Finite Group Databasebeta

← #10829981

Subgroups — #10829981

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic
2C3×#2582C211yesmaximal, characteristic, complemented
3C2×#5031C211yesmaximal, characteristic, complemented
4C2×C4×#215C211yesmaximal, characteristic
5#71634C211yesmaximal, characteristic, complemented
6#69690C212yesmaximal, complemented
7#69689C212yesmaximal
8#69632C212yesmaximal, complemented
9#69628C212yesmaximal
10#69584C211yesmaximal, characteristic
11#69579C211yesmaximal, characteristic, complemented
12#24189—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#946—21no—
16C22×#216C2212yes—
17C22×#215C2211yescharacteristic
18#5584—24no—
19#5584—24no—
20C2×#898—22no—
21C2×#892—21no—
22C2×#875C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2212yes—
26C3×#398—22no—
27C3×#391—21no—
28C3×#391C2212yes—
29C3×#389—21no—
30C3×#389C2212yes—
31C3×#384C2211yescharacteristic
32C2×#835C2211yescharacteristic
33#5031—24no—
34C4×#215—24no—
35C2×#810C2212yes—
36C2×#810C2212yes—
37C2×#810C2212yes—
38C2×#809C2211yescharacteristic
39C2×#809C2212yes—
40C2×#808C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#556—31no—
47C2×#462—31no—
48C2×C4×SD16—31no—
49#2582D311yescharacteristic, complemented
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1956—31no—
54#1878—32no—
55#1865—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×Dic6C2312yes—
62C3×C4×Q8—22no—
63C3×C4×Q8—22no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C4⋊C4D412yes—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C42—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#946—28no—
78C2×#216—24no—
79C2×#216—24no—
80C2×#216—42no—
81C2×#215—22no—
82C2×#215—22no—
83C2×#215—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—22no—
91C2×C4×Dic3—22no—
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2312yes—
94C22×C3⋊C8C2311yescharacteristic
95#898—216no—
96#892—28no—
97#875—24no—
98#875—24no—
99C4×D12—22no—
100C4×D12D414yes—
101#869—22no—
102#869D414yes—
103C4×Dic6—24no—
104C4×Dic6—24no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—22no—
108#835—22no—
109#810—24no—
110#810—24no—
111#810—42no—
112#809—22no—
113#809—42no—
114#808—22no—
115C4⋊(C3⋊C8)—22no—
116C4×C3⋊C8—22no—
117C22×Q16—32no—
118C22×SD16—31no—
119#556—64no—
120#556—64no—
121C2×C4⋊Q8—31no—
122C2×#126—62no—
123C2×#124—32no—
124C2×C22⋊Q8—61no—
125C2×C4×Q8—32no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—31no—
129C2×#117—61no—
130#462—64no—
131C4×SD16—64no—
132C2×#106—31no—
133C2×C4⋊C8—31no—
134C2×#103—31no—
135C2×#103—32no—
136C2×#103—32no—
137C2×#103—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C4×C8—31no—
141#398—22no—
142#391—21no—
143#391C2×D312yes—
144#389—21no—
145#389C2×D312yes—
146#384C2×D311yescharacteristic
147#340—32no—
148#336—32no—
149#336—32no—
150C23×D3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8#4112yes—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—22no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12C2×D412yes—
176C2×C4×D3—24no—
177C2×C4×D3—42no—
178C2×C4×D3—42no—
179C2×Dic6—22no—
180C2×Dic6—22no—
181C2×Dic6—22no—
182C2×Dic6—22no—
183C2×Dic6—22no—
184C2×Dic6C2×D414yes—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4—81no—
194C3×C4⋊C4—82no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—42no—
198C3×C42C2×D412yes—
199C3×C42C2×D412yes—
200#216—48no—
201#215—28no—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—24no—
205C4⋊Dic3—24no—
206C4⋊Dic3—24no—
207C4⋊Dic3—21no—
208C4⋊Dic3C2×D412yes—
209Dic3⋊C4—28no—
210C4×Dic3—28no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8#4112yes—
220C22×Q8—32no—
221C22×Q8#6711yescharacteristic
222C22×Q8C22×D311yescharacteristic
223C22×D4—31no—
224C23×C4—61no—
225C2×Q16—64no—
226C2×Q16—64no—
227C2×Q16—122no—
228C2×SD16—62no—
229C2×SD16—62no—
230C2×SD16—62no—
231C22×C8—31no—
232C22×C8—32no—
233C22×C8—31no—
234C4⋊Q8—34no—
235C4⋊Q8—62no—
236#126—616no—
237#124—64no—
238#124—64no—
239C22⋊Q8—68no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—64no—
245C4×Q8—64no—
246C4×D4—34no—
247C4×D4—62no—
248#117—68no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—61no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4#6711yescharacteristic
258C2×C4⋊C4#6712yes—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—61no—
263C2×C22⋊C4—62no—
264C2×C42—21no—
265C2×C42—62no—
266C2×C42C22×D312yes—
267C2×C42C22×D311yescharacteristic
268#106—62no—
269C4⋊C8—62no—
270#103—64no—
271#103—64no—
272#103—62no—
273#103—122no—
274#102—62no—
275#102—122no—
276C4×C8—62no—
277#95—22no—
278#95—22no—
279#95—22no—
280C23×C3C22×D411yescharacteristic
281C22×D3—22no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C22×D3—41no—
286C22×D3—41no—
287C22×D3—41no—
288C3×Q8—22no—
289C3×Q8—22no—
290C3×Q8—41no—
291C3×Q8—41no—
292C3×Q8—44no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4#12012yes—
311C2×C3×C4#12012yes—
312C2×C3×C4#13512yes—
313C2×C3×C4#13712yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319C2×C3×C42−1+411yescharacteristic
320C2×Dic3—22no—
321C2×Dic3—24no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326D12—22no—
327D12—22no—
328D12—41no—
329D12—41no—
330C4×D3—48no—
331Dic6—24no—
332Dic6—24no—
333Dic6—42no—
334Dic6—42no—
335C3⋊C8—22no—
336C3⋊C8—41no—
337C3⋊C8—42no—
338C24—61no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—34no—
348C2×Q8—62no—
349C2×Q8—62no—
350C2×Q8—62no—
351C2×Q8—62no—
352C2×Q8—62no—
353C2×Q8#23912yes—
354C2×D4—32no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—64no—
366C22×C4—122no—
367C22×C4—122no—
368C22×C4C2×#6712yes—
369C22×C4C2×#6712yes—
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6711yescharacteristic
372C22×C4C23×D311yescharacteristic
373Q16—128no—
374SD16—68no—
375C2×C8—32no—
376C2×C8—61no—
377C2×C8—61no—
378C2×C8—61no—
379C2×C8—62no—
380C2×C8—62no—
381C2×C8—61no—
382C2×C8—62no—
383C2×C8—61no—
384C4⋊C4—22no—
385C4⋊C4—24no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—32no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—68no—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C4⋊C4—61no—
397C4⋊C4—81no—
398C4⋊C4—82no—
399C22⋊C4—68no—
400C22⋊C4—68no—
401C22⋊C4—64no—
402C42—22no—
403C42—22no—
404C42—42no—
405C42—68no—
406C42D3×D412yes—
407C42D3×D412yes—
408C22×C3C2×#12011yescharacteristic
409C22×C3#54811yescharacteristic
410C22×C3#55011yescharacteristic
411C22×C3C2×#13511yescharacteristic
412C22×C3C2×#13711yescharacteristic
413C22×C3#57711yescharacteristic
414C22×C3#57811yescharacteristic
415C2×D3—42no—
416C2×D3—42no—
417C2×D3—42no—
418C2×D3—81no—
419C2×D3—81no—
420C2×D3—81no—
421C2×D3—81no—
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—82no—
427C3×C4—81no—
428C3×C4#45214yes—
429C3×C4#52511yescharacteristic
430C3×C4#53711yescharacteristic
431C3×C4#54611yescharacteristic
432C3×C4#54711yescharacteristic
433Dic3—44no—
434Dic3—42no—
435C23—62no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23—121no—
441C23—121no—
442C23C22×#6711yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—44no—
446Q8—41no—
447Q8—41no—
448Q8—64no—
449Q8—64no—
450Q8—122no—
451Q8—122no—
452D4—62no—
453D4—62no—
454D4—121no—
455D4—121no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—62no—
474C2×C4—64no—
475C2×C4—121no—
476C2×C4—122no—
477C2×C4—121no—
478C2×C4—122no—
479C2×C4—128no—
480C2×C4#93712yes—
481C2×C4#93712yes—
482C2×C4#94212yes—
483C2×C4#95012yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×D3×D411yescharacteristic
486C2×C4C2×#23911yescharacteristic
487C2×C4#100711yescharacteristic
488C2×C4D3×#4111yescharacteristic
489C2×C4#100911yescharacteristic
490C8—62no—
491C8—121no—
492C8—122no—
493C2×C3#240711yescharacteristic
494C2×C3C2×#45211yescharacteristic
495C2×C3#301311yescharacteristic
496C2×C3#336411yescharacteristic
497C2×C3#336911yescharacteristic
498C2×C3#339411yescharacteristic
499C2×C3#339811yescharacteristic
500D3—82no—
501C22—122no—
502C22—122no—
503C22—122no—
504C22—241no—
505C22—241no—
506C22—241no—
507C22—241no—
508C22C2×#93711yescharacteristic
509C22C2×#94211yescharacteristic
510C22#581711yescharacteristic
511C22C2×#95011yescharacteristic
512C22#583411yescharacteristic
513C22#583611yescharacteristic
514C22#583811yescharacteristic
515C4—22no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—81no—
520C4—82no—
521C4—122no—
522C4—124no—
523C4#504714yes—
524C4#562711yescharacteristic
525C4#562911yescharacteristic
526C4#563011yescharacteristic
527C4#563111yescharacteristic
528C3#2418911yescharacteristic, complemented, Sylow
529C2—242no—
530C2C2×#504711yescharacteristic
531C2#8281211yescharacteristic
532C2#8342711yescharacteristic
533C2#8368611yescharacteristic
534C2#8368911yescharacteristic
535C2#8370411yescharacteristic
536C2#8371011yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2582C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2582D3
hypocenterC3#24189
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2582

Complements

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

NH ≅ G/Ntype
2. C3×#2582C2semidirect
3. C2×#5031C2semidirect
5. #71634C2semidirect
6. #69690C2semidirect
8. #69632C2semidirect
11. #69579C2semidirect
49. #2582D3semidirect
528. C3#24189semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
C3×#25822
C2×#50312
C2×C4×#2152
#716342
#696902
#696892
#696322
#696282
#695842
#695792
#241893