Finite Group Databasebeta

← #10849335

Subgroups — #10849335

Order 768.

1092 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2C3×#2567C211yesmaximal, characteristic, complemented
3C2×#5087C211yesmaximal, characteristic, complemented
4C2×#5089C211yesmaximal, characteristic, complemented
5#71635C211yesmaximal, characteristic, complemented
6#69631C212yesmaximal, complemented
7#69628C212yesmaximal, complemented
8#69627C212yesmaximal
9#69625C212yesmaximal, complemented
10#69488C211yesmaximal, characteristic, complemented
11#69474C211yesmaximal, characteristic, complemented
12#25660—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947—21no—
16#5585—24no—
17#5585—24no—
18C2×#894—21no—
19C2×#894—21no—
20C2×#893—21no—
21C2×#875C2211yescharacteristic
22C2×#875C2211yescharacteristic
23C2×#874C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C3×#395—21no—
28C3×#395—21no—
29C3×#391—21no—
30C3×#389C2212yes—
31C3×#389C2212yes—
32C3×#384—21no—
33C3×C4×C4⋊C4C2211yescharacteristic
34#5089—24no—
35#5087—24no—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2212yes—
39C2×#809C2212yes—
40C2×#809C2212yes—
41C2×#809C2212yes—
42C2×#807C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×#803C2211yescharacteristic
45#4529C2212yes—
46#4524C2212yes—
47#4524C2212yes—
48C2×#559—31no—
49C2×#489—31no—
50C2×#474—31no—
51#2567D311yescharacteristic, complemented
52#2042—32no—
53#2041—32no—
54#1985—31no—
55#1983—31no—
56#1863—32no—
57#1862—32no—
58#1762—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8D411yescharacteristic
61C22×D12—21no—
62C22×D12—21no—
63C22×D12C2311yescharacteristic
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C22×C4×D3—21no—
67C22×Dic6C2311yescharacteristic
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2311yescharacteristic
84#947—28no—
85C2×(C2×D3)⋊C4—21no—
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×Dic3⋊C4—21no—
93C2×C4×Dic3—21no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2312yes—
96#894—28no—
97#894—28no—
98#893—28no—
99#875—22no—
100#875—22no—
101#875—22no—
102#875D414yes—
103#874—22no—
104#874—22no—
105C4×D12—22no—
106C4×D12—22no—
107C4×D12—22no—
108C4×D12D414yes—
109C4×Dic6—22no—
110C4×Dic6—22no—
111C3×#95—21no—
112C3×#95—22no—
113C3×#95—22no—
114#810—24no—
115#810—24no—
116#809—24no—
117#809—24no—
118#809—24no—
119#809—24no—
120#807—24no—
121C4⋊(C3⋊C8)—22no—
122#803—22no—
123#559—64no—
124#559—64no—
125C2×C4⋊Q8C2×D311yescharacteristic
126C2×#127—31no—
127C2×#124—31no—
128C2×#124—31no—
129C2×#124—61no—
130C2×#123—61no—
131C2×#123—61no—
132C2×C4×Q8—31no—
133C2×C4×Q8C2×D311yescharacteristic
134C2×C4×D4—31no—
135C2×C4×D4—31no—
136C2×C4×D4—61no—
137#489—64no—
138#474—64no—
139C2×#107—32no—
140C2×C4⋊C8—31no—
141C2×#103—32no—
142C2×#103—32no—
143C2×#102—32no—
144C2×#102—32no—
145C2×#102—32no—
146C2×#102—32no—
147C2×#97—31no—
148#395—21no—
149#395—21no—
150#391—21no—
151#389C2×D312yes—
152#389C2×D312yes—
153#384—21no—
154C4×C4⋊C4C2×D311yescharacteristic
155#340—32no—
156#336—32no—
157#336—32no—
158C23×D3—21no—
159C23×D3—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—41no—
163C2×C3×Q8—42no—
164C2×C3×Q8—82no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—21no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—24no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—24no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12—82no—
187C2×D12C2×D412yes—
188C2×D12C2×D412yes—
189C2×C4×D3—24no—
190C2×C4×D3—24no—
191C2×C4×D3—82no—
192C2×C4×D3—82no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—21no—
196C2×Dic6C2×D412yes—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—82no—
205C3×C4⋊C4—82no—
206C3×C4⋊C4#4114yes—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42—21no—
211C3×C42—42no—
212C3×C42C2×D412yes—
213C3×C42C2×D412yes—
214(C2×D3)⋊C4—24no—
215(C2×D3)⋊C4—24no—
216(C2×D3)⋊C4—24no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—24no—
219(C2×D3)⋊C4—24no—
220C4⋊Dic3—21no—
221C4⋊Dic3C2×D412yes—
222Dic3⋊C4—24no—
223C4×Dic3—24no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—22no—
230C22×Q8—21no—
231C22×Q8—31no—
232C22×Q8#6711yescharacteristic
233C22×D4—31no—
234C22×D4—31no—
235C22×D4—61no—
236C22×D4—61no—
237C23×C4—61no—
238C23×C4—61no—
239C22×C8—32no—
240C22×C8—32no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243#127—62no—
244#127—62no—
245#124—34no—
246#124—62no—
247#124—62no—
248#124—68no—
249#124—62no—
250#123—68no—
251#123—68no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8—62no—
255C4×Q8—62no—
256C4×D4—34no—
257C4×D4—62no—
258C4×D4—62no—
259C4×D4—68no—
260C4×D4—62no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—31no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4#6711yescharacteristic
267C2×C4⋊C4#6711yescharacteristic
268C2×C4⋊C4#6711yescharacteristic
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D312yes—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273C2×C22⋊C4—61no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—61no—
276C2×C22⋊C4—61no—
277C2×C42—21no—
278C2×C42—61no—
279C2×C42C22×D311yescharacteristic
280C2×C42C22×D311yescharacteristic
281C2×C42C22×D311yescharacteristic
282#107—64no—
283C4⋊C8—62no—
284#103—64no—
285#103—64no—
286#102—64no—
287#102—64no—
288#102—64no—
289#102—64no—
290#97—62no—
291#95—21no—
292#95—22no—
293#95—22no—
294C23×C3C22×D411yescharacteristic
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—41no—
300C22×D3—41no—
301C22×D3—41no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—42no—
305C3×Q8—44no—
306C3×Q8—84no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—24no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4#12012yes—
326C2×C3×C4#12012yes—
327C2×C3×C4#13712yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C2×#4112yes—
331C2×C3×C42−1+412yes—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336D12—22no—
337D12—22no—
338D12—24no—
339D12—48no—
340D12—41no—
341D12—41no—
342D12—42no—
343D12—84no—
344C4×D3—84no—
345C4×D3—84no—
346Dic6—24no—
347Dic6—41no—
348Dic6—41no—
349C3⋊C8—42no—
350C3⋊C8—42no—
351C24—61no—
352C24—61no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—32no—
356C2×Q8—41no—
357C2×Q8—42no—
358C2×Q8—61no—
359C2×Q8—62no—
360C2×Q8—62no—
361C2×Q8—82no—
362C2×D4—32no—
363C2×D4—32no—
364C2×D4—64no—
365C2×D4—64no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—122no—
373C2×D4—122no—
374C2×D4—242no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—64no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—64no—
381C22×C4—242no—
382C22×C4—242no—
383C22×C4C2×#6711yescharacteristic
384C22×C4C2×#6711yescharacteristic
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6711yescharacteristic
388C22×C4C2×#6711yescharacteristic
389C22×C4C23×D311yescharacteristic
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8—62no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—42no—
404C4⋊C4—64no—
405C4⋊C4—61no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4#23914yes—
409C22⋊C4—64no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—64no—
414C22⋊C4—64no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—42no—
420C42—64no—
421C42D3×D412yes—
422C42D3×D412yes—
423C22×C3C2×#12011yescharacteristic
424C22×C3#54812yes—
425C22×C3C2×#13711yescharacteristic
426C22×C3#57511yescharacteristic
427C22×C3#57612yes—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—82no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—82no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—44no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—82no—
446C3×C4#53211yescharacteristic
447C3×C4#53611yescharacteristic
448C3×C4#54712yes—
449Dic3—42no—
450Dic3—42no—
451C23—62no—
452C23—62no—
453C23—121no—
454C23—121no—
455C23—122no—
456C23—122no—
457C23—121no—
458C23—121no—
459C23—122no—
460C23—122no—
461C23C22×#6711yescharacteristic
462Q8—44no—
463Q8—64no—
464Q8—84no—
465Q8—121no—
466Q8—121no—
467D4—62no—
468D4—62no—
469D4—64no—
470D4—128no—
471D4—121no—
472D4—121no—
473D4—122no—
474D4—244no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—244no—
498C2×C4—244no—
499C2×C4#93712yes—
500C2×C4#93712yes—
501C2×C4#94212yes—
502C2×C4C2×D3×D411yescharacteristic
503C2×C4C2×D3×D411yescharacteristic
504C2×C4C2×#23912yes—
505C2×C4#100712yes—
506C8—122no—
507C8—122no—
508C2×C3#243211yescharacteristic
509C2×C3#300711yescharacteristic
510C2×C3#300811yescharacteristic
511C2×C3#337012yes—
512C2×C3#337512yes—
513D3—82no—
514D3—82no—
515C22—122no—
516C22—122no—
517C22—122no—
518C22—122no—
519C22—122no—
520C22—122no—
521C22—242no—
522C22—242no—
523C22—242no—
524C22—242no—
525C22C2×#93711yescharacteristic
526C22C2×#94211yescharacteristic
527C22#581712yes—
528C22#582211yescharacteristic
529C22#583512yes—
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—41no—
535C4—41no—
536C4—44no—
537C4—82no—
538C4—122no—
539C4—122no—
540C4#572411yescharacteristic
541C4#572511yescharacteristic
542C4#572612yes—
543C3#2566011yescharacteristic, complemented, Sylow
544C2—242no—
545C2—242no—
546C2#8273511yescharacteristic
547C2#8325511yescharacteristic
548C2#8338311yescharacteristic
549C2#8363912yes—
550C2#8370812yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2567C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#2567D3
hypocenterC3#25660
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
2. C3×#2567C2semidirect
3. C2×#5087C2semidirect
4. C2×#5089C2semidirect
5. #71635C2semidirect
6. #69631C2semidirect
7. #69628C2semidirect
9. #69625C2semidirect
10. #69488C2semidirect
11. #69474C2semidirect
51. #2567D3semidirect
543. C3#25660semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C3×#25672
C2×#50872
C2×#50892
#716352
#696312
#696282
#696272
#696252
#694882
#694742
#256603