Finite Group Databasebeta

← #10883540

Subgroups — #10883540

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3067C211yesmaximal, characteristic, complemented
2#83914C211yesmaximal, characteristic, complemented
3#83910C211yesmaximal, characteristic, complemented
4#83794C211yesmaximal, characteristic, complemented
5#72118C212yesmaximal, complemented
6#72116C212yesmaximal, complemented
7#72013C211yesmaximal, characteristic, complemented
8#70711C211yesmaximal, characteristic, complemented
9#70698C211yesmaximal, characteristic, complemented
10#70136C212yesmaximal, complemented
11#70132C212yesmaximal, complemented
12#28180—31nomaximal, Sylow
13C3×#575C2212yes—
14C2×C3×#135C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16#5828C2211yescharacteristic
17#5828C2212yes—
18#5819C2212yes—
19C22×D3×C8C2211yescharacteristic
20C3×#473—21no—
21C3×#472—21no—
22C3×#471—22no—
23C2×C3×#108C2211yescharacteristic
24C3×#413C2212yes—
25#5236C2212yes—
26#5236C2212yes—
27#5234C2211yescharacteristic
28#5233C2212yes—
29#5197—22no—
30#5196—22no—
31#5188—21no—
32#5171—22no—
33#5157—21no—
34#5127—22no—
35#5126C2211yescharacteristic
36#5126C2212yes—
37#5119C2211yescharacteristic
38C8×#67—22no—
39C2×#819C2211yescharacteristic
40#5106C2211yescharacteristic
41#5101C2211yescharacteristic
42#4559C2212yes—
43#4559C2212yes—
44#4558C2212yes—
45#4557C2212yes—
46#3067D311yescharacteristic, complemented
47#3050—31no—
48#2957—31no—
49#2929—31no—
50#2080—32no—
51#2010—32no—
52#1891—32no—
53#1881—32no—
54#1833—31no—
55#1814—31no—
56#1810—31no—
57C2×C3×#41C2312yes—
58C2×C3×#41C2311yescharacteristic
59C22×C4×D3C2311yescharacteristic
60C3×#137—22no—
61C3×#137—22no—
62C3×AGL(1,ℤ/8)—22no—
63C3×AGL(1,ℤ/8)—22no—
64C3×#135—21no—
65C3×#135—21no—
66C3×#135—21no—
67C3×#135—21no—
68C3×#135—42no—
69C2×C3×Q16D411yescharacteristic, complemented
70C2×C3×SD16D412yescomplemented
71C2×C3×D8D411yescharacteristic, complemented
72C3×#131—21no—
73C3×#131—21no—
74C3×#131—21no—
75C3×#131—21no—
76C2×C3×M16—21no—
77C2×C3×M16C2311yescharacteristic
78C2×C3×M16C2312yes—
79C22×C3×C8—21no—
80C22×C3×C8C2311yescharacteristic
81#948—24no—
82#948—42no—
83#946—21no—
84#946—22no—
85#938—21no—
86#938—22no—
87#937—21no—
88#937—22no—
89#935—21no—
90#935—22no—
91C4×#67—22no—
92C4×#67—42no—
93#926C2312yes—
94#926C2311yescharacteristic
95C2×#208—42no—
96C2×#208C2312yes—
97C2×#208C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99C2×D3×C8—21no—
100C2×D3×C8—21no—
101C2×D3×C8—21no—
102C2×D3×C8—21no—
103C2×D3×C8—21no—
104C2×D3×C8—21no—
105C3×#108—21no—
106C3×#108—21no—
107C3×#101—22no—
108C3×#100—22no—
109#837—24no—
110#837—42no—
111#835—22no—
112#832—22no—
113#830—22no—
114(C2×D3)⋊C8—22no—
115(C2×D3)⋊C8—42no—
116#819—21no—
117#819—21no—
118#818—21no—
119#817—21no—
120#815D412yescomplemented
121Dic3⋊C8—22no—
122C8×Dic3D412yescomplemented
123#575C2×D312yes—
124C2×#135C2×D311yescharacteristic
125#568—32no—
126C2×#131C2×D311yescharacteristic
127C23×C8—31no—
128#525—31no—
129#525—32no—
130#473—21no—
131#472—21no—
132#471—22no—
133#467—61no—
134#466—61no—
135#465—62no—
136#444—62no—
137#443—62no—
138#435—62no—
139C8×D4—62no—
140C2×#108—31no—
141C2×#108C2×D311yescharacteristic
142#427—31no—
143#424—31no—
144#421—32no—
145#421—32no—
146#416—32no—
147#413C2×D312yes—
148#408—31no—
149#407—31no—
150#407—32no—
151#405—31no—
152#344—32no—
153#344—32no—
154#341—32no—
155H(3,ℤ/4)—32no—
156C23×D3#4111yescharacteristic
157C3×#41—22no—
158C3×#41—21no—
159C3×#41—21no—
160C3×#41—22no—
161C3×#41—22no—
162C3×#41—21no—
163C3×#41—22no—
164C3×#41—21no—
165C2×C3×Q8C2×D412yes—
166C2×C3×Q8#4111yescharacteristic
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4C2×D412yes—
170C2×C3×D4#4111yescharacteristic
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4C2411yescharacteristic
174C2×#67—22no—
175C2×#67—42no—
176C22×Dic3#4111yescharacteristic
177C2×C4×D3—21no—
178C2×C4×D3—21no—
179C2×C4×D3—21no—
180C2×C4×D3—21no—
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C3×Q16—21no—
184C3×Q16—21no—
185C3×Q16—42no—
186C3×SD16—22no—
187C3×SD16—22no—
188C3×SD16—42no—
189C3×SD16—42no—
190C3×D8—21no—
191C3×D8—21no—
192C3×D8—42no—
193C3×M16—21no—
194C3×M16—22no—
195C3×M16—21no—
196C3×M16—22no—
197C3×M16—41no—
198C3×M16—41no—
199C2×C3×C8—21no—
200C2×C3×C8—22no—
201C2×C3×C8—41no—
202C2×C3×C8—41no—
203C2×C3×C8C2×D411yescharacteristic
204C2×C3×C8C2×D411yescharacteristic
205C2×C3×C8C2×D411yescharacteristic
206C2×C3×C8C2×D411yescharacteristic
207C2×C3×C8#4111yescharacteristic
208C2×C3×C8#4111yescharacteristic
209#217—22no—
210#217—21no—
211#217—22no—
212#217—42no—
213(C2×D3)⋊C4—22no—
214(C2×D3)⋊C4—42no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217Dic3⋊C4—22no—
218Dic3⋊C4—42no—
219C4×Dic3—22no—
220C4×Dic3C2×D412yes—
221#208—21no—
222#208—21no—
223#208—22no—
224#208—22no—
225#208—44no—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—44no—
231C2×C3⋊C8C2×D412yes—
232D3×C8—41no—
233D3×C8—41no—
234D3×C8—41no—
235D3×C8—41no—
236C2×#41C22×D311yescharacteristic
237C2×#41C22×D312yes—
238C23×C4—31no—
239#137—22no—
240#137—22no—
241AGL(1,ℤ/8)—22no—
242AGL(1,ℤ/8)—22no—
243#135—21no—
244#135—21no—
245#135—21no—
246#135—21no—
247#135—42no—
248C2×Q16#6711yescharacteristic, complemented
249C2×SD16#6712yescomplemented
250C2×D8#6711yescharacteristic, complemented
251#131—21no—
252#131—21no—
253#131—21no—
254#131—21no—
255#131—64no—
256#131—122no—
257C2×M16—21no—
258C2×M16—31no—
259C2×M16—32no—
260C2×M16—122no—
261C2×M16C22×D311yescharacteristic
262C2×M16C22×D312yes—
263C22×C8—21no—
264C22×C8—31no—
265C22×C8—61no—
266C22×C8—61no—
267C22×C8—61no—
268C22×C8—61no—
269C22×C8—61no—
270C22×C8—61no—
271C22×C8C22×D311yescharacteristic
272#123—61no—
273#123—62no—
274C22⋊Q8—61no—
275C22⋊Q8—62no—
276#121—61no—
277#121—62no—
278#120—61no—
279#120—62no—
280C4×D4—62no—
281C4×D4—122no—
282#117—31no—
283#117—32no—
284#108—21no—
285#108—21no—
286#108—61no—
287#108—61no—
288#107—61no—
289#106—61no—
290C4⋊C8—62no—
291#104—64no—
292#104—122no—
293#103—62no—
294#102—62no—
295#101—22no—
296#100—22no—
297C22⋊C8—62no—
298C22⋊C8—62no—
299C22⋊C8—122no—
300#97—32no—
301C4×C8—32no—
302C23×C3—21no—
303C23×C3—22no—
304C23×C3C2×#4111yescharacteristic
305C22×D3—21no—
306C22×D3—21no—
307C22×D3—21no—
308C22×D3—42no—
309C3×Q8—22no—
310C3×Q8—21no—
311C3×Q8—22no—
312C3×Q8—21no—
313C3×D4—22no—
314C3×D4—21no—
315C3×D4—22no—
316C3×D4—21no—
317C3×D4—42no—
318C3×D4—41no—
319C3×D4—41no—
320C3×D4—42no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4C2×#4111yescharacteristic
331C2×C3×C4H(5,𝔽2)11yescharacteristic
332C2×C3×C4H(5,𝔽2)11yescharacteristic
333#67—44no—
334#67—84no—
335C2×Dic3—22no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—42no—
340C4×D3—41no—
341C4×D3—41no—
342C4×D3—41no—
343C4×D3—41no—
344C3×C8—41no—
345C3×C8—42no—
346C3×C8#12111yescharacteristic
347C3×C8#12111yescharacteristic
348C3×C8#12111yescharacteristic
349C3×C8#12111yescharacteristic
350C3⋊C8—22no—
351C3⋊C8—41no—
352C3⋊C8—42no—
353C24—31no—
354#41—22no—
355#41—21no—
356#41—22no—
357#41—22no—
358#41—22no—
359#41—21no—
360#41—21no—
361#41—21no—
362C2×Q8#23711yescharacteristic
363C2×Q8C2×#6712yes—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—62no—
367C2×D4—122no—
368C2×D4#23711yescharacteristic
369C2×D4C2×#6712yes—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—31no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4C23×D311yescharacteristic
380Q16—21no—
381Q16—21no—
382Q16—42no—
383SD16—22no—
384SD16—22no—
385SD16—42no—
386SD16—42no—
387D8—21no—
388D8—21no—
389D8—42no—
390M16—21no—
391M16—22no—
392M16—21no—
393M16—22no—
394M16—41no—
395M16—41no—
396M16—61no—
397M16—61no—
398M16—62no—
399M16—62no—
400M16—124no—
401C2×C8—21no—
402C2×C8—22no—
403C2×C8—32no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—62no—
409C2×C8—61no—
410C2×C8—124no—
411C2×C8—121no—
412C2×C8—121no—
413C2×C8—121no—
414C2×C8—121no—
415C2×C8D3×D411yescharacteristic
416C2×C8D3×D411yescharacteristic
417C2×C8#23711yescharacteristic
418C2×C8#23711yescharacteristic
419C2×C8C2×#6711yescharacteristic
420C2×C8C2×#6711yescharacteristic
421C4⋊C4—62no—
422C4⋊C4—61no—
423C4⋊C4—62no—
424C4⋊C4—122no—
425C22⋊C4—62no—
426C22⋊C4—62no—
427C22⋊C4—61no—
428C22⋊C4—62no—
429C22⋊C4—122no—
430C22⋊C4—122no—
431C42—32no—
432C42—62no—
433C22×C3—41no—
434C22×C3—42no—
435C22×C3—41no—
436C22×C3—42no—
437C22×C3—82no—
438C22×C3#52511yescharacteristic
439C22×C3#54611yescharacteristic
440C22×C3#55311yescharacteristic
441C2×D3—22no—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—41no—
445C2×D3—41no—
446C2×D3—41no—
447C2×D3—41no—
448C3×C4—41no—
449C3×C4—42no—
450C3×C4C2×#12111yescharacteristic
451C3×C4#53511yescharacteristic
452C3×C4#54611yescharacteristic
453C3×C4#55311yescharacteristic
454Dic3—42no—
455Dic3—41no—
456Dic3—82no—
457C23—21no—
458C23—22no—
459C23—61no—
460C23—61no—
461C23—61no—
462C23—122no—
463C23D3×#4111yescharacteristic
464Q8—22no—
465Q8—21no—
466Q8—22no—
467Q8—21no—
468D4—22no—
469D4—21no—
470D4—22no—
471D4—21no—
472D4—41no—
473D4—42no—
474D4—41no—
475D4—42no—
476D4—124no—
477D4—244no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—61no—
485C2×C4—62no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—122no—
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23711yescharacteristic
495C2×C4C2×#23711yescharacteristic
496C2×C4#100411yescharacteristic
497C2×C4#100911yescharacteristic
498C2×C4C22×#6711yescharacteristic
499C8—41no—
500C8—42no—
501C8—62no—
502C8—121no—
503C8—122no—
504C8#93911yescharacteristic
505C8#93911yescharacteristic
506C8#93911yescharacteristic
507C8#93911yescharacteristic
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—81no—
512C2×C3—82no—
513C2×C3#253511yescharacteristic
514D3—42no—
515C22—41no—
516C22—42no—
517C22—41no—
518C22—42no—
519C22—62no—
520C22—62no—
521C22—62no—
522C22—82no—
523C22—121no—
524C22—121no—
525C22—121no—
526C22—121no—
527C22#560611yescharacteristic
528C22#561411yescharacteristic
529C22#582911yescharacteristic
530C4—42no—
531C4—41no—
532C4—122no—
533C4—121no—
534C4—242no—
535C4#560711yescharacteristic
536C4#561811yescharacteristic
537C4C2×#93911yescharacteristic
538C4#580611yescharacteristic
539C3#2818011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—21no—
543C2—81no—
544C2—82no—
545C2—122no—
546C2#8266611yescharacteristic
subgroupG/H
centerC4#5607
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3067C2
socleC2×C3#2535
hypercenter#3067D3
hypocenterC3#28180
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #3067

Complements

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

NH ≅ G/Ntype
1. C3×#3067C2semidirect
2. #83914C2semidirect
3. #83910C2semidirect
4. #83794C2semidirect
5. #72118C2semidirect
6. #72116C2semidirect
7. #72013C2semidirect
8. #70711C2semidirect
9. #70698C2semidirect
10. #70136C2semidirect
11. #70132C2semidirect
46. #3067D3semidirect
69. C2×C3×Q16D4semidirect
70. C2×C3×SD16D4semidirect
71. C2×C3×D8D4semidirect
120. #815D4semidirect
122. C8×Dic3D4semidirect
248. C2×Q16#67semidirect
249. C2×SD16#67semidirect
250. C2×D8#67semidirect
539. C3#28180semidirect

Maximal subgroups

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

subgroupindex
C3×#30672
#839142
#839102
#837942
#721182
#721162
#720132
#707112
#706982
#701362
#701322
#281803