Finite Group Databasebeta

← #10900883

Subgroups — #10900883

Order 768.

1260 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×#5816C211yesmaximal, characteristic
2C2×C3×#451C211yesmaximal, characteristic, complemented
3#82787C211yesmaximal, characteristic, complemented
4C2×#5047C211yesmaximal, characteristic
5#72132C211yesmaximal, characteristic, complemented
6#72073C212yesmaximal, complemented
7#72072C212yesmaximal, complemented
8#70284C212yesmaximal, complemented
9#70279C212yesmaximal, complemented
10#69960C211yesmaximal, characteristic
11#69911C211yesmaximal, characteristic, complemented
12#29633—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C22×C3×Q16C2212yes—
16C2×C3×C22⋊Q8C2211yescharacteristic
17#5816—24no—
18#5816—24no—
19C2×#947C2212yes—
20C2×#946—21no—
21C2×#946—21no—
22C2×#944C2212yes—
23C22×#216C2211yescharacteristic
24C22×#215C2211yescharacteristic
25C2×#929—22no—
26C2×#926C2211yescharacteristic
27C3×#451—24no—
28C2×C3×#103C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30#5229C2212yes—
31C2×#835C2212yes—
32C2×#835C2212yes—
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#821C2212yes—
36C2×#816C2212yes—
37#5047—24no—
38#4986—21no—
39#4981—21no—
40#4972C2212yes—
41C2×#810C2211yescharacteristic
42C2×#809C2211yescharacteristic
43#4952—22no—
44#4550C2212yes—
45#4550C2212yes—
46C2×#536—31no—
47C2×#452—31no—
48C2×#451D311yescharacteristic, complemented
49#2625—31no—
50#2043—32no—
51#2042—32no—
52#2016—32no—
53#2015—32no—
54#1818—31no—
55#1788—31no—
56#1787—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8—42no—
60C22×C3×Q8—42no—
61C22×C3×Q8D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×Q8C2312yes—
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C22×#67—21no—
67C2×#235—22no—
68C2×#235—42no—
69C2×#235—42no—
70C2×#235D414yes—
71C22×D12C2311yescharacteristic
72C22×C4×D3—21no—
73C22×Dic6C2311yescharacteristic
74C2×C3×Q16—24no—
75C2×C3×Q16—42no—
76C2×C3×Q16—42no—
77C22×C3×C8C2312yes—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×#217—21no—
84C2×#217D412yes—
85#947—24no—
86#947—24no—
87#946—28no—
88#946—28no—
89#944—24no—
90#944—24no—
91C2×#216—22no—
92C2×#216—42no—
93C2×#215—22no—
94C2×#215—42no—
95#929—216no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—22no—
98#926—22no—
99#926—22no—
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3D411yescharacteristic
102C2×C4⋊Dic3C2312yes—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×C4×Dic3C2312yes—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C3×#103—24no—
110C3×C22⋊C8—22no—
111#835—42no—
112#835—42no—
113C3⋊#95—21no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117#830—22no—
118#821—42no—
119#816—42no—
120#810—22no—
121#809—22no—
122C22×#41—31no—
123C23×Q8C2×D311yescharacteristic
124C22×Q16—31no—
125C22×Q16C2×D312yes—
126C22×SD16—31no—
127#536—64no—
128#536—64no—
129C2×C4⋊Q8—32no—
130C2×#124—32no—
131C2×#123—62no—
132C2×C22⋊Q8—61no—
133C2×C22⋊Q8—61no—
134C2×C22⋊Q8C2×D311yescharacteristic
135C2×#117—31no—
136#452—64no—
137#451—24no—
138C2×#103—31no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103C2×D312yes—
143C2×#102—31no—
144C2×#102—32no—
145C2×C22⋊C8—31no—
146C2×C22⋊C8C2×D311yescharacteristic
147#400—61no—
148#398—61no—
149#397—62no—
150#391—32no—
151#389—32no—
152#380—31no—
153#336—32no—
154#336—32no—
155C24×C3C2×D411yescharacteristic
156C23×D3—21no—
157C2×C3×Q8—24no—
158C2×C3×Q8—22no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—48no—
168C2×C3×Q8—44no—
169C2×C3×Q8—44no—
170C2×C3×Q8—44no—
171C2×C3×Q8C2×D414yes—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—44no—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C2×#67—24no—
184C2×#67—44no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3C2×D412yes—
188C22×Dic3C2×D412yes—
189#235—44no—
190#235—44no—
191#235—48no—
192C2×D12—22no—
193C2×D12—22no—
194C2×D12—21no—
195C2×D12C2×D412yes—
196C2×C4×D3—24no—
197C2×C4×D3—44no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—21no—
201C2×Dic6C2×D412yes—
202C3×Q16—48no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—42no—
209C3×C22⋊C4—42no—
210#217—24no—
211#217—81no—
212#216—44no—
213#215—44no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—28no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3—22no—
220Dic3⋊C4—28no—
221Dic3⋊C4—28no—
222Dic3⋊C4—82no—
223C4×Dic3—22no—
224C4×Dic3—22no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8—22no—
229C2×#41—34no—
230C2×#41—62no—
231C2×#41—122no—
232C2×#41—122no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—31no—
236C22×Q8—42no—
237C22×Q8—42no—
238C22×Q8#6714yes—
239C22×Q8C22×D311yescharacteristic
240C22×Q8C22×D312yes—
241C22×D4—31no—
242C22×D4—61no—
243C23×C4—21no—
244C23×C4—61no—
245C23×C4C22×D311yescharacteristic
246C2×Q16—24no—
247C2×Q16—42no—
248C2×Q16—42no—
249C2×Q16—62no—
250C2×Q16—122no—
251C2×SD16—62no—
252C2×SD16—122no—
253C22×C8—31no—
254C22×C8—31no—
255C22×C8C22×D312yes—
256C4⋊Q8—64no—
257C4⋊Q8—64no—
258#124—64no—
259#124—64no—
260#123—616no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—68no—
264C22⋊Q8—68no—
265#117—62no—
266#117—62no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280C2×C42—32no—
281#103—24no—
282#103—62no—
283#103—122no—
284#103—122no—
285#103—122no—
286#102—62no—
287#102—122no—
288C22⋊C8—22no—
289C22⋊C8—62no—
290#95—62no—
291#95—62no—
292#95—62no—
293#95—61no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3#12012yes—
299C23×C3C22×D411yescharacteristic
300C22×D3—22no—
301C22×D3—42no—
302C22×D3—41no—
303C22×D3—42no—
304C22×D3—41no—
305C3×Q8—24no—
306C3×Q8—24no—
307C3×Q8—48no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×Q8—41no—
311C3×Q8—41no—
312C3×Q8—42no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—48no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4#12012yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4H(5,𝔽2)12yes—
330C2×C3×C4H(5,𝔽2)12yes—
331#67—48no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—22no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342D12—22no—
343D12—22no—
344D12—42no—
345C4×D3—48no—
346Dic6—22no—
347Dic6—22no—
348Dic6—42no—
349C3×C8—42no—
350C3⋊C8—41no—
351C3⋊C8—41no—
352C24—61no—
353C24D3×D411yescharacteristic
354#41—124no—
355#41—124no—
356#41—128no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—32no—
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—61no—
373C2×Q8—62no—
374C2×Q8—62no—
375C2×Q8C2×#6714yes—
376C2×D4—32no—
377C2×D4—64no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—61no—
381C2×D4—124no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—44no—
391C22×C4—61no—
392C22×C4—64no—
393C22×C4—62no—
394C22×C4—124no—
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D412yes—
397C22×C4C2×#6712yes—
398C22×C4C23×D311yescharacteristic
399Q16—48no—
400Q16—124no—
401SD16—124no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—62no—
408C2×C8—62no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—42no—
412C4⋊C4—68no—
413C4⋊C4—68no—
414C4⋊C4—62no—
415C4⋊C4—62no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—242no—
419C22⋊C4—42no—
420C22⋊C4—68no—
421C22⋊C4—68no—
422C22⋊C4—64no—
423C22⋊C4—241no—
424C42—62no—
425C42—62no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3C2×#12011yescharacteristic
432C22×C3#53512yes—
433C22×C3#57711yescharacteristic
434C22×C3#57812yes—
435C22×C3#57811yescharacteristic
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—42no—
439C2×D3—82no—
440C2×D3—82no—
441C3×C4—22no—
442C3×C4—44no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4#53412yes—
446C3×C4#53411yescharacteristic
447C3×C4#53411yescharacteristic
448Dic3—42no—
449Dic3—42no—
450Dic3—42no—
451Dic3—82no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—62no—
457C23—122no—
458C23—121no—
459C23—121no—
460C23—122no—
461C23#93712yes—
462C23C2×D3×D411yescharacteristic
463Q8—24no—
464Q8—24no—
465Q8—48no—
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—41no—
470Q8—41no—
471Q8—62no—
472Q8—62no—
473Q8—122no—
474D4—62no—
475D4—62no—
476D4—128no—
477D4—122no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—64no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—62no—
497C2×C4—128no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4#93712yes—
503C2×C4C2×D3×D411yescharacteristic
504C2×C4#100412yes—
505C2×C4#100412yes—
506C2×C4C22×#6711yescharacteristic
507C8—42no—
508C8—121no—
509C8—121no—
510C2×C3—42no—
511C2×C3#261311yescharacteristic
512C2×C3#301111yescharacteristic
513C2×C3#301311yescharacteristic
514C2×C3#322312yes—
515C2×C3#322512yes—
516D3—82no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—42no—
521C22—42no—
522C22—122no—
523C22—122no—
524C22—122no—
525C22—242no—
526C22—242no—
527C22#559212yes—
528C22#576612yes—
529C22#577811yescharacteristic
530C22#577911yescharacteristic
531C22C2×#93711yescharacteristic
532C4—22no—
533C4—44no—
534C4—81no—
535C4—81no—
536C4—122no—
537C4—122no—
538C4—122no—
539C4—242no—
540C4#558911yescharacteristic
541C4#558911yescharacteristic
542C4#580412yes—
543C3#2963311yescharacteristic, complemented, Sylow
544C2—42no—
545C2—242no—
546C2#8018012yes—
547C2#8020312yes—
548C2#8260711yescharacteristic
549C2#8388411yescharacteristic
550C2#8398411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#451C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenterC2×#451D3
hypocenterC3#29633
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
2. C2×C3×#451C2semidirect
3. #82787C2semidirect
5. #72132C2semidirect
6. #72073C2semidirect
7. #72072C2semidirect
8. #70284C2semidirect
9. #70279C2semidirect
11. #69911C2semidirect
48. C2×#451D3semidirect
543. C3#29633semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
C2×C3×#4512
#827872
C2×#50472
#721322
#720732
#720722
#702842
#702792
#699602
#699112
#296333