Finite Group Databasebeta

← #10945023

Subgroups — #10945023

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5827C211yesmaximal, characteristic
2C2×#5239C211yesmaximal, characteristic
3C2×#4721C211yesmaximal, characteristic, complemented
4#73951C211yesmaximal, characteristic, complemented
5C3×#1788C211yesmaximal, characteristic
6#70239C212yesmaximal, complemented
7#70238C212yesmaximal, complemented
8#70171C212yesmaximal, complemented
9#70170C212yesmaximal, complemented
10#69911C211yesmaximal, characteristic, complemented
11#69502C211yesmaximal, characteristic, complemented
12#33016—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14#5827—24no—
15#5827—24no—
16C2×Q8×Dic3C2212yes—
17C22×#216C2211yescharacteristic
18C2×D4×Dic3—22no—
19C2×D4×Dic3C2212yes—
20C22×#214C2211yescharacteristic
21C2×#926—21no—
22C2×#926C2211yescharacteristic
23C22×C4⋊Dic3—21no—
24C22×C4⋊Dic3C2211yescharacteristic
25C2×#924C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C3×#380C2211yescharacteristic
30#5239—24no—
31C2×#835C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#830C2211yescharacteristic
34C2×#817C2212yes—
35C2×#816C2212yes—
36#4984C2212yes—
37#4973C2212yes—
38#4953—22no—
39#4721—24no—
40#4661—21no—
41#4659—21no—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#520—31no—
47C2×#483—31no—
48C2×#452—31no—
49#2625—31no—
50#1878—32no—
51#1877—32no—
52#1865—32no—
53#1864—32no—
54#1824—31no—
55#1818—31no—
56#1788D311yescharacteristic
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C2×C3×#41D414yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C23×Dic3—21no—
67C23×Dic3—22no—
68C22×Dic6C2311yescharacteristic
69C22×C3×C8C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—21no—
72C2×#217—22no—
73C2×#217—21no—
74C2×#217D412yes—
75Q8×Dic3—24no—
76Q8×Dic3—24no—
77C2×#216—22no—
78C2×#216—24no—
79D4×Dic3—216no—
80D4×Dic3—24no—
81D4×Dic3—24no—
82C2×#214—22no—
83C2×#214—24no—
84#926—28no—
85#926—22no—
86#926—22no—
87C2×C4⋊Dic3—28no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3D411yescharacteristic
94C2×C4⋊Dic3D414yes—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2312yes—
97#924—22no—
98#924—22no—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3—22no—
101C2×C4×Dic3C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C3×#103—24no—
105C3×#102—24no—
106C3×C22⋊C8—22no—
107C3×#95—21no—
108#835—22no—
109#832—22no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#830—22no—
114#817—24no—
115#816—24no—
116C22×#41C2×D311yescharacteristic
117C22×Q16—31no—
118C22×SD16—31no—
119C2×C22⋊Q8—31no—
120#520—64no—
121#520—64no—
122C2×C4×Q8—32no—
123C2×C4×D4—32no—
124C2×C4×D4—62no—
125C2×#117—31no—
126C2×#117—61no—
127C22×C4⋊C4—31no—
128C22×C4⋊C4—61no—
129#483—64no—
130#452—64no—
131C2×#106—32no—
132C2×#103—31no—
133C2×#103—32no—
134C2×#103C2×D312yes—
135C2×#102—31no—
136C2×#102C2×D312yes—
137C2×C22⋊C8—31no—
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380C2×D311yescharacteristic
145#340—32no—
146#340—32no—
147#336—32no—
148#336—32no—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C3×#41—216no—
152C3×#41—28no—
153C3×#41—28no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D412yes—
157C2×C3×Q8#4114yes—
158C2×C3×D4—28no—
159C2×C3×D4—24no—
160C2×C3×D4—21no—
161C2×C3×D4—22no—
162C2×C3×D4C2×D412yes—
163C2×C3×D4#4114yes—
164C22×C3×C4—28no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—22no—
175C22×Dic3—28no—
176C22×Dic3—24no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—84no—
180C22×Dic3—82no—
181C22×Dic3C2×D412yes—
182C22×Dic3C2×D412yes—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C22⋊C4—81no—
193#217—28no—
194#217—24no—
195#217—42no—
196#216—44no—
197#214—44no—
198C4⋊Dic3—28no—
199C4⋊Dic3—22no—
200C4⋊Dic3—24no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—82no—
206C4⋊Dic3C2×D414yes—
207Dic3⋊C4—42no—
208C4×Dic3—28no—
209C4×Dic3—22no—
210C4×Dic3—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—22no—
218C2×#41D1214yes—
219C22×Q8—31no—
220C22×Q8C22×D311yescharacteristic
221C22×D4—21no—
222C22×D4C22×D311yescharacteristic
223C23×C4—21no—
224C23×C4—62no—
225C23×C4—61no—
226C23×C4C22×D311yescharacteristic
227C2×Q16—62no—
228C2×Q16—64no—
229C2×SD16—62no—
230C2×SD16—64no—
231C22×C8—31no—
232C22×C8—31no—
233C22×C8C22×D312yes—
234C22⋊Q8—62no—
235C22⋊Q8—62no—
236C4×Q8—64no—
237C4×Q8—64no—
238C4×D4—64no—
239C4×D4—616no—
240C4×D4—64no—
241#117—68no—
242#117—62no—
243#117—62no—
244C2×C4⋊C4—34no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—68no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4C22×D312yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—32no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—61no—
260C2×C42—32no—
261C2×C42—62no—
262#106—64no—
263#103—24no—
264#103—64no—
265#103—62no—
266#102—24no—
267#102—62no—
268C22⋊C8—22no—
269C22⋊C8—62no—
270#95—21no—
271#95—62no—
272#95—62no—
273#95—62no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—24no—
277C23×C3—21no—
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3#12012yes—
284C23×C3C22×D411yescharacteristic
285C3×Q8—22no—
286C3×Q8—24no—
287C3×Q8#12314yes—
288C3×D4—216no—
289C3×D4—22no—
290C3×D4—24no—
291C3×D4#12314yes—
292C2×C3×C4—216no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4#12012yes—
303C2×C3×C4C22×D411yescharacteristic
304C2×C3×C4C22×D411yescharacteristic
305C2×C3×C4C2×#4112yes—
306C2×C3×C4C2×#4112yes—
307C2×Dic3—22no—
308C2×Dic3—22no—
309C2×Dic3—24no—
310C2×Dic3—22no—
311C2×Dic3—24no—
312C2×Dic3—22no—
313C2×Dic3—44no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—84no—
320C2×Dic3—88no—
321Dic6—41no—
322Dic6—41no—
323Dic6—42no—
324C3×C8—42no—
325C3⋊C8—41no—
326C3⋊C8—41no—
327C24—21no—
328C24D3×D411yescharacteristic
329#41—216no—
330#41—28no—
331#41—28no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8C2×D1212yes—
339C2×Q8#23714yes—
340C2×D4—28no—
341C2×D4—24no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4C2×D1212yes—
345C2×D4#23714yes—
346C22×C4—28no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—62no—
355C22×C4—68no—
356C22×C4—64no—
357C22×C4—61no—
358C22×C4—61no—
359C22×C4—244no—
360C22×C4—242no—
361C22×C4C2×D1212yes—
362C22×C4D3×D411yescharacteristic
363C22×C4D3×D412yes—
364C22×C4C23×D311yescharacteristic
365Q16—124no—
366SD16—124no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—61no—
371C2×C8—61no—
372C2×C8—62no—
373C2×C8—62no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—34no—
377C4⋊C4—68no—
378C4⋊C4—64no—
379C4⋊C4—62no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—122no—
385C4⋊C4—242no—
386C22⋊C4—68no—
387C22⋊C4—64no—
388C22⋊C4—81no—
389C22⋊C4—122no—
390C42—68no—
391C42—62no—
392C42—62no—
393C22×C3—22no—
394C22×C3—24no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—42no—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#52512yes—
405C22×C3#57712yes—
406C22×C3#57711yescharacteristic
407C22×C3#57811yescharacteristic
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—82no—
411C3×C4C2×#12311yescharacteristic
412C3×C4C2×#12311yescharacteristic
413C3×C4#52512yes—
414Dic3—42no—
415Dic3—44no—
416Dic3—42no—
417Dic3—81no—
418Dic3—81no—
419C23—22no—
420C23—22no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—24no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23#88212yes—
429C23C2×D3×D411yescharacteristic
430Q8—22no—
431Q8—24no—
432Q8—121no—
433Q8—121no—
434Q8—122no—
435Q8#88614yes—
436D4—216no—
437D4—22no—
438D4—24no—
439D4#88614yes—
440C2×C4—216no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—62no—
451C2×C4—64no—
452C2×C4—62no—
453C2×C4—64no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—124no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—122no—
462C2×C4—244no—
463C2×C4—248no—
464C2×C4#88212yes—
465C2×C4C22×D1211yescharacteristic
466C2×C4C2×D3×D411yescharacteristic
467C2×C4C2×#23712yes—
468C2×C4C2×#23712yes—
469C8—42no—
470C8—121no—
471C8—121no—
472C2×C3—42no—
473C2×C3—44no—
474C2×C3#284911yescharacteristic
475C2×C3#300911yescharacteristic
476C2×C3#301311yescharacteristic
477C2×C3#312012yes—
478C2×C3#312512yes—
479C22—22no—
480C22—22no—
481C22—24no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—42no—
487C22—44no—
488C22—42no—
489C22C2×#88211yescharacteristic
490C22#558612yes—
491C22#577012yes—
492C22#577511yescharacteristic
493C22#577911yescharacteristic
494C4—22no—
495C4—24no—
496C4—82no—
497C4—122no—
498C4—122no—
499C4—124no—
500C4—241no—
501C4—241no—
502C4C2×#88611yescharacteristic
503C4C2×#88611yescharacteristic
504C4#558612yes—
505C3#3301611yescharacteristic, complemented, Sylow
506C2—42no—
507C2—44no—
508C2#7384711yescharacteristic
509C2#7883511yescharacteristic
510C2#7888711yescharacteristic
511C2#8007512yes—
512C2#8021612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1788C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenter#1788D3
hypocenterC3#33016
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
3. C2×#4721C2semidirect
4. #73951C2semidirect
6. #70239C2semidirect
7. #70238C2semidirect
8. #70171C2semidirect
9. #70170C2semidirect
10. #69911C2semidirect
11. #69502C2semidirect
505. C3#33016semidirect

Maximal subgroups

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

subgroupindex
C2×#58272
C2×#52392
C2×#47212
#739512
C3×#17882
#702392
#702382
#701712
#701702
#699112
#695022
#330163