Finite Group Databasebeta

← #10945029

Subgroups — #10945029

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5523C211yesmaximal, characteristic
2C3×#2625C211yesmaximal, characteristic, complemented
3C2×#5047C211yesmaximal, characteristic
4C2×#5044C211yesmaximal, characteristic, complemented
5#71584C211yesmaximal, characteristic, complemented
6#69690C212yesmaximal, complemented
7#69687C212yesmaximal, complemented
8#69631C212yesmaximal, complemented
9#69625C212yesmaximal, complemented
10#69522C211yesmaximal, characteristic
11#69501C211yesmaximal, characteristic, complemented
12#33017—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C22×#216C2211yescharacteristic
17C22×#215C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19#5523—24no—
20#5523—24no—
21C2×#892—21no—
22C2×D3×C4⋊C4—21no—
23C2×#67⋊C4—22no—
24C2×C4×D12C2212yes—
25C2×C4×Dic6C2212yes—
26C3×#400—21no—
27C3×#398—21no—
28C3×#397—22no—
29C3×#391C2212yes—
30C3×#389C2212yes—
31C3×#380C2211yescharacteristic
32C2×#835C2212yes—
33C2×#830C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5047—24no—
36#5044—24no—
37C2×#810C2211yescharacteristic
38C2×#810C2212yes—
39C2×#809C2211yescharacteristic
40C2×#809C2212yes—
41C2×#807C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#520—31no—
47C2×#484—31no—
48C2×#452—31no—
49#2625D311yescharacteristic, complemented
50#1878—32no—
51#1877—32no—
52#1863—32no—
53#1862—32no—
54#1823—31no—
55#1818—31no—
56#1788—31no—
57C22×C3×Q8C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C22×#67—21no—
60C23×Dic3—21no—
61C2×#235—22no—
62C2×#235—24no—
63C2×#235—24no—
64C2×#235D414yes—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C22×C4×D3—22no—
68C22×Dic6C2311yescharacteristic
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×#117—22no—
72C3×#117—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4D412yes—
82C2×C3×C42C2312yes—
83C2×#216—22no—
84C2×#216—24no—
85C2×#215—22no—
86C2×#215—24no—
87C2×(C2×D3)⋊C4—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3D414yes—
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C2×C4×Dic3—22no—
94C22×C3⋊C8C2312yes—
95C22×C3⋊C8C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97#892—28no—
98D3×C4⋊C4—28no—
99#67⋊C4—216no—
100C4×D12—24no—
101C4×D12—24no—
102C4×Dic6—24no—
103C4×Dic6—24no—
104C3×#95—21no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—22no—
108#835—24no—
109#830—22no—
110#830—22no—
111#810—22no—
112#810—24no—
113#809—22no—
114#809—24no—
115#807—24no—
116C22×#41—31no—
117C22×Q16—31no—
118C22×SD16—31no—
119C2×C22⋊Q8C2×D311yescharacteristic
120#520—64no—
121#520—64no—
122C2×C4×Q8—32no—
123C2×C4×D4—32no—
124C2×C4×D4—62no—
125C2×#117—61no—
126C2×#117C2×D311yescharacteristic
127C22×C4⋊C4—31no—
128C22×C4⋊C4—61no—
129#484—64no—
130#452—64no—
131C2×#107—32no—
132C2×#103—32no—
133C2×#103—31no—
134C2×#103—32no—
135C2×#102—31no—
136C2×#102—32no—
137C2×C22⋊C8—31no—
138C2×C22⋊C8—31no—
139#400—21no—
140#398—21no—
141#397—22no—
142#391C2×D312yes—
143#389C2×D312yes—
144#380C2×D311yescharacteristic
145#340—32no—
146#340—32no—
147#336—32no—
148#336—32no—
149C24×C3C2×D411yescharacteristic
150C23×D3—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C2×#67—28no—
167C2×#67—24no—
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3—24no—
171C22×Dic3—81no—
172C22×Dic3—81no—
173#235—216no—
174#235—28no—
175#235—28no—
176C2×D12—21no—
177C2×D12—22no—
178C2×D12C2×D412yes—
179C2×D12#4114yes—
180C2×C4×D3—28no—
181C2×C4×D3—24no—
182C2×C4×D3—28no—
183C2×C4×D3—82no—
184C2×C4×D3—82no—
185C2×Dic6—21no—
186C2×Dic6—22no—
187C2×Dic6C2×D412yes—
188C2×Dic6#4114yes—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—42no—
196C3×C4⋊C4—82no—
197C3×C22⋊C4—24no—
198C3×C22⋊C4—42no—
199C3×C22⋊C4—81no—
200C3×C42—22no—
201C3×C42—22no—
202#216—44no—
203#215—44no—
204(C2×D3)⋊C4—28no—
205C4⋊Dic3—24no—
206C4⋊Dic3—22no—
207C4⋊Dic3C2×D414yes—
208Dic3⋊C4—28no—
209C4×Dic3—28no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×#41—34no—
218C2×#41—64no—
219C2×#41—64no—
220C2×#41—62no—
221C22×Q8—31no—
222C22×Q8C22×D311yescharacteristic
223C22×D4—31no—
224C22×D4—61no—
225C23×C4—61no—
226C23×C4—61no—
227C23×C4—62no—
228C23×C4C22×D311yescharacteristic
229C2×Q16—64no—
230C2×Q16—62no—
231C2×SD16—64no—
232C2×SD16—62no—
233C22×C8—31no—
234C22×C8—31no—
235C22×C8—32no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C4×Q8—64no—
239C4×Q8—64no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—616no—
243#117—22no—
244#117—22no—
245#117—68no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—34no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—68no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4#6711yescharacteristic
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4#6712yes—
262C2×C42—62no—
263C2×C42C22×D312yes—
264#107—64no—
265#103—64no—
266#103—62no—
267#103—64no—
268#102—64no—
269#102—62no—
270C22⋊C8—62no—
271C22⋊C8—62no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276C23×C3—22no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—22no—
280C23×C3#12012yes—
281C23×C3C22×D411yescharacteristic
282C22×D3—22no—
283C22×D3—24no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—42no—
287C3×Q8—41no—
288C3×Q8—41no—
289C3×Q8—42no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—42no—
307C2×C3×C4#12012yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C2×#4112yes—
311C2×C3×C4C2×#4112yes—
312#67—216no—
313C2×Dic3—22no—
314C2×Dic3—24no—
315C2×Dic3—24no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—84no—
319D12—22no—
320D12—24no—
321D12#12314yes—
322C4×D3—216no—
323C4×D3—88no—
324Dic6—22no—
325Dic6—24no—
326Dic6#12314yes—
327C3⋊C8—41no—
328C3⋊C8—41no—
329C3⋊C8—42no—
330C24—61no—
331C24C2×#6711yescharacteristic
332#41—68no—
333#41—68no—
334#41—616no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—32no—
340C2×Q8—34no—
341C2×Q8—61no—
342C2×Q8—62no—
343C2×D4—32no—
344C2×D4—34no—
345C2×D4—68no—
346C2×D4—64no—
347C2×D4—61no—
348C2×D4—62no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—68no—
355C22×C4—68no—
356C22×C4—61no—
357C22×C4—64no—
358C22×C4—62no—
359C22×C4—64no—
360C22×C4—242no—
361C22×C4—242no—
362C22×C4—241no—
363C22×C4—241no—
364C22×C4D3×D412yes—
365C22×C4D3×D412yes—
366C22×C4C2×#6712yes—
367C22×C4C2×#6712yes—
368C22×C4C2×#6711yescharacteristic
369C22×C4C23×D311yescharacteristic
370Q16—124no—
371SD16—124no—
372C2×C8—61no—
373C2×C8—62no—
374C2×C8—61no—
375C2×C8—62no—
376C2×C8—62no—
377C2×C8—62no—
378C2×C8—62no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—34no—
386C4⋊C4—42no—
387C4⋊C4—68no—
388C4⋊C4—64no—
389C4⋊C4—62no—
390C4⋊C4—82no—
391C22⋊C4—24no—
392C22⋊C4—42no—
393C22⋊C4—68no—
394C22⋊C4—81no—
395C42—22no—
396C42—22no—
397C42—68no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#52512yes—
405C22×C3#57611yescharacteristic
406C22×C3#57711yescharacteristic
407C22×C3#57812yes—
408C2×D3—24no—
409C2×D3—44no—
410C2×D3—44no—
411C2×D3—42no—
412C2×D3—42no—
413C3×C4—22no—
414C3×C4—42no—
415C3×C4—42no—
416C3×C4—81no—
417C3×C4—82no—
418C3×C4—81no—
419C3×C4C2×#12311yescharacteristic
420C3×C4C2×#12311yescharacteristic
421C3×C4#52512yes—
422Dic3—24no—
423Dic3—44no—
424C23—22no—
425C23—21no—
426C23—21no—
427C23—22no—
428C23—62no—
429C23—64no—
430C23—121no—
431C23—121no—
432C23—122no—
433C23#93712yes—
434C23C22×#6711yescharacteristic
435Q8—34no—
436Q8—41no—
437Q8—41no—
438Q8—42no—
439Q8—62no—
440Q8—64no—
441D4—34no—
442D4—616no—
443D4—62no—
444D4—64no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—616no—
463C2×C4—62no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—248no—
469C2×C4—244no—
470C2×C4#93712yes—
471C2×C4C2×D3×D411yescharacteristic
472C2×C4C2×D3×D411yescharacteristic
473C2×C4D3×#4112yes—
474C2×C4D3×#4112yes—
475C8—121no—
476C8—121no—
477C8—122no—
478C2×C3—42no—
479C2×C3#284911yescharacteristic
480C2×C3#301111yescharacteristic
481C2×C3#301211yescharacteristic
482C2×C3#312012yes—
483C2×C3#312412yes—
484D3—44no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—42no—
489C22—42no—
490C22—64no—
491C22—124no—
492C22—122no—
493C22—122no—
494C22—124no—
495C22C2×#93711yescharacteristic
496C22#582812yes—
497C22#583511yescharacteristic
498C22#583711yescharacteristic
499C22#583812yes—
500C4—22no—
501C4—42no—
502C4—42no—
503C4—64no—
504C4—82no—
505C4—81no—
506C4—81no—
507C4—124no—
508C4D3×#12311yescharacteristic
509C4D3×#12311yescharacteristic
510C4#565612yes—
511C3#3301711yescharacteristic, complemented, Sylow
512C2—42no—
513C2—124no—
514C2#8291211yescharacteristic
515C2#8341111yescharacteristic
516C2#8342411yescharacteristic
517C2#8344612yes—
518C2#8346112yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2625C2
socleC23×C3C22×D4
prime radicalC22×#235C22
hypercenter#2625D3
hypocenterC3#33017
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2625

Complements

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

NH ≅ G/Ntype
2. C3×#2625C2semidirect
4. C2×#5044C2semidirect
5. #71584C2semidirect
6. #69690C2semidirect
7. #69687C2semidirect
8. #69631C2semidirect
9. #69625C2semidirect
11. #69501C2semidirect
49. #2625D3semidirect
511. C3#33017semidirect

Maximal subgroups

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

subgroupindex
C2×#55232
C3×#26252
C2×#50472
C2×#50442
#715842
#696902
#696872
#696312
#696252
#695222
#695012
#330173