Finite Group Databasebeta

← #10883579

Subgroups — #10883579

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2955C211yesmaximal, characteristic, complemented
2C2×#5142C211yesmaximal, characteristic, complemented
3#83799C211yesmaximal, characteristic, complemented
4#78561C211yesmaximal, characteristic, complemented
5#72119C212yesmaximal, complemented
6#72013C211yesmaximal, characteristic, complemented
7#72012C211yesmaximal, characteristic, complemented
8#70731C212yesmaximal, complemented
9#70408C212yesmaximal, complemented
10#70407C212yesmaximal, complemented
11#69896C211yesmaximal, characteristic, complemented
12#28183—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14C2×C3×#131C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16#5830C2212yes—
17C2×#949C2212yes—
18C22×D24C2211yescharacteristic
19#5521C2212yes—
20C3×#444—22no—
21C3×#443—22no—
22C3×#430C2211yescharacteristic
23C3×#421C2211yescharacteristic
24C3×#421C2211yescharacteristic
25C3×#405C2211yescharacteristic
26#5240C2212yes—
27#5236C2212yes—
28#5142—22no—
29#5142—22no—
30#5135—22no—
31#5131C2212yes—
32#5131C2212yes—
33C2×#823C2212yes—
34#5115—22no—
35C2×#819C2211yescharacteristic
36#5106C2211yescharacteristic
37#5106C2211yescharacteristic
38#4711—22no—
39C4×D24—22no—
40#4559C2212yes—
41#4557C2212yes—
42#4556C2212yes—
43#4531C2212yes—
44C2×#472—31no—
45#3058—31no—
46#2955D311yescharacteristic, complemented
47#2948—31no—
48#2081—32no—
49#2008—32no—
50#1911—32no—
51#1904—32no—
52#1833—31no—
53#1814—31no—
54#1812—31no—
55C2×C3×#41C2311yescharacteristic
56C2×C3×#41C2311yescharacteristic
57C22×D12C2312yes—
58C3×#137—21no—
59C3×#137—21no—
60C3×AGL(1,ℤ/8)—21no—
61C3×AGL(1,ℤ/8)—21no—
62C3×#135—22no—
63C3×#131—21no—
64C3×#131—21no—
65C3×#131—21no—
66C3×#131—21no—
67C3×#131—21no—
68C3×#131—21no—
69C3×#131—21no—
70C3×#131—21no—
71C2×C3×M16—21no—
72C2×C3×M16—21no—
73C2×C3×M16C2311yescharacteristic
74C2×C3×M16C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C22×C3×C8—21no—
77C22×C3×C8—21no—
78C22×C3×C8C2311yescharacteristic
79C3×#117C2311yescharacteristic
80#949—24no—
81#949—42no—
82#947—22no—
83C2×#215—22no—
84#940—22no—
85#937—42no—
86C2×#213—22no—
87#930—24no—
88#926C2312yes—
89#926C2312yes—
90C2×#208C2312yes—
91C2×D24—21no—
92C2×D24—21no—
93C2×D24—21no—
94C2×D24—21no—
95C2×D24D412yes—
96C2×D24D412yes—
97#893—42no—
98D3×C22⋊C4—42no—
99C4×D12—24no—
100C3×#108—22no—
101C3×#104—22no—
102C3×#104—22no—
103C3×#97D412yescomplemented
104C3×C4×C8D412yescomplemented
105#837—42no—
106#834—42no—
107#823—24no—
108#821—24no—
109#821—24no—
110#819—21no—
111#819—21no—
112#818—21no—
113#818—21no—
114#817—21no—
115#817—21no—
116#575C2×D311yescharacteristic
117C2×AGL(1,ℤ/8)—32no—
118C22×D8—31no—
119C2×#131C2×D311yescharacteristic
120C2×#131C2×D311yescharacteristic
121#535—32no—
122#518—32no—
123#472—62no—
124#472—62no—
125#469—62no—
126#466—62no—
127#444—22no—
128#443—22no—
129#442—62no—
130C4×D8—62no—
131#430C2×D311yescharacteristic
132C2×#108—31no—
133#427—31no—
134#427—31no—
135#421—32no—
136#421C2×D311yescharacteristic
137#421C2×D311yescharacteristic
138#418—32no—
139#418—32no—
140C2×#100—32no—
141#410—32no—
142#405C2×D311yescharacteristic
143#344—32no—
144#344—32no—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)—32no—
147C23×D3—22no—
148C3×#41—21no—
149C3×#41—21no—
150C3×#41—21no—
151C3×#41—21no—
152C3×#41—21no—
153C3×#41—21no—
154C3×#41—21no—
155C3×#41—21no—
156C2×C3×Q8#4111yescharacteristic
157C2×C3×Q8#4111yescharacteristic
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4#4111yescharacteristic
161C2×C3×D4#4111yescharacteristic
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4C2411yescharacteristic
165C2×#67—44no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12C2×D414yes—
172C2×C4×D3—44no—
173C3×Q16—22no—
174C3×SD16—22no—
175C3×SD16—22no—
176C3×D8—22no—
177C3×M16—21no—
178C3×M16—21no—
179C3×M16—21no—
180C3×M16—21no—
181C3×M16—21no—
182C3×M16—21no—
183C3×M16—41no—
184C3×M16—41no—
185C3×M16—41no—
186C3×M16—41no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—41no—
190C2×C3×C8—41no—
191C2×C3×C8—41no—
192C2×C3×C8—41no—
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8C2×D412yes—
196C2×C3×C8C2×D411yescharacteristic
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8#4111yescharacteristic
199C2×C3×C8#4111yescharacteristic
200C3×C4⋊C4—21no—
201C3×C22⋊C4—21no—
202C3×C42C2×D412yes—
203#217—22no—
204#217—22no—
205#215—44no—
206#213—44no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—44no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4×Dic3—22no—
212C4×Dic3—22no—
213#208—22no—
214#208—22no—
215C2×C3⋊C8—22no—
216D24—22no—
217D24—22no—
218D24—22no—
219D24—22no—
220C2×#41C22×D311yescharacteristic
221C2×#41C22×D311yescharacteristic
222C22×D4—32no—
223#137—21no—
224#137—21no—
225AGL(1,ℤ/8)—21no—
226AGL(1,ℤ/8)—21no—
227AGL(1,ℤ/8)—64no—
228AGL(1,ℤ/8)—122no—
229#135—22no—
230C2×SD16—62no—
231C2×D8—32no—
232C2×D8—32no—
233C2×D8—62no—
234C2×D8—61no—
235C2×D8—61no—
236C2×D8—61no—
237C2×D8—61no—
238#131—21no—
239#131—21no—
240#131—21no—
241#131—21no—
242#131—21no—
243#131—21no—
244#131—21no—
245#131—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—32no—
249C2×M16C22×D311yescharacteristic
250C2×M16C22×D311yescharacteristic
251C2×M16C22×D311yescharacteristic
252C22×C8—21no—
253C22×C8—21no—
254C22×C8C22×D311yescharacteristic
255#127—62no—
256#124—62no—
257#121—64no—
258#120—122no—
259C4×D4—64no—
260C4×D4—122no—
261#117—32no—
262#117—32no—
263#117C22×D311yescharacteristic
264C2×C22⋊C4—122no—
265#108—22no—
266#108—61no—
267#108—61no—
268#107—61no—
269#107—61no—
270#106—61no—
271#106—61no—
272#104—22no—
273#104—22no—
274#104—122no—
275#102—64no—
276#102—64no—
277#100—64no—
278#99—122no—
279#97#6712yescomplemented
280C4×C8#6712yescomplemented
281C23×C3—21no—
282C23×C3—21no—
283C23×C3H(5,𝔽2)11yescharacteristic
284C22×D3—24no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—162no—
288C3×Q8—21no—
289C3×Q8—21no—
290C3×Q8—21no—
291C3×Q8—21no—
292C3×D4—21no—
293C3×D4—21no—
294C3×D4—21no—
295C3×D4—21no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—41no—
300C2×C3×C4—22no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C4H(5,𝔽2)11yescharacteristic
313#67—164no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316D12—24no—
317D12—24no—
318D12—42no—
319D12—42no—
320C4×D3—164no—
321C3×C8—22no—
322C3×C8—41no—
323C3×C8—41no—
324C3×C8#12111yescharacteristic
325C3×C8#12111yescharacteristic
326C3×C8#12111yescharacteristic
327C3×C8#12111yescharacteristic
328C3⋊C8—42no—
329C24—62no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334#41—21no—
335#41—21no—
336#41—21no—
337#41—21no—
338C2×Q8#23511yescharacteristic
339C2×Q8#23511yescharacteristic
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—34no—
343C2×D4—62no—
344C2×D4—62no—
345C2×D4—62no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—124no—
349C2×D4#23511yescharacteristic
350C2×D4#23511yescharacteristic
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—124no—
354C22×C4C23×D311yescharacteristic
355Q16—22no—
356SD16—22no—
357SD16—22no—
358SD16—124no—
359D8—22no—
360D8—62no—
361D8—62no—
362D8—62no—
363D8—62no—
364D8—124no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—62no—
376M16—62no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—62no—
384C2×C8D3×D411yescharacteristic
385C2×C8D3×D411yescharacteristic
386C2×C8#23711yescharacteristic
387C2×C8#23711yescharacteristic
388C2×C8C2×#6712yes—
389C2×C8C2×#6711yescharacteristic
390C2×C8C2×#6711yescharacteristic
391C4⋊C4—21no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C22⋊C4—21no—
395C22⋊C4—62no—
396C22⋊C4—62no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—62no—
400C42—62no—
401C42C2×#6712yes—
402C22×C3—41no—
403C22×C3—42no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—81no—
407C22×C3#53511yescharacteristic
408C22×C3#54611yescharacteristic
409C22×C3#55311yescharacteristic
410C2×D3—44no—
411C2×D3—84no—
412C2×D3—84no—
413C2×D3—162no—
414C3×C4—42no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4C2×#12111yescharacteristic
418C3×C4#52511yescharacteristic
419C3×C4#54611yescharacteristic
420C3×C4#55311yescharacteristic
421Dic3—82no—
422Dic3—82no—
423C23—21no—
424C23—21no—
425C23—64no—
426C23—122no—
427C23—122no—
428C23—482no—
429C23#100911yescharacteristic
430Q8—21no—
431Q8—21no—
432Q8—21no—
433Q8—21no—
434D4—21no—
435D4—21no—
436D4—21no—
437D4—21no—
438D4—41no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—64no—
443D4—64no—
444D4—122no—
445D4—122no—
446D4—484no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—484no—
457C2×C4C2×#23511yescharacteristic
458C2×C4C2×#23511yescharacteristic
459C2×C4C2×D3×D411yescharacteristic
460C2×C4C2×#23711yescharacteristic
461C2×C4#100911yescharacteristic
462C2×C4C22×#6711yescharacteristic
463C8—22no—
464C8—41no—
465C8—41no—
466C8—122no—
467C8#93811yescharacteristic
468C8#93811yescharacteristic
469C8#93811yescharacteristic
470C8#93811yescharacteristic
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—81no—
475C2×C3—81no—
476C2×C3#253511yescharacteristic
477D3—84no—
478C22—41no—
479C22—42no—
480C22—41no—
481C22—41no—
482C22—81no—
483C22—124no—
484C22—244no—
485C22—244no—
486C22—482no—
487C22#555011yescharacteristic
488C22#556511yescharacteristic
489C22#583011yescharacteristic
490C4—42no—
491C4—41no—
492C4—41no—
493C4—242no—
494C4—242no—
495C4#555011yescharacteristic
496C4#556511yescharacteristic
497C4#579111yescharacteristic
498C4C2×#93811yescharacteristic
499C3#2818311yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—81no—
504C2—81no—
505C2—244no—
506C2#8241511yescharacteristic
subgroupG/H
centerC2#82415
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2955C2
socleC2×C3#2535
hypercenter#2955D3
hypocenterC3#28183
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2955

Complements

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

NH ≅ G/Ntype
1. C3×#2955C2semidirect
2. C2×#5142C2semidirect
3. #83799C2semidirect
4. #78561C2semidirect
5. #72119C2semidirect
6. #72013C2semidirect
7. #72012C2semidirect
8. #70731C2semidirect
9. #70408C2semidirect
10. #70407C2semidirect
11. #69896C2semidirect
46. #2955D3semidirect
103. C3×#97D4semidirect
104. C3×C4×C8D4semidirect
279. #97#67semidirect
280. C4×C8#67semidirect
499. C3#28183semidirect

Maximal subgroups

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

subgroupindex
C3×#29552
C2×#51422
#837992
#785612
#721192
#720132
#720122
#707312
#704082
#704072
#698962
#281833