Finite Group Databasebeta

← #10795101

Subgroups — #10795101

Order 768.

1342 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5522C211yesmaximal, characteristic
2C2×C3×#433C211yesmaximal, characteristic, complemented
3C2×#5000C211yesmaximal, characteristic, complemented
4#70476C212yesmaximal, complemented
5#70475C212yesmaximal, complemented
6#69956C212yesmaximal
7#69910C212yesmaximal, complemented
8#69830C211yesmaximal, characteristic, complemented
9#69807C211yesmaximal, characteristic, complemented
10#69753C211yesmaximal, characteristic, complemented
11C4×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
12#20720—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C4×Dic3C2211yescharacteristic
16#5522—24no—
17#5522—24no—
18C2×#893C412yes—
19C2×#887C412yes—
20C2×#67⋊C4C412yes—
21C2×#67⋊C4C412yes—
22C2×#872C2211yescharacteristic
23C2×#872C2211yescharacteristic
24C2×C42×D3C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C3×#433—24no—
27C2×C3×C4⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31C2×#830C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36#5000—24no—
37C2×C4⋊(C3⋊C8)C2212yes—
38C2×#803C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4550C2212yes—
41#4550C2212yes—
42#4550C2212yes—
43#4550C2212yes—
44C2×C4×#41—31no—
45C2×#433—31no—
46C2×#433D311yescharacteristic, complemented
47#1817—32no—
48#1798—32no—
49#1797—32no—
50#1780—32no—
51#1750—31no—
52#1750—31no—
53#1749—31no—
54C4×C22⋊C8—31no—
55C23×C3×C4C2311yescharacteristic
56C22×#67C2×C411yescharacteristic
57C22×#67C2×C411yescharacteristic
58C23×Dic3C2×C412yes—
59C2×#235—22no—
60C2×#235—22no—
61C2×#235—24no—
62C2×#235—24no—
63C22×D12C2×C411yescharacteristic
64C22×C4×D3C2×C412yes—
65C22×C4×D3C2×C412yes—
66C22×C4×D3C2311yescharacteristic
67C22×C4×D3C2311yescharacteristic
68C22×Dic6C2×C411yescharacteristic
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C3×#117D414yes—
72C3×#117D414yes—
73C2×C3×C4⋊C4C2×C412yes—
74C2×C3×C22⋊C4C2×C412yes—
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×(C2×D3)⋊C4C2×C412yes—
78C2×(C2×D3)⋊C4C2×C412yes—
79C2×Dic3⋊C4C2×C412yes—
80C2×Dic3⋊C4C2×C412yes—
81C2×C4×Dic3—22no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3C2×C412yes—
84C2×C4×Dic3C2×C412yes—
85C2×C4×Dic3C2×C412yes—
86C2×C4×Dic3C2311yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C22×C3⋊C8C2312yes—
89C22×C3⋊C8C2312yes—
90#893—28no—
91#887—28no—
92#67⋊C4—28no—
93#67⋊C4—28no—
94#872—22no—
95#872—22no—
96#872—22no—
97#872—22no—
98C42×D3—22no—
99C42×D3—22no—
100C42×D3—22no—
101C42×D3—22no—
102C3×C4⋊C8—24no—
103C3×C22⋊C8—24no—
104C3×#97—22no—
105C3×C4×C8—22no—
106#830—24no—
107(C2×D3)⋊C8—24no—
108(C2×D3)⋊C8—24no—
109(C2×D3)⋊C8—24no—
110(C2×D3)⋊C8—24no—
111C4⋊(C3⋊C8)—24no—
112#803—22no—
113C4×C3⋊C8—22no—
114C22×#41—31no—
115C4×#41—64no—
116C4×#41—64no—
117C2×C4×Q8—32no—
118C2×C4×D4—32no—
119C2×C4×D4—32no—
120C2×C4×D4—32no—
121C2×#117—31no—
122C2×#117—31no—
123C2×#117C2×D311yescharacteristic
124C22×C42—31no—
125C22×C42—31no—
126C22×C42—31no—
127#433—24no—
128#433—64no—
129C2×C4⋊C8—32no—
130C2×C4⋊C8C2×D312yes—
131C2×C22⋊C8—32no—
132C2×C22⋊C8—32no—
133C2×C22⋊C8—32no—
134C2×C22⋊C8—32no—
135C2×C22⋊C8—32no—
136C2×C22⋊C8C2×D312yes—
137C2×#97—31no—
138C2×#97C2×D311yescharacteristic
139C2×C4×C8—31no—
140C2×C4×C8C2×D311yescharacteristic
141#336—32no—
142#336—32no—
143#336—32no—
144#336—32no—
145C24×C3C22×C411yescharacteristic
146C23×D3C22×C411yescharacteristic
147C23×D3C22×C411yescharacteristic
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4C22×C411yescharacteristic
151C22×C3×C4C22×C412yes—
152C22×C3×C4C22×C412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2411yescharacteristic
156C2×#67—22no—
157C2×#67—22no—
158C2×#67—22no—
159C2×#67—22no—
160C2×#67—22no—
161C2×#67—22no—
162C22×Dic3—24no—
163C22×Dic3—24no—
164C22×Dic3—42no—
165C22×Dic3C22×C412yes—
166C22×Dic3C22×C411yescharacteristic
167C22×Dic3C22×C411yescharacteristic
168C22×Dic3C22×C412yes—
169#235—216no—
170#235—216no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×C4×D3—21no—
175C2×C4×D3—21no—
176C2×C4×D3—21no—
177C2×C4×D3—21no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—24no—
181C2×C4×D3—24no—
182C2×C4×D3—24no—
183C2×C4×D3—24no—
184C2×C4×D3—42no—
185C2×C4×D3—42no—
186C2×C4×D3#4114yes—
187C2×C4×D3#4114yes—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C3×C4⋊C4C22⋊C418yes—
198C3×C22⋊C4C22⋊C418yes—
199C3×C42C2×D412yes—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202C3×C42C2×D412yes—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—24no—
205Dic3⋊C4—24no—
206Dic3⋊C4—24no—
207C4×Dic3—24no—
208C4×Dic3—24no—
209C4×Dic3—24no—
210C4×Dic3—21no—
211C4×Dic3—21no—
212C4×Dic3—21no—
213C4×Dic3—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×#41—64no—
221C2×#41—64no—
222C2×#41—62no—
223C2×#41—62no—
224C22×Q8—31no—
225C22×D4—31no—
226C22×D4—31no—
227C22×D4—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—32no—
232C23×C4—32no—
233C23×C4C22×D311yescharacteristic
234C22×C8—32no—
235C22×C8—32no—
236C22×C8C22×D312yes—
237C22×C8C22×D312yes—
238C4×Q8—68no—
239C4×D4—68no—
240C4×D4—68no—
241C4×D4—68no—
242#117—62no—
243#117—62no—
244#117—62no—
245#117—62no—
246#117D1214yes—
247#117#6714yes—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4C4×D312yes—
251C2×C22⋊C4—32no—
252C2×C22⋊C4—32no—
253C2×C22⋊C4C4×D312yes—
254C2×C42—32no—
255C2×C42—32no—
256C2×C42—32no—
257C2×C42—31no—
258C2×C42—31no—
259C2×C42—62no—
260C2×C42—62no—
261C2×C42—62no—
262C2×C42—62no—
263C2×C42—62no—
264C2×C42—62no—
265C2×C42C22×D311yescharacteristic
266C2×C42C22×D311yescharacteristic
267C4⋊C8—24no—
268C4⋊C8—64no—
269C22⋊C8—24no—
270C22⋊C8—64no—
271C22⋊C8—64no—
272C22⋊C8—64no—
273C22⋊C8—64no—
274C22⋊C8—64no—
275#97—22no—
276#97—62no—
277C4×C8—22no—
278C4×C8—62no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3C2×C22⋊C412yes—
284C23×C3C23×C411yescharacteristic
285C22×D3—21no—
286C22×D3—21no—
287C22×D3—21no—
288C22×D3—21no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3#11712yes—
294C22×D3#11712yes—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4C2×C22⋊C412yes—
304C2×C3×C4C2×C22⋊C414yes—
305C2×C3×C4C2×C22⋊C414yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4112yes—
309C2×C3×C4C2×#4112yes—
310#67—28no—
311#67—28no—
312C2×Dic3—22no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—48no—
322C2×Dic3#11712yes—
323C2×Dic3#11712yes—
324C2×Dic3#13114yes—
325D12—28no—
326C4×D3—24no—
327C4×D3—24no—
328C4×D3—24no—
329C4×D3—24no—
330C4×D3—48no—
331C4×D3—48no—
332Dic6—28no—
333C3×C8—42no—
334C3×C8—42no—
335C3⋊C8—42no—
336C3⋊C8—42no—
337C24—31no—
338C24—31no—
339C24C2×C4×D311yescharacteristic
340#41—616no—
341#41—616no—
342C2×Q8—62no—
343C2×Q8—62no—
344C2×Q8—62no—
345C2×D4—62no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—62no—
349C2×D4—62no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—62no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—32no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—34no—
360C22×C4—34no—
361C22×C4—32no—
362C22×C4—64no—
363C22×C4—64no—
364C22×C4—64no—
365C22×C4—61no—
366C22×C4—61no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—64no—
372C22×C4—64no—
373C22×C4—64no—
374C22×C4—122no—
375C22×C4—122no—
376C22×C4—122no—
377C22×C4C2×C4×D311yescharacteristic
378C22×C4C2×C4×D312yes—
379C22×C4C2×C4×D312yes—
380C22×C4C2×D1212yes—
381C22×C4C2×#6712yes—
382C22×C4C23×D311yescharacteristic
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8—62no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C4⋊C4—64no—
396C4⋊C4—64no—
397C4⋊C4(C2×D3)⋊C418yes—
398C22⋊C4—64no—
399C22⋊C4—64no—
400C22⋊C4(C2×D3)⋊C418yes—
401C42—64no—
402C42—64no—
403C42—64no—
404C42—61no—
405C42—61no—
406C42—61no—
407C42—61no—
408C42C2×D1212yes—
409C42C2×D1212yes—
410C42C2×#6712yes—
411C42C2×#6712yes—
412C22×C3—24no—
413C22×C3—22no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—42no—
417C22×C3C22×C22⋊C411yescharacteristic
418C22×C3C2×#11711yescharacteristic
419C22×C3C2×#11711yescharacteristic
420C22×C3C2×#13112yes—
421C22×C3#56812yes—
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—24no—
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—41no—
430C2×D3—41no—
431C2×D3—42no—
432C3×C4—24no—
433C3×C4—24no—
434C3×C4—22no—
435C3×C4#52512yes—
436C3×C4#52512yes—
437Dic3—24no—
438Dic3—22no—
439Dic3—22no—
440Dic3—42no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—32no—
446C23—32no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—61no—
452C23—61no—
453C23—61no—
454C23—61no—
455C23C2×(C2×D3)⋊C412yes—
456C23C22×C4×D311yescharacteristic
457Q8—68no—
458D4—68no—
459D4—68no—
460D4—68no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—34no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—62no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—62no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—64no—
479C2×C4—64no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—128no—
486C2×C4—128no—
487C2×C4—128no—
488C2×C4C2×(C2×D3)⋊C412yes—
489C2×C4C2×(C2×D3)⋊C414yes—
490C2×C4C2×(C2×D3)⋊C414yes—
491C2×C4C22×D1211yescharacteristic
492C2×C4D3×#4112yes—
493C2×C4D3×#4112yes—
494C2×C4C22×#6711yescharacteristic
495C8—42no—
496C8—42no—
497C8—122no—
498C8—122no—
499C2×C3—42no—
500C2×C3#228011yescharacteristic
501C2×C3#288012yes—
502C2×C3#292211yescharacteristic
503C2×C3#292311yescharacteristic
504C2×C3#292512yes—
505D3—42no—
506D3—42no—
507C22—24no—
508C22—22no—
509C22—41no—
510C22—41no—
511C22—42no—
512C22—62no—
513C22—62no—
514C22—64no—
515C22—62no—
516C22—62no—
517C22—121no—
518C22—121no—
519C22—122no—
520C22—122no—
521C22—122no—
522C22D3×#11711yescharacteristic
523C22D3×#11711yescharacteristic
524C22D3×#13112yes—
525C22#575012yes—
526C22C22×(C2×D3)⋊C411yescharacteristic
527C4—24no—
528C4—24no—
529C4—22no—
530C4—64no—
531C4—62no—
532C4—62no—
533C4—122no—
534C4#552412yes—
535C4#582812yes—
536C3#2072011yescharacteristic, complemented, Sylow
537C2—42no—
538C2—122no—
539C2—122no—
540C2#7937211yescharacteristic
541C2#7937311yescharacteristic
542C2#7937512yes—
543C2#8226711yescharacteristic
544C2#8379312yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#433C2
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenterC2×#433D3
hypocenterC3#20720
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
2. C2×C3×#433C2semidirect
3. C2×#5000C2semidirect
4. #70476C2semidirect
5. #70475C2semidirect
7. #69910C2semidirect
8. #69830C2semidirect
9. #69807C2semidirect
10. #69753C2semidirect
11. C4×(C2×D3)⋊C8C2semidirect
46. C2×#433D3semidirect
536. C3#20720semidirect

Maximal subgroups

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

subgroupindex
C2×#55222
C2×C3×#4332
C2×#50002
#704762
#704752
#699562
#699102
#698302
#698072
#697532
C4×(C2×D3)⋊C82
#207203