Finite Group Databasebeta

← #10883580

Subgroups — #10883580

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×#4897C211yesmaximal, characteristic, complemented
2#79183C211yesmaximal, characteristic, complemented
3#78579C211yesmaximal, characteristic, complemented
4#78561C211yesmaximal, characteristic, complemented
5C3×#1833C211yesmaximal, characteristic, complemented
6#70719C211yesmaximal, characteristic, complemented
7#70664C211yesmaximal, characteristic, complemented
8#69028C212yesmaximal, complemented
9#69020C212yesmaximal, complemented
10#68705C212yesmaximal, complemented
11#68704C212yesmaximal, complemented
12#28183—31nomaximal, Sylow
13#5748C2211yescharacteristic
14C2×#908C2212yes—
15C22×D24C2211yescharacteristic
16C2×#901C2211yescharacteristic
17C2×#901C2211yescharacteristic
18#5525C2212yes—
19#5521C2212yes—
20C2×C3×#108C2211yescharacteristic
21C3×#427C2211yescharacteristic
22C3×#405C2211yescharacteristic
23#5139C2212yes—
24#5131C2212yes—
25C2×#823C2212yes—
26#5125C2211yescharacteristic
27#5106C2211yescharacteristic
28#5006C2212yes—
29#5005C2211yescharacteristic
30#5005C2211yescharacteristic
31#5001C2212yes—
32#4897—22no—
33#4897—22no—
34#4883—22no—
35#4865—22no—
36#4717—22no—
37#4711—22no—
38#4700—22no—
39C4×D24—22no—
40C3×#344C2212yes—
41C3×H(3,ℤ/4)C2212yes—
42#4556C2212yes—
43#4531C2212yes—
44C2×#472—31no—
45#3058—31no—
46#2955—31no—
47#2948—31no—
48#2081—32no—
49#2008—32no—
50#1911—32no—
51#1904—32no—
52#1833D311yescharacteristic, complemented
53#1814—31no—
54#1812—31no—
55C2×#235C2311yescharacteristic
56C2×#235C2311yescharacteristic
57C22×D12C2312yes—
58C2×C3×M16C2312yes—
59C2×C3×M16C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×#117C2312yes—
62C3×#117C2311yescharacteristic
63#926C2312yes—
64C2×#208C2311yescharacteristic
65C2×#208C2311yescharacteristic
66#909—21no—
67#909—21no—
68#908—21no—
69#908—21no—
70#908—24no—
71#908—42no—
72#904—22no—
73C2×D24—21no—
74C2×D24—21no—
75C2×D24—21no—
76C2×D24—21no—
77C2×D24—22no—
78C2×D24D412yes—
79C2×D24D412yes—
80C2×#204—22no—
81#901—21no—
82#901—21no—
83#901—21no—
84#901—21no—
85#901—21no—
86#901—21no—
87#901—21no—
88#901—21no—
89C2×ΓL(1,𝔽25)—21no—
90C2×ΓL(1,𝔽25)—21no—
91C2×D3×C8—21no—
92C2×D3×C8—21no—
93#895—24no—
94#893—42no—
95#882—42no—
96D3×C22⋊C4—42no—
97#875—22no—
98#874—22no—
99C4×D12—24no—
100C3×#108—21no—
101C3×#108—21no—
102C3×#107—21no—
103C3×#106—21no—
104C3×#97D412yescomplemented
105C3×C4×C8D412yescomplemented
106#825—42no—
107#823—24no—
108#821—24no—
109#819—22no—
110#818—21no—
111#817—21no—
112#809—24no—
113#806—42no—
114#805—22no—
115#805—22no—
116#575—31no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8—31no—
119C2×#131—31no—
120C2×#131—31no—
121#535—32no—
122#518—32no—
123#472—62no—
124#472—62no—
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#427C2×D311yescharacteristic
135#421—31no—
136#421—31no—
137#421—32no—
138#418—32no—
139#418—32no—
140C2×#100—32no—
141#410—32no—
142#405C2×D311yescharacteristic
143#344—32no—
144#344C2×D312yes—
145H(3,ℤ/4)—32no—
146H(3,ℤ/4)C2×D312yes—
147C23×D3—22no—
148C22×C3×C4C2411yescharacteristic
149C2×#67—21no—
150C2×#67—21no—
151#235—21no—
152#235—21no—
153#235—21no—
154#235—21no—
155#235—21no—
156#235—21no—
157#235—21no—
158#235—21no—
159C2×D12—22no—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—44no—
165C2×D12C2×D414yes—
166C2×D12#4111yescharacteristic
167C2×D12#4111yescharacteristic
168C2×C4×D3—21no—
169C2×C4×D3—21no—
170C2×C4×D3—44no—
171C2×Dic6#4111yescharacteristic
172C2×Dic6#4111yescharacteristic
173C3×M16—21no—
174C3×M16—21no—
175C3×M16—22no—
176C3×M16—22no—
177C2×C3×C8—22no—
178C2×C3×C8C2×D412yes—
179C2×C3×C8C2×D411yescharacteristic
180C2×C3×C8C2×D411yescharacteristic
181C2×C3×C8C2×D411yescharacteristic
182C2×C3×C8C2×D411yescharacteristic
183C2×C3×C8#4111yescharacteristic
184C2×C3×C8#4111yescharacteristic
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—22no—
189C3×C42—22no—
190C3×C42C2×D412yes—
191#217—22no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—44no—
194C4⋊Dic3—22no—
195C4×Dic3—22no—
196#208—21no—
197#208—21no—
198#208—21no—
199#208—21no—
200C2×C3⋊C8—21no—
201C2×C3⋊C8—21no—
202Dic12—22no—
203D24—22no—
204D24—22no—
205D24—22no—
206D24—22no—
207D24—22no—
208D24—44no—
209#204—22no—
210#204—22no—
211#204—44no—
212ΓL(1,𝔽25)—41no—
213ΓL(1,𝔽25)—41no—
214ΓL(1,𝔽25)—41no—
215ΓL(1,𝔽25)—41no—
216D3×C8—41no—
217D3×C8—41no—
218D3×C8—41no—
219D3×C8—41no—
220C2×#41—31no—
221C2×#41—31no—
222C22×D4—32no—
223#137—61no—
224#137—61no—
225AGL(1,ℤ/8)—61no—
226AGL(1,ℤ/8)—61no—
227AGL(1,ℤ/8)—64no—
228AGL(1,ℤ/8)—122no—
229#135—62no—
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—61no—
239#131—61no—
240#131—61no—
241#131—61no—
242#131—61no—
243#131—61no—
244#131—61no—
245#131—61no—
246C2×M16—31no—
247C2×M16—31no—
248C2×M16—61no—
249C2×M16—61no—
250C2×M16C22×D312yes—
251C2×M16C22×D311yescharacteristic
252C22×C8—61no—
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—32no—
262#117C22×D312yes—
263#117C22×D311yescharacteristic
264C2×C22⋊C4—122no—
265#108—21no—
266#108—21no—
267#108—62no—
268#107—21no—
269#107—61no—
270#106—21no—
271#106—61no—
272#104—62no—
273#104—62no—
274#104—122no—
275#102—64no—
276#102—64no—
277#100—64no—
278#99—122no—
279#97D1212yescomplemented
280C4×C8D1212yescomplemented
281C23×C3H(5,𝔽2)11yescharacteristic
282C22×D3—21no—
283C22×D3—24no—
284C22×D3—21no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—162no—
288C2×C3×C4—22no—
289C2×C3×C4—42no—
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4H(5,𝔽2)11yescharacteristic
296#67—41no—
297#67—41no—
298#67—41no—
299#67—41no—
300C2×Dic3—21no—
301C2×Dic3—21no—
302C2×Dic3—42no—
303D12—24no—
304D12—21no—
305D12—21no—
306D12—24no—
307D12—21no—
308D12—21no—
309D12—42no—
310D12—42no—
311D12—164no—
312C4×D3—41no—
313C4×D3—41no—
314C4×D3—41no—
315C4×D3—41no—
316C4×D3—164no—
317Dic6—21no—
318Dic6—21no—
319Dic6—21no—
320Dic6—21no—
321C3×C8—22no—
322C3×C8—42no—
323C3×C8#12111yescharacteristic
324C3×C8#12111yescharacteristic
325C3×C8#12111yescharacteristic
326C3×C8#12111yescharacteristic
327C3⋊C8—41no—
328C3⋊C8—41no—
329C24—62no—
330#41—61no—
331#41—61no—
332#41—61no—
333#41—61no—
334#41—61no—
335#41—61no—
336#41—61no—
337#41—61no—
338C2×Q8—31no—
339C2×Q8—31no—
340C2×D4—34no—
341C2×D4—31no—
342C2×D4—31no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—62no—
346C2×D4—62no—
347C2×D4—62no—
348C2×D4—62no—
349C2×D4—62no—
350C2×D4—124no—
351C22×C4—61no—
352C22×C4—61no—
353C22×C4—124no—
354C22×C4C23×D311yescharacteristic
355Q16—62no—
356SD16—62no—
357SD16—62no—
358SD16—124no—
359D8—62no—
360D8—62no—
361D8—62no—
362D8—62no—
363D8—62no—
364D8—124no—
365M16—21no—
366M16—21no—
367M16—22no—
368M16—22no—
369M16—61no—
370M16—61no—
371M16—61no—
372M16—61no—
373M16—121no—
374M16—121no—
375M16—121no—
376M16—121no—
377C2×C8—22no—
378C2×C8—61no—
379C2×C8—61no—
380C2×C8—121no—
381C2×C8—121no—
382C2×C8—121no—
383C2×C8—121no—
384C2×C8C2×D1212yes—
385C2×C8C2×D1211yescharacteristic
386C2×C8C2×D1211yescharacteristic
387C2×C8D3×D411yescharacteristic
388C2×C8D3×D411yescharacteristic
389C2×C8#23911yescharacteristic
390C2×C8#23911yescharacteristic
391C4⋊C4—21no—
392C4⋊C4—22no—
393C4⋊C4—62no—
394C22⋊C4—21no—
395C22⋊C4—22no—
396C22⋊C4—62no—
397C22⋊C4—124no—
398C22⋊C4—124no—
399C42—22no—
400C42—62no—
401C42C2×D1212yes—
402C22×C3—41no—
403C22×C3#53511yescharacteristic
404C22×C3#54611yescharacteristic
405C22×C3#55311yescharacteristic
406C2×D3—42no—
407C2×D3—41no—
408C2×D3—41no—
409C2×D3—44no—
410C2×D3—84no—
411C2×D3—84no—
412C2×D3—81no—
413C2×D3—162no—
414C3×C4—42no—
415C3×C4—82no—
416C3×C4C2×#12111yescharacteristic
417C3×C4#52511yescharacteristic
418C3×C4#54611yescharacteristic
419C3×C4#55311yescharacteristic
420Dic3—41no—
421Dic3—41no—
422Dic3—82no—
423C23—61no—
424C23—64no—
425C23—61no—
426C23—122no—
427C23—122no—
428C23—482no—
429C23#100911yescharacteristic
430Q8—61no—
431Q8—61no—
432Q8—61no—
433Q8—61no—
434D4—61no—
435D4—64no—
436D4—61no—
437D4—64no—
438D4—61no—
439D4—61no—
440D4—121no—
441D4—121no—
442D4—121no—
443D4—121no—
444D4—122no—
445D4—122no—
446D4—484no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—122no—
456C2×C4—484no—
457C2×C4C22×D1211yescharacteristic
458C2×C4C2×D3×D411yescharacteristic
459C2×C4C2×#23911yescharacteristic
460C2×C4D3×#4111yescharacteristic
461C2×C4D3×#4111yescharacteristic
462C2×C4#100911yescharacteristic
463C8—22no—
464C8—42no—
465C8—121no—
466C8—121no—
467C8#89511yescharacteristic
468C8#89511yescharacteristic
469C8#89511yescharacteristic
470C8#89511yescharacteristic
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3#253511yescharacteristic
475D3—81no—
476D3—81no—
477D3—84no—
478C22—41no—
479C22—122no—
480C22—121no—
481C22—121no—
482C22—124no—
483C22—244no—
484C22—241no—
485C22—244no—
486C22—482no—
487C22#552511yescharacteristic
488C22#567611yescharacteristic
489C22#569511yescharacteristic
490C4—42no—
491C4—82no—
492C4—121no—
493C4—121no—
494C4—242no—
495C4C2×#89511yescharacteristic
496C4#552411yescharacteristic
497C4#567611yescharacteristic
498C4#569511yescharacteristic
499C3#2818311yescharacteristic, complemented, Sylow
500C2—21no—
501C2—21no—
502C2—21no—
503C2—241no—
504C2—244no—
505C2—241no—
506C2#7423911yescharacteristic
subgroupG/H
centerC2#74239
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1833C2
socleC2×C3#2535
hypercenter#1833D3
hypocenterC3#28183
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1833

Complements

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

NH ≅ G/Ntype
1. C2×#4897C2semidirect
2. #79183C2semidirect
3. #78579C2semidirect
4. #78561C2semidirect
5. C3×#1833C2semidirect
6. #70719C2semidirect
7. #70664C2semidirect
8. #69028C2semidirect
9. #69020C2semidirect
10. #68705C2semidirect
11. #68704C2semidirect
52. #1833D3semidirect
104. C3×#97D4semidirect
105. C3×C4×C8D4semidirect
279. #97D12semidirect
280. C4×C8D12semidirect
499. C3#28183semidirect

Maximal subgroups

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

subgroupindex
C2×#48972
#791832
#785792
#785612
C3×#18332
#707192
#706642
#690282
#690202
#687052
#687042
#281833