#386—61no—
135#380—31no—
136#340—32no—
137#340—32no—
138#340—32no—
139#340—32no—
140#336C2×D311yescharacteristic
141#336C2×D311yescharacteristic
142#336C2×D312yes—
143C24×C3C2×D411yescharacteristic
144C23×D3—21no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—41no—
153C22×C3×C4—41no—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D414yes—
160C22×C3×C4C2411yescharacteristic
161C2×#67—42no—
162C2×#67—42no—
163C2×#67—42no—
164C22×Dic3—22no—
165C22×Dic3—22no—
166C22×Dic3—21no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—21no—
172C2×D12—21no—
173C2×Dic6—24no—
174C2×Dic6—24no—
175C2×Dic6—42no—
176C2×Dic6—42no—
177C2×Dic6—42no—
178C2×Dic6—42no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—22no—
189#217—28no—
190#217—42no—
191(C2×D3)⋊C4—42no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—42no—
194(C2×D3)⋊C4—42no—
195(C2×D3)⋊C4—42no—
196C4⋊Dic3—24no—
197C4⋊Dic3—24no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3C2×D414yes—
205C4⋊Dic3C2×D414yes—
206Dic3⋊C4—28no—
207Dic3⋊C4—28no—
208Dic3⋊C4—42no—
209Dic3⋊C4—42no—
210C4×Dic3—28no—
211C22×Q8—61no—
212C22×Q8—61no—
213C22×D4—31no—
214C22×D4—61no—
215C23×C4D1211yescharacteristic
216C23×C4D1211yescharacteristic
217C23×C4C22×D311yescharacteristic
218C22×C8C22×D312yes—
219C22×C8C22×D312yes—
220C4⋊Q8—64no—
221C4⋊Q8—64no—
222#125—64no—
223#125—64no—
224#123—124no—
225C22⋊Q8—68no—
226C22⋊Q8—68no—
227#121—64no—
228C4×Q8—68no—
229C4×Q8—68no—
230#117—38no—
231#117—64no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—32no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—62no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C22⋊C4—61no—
248C2×C22⋊C4—61no—
249C2×C22⋊C4—61no—
250C2×C42—24no—
251C2×C42—62no—
252C2×C42D1211yescharacteristic
253C2×C42D1211yescharacteristic
254C2×C42C22×D311yescharacteristic
255C2×C42C22×D311yescharacteristic
256#107—64no—
257#106—64no—
258#102—64no—
259#102—64no—
260C22⋊C8—24no—
261#95—61no—
262#95—61no—
263#95—61no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3#12012yes—
271C23×C3C22×D411yescharacteristic
272C22×D3—41no—
273C22×D3—41no—
274C22×D3—41no—
275C22×D3—41no—
276C22×D3—41no—
277C22×D3—41no—
278C22×D3—41no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—44no—
294C2×C3×C4—44no—
295C2×C3×C4#12012yes—
296C2×C3×C4#13512yes—
297C2×C3×C4AGL(1,ℤ/8)12yes—
298C2×C3×C4C22×D412yes—
299C2×C3×C42−1+412yes—
300C2×C3×C42−1+411yescharacteristic
301C2×C3×C42−1+411yescharacteristic
302#67—84no—
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318D12—42no—
319D12—42no—
320Dic6—48no—
321Dic6—48no—
322C3×C8—42no—
323C3×C8—42no—
324C24—61no—
325C24C2×D1211yescharacteristic
326C2×Q8—64no—
327C2×Q8—64no—
328C2×Q8—122no—
329C2×Q8—122no—
330C2×Q8—122no—
331C2×Q8—122no—
332C2×D4—62no—
333C2×D4—62no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4—122no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4—61no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4C2×D1211yescharacteristic
354C22×C4C2×D1211yescharacteristic
355C22×C4C2×D1211yescharacteristic
356C22×C4C2×D1211yescharacteristic
357C22×C4C2×D1211yescharacteristic
358C22×C4D3×D414yes—
359C22×C4C23×D311yescharacteristic
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C4⋊C4—34no—
367C4⋊C4—34no—
368C4⋊C4—68no—
369C4⋊C4—68no—
370C4⋊C4—64no—
371C4⋊C4—64no—
372C4⋊C4—62no—
373C4⋊C4—62no—
374C4⋊C4—62no—
375C4⋊C4—62no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C4⋊C4—122no—
379C4⋊C4—122no—
380C22⋊C4—68no—
381C22⋊C4—122no—
382C22⋊C4—122no—
383C22⋊C4—122no—
384C22⋊C4—122no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—68no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3C2×#12011yescharacteristic
400C22×C3#53611yescharacteristic
401C22×C3#53611yescharacteristic
402C22×C3C2×#13511yescharacteristic
403C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
404C22×C3#57511yescharacteristic
405C22×C3#57511yescharacteristic
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C2×D3—81no—
411C2×D3—81no—
412C2×D3—81no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4#53612yes—
419C3×C4#53612yes—
420Dic3—44no—
421Dic3—44no—
422Dic3—81no—
423Dic3—81no—
424Dic3—81no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—121no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23—121no—
436C23—121no—
437C23—121no—
438C23#88212yes—
439C23C22×D1211yescharacteristic
440Q8—128no—
441Q8—128no—
442D4—122no—
443D4—122no—
444D4—244no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4#88212yes—
477C2×C4#90412yes—
478C2×C4#90812yes—
479C2×C4C2×D3×D412yes—
480C2×C4#101012yes—
481C2×C4#101011yescharacteristic
482C2×C4#101011yescharacteristic
483C8—42no—
484C8—42no—
485C2×C3—42no—
486C2×C3#261411yescharacteristic
487C2×C3#300211yescharacteristic
488C2×C3#300611yescharacteristic
489C2×C3#322811yescharacteristic
490C2×C3#323311yescharacteristic
491C2×C3#323511yescharacteristic
492C2×C3#324011yescharacteristic
493D3—161no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—241no—
502C22—241no—
503C22—241no—
504C22—241no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22C2×#88211yescharacteristic
509C22#552611yescharacteristic
510C22#552611yescharacteristic
511C22C2×#90411yescharacteristic
512C22C2×#90811yescharacteristic
513C22#574811yescharacteristic
514C22#574811yescharacteristic
515C4—22no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—124no—
521C4—124no—
522C4—241no—
523C4—241no—
524C4—241no—
525C4#564812yes—
526C4#564812yes—
527C3#2964411yescharacteristic, complemented, Sylow
528C2—42no—
529C2—481no—
530C2#7413511yescharacteristic
531C2#7861011yescharacteristic
532C2#7863911yescharacteristic
533C2#7927011yescharacteristic
534C2#7927511yescharacteristic
535C2#7931911yescharacteristic
536C2#7932411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1817C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1817D3
hypocenterC3#29644
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
4. C3×#1817C2semidirect
5. #69926C2semidirect
6. #68679C2semidirect
7. #68678C2semidirect
8. #68678C2semidirect
9. #68634C2semidirect
10. #68633C2semidirect
11. #68632C2semidirect
56. #1817D3semidirect
527. C3#29644semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
#820802
C2×#47372
C3×#18172
#699262
#686792
#686782
#686342
#686332
#686322
#296443