Finite Group Databasebeta

← #10883577

Subgroups — #10883577

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
1C2×C3×#472C211yesmaximal, characteristic, complemented
2#83836C211yesmaximal, characteristic, complemented
3#83823C211yesmaximal, characteristic, complemented
4#83792C211yesmaximal, characteristic, complemented
5#72099C212yesmaximal, complemented
6#72089C212yesmaximal, complemented
7#72012C211yesmaximal, characteristic, complemented
8#70719C211yesmaximal, characteristic, complemented
9#70698C211yesmaximal, characteristic, complemented
10#70078C212yesmaximal, complemented
11#70071C212yesmaximal, complemented
12#28183—31nomaximal, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C22×C3×D8C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16#5822C2211yescharacteristic
17#5805C2212yes—
18#5798C2212yes—
19C2×#901C2211yescharacteristic
20C3×#472—22no—
21C3×#472—22no—
22C2×C3×#108C2211yescharacteristic
23C2×C3×#100C2212yes—
24#5240C2212yes—
25#5236C2211yescharacteristic
26#5215C2212yes—
27#5215C2212yes—
28#5159—22no—
29#5157—22no—
30#5152—22no—
31Dic3×D8—22no—
32#5145—22no—
33#5144—22no—
34#5139C2211yescharacteristic
35#5139C2212yes—
36#5124C2211yescharacteristic
37#5106C2211yescharacteristic
38#5106C2211yescharacteristic
39#5101C2211yescharacteristic
40#4559C2212yes—
41#4557C2212yes—
42#4557C2212yes—
43#4556C2212yes—
44C2×#472D311yescharacteristic, complemented
45#3058—31no—
46#2955—31no—
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
56C22×C3×D4C2312yes—
57C2×#235C2311yescharacteristic
58C3×AGL(1,ℤ/8)—24no—
59C3×AGL(1,ℤ/8)—42no—
60C2×C3×SD16—22no—
61C2×C3×D8—22no—
62C2×C3×D8—21no—
63C2×C3×D8—21no—
64C2×C3×D8—21no—
65C2×C3×D8—21no—
66C2×C3×D8D412yes—
67C2×C3×D8D412yes—
68C3×#131—21no—
69C3×#131—21no—
70C3×#131—21no—
71C3×#131—21no—
72C2×C3×M16—21no—
73C2×C3×M16C2311yescharacteristic
74C2×C3×M16C2312yes—
75C22×C3×C8—21no—
76C22×C3×C8C2311yescharacteristic
77#953—42no—
78C2×#217—42no—
79#951—21no—
80#950—22no—
81#949—21no—
82#942—21no—
83#940—22no—
84#938—24no—
85#936—22no—
86D4×Dic3—24no—
87D4×Dic3—42no—
88#932—21no—
89#926C2312yes—
90#926C2312yes—
91#926C2311yescharacteristic
92C2×#208C2311yescharacteristic
93C2×#208C2311yescharacteristic
94#901—21no—
95#901—21no—
96#901—21no—
97#901—21no—
98C2×ΓL(1,𝔽25)—21no—
99C2×D3×C8—21no—
100C3×#108—21no—
101C3×#108—21no—
102C3×#100—24no—
103#837—22no—
104#834—42no—
105#832—24no—
106#832—24no—
107#825—22no—
108#825—42no—
109#822—22no—
110#818—21no—
111#818—21no—
112#817—21no—
113#817—21no—
114#815D412yescomplemented
115C8×Dic3D412yescomplemented
116#575—31no—
117C2×AGL(1,ℤ/8)C2×D312yes—
118C22×D8C2×D311yescharacteristic
119C2×#131—31no—
120C2×#131C2×D311yescharacteristic
121#535—32no—
122#518—32no—
123#472—22no—
124#472—22no—
125#469—62no—
126#466—62no—
127#444—62no—
128#443—62no—
129#442—62no—
130C4×D8—62no—
131#430—31no—
132C2×#108C2×D311yescharacteristic
133#427—31no—
134#427—31no—
135#421—31no—
136#421—31no—
137#421—32no—
138#418—32no—
139#418—32no—
140C2×#100C2×D312yes—
141#410—32no—
142#405—31no—
143#344—32no—
144#344—32no—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)—32no—
147C24×C3—22no—
148C3×#41—21no—
149C3×#41—21no—
150C3×#41—21no—
151C3×#41—21no—
152C2×C3×Q8#4111yescharacteristic
153C2×C3×D4—21no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4C2×D414yes—
160C2×C3×D4#4111yescharacteristic
161C22×C3×C4—21no—
162C22×C3×C4C2411yescharacteristic
163C2×#67—21no—
164C2×#67—44no—
165C22×Dic3—44no—
166#235—21no—
167#235—21no—
168#235—21no—
169#235—21no—
170C2×D12#4111yescharacteristic
171C2×C4×D3—21no—
172C2×Dic6#4111yescharacteristic
173C3×SD16—44no—
174C3×D8—22no—
175C3×D8—22no—
176C3×D8—22no—
177C3×D8—22no—
178C3×D8—44no—
179C3×M16—21no—
180C3×M16—21no—
181C3×M16—22no—
182C3×M16—22no—
183C3×M16—41no—
184C3×M16—41no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8—41no—
188C2×C3×C8—41no—
189C2×C3×C8C2×D411yescharacteristic
190C2×C3×C8C2×D411yescharacteristic
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8#4111yescharacteristic
194C2×C3×C8#4111yescharacteristic
195#217—21no—
196#217—22no—
197#217—22no—
198#217—44no—
199#217—44no—
200#216—22no—
201#215—22no—
202#214—22no—
203#213—22no—
204C4⋊Dic3—21no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4×Dic3—22no—
208C4×Dic3—22no—
209C4×Dic3C2×D412yes—
210#208—21no—
211#208—21no—
212#208—21no—
213#208—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8C2×D412yes—
216ΓL(1,𝔽25)—41no—
217ΓL(1,𝔽25)—41no—
218D3×C8—41no—
219D3×C8—41no—
220C2×#41—31no—
221C2×#41C22×D311yescharacteristic
222C22×D4C22×D312yes—
223#137—61no—
224#137—61no—
225AGL(1,ℤ/8)—24no—
226AGL(1,ℤ/8)—42no—
227AGL(1,ℤ/8)—61no—
228AGL(1,ℤ/8)—61no—
229#135—62no—
230C2×SD16—22no—
231C2×D8—22no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8#6712yes—
237C2×D8#6712yes—
238#131—21no—
239#131—21no—
240#131—21no—
241#131—21no—
242#131—61no—
243#131—61no—
244#131—61no—
245#131—61no—
246C2×M16—21no—
247C2×M16—31no—
248C2×M16—31no—
249C2×M16—61no—
250C2×M16C22×D311yescharacteristic
251C2×M16C22×D312yes—
252C22×C8—21no—
253C22×C8—61no—
254C22×C8C22×D311yescharacteristic
255#127—62no—
256#124—62no—
257#121—64no—
258#120—122no—
259C4×D4—64no—
260C4×D4—122no—
261#117—31no—
262#117—32no—
263#117—32no—
264C2×C22⋊C4—122no—
265#108—21no—
266#108—21no—
267#108—62no—
268#107—61no—
269#107—61no—
270#106—61no—
271#106—61no—
272#104—62no—
273#104—62no—
274#104—122no—
275#102—64no—
276#102—64no—
277#100—24no—
278#99—122no—
279#97—32no—
280C4×C8—32no—
281C23×C3—24no—
282C23×C3—21no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—162no—
286C23×C3H(5,𝔽2)11yescharacteristic
287C22×D3—21no—
288C3×Q8—21no—
289C3×Q8—21no—
290C3×D4—24no—
291C3×D4—21no—
292C3×D4—21no—
293C3×D4—24no—
294C3×D4—41no—
295C3×D4—41no—
296C3×D4—42no—
297C3×D4—42no—
298C2×C3×C4—21no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×D411yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4H(5,𝔽2)11yescharacteristic
307#67—41no—
308#67—41no—
309#67—164no—
310C2×Dic3—22no—
311C2×Dic3—21no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—164no—
315D12—21no—
316D12—21no—
317C4×D3—41no—
318C4×D3—41no—
319Dic6—21no—
320Dic6—21no—
321C3×C8—41no—
322C3×C8—42no—
323C3×C8#12111yescharacteristic
324C3×C8#12111yescharacteristic
325C3×C8#12111yescharacteristic
326C3×C8#12111yescharacteristic
327C3⋊C8—22no—
328C3⋊C8—41no—
329C24—22no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334#41—61no—
335#41—61no—
336#41—61no—
337#41—61no—
338C2×Q8—31no—
339C2×Q8#23711yescharacteristic
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—31no—
347C2×D4—61no—
348C2×D4—124no—
349C2×D4#23711yescharacteristic
350C2×D4C2×#6714yes—
351C22×C4—21no—
352C22×C4—61no—
353C22×C4—124no—
354C22×C4C23×D311yescharacteristic
355Q16—62no—
356SD16—44no—
357SD16—62no—
358SD16—62no—
359D8—22no—
360D8—22no—
361D8—22no—
362D8—22no—
363D8—44no—
364D8—62no—
365M16—21no—
366M16—22no—
367M16—21no—
368M16—22no—
369M16—41no—
370M16—41no—
371M16—61no—
372M16—61no—
373M16—61no—
374M16—61no—
375M16—121no—
376M16—121no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—32no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—61no—
383C2×C8—121no—
384C2×C8—121no—
385C2×C8D3×D411yescharacteristic
386C2×C8D3×D411yescharacteristic
387C2×C8#23711yescharacteristic
388C2×C8#23711yescharacteristic
389C2×C8C2×#6711yescharacteristic
390C2×C8C2×#6711yescharacteristic
391C4⋊C4—61no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C22⋊C4—61no—
395C22⋊C4—62no—
396C22⋊C4—62no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—32no—
400C42—62no—
401C42—62no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—44no—
405C22×C3—84no—
406C22×C3—81no—
407C22×C3—84no—
408C22×C3—162no—
409C22×C3#53511yescharacteristic
410C22×C3#54611yescharacteristic
411C22×C3#55311yescharacteristic
412C2×D3—42no—
413C2×D3—41no—
414C3×C4—41no—
415C3×C4C2×#12111yescharacteristic
416C3×C4#52511yescharacteristic
417C3×C4#54611yescharacteristic
418C3×C4#55311yescharacteristic
419Dic3—42no—
420Dic3—41no—
421Dic3—82no—
422Dic3—82no—
423C23—24no—
424C23—21no—
425C23—42no—
426C23—42no—
427C23—61no—
428C23—162no—
429C23#100911yescharacteristic
430Q8—21no—
431Q8—21no—
432Q8—61no—
433Q8—61no—
434D4—24no—
435D4—21no—
436D4—24no—
437D4—21no—
438D4—41no—
439D4—41no—
440D4—42no—
441D4—42no—
442D4—61no—
443D4—61no—
444D4—121no—
445D4—121no—
446D4—484no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—61no—
451C2×C4—62no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—484no—
457C2×C4C2×D3×D411yescharacteristic
458C2×C4C2×#23711yescharacteristic
459C2×C4C2×#23711yescharacteristic
460C2×C4#100411yescharacteristic
461C2×C4D3×#4111yescharacteristic
462C2×C4C22×#6711yescharacteristic
463C8—41no—
464C8—42no—
465C8—62no—
466C8—121no—
467C8#93911yescharacteristic
468C8#93911yescharacteristic
469C8#93911yescharacteristic
470C8#93911yescharacteristic
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—81no—
475C2×C3—84no—
476C2×C3#253511yescharacteristic
477D3—81no—
478C22—41no—
479C22—41no—
480C22—44no—
481C22—84no—
482C22—81no—
483C22—84no—
484C22—122no—
485C22—121no—
486C22—162no—
487C22#560711yescharacteristic
488C22#561811yescharacteristic
489C22#582911yescharacteristic
490C4—41no—
491C4—122no—
492C4—121no—
493C4—242no—
494C4—242no—
495C4#560611yescharacteristic
496C4#561411yescharacteristic
497C4C2×#93911yescharacteristic
498C4#580611yescharacteristic
499C3#2818311yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—84no—
504C2—81no—
505C2—241no—
506C2#8266611yescharacteristic
subgroupG/H
centerC2#82666
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#472C2
socleC2×C3#2535
hypercenterC2×#472D3
hypocenterC3#28183
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. C2×C3×#472C2semidirect
2. #83836C2semidirect
3. #83823C2semidirect
4. #83792C2semidirect
5. #72099C2semidirect
6. #72089C2semidirect
7. #72012C2semidirect
8. #70719C2semidirect
9. #70698C2semidirect
10. #70078C2semidirect
11. #70071C2semidirect
44. C2×#472D3semidirect
114. #815D4semidirect
115. C8×Dic3D4semidirect
499. C3#28183semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4722
#838362
#838232
#837922
#720992
#720892
#720122
#707192
#706982
#700782
#700712
#281833